The Hall of Mirrors: How Einstein's concept of Relativity is invalid

Statement

I have decided to publish this now, even though it is not quite finished. This is because: 1) it relieves me, in the short-term, of the effort required to finish it, 2) the unfinished sections are implicitly covered anyway. In other words, any reader who can follow the argument as currently written, does not need the other sections.

Also, I am getting tired. What started out with a sense that ‘there is something wrong here’, and then endeavouring to establish what it is, has been intriguing. However, I have a life to live which extends beyond that.

Paul

Dedicated to myself. Thanks for finishing this Mr Tambourine Man.

This is provided free. If you have in any way appreciated it, or even if not, please make a donation to Barnardo’s. You will then help those who did not have a good start in life reach their potential.

Hall of Mirrors: Introduction

This paper is concerned with Einstein’s concept of Relativity, and how it is invalid. It does not examine any repercussions which might arise from this.

For physics to be a science based on knowledge, rather than belief, and provide a coherent explanation (aka the Theory of Everything), it must be in accord with the inherent nature of the physical existence of which we are a part. The key point being that we are confined to an existentially closed system, as we cannot transcend our own existence, and can only know what it is possible for us to know from within it.

Given a closed system, the pursuit of due scientific process within it does not entail presumptions, if they are consistent with that system. Indeed, such an approach is a necessity. Not imposing such limitations, let alone using invalid ones, while superficially appearing a more intellectual form of analysis, will, other than by fortuitous coincidence, result in a construct of belief, not knowledge.

Furthermore, what physically exists can only be known to us through representations of it. While the physical processes involved in that need to be understood, physics is not a science of representations of what happened, but knowledge explaining what actually occurred. Reality, and representations thereof, must not be conflated. Neither should concepts such as space, x, y, z, and at a time, t, be reified and attributed with an independence disconnected from reality.

While the initial argument is expressed in generic terms and may appear simplistic, it actually enables exposure of fundamental facts about our physical existence. It could be referred to, colloquially, as a case of ‘seeing the wood by ignoring the trees’.

In the critique, the normal approach of references has not been followed. Since the author found that the writers explained themselves adequately, so there was no need to re-phrase them with the support of a complex system of referencing, which would lose the immediacy of what was written. More importantly, the author found that many derivative descriptions bore little relationship to what had actually been written a 100 years ago. Hence, quotes are given at the start of sections, and then selections therefrom used within the commentary in order to, in effect, give the original writers their ‘voice’ back. Unless otherwise stated, quotes in italics are Einstein.

In many cases whether what was actually said was correct, or not, either on the basis of what was known then, or known now, is irrelevant. Since a central issue is what was actually written at the time, and then how that was developed.

Paul

Hall of Mirrors: Contents

Published

Section 1: Introduction

Section 2.1: The logic of physical existence

Section 2.2: Summary of the logic of physical existence

Section 3: The context of the 1905 paper

Section 3.2: Calculation of movement

Section 3.3: Lorentz

Section 3.4: Poincaré

Section 3.5: Einstein

Section 4: The Reconciliation Argument

Section 4.1: Introduction

Section 4.4: The Argument

Section 6.1: The Special and General Theory

Section 6.2: General Theory examples

Section 6.3: Einstein’s attempt to define physical existence

Section 7: Conclusions

References

Section 3.1: Quotes

Section 4.2: Einstein quotes

Section 4.3: Other Einstein quotes

To be published
Section 4.5: Time

Section 4.6: Space

Section 5: Minkowski

Hall of Mirrors: Section 2.1: The logic of physical existence

This section explains the nature of our existentially closed system.

2.1.1 The nature of the closed system
By definition, there is always the possibility of a mutually exclusive alternative, ie if A, there must be the possibility of not-A. Assuming A to be our ‘being’, whatever that is, then while there could be an alternative, that is irrelevant as we cannot be of both A and not-A. We cannot transcend our own existence, and can never have knowledge, as opposed to belief, of any other possibility. Which means we are confined to an existentially closed system, and the issue becomes how to determine validity from within that, given its inherent nature.

With a closed system, the pursuit of due scientific process within it does not entail presumptions, if they are consistent with its nature. Indeed, such an approach is a necessity, because the consequences are that there is a ‘boundary’ beyond which nothing is knowable, and there are fundamental ‘rules’ concerning how it occurs. Not imposing such parameters, let alone using invalid ones, though appearing a more intellectual form of analysis, will, other than by fortuitous coincidence, result in a construct of belief rather than knowledge.

Without such constraints, all statements would have to be regarded as equally valid, since no reference is available for verification, ie it would be impossible to differentiate knowledge and belief. It is irrelevant that some judgements may seem more plausible, either the judgement is based on something that is in accord with the nature of our physical existence, or it is not. This process must involve a proper system of referencing in order to identify actuals, and, by definition, no extrinsic (‘absolute’) reference is available.

So, the issues are to identify what feature enables verification of whether any given ‘something’ is of our existentially closed system, or not, and to understand how knowledge can be developed solely on that basis. We must accept, when it is the case, that being unknowable is the proper scientific decision, rather than invoking a belief system in an attempt to solve the unsolvable.

2.1.2 The nature of knowledge
The epistemological problem is how to develop knowledge from within a closed system. Most statements are a form of judgement involving two caveats. First, the structure of it is fundamentally of the form ‘with respect to’. Second, there must be a context. Sometimes the latter is not overtly identified as it was not realised; but mainly it is not acknowledged because it is obvious, and hence unnecessarily repetitive. Put simply, something is defined by comparing, and differentiating, it from something else, within a specified circumstance.

As we are constrained in a closed system, we are able to determine actuals, albeit only within it, since there is a finite number of possibilities. Logically, the ensuing process of comparison entails referencing any given manifestation of an attribute with every other one. Any specific example could be the reference, as the purpose is to identify differences, ie so long as a constancy of reference is maintained it is irrelevant what it is. The critical requirement is constancy, which could involve adjustments to compensate for known variations.

With a closed system, even if the correct procedures have been maintained, our knowledge can only ever be regarded as ‘proven up to that point’. This reflects the nature of the process, and does not imply any degree of subjectivity, or that there is no limit to what is knowable. Ultimately, the knowledge established could be accepted as the equivalent of our closed system, ie a correct, and comprehensive, explanation of that from within it. But as we are confined to that system, we will only become aware of that by default, ie when no new knowledge is revealed by the process of comparison.

2.1.3 The nature of physical existence
Sensorial detectability is the feature that enables determination of whether something is of our existentially closed system, or not. It is concerned with the processes involving interactions between whatever existed, and certain physically existent phenomena, which result in the creation of representations of what occurred. There is an exception, in that this cannot be limited solely to direct experience, ie the presumption that all the sensory systems available, and not just those of humans, are able to detect everything. But such instances must only be confirmed on the basis of logical deduction from what has been verifiably established by direct experience, and not an assertion based on belief.

Therefore, physical existence (ie our closed system) at any given time comprises whatever is physically existent, and physically existent representations of what previously existed. Though these representations only result from interactions with what occurred, they then exist independently of that. Moreover, they are independent of the sensory processes which have evolved to detect them. That is, physical existence occurs irrespective of those processes, neither can they affect it as it has already happened. If all sensory processes were eliminated then there would just be no awareness of it, but it would still exist. Which was the circumstance which prevailed before these sensory systems developed.

To be physically existent must, by definition, entail physical unambiguity, involving a physically existent state of any given ‘entity’, or representation, because different physical versions of such cannot co-exist. Neither can more than one physically existent state occupy the same spatial position at the same time. Thus, at any given time, unique physically existent states of some ‘substance’ exist independently of sensory detection.

The next consideration is whether alteration is involved, or not. If not, then physical existence would be the same physically existent ‘substance’ in the same physically existent state, ad infinitum, ie a solid state. As comparisons at different times reveal differences, there must be alteration. That is, at any given time, unique physically existent states of some ‘substance’ exist independently of sensory detection, and have altered from the preceding existent version thereof. [At the generic level, what constitutes ‘substance’, ‘physically existent state’, ‘alteration’, etc, is irrelevant detail].

The issue then becomes reconciling the apparent contradiction between being physically existent, which necessitates substance in a definitive state and spatial position, at any given time, and alteration, which involves changes to that. The resolution of that conundrum lies in the nature of sequence, whereby any change results in the next step in the sequence, and the cessation of the preceding one. Therefore, physical existence is an existential sequence, of which there can only ever be one occurrence at a time, ie a reality. The frequency of alteration being determined by the quickest type(s) of attribute change, and the overall rate of alteration is one unit of duration, assuming constancy.

Similarly, that underlying unambiguity means that the sequence itself must progress incrementally. Every physically existent state, and its spatial position, must comprise whatever constitutes the next in the sequence, and that must involve only one degree of difference (a change) in any given attribute of that state. When considering any given existent state, and comparing it to its previous existence, change will have occurred in at least one of its attributes, but not necessarily all of them. There is no implication that all possible degrees of attribute alteration progress at the same rate.

Invoking more than one degree of change raises the issue as to how many would then be the determinant, since either there is a change in any given attribute, or there is not. Being similar is irrelevant. Any given ‘entity’, or representation, is not the same if there has been any change, irrespective of how few have occurred, or what was involved, ie it is not of the same reality. The differentiation of realities is a function of one degree of change in any given attribute, not some arbitrary percentage, and/or type. Any other circumstance presupposes that a reality both exists and is altering concurrently, which is an impossibility as physical existence is unambiguous. This means, any given reality is purely a spatial phenomenon. The notion of duration (as opposed to ‘at a time’) arises from physical existence appearing to have a continuity, whereas it is actually unambiguous of discrete realities.

Hence, the usual, understandable, view that our existence involves ‘entities’ which persist whilst also undergoing alteration, is incorrect. This also applies to representations. Though in that case, there is a tendency for them not to be thought of as being physically existent, and/or independently so, in the first place. Also, there is very little recognition, given their inherent physical characteristics (as explained below), that they actually do not maintain the same physically existent state over more than one unit of duration.

The misconception that entities or representations exist in the same state over many durations is a reflection of the continued manifestation of certain superficial physical attributes. We tend to only deem them to have changed when the extent of the alteration reaches an arbitrary point. This perspective represents a higher-level conceptualisation of physical existence.

An analogy would be a film, where an illusion of continuity is created, but actually only one frame is being shown at a time. While the effect is similar, the difference is that the frames shown, and the frames as yet unshown, all exist at the same time. Whereas, with physical existence only a unique physically existent state occurs at any given time, whilst what did happen has ceased to exist, and what will occur does not yet exist. The implication being that whatever constitutes the elementary components (‘substance’) which comprise physical existence must be what continue to exist, ie each reality is a different configuration of these.

Apart from the concept of entity/representation as usually understood being invalid, so too is the implication that there is a difference between them at the fundamental level, as they are both a discrete physically existent state of some form of substance. However, the objective is to develop knowledge, and the language that has evolved on the basis of this misconception must be used for that purpose. Furthermore, without the, albeit false, notion of ‘something which alters’ that enables our arrangement of what occurs into some form of order for the sake of comprehension, there are then only the elementary components. Attempting to develop an analysis based on those would be impossible, given the complexity of what is involved, and the speed at which alteration occurs. Some degree of higher-level conceptualisation is inevitable, but that must be understood, and not conflated with reality.

The important points are that physical existence (our closed system):
-comprises whatever is physically existent, and existent representations of what happened
-exists independently of the sensory processes which have evolved to detect it
-involves physical substance in a unique physically existent state and spatial position, ie it is
an existential sequence involving only one occurrence (a reality) at a time
-has a frequency of transition in realities which is determined by the quickest form(s) of change in any given attribute, and a rate of alteration of one unit of duration
-involves realities that are purely a spatial phenomenon. The notion of a greater duration reflects the sequence of realities, not any given reality.
-comprises whatever constitutes its elementary components, each reality being a different configuration thereof.

Overall, what seems to be ‘something’, whether an entity or a representation, does not continue to exist in the same physically existent state, and the two must not be conflated.

2.1.4 The nature of the future
With respect to what is referred to as the future, there is no validity in any notion of change to it, or that it can have some influence, as there is nothing in existence to affect, or cause an effect. This interpretation implies that it is already existent, which it is not. The proper depiction is: the circumstance where a subsequent existent state in the sequence occurred that was different to what would otherwise have occurred, had the causal factors been different. Which is meaningless, since every step in the sequence is a function of preceding conditions, and any outcome would be different if they had been different. Obviously, given proven trends in the relevant conditions, it is possible to extrapolate what alteration any given existent state might undergo with a significant degree of certainty, but that does not mean it has become physically existent.

2.1.5 The nature of sensory detection
We can only become aware of what existed from what is manifested to us via certain physical processes (light, sound, etc). The resultant representations are independently physically existent, and potentially receivable sensorially. Knowledge can then be developed either on the basis of validated direct experience, or by logical deduction therefrom. This includes any proven sensory process, and is not limited to human experience. The caveat of potentiality relates to issues with the physical processes involved, as explained below. This paper is only concerned with the process up to the point of interception of a representation (ie receipt) by the appropriate sensory system, and not what happens subsequently.

When any representation is received, which necessitates being in its line of travel, it ceases to exist in that state. That applies to any interaction, not just one with the appropriate sensory system. Receipt entails detection of a representation of what has already occurred. Hence, no hypothesis which purports to explain physical existence can rely on any implication that the receipt itself, or its subsequent processing, has any effect on it. Though that process is important in itself, and involves many variable factors, but that is a different issue.

The common physical traits in the different types of representations led to the evolution of sensory systems, as the ability to receive, and then process, them afforded the advantage of an awareness of the environment. These are consistency, and regularity, of occurrence, stability of physically existent state, and the omnipresence of the physical phenomena which enable them. Therefore, when representations are considered at a higher-level than their physically existent state, it can be assumed any interaction with any given existent state results in a range of representations of it which are similar. Also, interactions with other entities which have superficially common characteristics result in similar outcomes. Subsequently, they are not unduly affected by most physical effects encountered whilst travelling.

However, functionally they are just representations of whatever existed, which while sufficient in the context of survival, are deficient when used to establish what they depict. This relates to the physical processes involved in their creation and transmission:

1 The relationship between the occurrence and any type of representational effect:
Every occurrence is too complex, ie too numerous, frequent, varied, for the capabilities of any of the phenomena which enable representations to cope with. Therefore, they will be deficient as a depiction of what actually happened. Another possibility is that the occurrence involved a physical property which precluded some aspect of the interaction with any given phenomenon, so no representation of that particular characteristic was generated. Though it might be deducible from other direct experiences. Or, a representational type exists, but no sensory system has evolved, or indirect method developed, to detect it.

2 How the physically existent state of any representational effect is altered by prevailing physical circumstances:
Any representation could be affected by different aspects of the physical environment it interacts with whilst travelling, and thereby altered from its original state. This could involve delay, distortion, partial elimination, diversion from the original line of travel, etc.

3 A representation is never received:
No appropriate recipient sensory system was in the line of travel whilst it existed.

Therefore, apart from what is required to interpret what was received and then develop a commonly agreed depiction thereof, some reverse engineering is then required in order to equate that with what occurred in reality. But this is not a concern of this paper.

2.1.6 The nature of measurement systems
The underlying purpose of measurement systems is to identify differences. This necessitates the use of a reference that must have consistency in order to ensure the comparability of all the outcomes, validity, ie be in accordance with physical existence, and practicality, ie usable as a reference. As we are in a closed system the process of comparison can identify actuals, albeit only within the system. In this circumstance, any given attribute could be chosen as the reference, and then compared to everything else. But that would be impractical.

That is why measurement systems use a reference which is a generic conceptualisation of the characteristic being quantified. This is then embodied in the measuring devices so that they all function as if they were one device. The reference, per se, is not physically existent. Deficiencies in their construction, or susceptibility to external influences which affect their functioning, must not be confused with variations in reality. For example, in different temperatures metal rulers change in length, but the distance has not altered, or when timing devices cannot maintain a constant rate of ‘tick’ due to some external physical effect, it does not mean that the actual rate of alteration has varied.

2.1.7 The nature of time
Alteration in a physically existent state can involve any form of change in any given attribute. The change itself did not exist; it was a difference between two consecutive states in a sequence. The rates for different types of change will vary, and possibly even for identical attributes if physical circumstances alter. These need to be quantified, both in respect of any particular sequence, and for comparison between sequences. The measurement system for this is known as timing.

Identifying rates of alteration involves counting, and then comparing, the number of changes in different sequences over the same duration. This could involve any existent state, attribute, and either concurrent sequences or otherwise, as it is only the rate of change which is of concern, not what was involved. Therefore, the requirement for this measurement system is a reference with a constant, high frequency, change.

As with all measurement systems, the reference for timing is a generic conceptualisation of what is being quantified. If timing, and language, had been expressed in terms of the reference being a constant numeric sequence of ‘ticks’, this would have emphasised the true nature of the system, and the concept of ‘time’ would have been better understood. That is, ‘at any given tick’ (when), and number of ‘ticks’ (duration). The nomenclature used highlights this with the terms minutes, hours, days, etc, which reflect an early timing device, ie the movement of the earth.

Timing devices are a physical manifestation of the conceptualised reference, and should function as if they were one device, ie synchronised. What causes the ‘tick’ is irrelevant. For example, frequency of crystal oscillation is the actual reference with a quartz device. Timing devices are independent of what is being calculated, ie any physical effect which affects their function does not mean there has been a variation in reality. Conversely, any actual variation in any given rate of alteration in any given attribute does not mean that the reference has altered, as it is, by definition, a hypothetical constant.

2.1.8 The nature of distance
Distance is the spatial difference between any given pair of physically existent states at any given time. It is a spatial quantity reflecting a unique occurrence in terms of when, and what was involved. For there to be a distance whatever is under consideration must exist at the same time, since something that is non-existent cannot have a spatial position, ie they must be part of the same reality. As distance is a difference, it does not physically exist, only the states did in their respective spatial positions at that time.

When measuring distance, in effect a grid of spatial positions is used as the reference. Its optimum unit ‘mesh’ size would be the equivalent of the size of the smallest type of substance(s) in physical existence. A distance can only be expressed properly in terms of spatial units, by comparing spatial positions and identifying the difference. However, it can be expressed in terms of duration, ie x = vt. This is hypothetical, as it is the duration which would have been incurred had any given state, at a known velocity, been able to travel that difference in either direction. But a distance is only manifest once, so misinterpretation of this method of expression can result in the reification of distance, and time.

A distance measurement usually involves the two nearest dimensions of the states involved, but could include a range of dimensions. Anything greater in size than one spatial unit will have a spatial footprint, ie a set of spatial positions ‘occupied’. Dimension is a specific aspect of the ‘spatial footprint’ of a physically existent state, and relates to possible axes of it. Three is the minimum number of spatial dimensions that is ontologically correct at the highest level of conceptualisation. At the existential level, the number of dimensions is half the number of directions a substance the size of one spatial unit could travel on this footprint.

2.1.9 The nature of movement
Movement is alteration in distance. It is revealed when discrete distances in sequences are compared retrospectively. As calculating movement, as opposed to distance, involves sequence, any given entity/representation involved has to be deemed to be the ‘same’ thing over the duration of the measurement, which at the existential level it is not. The comparison of these discrete distances enables the calculation of velocity, which involves speed, ie rate of alteration in distance, and divergence, ie rate of alteration in direction.

For movement the reference needs to be something which is constantly ‘stationary’. But everything is moving. However, as we are in a closed system, that issue can be overcome because the options are finite, and therefore being designated the reference renders whatever it is, in effect, ‘stationary’. In practice, this involves spatial points, identified by something, colloquially known as the start and finish. Calculating movement with respect to such points reveals the actual movement, within the closed system. Whereas, measurements made with respect to something which has a known difference in velocity, vis a vis a valid reference, ie a movement of its own, reveals a derivative movement. The relationship between different movements involves the factor √(1 – v²/V²), where V is the faster moving entity.

The time of occurrence, and spatial position at that time, can only be determined from when, and where, the representations were received. However, as different entities cannot occupy the same spatial position there is always a distance between them, which means there is always a delay between occurrence and the representation being created, and when it was received. Hence, the distance travelled, and the duration thereby incurred, is a function of the spatial position of the occurrence with respect to the position of what became the ‘receiving’ entity when the representation was received. Not the distance between the event and that entity when it happened, or when the representation was received. Therefore, alterations in spatial position whilst any representation is travelling are critical.

The movement of the ‘emitting’ entity is deduced from the receipt of representations of it by the other entity. While an event occurred at a specific time, due to differing spatial positions, when the representation of it was received by various entities will likely vary. This could involve the original differences in spatial position, with respect to the occurrence, remaining the same, or, during the delay there is some alteration in those spatial positions. The circumstance becomes more complicated when a sequence of occurrences is involved.

In a relationship involving a representation, then the path of a representation, at any given t, would be defined by a distance starting from a point on the path of the ‘emitting’ entity at that t (which is also the start of the distance (x) between the entities at that t), to a point on the path of the ‘receiving’ entity, which would be beyond the end point of distance x. The relationship between ‘distance x’ and ‘distance transmission’, at any given t, again involves the factor √(1 – v²/V²). With a representation based on light, V would be the speed of light in that circumstance, but not c, as that is light speed in a hypothetical circumstance.

The Doppler effect is explained as follows:
Assuming a regularity in the sequence, and a constancy of velocity and direction of travel in the ‘emitting’, and ‘receiving’, entities, then the frequency of the sequence, when assessed with respect to the receiving of the representations, will be constant. However, if the relationship of distance, and/or direction, varies from occasion to occasion, then their relative spatial position could vary on each such occasion, and the frequency appear to differ from that emitted to that received.

This is because, as the spatial relationship alters, successive representations have to travel distances which alter at a varying rate. In this circumstance, a sequence measured with respect to the ‘receiving’ entity, either increases or decreases at a rate dependent upon the rate at which relative spatial position is altering whilst each representation is being transmitted. That is, it appears, for the ‘receiving’ entity, to be speeding up/slowing down, even though the actual emitted rate did not change. Without an understanding of this, it can be believed that ‘time’ is altering.

Hall of Mirrors: Section 2.2: Summary of the logic of physical existence

This section summarises the consequences for physics of our existentially closed system.

1 Knowledge can only consist of what it is possible for us to know from within the closed system, otherwise it is belief.

2 Development of that knowledge must proceed in a way that is in accord with the inherent nature of the closed system.

3 There must be a constancy, and validity, of reference points in order to identify actuals from within the closed system.

4 Verification as to whether any given something is of our existentially closed system is determined by sensorial detectability. This is not limited to human experience, and includes any verified sensory process, as well as what is deducible from proven direct experience.

5 Our existentially closed system (physical existence), at any given time, comprises what is physically existent, and existent representations of what previously existed.

6 These representations physically exist independently of whatever was involved in their formation, and of the sensory systems which have evolved to detect them.

7 To be physically existent must entail physical unambiguity, ie physical existence must involve physical substance in a unique physically existent state, and spatial position. Therefore, in respect of any given entity or representation:

-different physical versions of them cannot co-exist.

-no more than one physically existent ‘entity’ can occupy the same spatial position at the same time.

8 Given that to be physically existent is unambiguous, but involves alteration, physical existence must be an existential sequence, of which there can only ever be one occurrence (a reality) at a time. Any change in any attribute results in the next step in the sequence, and the cessation of the preceding one. Therefore:

8.1 Any circumstance which presupposes that a reality both exists, and is altering, concurrently, is invalid.

8.2 Any given reality is purely a spatial phenomenon. The notion of duration arises from physical existence being a sequence which is actually a succession of discrete realities.

8.3 Every physically existent state must comprise whatever constitutes the next in the sequence. Alteration must only involve one degree of change in any given attribute.

8.4 Some attributes will remain constant for more than one reality. There is no implication that all possible degrees of change progress at the same rate.

8.5 The frequency of alteration in the overall sequence is determined by the quickest type(s) of attribute change, and the rate, by definition and assuming constancy, is one unit of duration.

8.6 Physical existence comprises whatever constitutes its elementary components, which must be what continue to exist. Each reality being a different configuration of these.

9 The concept that entities or representations persist, and undergo alteration, is a misconception. Any given ‘thing’ actually involves a sequence of physically existent states. That perspective represents a higher-level conceptualisation of what happens, which is a sequence of discrete physically existent states with superficial similarities.

10 What existed is only knowable from the receipt of representations of it. What was involved, the time of its occurrence and spatial position, has to be discerned from the content of the representations, and when and where they were received.

11 What existed must not be conflated with its existent representations thereof.

12 No hypothesis that purports to explain physical existence can rely on any implication that what is involved in processing representations has any role in determining it.

13 There is always a delay between when the event occurred and the representation created, and when it was received, so times will vary depending on relative spatial position. With a sequence of occurrences, its frequency can appear to alter as relative spatial position varies.

14 While reliable in the context of survival, when representations are used to define what they depict, they are deficient. This relates to the physical processes involved in their creation and transmission.

15 Measurement systems enable the identification of differences. As we are in a closed system there is a finite number of options, so actuals can be identified. Any specific attribute could be designated the reference against which all other manifestations thereof are compared. But as that would be impractical, measurement systems use a generic conceptualisation of the attribute, which is embodied in the measuring devices so that they function as if they are one device. Deficiencies, or susceptibility to external influences, which affect their functioning must not be confused with variations in reality.

16 The measurement system for quantifying rates of alteration is known as timing. This involves counting, and then comparing, the number of changes in different sequences over the same duration. This could involve any existent state, attribute, and either concurrent sequences or otherwise, as it is only the rate of alteration which is of concern, not what was involved. Therefore, the requirement for the reference of this measurement system is a numeric sequence with a constant high-speed turnover.

17 Distance is the spatial difference between any given pair of physically existent states at any time. They must both exist at the same time, since a non-existent state cannot have a spatial position. As distance is a difference, it does not physically exist, only the states did, in their respective spatial positions at that time.

18 A distance can only properly be expressed in terms of spatial units, ie by comparing spatial positions and identifying differences. Alternatively, distance can be expressed in hypothetical terms, ie x = vt. This is the duration that would have been incurred had any given state, at a known velocity, been able to travel that difference in either direction. The issue being that this methodology can lead to the reification of distance, and time.

19 Movement is the manifestation of an alteration in distance revealed when distances are compared retrospectively. Therefore, calculating movement involves a sequence of states, and any given entity, or representation, involved has to be deemed to be the ‘same’ over the duration. The comparison of such measurements enables the calculation of velocity, which is the rate of alteration in distance, and divergence, the rate of alteration in direction.

20 For movement the reference needs to be something which is constantly ‘stationary’. But everything is moving. However, as we are in a closed system anything can be designated the reference, and everything else compared to it, which renders it, in effect, ‘stationary’. Calculating movement properly reveals the actual movement within the closed system. Whereas, measurements made with respect to something which has a known difference in velocity vis a vis a valid reference, ie a movement of its own, reveals a derivative movement. Comparison of movements involves √(1 – v²/V²), where V is the faster moving entity.

21 Actual movement must not be conflated with the movement of representations, especially when sequences are involved. As the delay whilst transmitted, and hence the possibility of alteration in spatial position, can result in illusions, eg the Doppler effect.

Hall of Mirrors: Section 3: The context of the 1905 paper

On the electrodynamics of moving bodies was published in 1905. Since its start point was the deliberations, and concepts, of others in the preceding years, this section quotes relevant passages from their papers in order to establish that context.

The validity of the arguments referred to in Section 3.1 is not fully evaluated in this paper, as that is only tangentially relevant to its purpose. The focus being how some concepts developed, which were then relied upon by Einstein in 1905.

Using comments by Einstein specifically about Lorentz, Section 3.5 reveals, in essence, how the concept of Relativity developed. More detailed arguments by Einstein explaining his new conceptualization of time and space, and justification for it, are examined later. These quotes are from post 1905, as in that paper this was not properly explained.

Hall of Mirrors: Section 3.2: Calculation of movement

3.2.1 Calculating movement: Configuration 1: Back and forth over fixed distance

Michelson (1881): The Relative Motion of the Earth and the Luminiferous Ether
Assuming the ether is at rest, the earth moving through it. Suppose the direction of a line joining the two points to coincide with the direction of earth’s motion. Then the time required for light to pass from one point to another on the earth’s surface, would depend on the direction in which it travels. Let:

V = velocity of light
v = speed of the earth with respect to the ether
D = distance between the two points
d = distance earth moves while light travels between points [away from each other]
d’ = distance earth moves while light travels between points [towards each other]
T = time for light to pass between points when travelling away from each other
T’= time for light to pass between points when travelling towards each other
To = time for light to pass between points when earth is at rest.

Then: T = (D + d)/V = d/v and T’ = (D – d’)/V = d’/v

Therefore: d = D[v/(V – v)] and d’ = D[v/(V + v)]

Therefore: T = D/(V – v) and T’ = D/(V + v). So: T + T’ = 2D[V/( V²– v²)]

If the light had travelled in a direction at right angles to the earth’s motion it would be entirely unaffected and the time of going and returning is 2To = 2D/V. Difference between T+T’ and 2To is 2DV[v²/(V²(V²– v²)]nearly or 2Tov²/ V² nearly. The distance light travels in the first case [T+T’] is greater than in the second [2To], by the quantity 2D(v²/V²).

Correct expressions are:
Total time when earth moving is: T+T’ = 2DV/(V²- v²)]

Total time when earth is stationary is: 2To = 2D/V

Difference between these is: 2Dv²/[V( V²- v²)]

On the same subject, Lorentz: If L is the distance of the points, V the velocity of light, and p that of earth, then: the time (if line of points parallel to direction of motion) is 2(L/V)(1 +p²/V²⁾, and if it is perpendicular to it 2(L/V)(1 + p²/2V²). Making a difference of Lp²/V³. So the rays that travel forth and back in the earth’s direction suffer a delay of Lp²/V³ in respect to the other [perpendicular back and forth] ray.

Example:
Let V = 10, v = 2, D = 6

When the earth and the light are moving in the same direction:
T = 0.75, the earth travels 1.5, the light travels 7.5
When the earth and the light are moving towards each other:
T’ = 0.5, the earth travels 1, light travels 5

When the earth does not move:
To = 0.6, the light travels 6.
Total time is 1.25 when earth moves, and 1.2 when stationary.
Total distance the light travels is 12.5 when the earth moves, and 12 when stationary.
Total distance the earth moves is 2.5.

Most of Michelson’s expressions are correct, though the ones that are ‘nearly’ correct are of concern, since they should be correct, not an approximation. The first expression of Lorentz is nearly correct, the other two are incorrect.

Comment:
This is a truism, ie if two entities are related in this way, then that is the answer. The fact that one is light is irrelevant, since it is still a physically existent entity, as is the earth.

Superficially, the deemed stationary reference appears to be the ether, as the speed of the earth is stated by Michelson to be with respect to the ether. But the ether is said to be at rest, with the earth moving through it, which does not necessarily mean it is stationary. Neither is any specific movement of the ether, or spatial point within it, stipulated. The fixed distance is of the earth, ie its velocity is defined by the earth movement. Light speed is deemed to be ‘independent’, ie a constant. Irrespective of what was declared, the ether is irrelevant in the circumstance as defined.

In the first part, ie when the light and earth move in the same direction: v has a ‘headstart’ of D, as the defined circumstance is fulfilled when V catches up with what was the finish point of D, before it moved forwards. So, when does Vt = D + vt?

In the second part, ie when they move in opposite directions: the two entities need to meet. One is travelling from what was the start of D and the other from what was the finish point of D. So, when does Vt = D – vt?

Irrespective of the ‘mathematical’ issues, which might reflect typographical mistakes(?), the important point, as Michelson noted, is that the distance travelled by the light is different in the two parts of the movement, and the total of that is different from the total when earth is stationary. Hence, the durations incurred are different.

There tends to be an incorrect presumption associated with this configuration. That is, while the times incurred travelling either way are expected to be different from one another, the total time taken going back and forth will be the same, whether or not the fixed distance moves. This is because the velocities and fixed distance are unchanged, which gives rise to the notion that when the fixed distance is moving, the differences travelling one way, as opposed to the other, will compensate for each other, and therefore be equivalent to twice that incurred when the fixed distance is stationary.

‘Time’ has not altered, neither has the fixed distance, and the light has a constant speed. It is just that by moving the fixed distance with respect to the light, the distance the light needs to travel alters, and hence the duration incurred varies. The potential for reifying ‘time’, and distance, by using x = vt as a means of expressing distance was noted in Section 2.1.8. The confusion that can arise by referring a movement to something which has a known movement of its own, ie a derivative movement, as opposed to something which is ‘stationary’, ie the actual, was noted in Section 2.1.9. The derivation of √(1 – v²/V²) is explained in Section 3.2.2.

End of Comment

In respect of a possible way of detecting a motion of the solar system through the ether, Maxwell wrote to the American astronomer David Todd (Director of the Nautical Almanac Office) in 1879, and enquired about the observational accuracy of the moons of Jupiter. Maxwell stated: Even if we were sure of the theory of aberration, we can only get differences of position of stars, and in the terrestrial methods of determining the velocity of light, the light comes back along the same path again, so that the velocity of the earth with respect to the ether would alter the time of the double passage by a quantity depending on the square of the ratio of the earth’s velocity to that of light, and this is quite too small to be observed.

Both Lorentz and Michelson alluded to Maxwell. Lorentz 1892: It was noted by Maxwell that if the ether remains at rest, then the motion of earth must have an influence on the time required by light to travel forth and back between two points regarded as fixed to earth. Michelson 1881: The difference depending on the square of the ratio of the two velocities, according to Maxwell, is far too small to measure.

Concerns about the experiments include questions as to what type of earth movement was referenced, and why? Then to explain how that was known at any given spatial position at any given time? The underlying issue being whether if the light had travelled in a direction at right angles to the earth’s motion is the equivalent of the earth being ‘stationary’ with respect to the light? Which is what the time of going and returning is 2D/V = 2To actually means. Lorentz referred to the perpendicular.

As hypothetical depictions of light movement these may suffice, especially since there are other serious concerns with these experiments anyway. The author assumes that by now there is a reasonable understanding of the variations involved in the relevant two, of the four, earth movements, ie its rotation on its axis, and its elliptical orbit round the sun. As well as the effect of refraction on light velocity as it passes through layers of the earth’s atmosphere. Hence, in the time taken for any given light from the sun to reach the earth, the spatial difference between where it would have been received had there not been a delay, and where it actually was, in respect of any given location at any given time, is understood. Should that be necessary to establish.

Then, leaving aside earth-bound sources of light, there is reflected light. Because it involves the same atomic process, the start speed of reflected light is the same as incident light, ie it is not affected by the velocity of any given entity involved, or that of the light previously. This was one of Lorentz’s key conclusions. After that, its velocity depends on the physical circumstances encountered during transmission, which, given the nature of light as a physically existent entity, are minimal. Therefore, on earth, given the distances involved and the homogeneity of the medium through which light travels, for practical purposes it can be assumed that invariably light travels at a constant speed in a straight line. But that speed is not c. Something moving towards the direction of travel of any given light must receive it before what would have been the case had it been moving away from, or stationary to, that light. That is, while the speed is actually constant, the outcome when compared to different movements is not.

In other words, there is no necessity for experiments, or discussions, in order to establish this as a generality. Furthermore, using light, per se, in an attempt to discern earth movement is pointless. As too would be using the ‘ether’, whatever that is. Neither provide an acceptable reference against which to establish movement in our existentially closed system, as explained in Section 2.1. Understanding the nature of light is essential, so that when received, when and where the occurrence happened, as depicted in a light-based representation, can be determined.

In conclusion, there is serious concern as to whether a mirror contraption attached to a stone base floating in mercury could have proved anything. Possibly, with current technology, such vanishingly small variations can be detected. But the general outcome was known anyway, so the exercise was not necessary, ie these experiments, and their results, were best ignored. But this is not what happened.

3.2.2 Calculating movement: Configuration 2: General
The above sub-section addressed a particular circumstance. This section addresses that relationship between two entities in general. Distance is the spatial difference between any given pair of physically existent states at any given time. It is a spatial quantity reflecting a unique occurrence in terms of when, and what was involved. For there to be a distance whatever is under consideration must exist at the same time, since something that is non-existent cannot have a spatial position. As distance is a difference, it does not physically exist, only the states do in their respective spatial positions at that time.

Distance can only be expressed properly in terms of spatial units, by comparing spatial positions and identifying the difference. However, it can be expressed in terms of duration, ie x = vt. This is hypothetical, as it is the duration which would have been incurred had any given entity, at a known velocity, been able to travel that difference in either direction. But any given distance is only manifest once, so misinterpretation of this method of expression can result in the reification of distance.

Movement is alteration in distance, revealed when discrete distances in sequences are compared retrospectively. As calculating movement, as opposed to distance, involves sequence, any given entity/representation involved has to be deemed to be the ‘same’ thing over the duration of the measurement. Comparison of these discrete distances enables the calculation of velocity, which involves speed, ie rate of alteration in distance, and divergence, ie rate of alteration in direction.

For movement the reference needs to be something which is constantly ‘stationary’. But everything is moving. However, as we are in a closed system and the options are finite, whatever is designated the reference against which everything else is compared is, in effect, ‘stationary’. Calculating movement with respect to a proper reference reveals the actual movement, whereas, measurements made with respect to something which has a known difference in velocity vis a vis that reference, ie a movement of its own, reveals a derivative movement. The comparison of different movements involves √(1 – v²/V²), where V is the faster moving entity.

Let it be assumed that both entities start from the same point at the same time, travel at a constant speed, and do so along the same line of dimension. Their distance apart on any occasion is denoted as x. When travelling in the same direction, the distance between them per t is V-v. When travelling in opposite directions, the distance between them per t is V+v. In all other circumstances, their direction of travel is diverging, either at an acute, obtuse, or right, angle. Whilst the two entities maintain a constant velocity, the change in relative spatial position remains constant. They are in a fixed spatial relationship with each other, ie the actual value of x varies at each t, but the rate at which it does so remains constant.

For the sake of illustration, let the entities start from the same point on the x axis, at the same time, and be deemed to finish whenever they reach a y axis at the same time. If this arrangement is represented as a right-angled triangle, then while the slower moving entity (v) travels along the x axis, the faster moving entity (V) travels along the hypotenuse. The difference between them on any occasion is manifested by the two finish points on the y axis. In Pythagorean terms: h² = y² + x². As v increases whilst V remains constant, then the angle of incline of the hypotenuse decreases, and x decreases. As V increases while v remains constant, then the angle of incline of the hypotenuse increases, and x increases.

When the circumstance is not designed to be representable by a right-angled triangle, but requires the two finish points be related vertically which determines their degree of divergence, the law of cosines applies. When the angle of divergence is obtuse, any increase in that angle, and/or any increase in either, or both, of the velocities, results in an increase in x, and vice versa. When the angle of divergence is acute, a variance in velocity can alter x either way, depending on the velocities.

Alternatively, the above relationship can be demonstrated as follows. Instead of referencing both constant movements to a common start point, let the slower moving entity travel along the x axis, with reference to the original start point, whilst the movement upwards of the faster moving entity is referenced to the slower moving entity. So, on each occasion, the faster moving entity travels V from its start point. But instead of travelling vertically, its directional path is determined by the intersection of that, and its distance from the slower moving entity on the x axis, on each occasion. The direction of travel of the faster moving entity, and hence the degree of divergence of the two entities, depends on the ratio V:v.

Yet another way of representing this relationship is to constrict it in terms of duration and spatial position. That is, the two entities must start at the same position, at the same time, and finish at the same position, which is different from the start, at the same time. Both movements are referenced to the same start point. So, whilst the slower moving entity travels along the x axis to the finish point, the faster moving entity must travel an extra distance to arrive at that point, in the same duration, ie it incurs a certain amount of ‘lateral displacement’, a distance V – v per t. As it is travelling at a constant velocity, the path of this displacement can be represented by a triangle with a base on the x axis of vt, and two sides equal to ½Vt. If both entities had to follow a path along the x axis, then in order to satisfy the same circumstances, the slower moving entity would have to be given a ‘head start’ of V – v per t.

The effect of relating V and v can be expressed as the ratio ‘V with v’: ‘V without v’. With v, x = t√(V² – v²), or as a hypothetical duration, x/√(V² – v²). Without v, x = Vt, or expressing that distance as a hypothetical duration, x/V.

The ratios of with/without v are:

Distance, (with v)/(without v) is: t√(V² – v²)/Vt = √(V² – v²)/V or √(1 – v²/V²)

or (without v)/(with v) is: V/√(V² – v²) or 1/√(1 – v²/V²)

Duration, (with v)/(without v) is: [X/√(V² – v²)]/X/V = V/√(V² – v²) or 1/√(1 – v²/V²)

or (without v)/(with v) is: √(V² – v²)/V or √(1 – v²/V²)

3.2.3 Calculating movement: Configuration 3: Receipt of representation
The above sub-sections concerned the relationship between entities in terms of occurrence, ie the fact that one of them could be light was irrelevant, as it is a physically existent entity in itself. The relationship being considered here involves the receipt of a representation of one of the entities, rather than the entity itself. The diagrammatical difference being that each measurement is taken when the representation intersects (is received) with the other entity, ie distance, and hence movement, is being deduced from the receipt of representations by the other entity.

The circumstance becomes more complicated when it involves a sequence of occurrences. In a relationship involving a representation, then the path of a representation, at any given t, would be defined by a distance starting from a point on the path of the ‘emitting’ entity at that t (which is also the start of the distance (x) between the entities at that t), to a point on the path of the ‘receiving’ entity, which would be beyond the end point of distance x. The relationship between ‘distance x’ and ‘distance transmission’, at any given t, again involves the factor √(1 – v²/V²).

Assuming a regularity in the sequence, and a constancy of velocity in the ‘emitting’, and ‘receiving’, entities, then the frequency of the sequence, when calculated with respect to the entity receiving the representations, will be constant. However, if the relationship of distance and/or direction varies from occasion to occasion, then their relative spatial position could vary on each such occasion, and the frequency appear to vary from that emitted to that received.

This is because, as the spatial relationship alters, then successive representations have to travel distances which alter at a varying rate. Then, a sequence calculated with respect to the ‘receiving’ entity, either increases or decreases at a rate dependent upon the rate at which relative spatial position is altering whilst the representation is being transmitted. That is, it appears, to the ‘receiving’ entity, to be speeding up, or slowing down, though the actual emitted rate did not change. This is the Doppler effect, which if not understood, can result in the belief that ‘time’ is altering.

Hall of Mirrors: Section 3.3: Lorentz

For this paper, how Lorentz interpreted the results of the experiments that resulted in his hypothesis, which supposedly resolved the ‘discrepancy’ between expectations and experimental outcomes, is the important issue. Rather than what was at the time, or is now, known to be correct. Because, Einstein’s interpretation of what Lorentz decided, along with other contemporary inputs, was the basis upon which he developed his ideas in 1905.

3.3.1 What Lorentz wrote
Lorentz’s start point was that certain differentials should occur. As explained in Sections 3.1 and 3.2, generically the notion was obvious and correct; the issue was its mathematical expression, ie, it did not need to be proven generically.

He did note possible practical issues with the experiments: One can argue that the length of the arms was too small to obtain any observable displacement of the fringes. And: Michelson thought that he is allowed to conclude that the ether wouldn’t remain at rest when the earth is moving, a conclusion whose correctness was soon questioned. And: Michelson has estimated the change of the phase differences, as expected by the theory, to double of the correct value; if we correct this error, we arrive at displacements, which could be hidden by the observational errors. And: Together with Morley, Mitcheson started again, nevertheless, during the rotation only displacements of at most 0.02 of the fringe-distance were obtained; they might stem from observational errors.

However, he did not discount the results of the Michelson experiments as either, the function of inadequate equipment, and/or the impossibility of accurately identifying the expected effect in the first place. Instead, he sought something which explained their apparent failure. Initially, Lorentz posited refraction, but then discounted that: however, the shift as required by Fresnel’s theory could not be observed.

But there was no need for such a search, in an attempt to identify some previously unknown factor, since it did not exist. Both Lorentz’s contraction, and Einstein’s Relativity, were attempts to solve a non-existent issue. Though in the case of Lorentz, it must be noted that there is the possibility that the effect of a force which causes a variation in the movement that normally prevails when everything is in a state of equilibrium, could also involve a variation in shape. If this is so, then that needs to be determined, and the resulting actual differentials in any given position, at any given time, identified, and not depicted by generic equations.

Lorentz introduced the caveat which determined his approach: is it allowed to assume that the ether shares the motion of earth, and thus Stokes’ aberration theory is the correct one? The difficulties, with which this theory is confronted when explaining aberration, seem too great to me for having that opinion, so I rather should try to remove the contradiction between Fresnel’s theory and Michelson’s result.

Therefore, his initial idea was: I have sought to explain this experiment without success, and eventually found only one way to reconcile the result with Fresnel’s theory. It consists of the assumption that the line joining two points of a solid body doesn’t conserve its length, when it is once in motion parallel to the direction of motion of earth, and afterwards it is brought normal to it. The difference would be removed when a = p²/2v². A change in length of the arms in the first experiment, and the stone plate in the second, is not inconceivable.

To justify that he wrote: What determines the size and shape of a solid body? Apparently, the intensity of molecular forces; any cause that could modify it, could modify the shape and size as well. We can assume that electric and magnetic forces act by intervention of the ether. It is not unnatural to assume the same for molecular forces, but then it can make a difference, whether the connecting line of two particles is moving parallel to the direction of motion or perpendicular to it.

However, he then cautioned: Since we know nothing about the nature of molecular forces, it is impossible to verify the hypothesis. We can, by introducing more or less plausible assumptions, calculate the influence of the motion of ponderable matter on electric and magnetic forces. We cannot ascribe great importance to this result; the transfer to molecular forces of what we have found for electrical forces, may be too risky for some. Moreover, if we want to do this, it remains undecided whether earth’s motion shortens the dimensions in one direction, as it was supposed before, or elongates the length perpendicular to it, by which assumption we could reach the same result. Anyway, changes of the molecular forces, and consequently of the body’s size of order 1 – p²/2v² are possible.

As stated above, even if there is a ‘contraction’ effect, it is safe to assume that the exact effect, both in terms of dimension and extent, would vary with the difference in how any given entity type is constituted, what type of force was involved, and its ‘severity’ level on any given occasion. However, even if, coincidentally, this is true, it is irrelevant to the process being examined in this paper. What Lorentz assumed was an effect that was omnipresent, ie it resulted in the same affect on every entity, on every occasion. Put simply, Lorentz proposed a consequence, without any real consideration of its cause, to resolve an issue which did not exist. But at least he afforded the possibility of a real cause, whereas Einstein did not.

Once whatever it was ceased, entities regained their previous ‘normal equilibrium’ shape and velocity, and the generic magnitude of that alteration apparently compensated for the difference which was not manifest by the experiments. However, that was because the same mathematics was used for everything, involving √(1 – v²/V²).

He noted one issue with this idea: it remains undecided whether earth’s motion shortens the dimensions in one direction, as it was supposed, or elongates the length perpendicular to it, by which assumption we could reach the same result. And he was also circumspect about the underlying assumptions underpinning how, but not why, this might occur: Since we know nothing about the nature of molecular forces, it is impossible to verify the hypothesis. And: We cannot ascribe great importance to this result; the transfer to molecular forces of what we have found for electrical forces, may be too risky for some.

3.3.2 What Lorentz wrote

Later, when Lorentz had developed his ideas, he wrote: That we cannot speak about an absolute rest of the ether is self-evident. When I say the ether is at rest, this only means that one part of this medium does not move against the other.

To come to the basic equations for the phenomena of electricity in moving bodies, I have assumed that small electrically charged molecules exist in all bodies, and that all electric processes are based on the location and motion of these “ions”. The periodically changing polarization which forms a light ray, according to Maxwell’s theory, become vibrations of the ions in this conception.

Now according to Maxwell, two kinds of deviations from the equilibrium state can exist in the ether. First, dielectric displacement in the spaces between the ions. We now want to assume that ether exists in the space where an ion is located, and that a dielectric displacement can happen here. Second, the magnetic force that depends on the current distribution, and satisfies [a condition of the magnetic force], whose applicability we also presuppose for the interior of ponderable matter. We also assume this relation for the interior of the ions as well as for the interspaces by which, in Maxwell’s theory, the dielectric displacement is connected with the temporal variation of the magnetic force.

From now on it will be assumed that the bodies to be considered are moving at a steady velocity of translation p, under which we will have to understand in almost all applications, the speed of the earth in its motion around the sun. We introduce another one [co-ordinate system] which is rigidly connected with ponderable matter and therefore shares any displacement. By a fixed point we now understand one point that has a steady position with respect to the new axis; in the same way, by rest or motion of a physical particle we shall mean the relative rest or the relative motion in relation to ponderable matter. Real velocity is thus: the velocity of the previously mentioned relative motion plus p.

To clearly define the meaning of the above, we will compare the considered system S₁ with a second one S₂. The latter should not be moved, and it arises from S₁ by increasing all the dimensions that have the direction of the x-axis (therefore the relevant dimensions of the ions as well), in the ratio √(V² – p²) to V, or: between the coordinates x, y, z of a point of S₁ and the coordinates x’, y’, z’ of the same corresponding point of S₂, we let remain the relations: x = x’√(1 – p²/V²), y = y’, z = z’. If we apply to all magnitudes, which are related to the second system, a prime so they can be distinguished, then: p’ = p√(1 – p²/V²).

With respect to the last variable [t], these are periodic functions, if the ions carry out oscillations with constant amplitude and a common oscillation interval T. It is easy to see, that in this case the equations are satisfied by values of [dielectric displacement and magnetic force at spatial co-ordinates] which also have the period T. Therefore, the important theorem is given: If ion oscillations of period T take place in a light source, then the dielectricd displacement and the magnetic force indicate the same periodicity at each point that shares the translation of the source.

This suggests a new variable instead of t: ie t’ = t – (p_x/ V²)x – (p_y/ V²)y – (p_z/ V²)z. The transition from one time to another is provided by this equation. The variable t’ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this the local time of this point, in contrast to the general time, t.

If a state of oscillation in the moving system shall correspond to a state in the stationary system, then the relative oscillation period in the first mentioned case must be equal to the oscillation period in the second case.

In the stationary system, no motion of light may take place at an arbitrary location. From that it follows that a surface which forms the border of a space filled with light within a stationary body, can have the same meaning when the body is moving. In the moving system, relative light rays of relative oscillation period T were mirrored and refracted by the same laws as rays of the oscillations period T in the stationary system.

If we assume, that the arm lying in the direction of earth’s motion, is shorter by L.(p²/2V²) than the other one, and simultaneously the translation would have an influence which follows from Fresnel’s theory, then the result of Michelson’s experiment is fully explained. Consequently, we have to imagine that the motion of a rigid body, e.g. a brass rod or of the stone plate used in later experiments, would have an influence on the dimensions which, depending on the orientation of the body, is different, and thus it would cause a contraction in the direction of motion in the ratio of 1 to √(1 – p²/v²).

If, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Σ’, if the velocity w is imparted to it, will of itself change into the system Σ. In other terms, the translation will produce the deformation.

As strange as this hypothesis would appear, it’s not far off if we assume that the molecular forces, similarly as we now definitely can say of the electrical and magnetic forces, are transmitted through the ether. Then the translation will change the action between two molecules or atoms, similarly to the attraction or repulsion between charged particles. Since the shape and the dimensions of a fixed body are, in the last instance, determined by the intensity of the molecular effects, then a change therein is inevitable.

It is noteworthy, the we are led exactly to the above presupposed changes of dimensions, when we first (without consideration of the molecular motion) assume that in a rigid body, which remains at its own, the forces, attractions or repulsions which act on an arbitrary molecule, are mutually in equilibrium.

In reality the molecules of a body are not at rest, but there exists a stationary motion in every “equilibrium state”. How this is of influence as regards the considered phenomenon, may remain undecided.

Comment

Lorentz deemed the ether to be at rest in the sense that parts of it did not have different movements. Its actual movement was not determined. He did not say the ether was immobile. That we cannot speak about an absolute rest of the ether is self-evident. When I say the ether is at rest, this only means that one part of this medium does not move against the other.

He altered the reference for movement from the ether, which in practice it was not anyway (see Section 3.1), to ponderable bodies: Instead of the co-ordinate system used above [co-ordinate system at rest in the ether], we introduce another one which is rigidly connected with ponderable matter and therefore shares any displacement. From now on it will be assumed that the bodies are moving at a steady velocity of translation p, under which we will have to understand in almost all applications, the speed of the earth in its motion around the sun. Real velocity is thus: the velocity of the previously mentioned relative motion plus p.

His point was not that the bodies were actually stationary, but that there is an omnipresent prevailing movement, ie a steady velocity of translation p. For reference purposes this could be, in effect, regarded as ‘stationary’, as everything on earth was affected. In addition, he asserted there was one state of ‘equilibrium’, irrespective of movement: In reality the molecules of a body are not at rest, but there exists a stationary motion in every “equilibrium state”. If, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium.

What is referred to as ‘stationary’ is in fact the circumstance of sharing an omnipresent state of movement. Which for the earth is considerable, as it involves a variable rotation around its axis, an elliptical orbit around the sun, and then, as an integral part of a galaxy, the orbit about its centre and a movement through outer space. That is, the concept of ‘moving’, as opposed to ‘stationary’, bodies can be something of a misnomer.

There was a revision to his original idea: I have assumed that small electrically charged molecules exist in all bodies, and that all electric processes are based on the location and motion of these “ions”. We now want to assume that ether exists in the space where an ion is located, and that a dielectric displacement can happen here. The magnetic force, whose applicability we also presuppose for the interior of ponderable matter, and for the interior of the ions, as well as for the interspaces. In Maxwell’s theory, the dielectric displacement is connected with the temporal variation of the magnetic force.

The resulting supposed contraction meant that, for the duration of the effect, the spatial position occupied, and its time of occurrence, was altered from what would have otherwise occurred. The reference for that being the usual state of equilibrium. For this, Lorentz used t’, or local time, instead of t, x’ instead of x, and p’ instead of p. The alteration was only deemed to involve a variation to a new steady velocity in one dimension (x axis), and no alteration in direction. The variable t’ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this variable the local time of this point, in contrast to the general time t. The transition from one time to another is provided by equation t’ = t – (p_x/V²)x – (p_y/V²)y – (p_z/V²)z. [Where p = translational velocity of ponderable matter, and V = velocity of light in the ether].

Lorentz was not suggesting that time and spatial position were somehow ‘variable’. It was his method of relating the state of a body in terms of when and where it was when ‘stationary’, to that which prevailed when it was caused to ‘move’. Inevitably this transformation involved the factor √(1 – v²/V²). That resulted in a range of statements which are all of the same magnitude, and so apparently compensated each other:

1 We let remain the relations: x = x’√(1 – p²/V²), y = y’, z = z’.

2 If we apply to all magnitudes, a prime can be distinguished: p’ = p√(1 – p²/V²).

3 The displacement would of course cause this configuration by itself, and would cause a contraction in the direction of motion in the ratio of 1 to √(1-p²/V²).

Or as Lorentz expressed it: It is noteworthy, the we are led exactly to the above presupposed changes of dimensions, when we first (without consideration of the molecular motion) assume that in a rigid body, which remains at its own, the forces, attractions or repulsions which act on an arbitrary molecule, are mutually in equilibrium. And: We could say that the displacement stemming from the changes of length, is compensated by Maxwell’s displacement. Whereas, what was actually noteworthy was that the outcomes are the same because the same mathematical process was applied in all circumstances.

The ‘contraction’ was said by Lorentz not to affect oscillation rates, or the start velocity of light, because: With respect to the last variable [t], these are periodic functions if the ions carry out oscillations with constant amplitude and a common oscillation interval T. It is easy to see that in this case the equations are satisfied by values of [dielectric displacement and magnetic force at spatial co-ordinates] which also have the period T. Therefore, the important theorem is given: If ion oscillations of period T take place in a light source, then the dielectric displacement and the magnetic force indicate the same periodicity at each point that shares the translation of the source.

And: In the stationary system, no motion of light may take place at an arbitrary location. From that it follows that a surface which forms the border of a space filled with light within a stationary body, can have the same meaning when the body is moving. In the moving system, relative light rays of relative oscillation period T were mirrored and refracted by the same laws as rays of the oscillations period T in the stationary system.

Lorentz provided some justification for the latest iteration of his idea: As strange as this hypothesis would appear, it’s not far off if we assume that the molecular forces, similarly as we now definitely can say of the electrical and magnetic forces, are transmitted through the ether. Then the translation will change the action between two molecules or atoms, similarly to the attraction or repulsion between charged particles. Since the shape and the dimensions of a fixed body are, in the last instance, determined by the intensity of the molecular effects, then a change therein is inevitable.

3.3.3 What Lorentz wrote

A few years later Lorentz made the following points:

The problem of determining the influence exerted on electric and optical phenomena by a translation admits of a comparatively simple solution, so long as only those terms need be taken into account which are proportional to the first power of the ratio between the velocity of translation w and the velocity of light c. Cases in which quantities of the second order, ie w²/ c² present more difficulties. We shall further transform these formulae by a change of variables: c²(c² – w²) = k², and understanding by l another numerical quantity, to be determined further on.

Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelsons’s negative result, the introduction of a new hypothesis has been required. Inventing a special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory if it were possible to show, by means of certain fundamental assumptions, and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. I believe to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light.

I now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming kl times, and those in perpendicular direction l times, smaller. Our assumption amounts to saying that in an electrostatic system, moving with a velocity, all electrons are flattened ellipsoids with their smaller axes in the direction of motion. In this deformation each element of volume is understood to preserve its charge.

I suppose the forces between uncharged particles are influenced by a translation in the same way as the electric forces in an electrostatic system, ie whatever the nature of the particles composing a ponderable body, so long as they do not move relatively to each other, we shall have between the forces acting in a system without, and the same system with, a translation the same relation [of electric force for stationary ions]. We may draw the conclusion that the system Σ’, if the velocity w is imparted to it, will of itself change into the system Σ.

The present hypothesis is more general because the only limitation imposed on the motion is that its velocity be smaller than that of light.

In a system without translation, those parts which are dark while the system is at rest, will remain so after it has been put in motion. If in two points of a system, rays of light of the same state of polarization are propagated in the same direction, the ratio between the amplitudes in these points may be shown not to be altered by a translation.

The above conclusions confirm the results formerly obtained by a similar train of reasoning in which the terms of the second order were neglected. They also contain an explanation of Michelson’s negative result, more general and of somewhat different form than the one previously given.

Comment

Overall, the objective of Lorentz was summed up by: It would be more satisfactory if it were possible to show by means of certain fundamental assumptions, and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. I believe to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light.

His start point was: I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming kl times, and those in perpendicular direction l times, smaller. We shall further transform these formulae by a change of variables: c²(c² – w²) = k². I take as new independent variables: x’ = klx, y’ = ly, z’ = lz. So, t’ = (l/k)t – (kl)(w/c²)x. The coefficient l, for small values of w, differs from unity no more than by an amount of the second order. The variable t’ may be called the local time; indeed, for k = 1, l = 1. [Where: the system as a whole moves in the direction of x with a constant velocity w, and, the velocity of light c]. Other sentences which illuminate the variables k & l are:

We want to say k, locations that are uniformly distributed over the sphere l.
The product kl is a function of w.
k and l differ from unity by quantities of the order w²/c².
The value of the constant must be unity, because we know already that for w=0, l=1.

Our assumption amounts to saying that in an electrostatic system, moving with a velocity, all electrons are flattened ellipsoids with their smaller axes in the direction of motion. In this deformation each element of volume is understood to preserve its charge.

His conclusion was: the influence of a translation on the dimensions (of the separate electrons and of a ponderable body as a whole) is confined to those that have the direction of the motion, these becoming k times smaller than they are in the state of rest.

From the perspective of this paper, the core of what he had written earlier is unchanged. The fundamental point being that electromagnetic actions are entirely independent of the motion of the system. That included light, ie its start speed was always the same. A mathematical consequence of his hypothesis was that: the only restriction as regards the velocity will be that it be smaller than that of light. Being a function of mathematics is not the same as evidence, which proves that no entity can travel faster than light.

It should be noted that in the Lorentz paper of 1895 the author found no reference to c as the speed of light, nor any reference to the hypothetical speed of light in a vacuum. In that paper, the speed of light was denoted as V. In the Lorentz paper of 1904, the speed of light is denoted as c, though no mention is made of the light being in a vacuum, and that c is ‘adjusted’ by w (translational velocity of the system) where appropriate.

Whether what he actually wrote was correct or not, given knowledge either at the time or now, Lorentz did not assert most of what was subsequently attributed to him by Einstein. See Section 3.5. As was his style, he concluded with the caveat: Our assumption about the contraction of the electrons cannot in itself be pronounced to be either plausible or inadmissible. What we know about the nature of electrons is very little.

Hall of Mirrors: Section 3.4: Poincaré

3.4.1 What Poincaré wrote
After pages of argument Poincaré states the following conclusions:

1. There is no absolute space, and we only conceive of relative motion.

2. There is no absolute time. When we say that two periods are equal, the statement has no meaning. We have no direct intuition of the simultaneity of two events occurring in two different places.

These statements are invalid, as explained in Section 2.1.

3.4.2 What Poincaré wrote
He then went on to assert that:
The principle of relativity [of motion], according to which the laws of physical phenomena must be the same for a stationary observer as for an observer carried along in a uniform motion of translation; so that we have not, and cannot have, any means of discerning whether or not we are carried along in such a motion.

This misinterpreted Galileo, and resulted in a false equivalence of a certain mode of motion. Also, as there is no physical relationship between reality and any given reference, the laws of physical phenomena cannot be affected by, or have an effect on, what actually happened. Physical existence has already occurred when its manifestation is received.

The principle of relativity of motion, as expressed by Poincaré, and subsequently Einstein, is an invalid concept. The laws of physical phenomena must be valid statements based on proven direct experience, or logical deduction therefrom, about actual physical events. Not as defined with respect to some arbitrary reference. These occur irrespective of whether they are sensed or not, or the state of motion of the entity receiving representations of them. That is, it is irrelevant what sensory system is involved, or whether the recipient entity is human/non-human, stationary/moving, in respect of what actually occurred.

The point Galileo made in 1632, when he described a thought experiment involving a ship, was to counter ideas that the earth was stationary. An argument for that being if the earth was moving, then objects would fall differently according to circumstances. Galileo imagined being in a ship’s cabin observing a variety of simple motions whilst the ship was ‘stationary’, as in with respect to the earth. He then suggested that if these motions were repeated with the ship moving at a uniform speed, the effects would be the same. The purpose of this was to demonstrate that if the earth is moving that would not necessarily be noticed, unless some other clue was available, as the motion of the earth is common to all things associated with it, which can give the false impression that it is stationary.

Observation was invoked for obvious reasons, and not because that had some different generic characteristic compared to all other sensory systems. Similarly, the proposed alternative motion to being ‘stationary’ (eg in the harbour) was a uniform translation (eg calm seas), because otherwise it would be known that the ship was moving. But more importantly, the experiments could be affected. That is, what sensory system was used was irrelevant, ie this was not about some characteristic of light, and the choice of alternative motion to ‘stationary’ was purely so that the experiments were not affected by that very movement, ie this was not concerned with ‘inertial frames of reference’.

Einstein’s version of Galileo’s ship analogy was: The pillar upon which the “theory of relativity” rests is the so-called principle of relativity: Picture two physicists. Let both be equipped with every physical instrument imaginable, and have a laboratory. Suppose the laboratory of one of the physicists is arranged somewhere in an open field, and that of the second in a railroad car traveling at constant velocity in a given direction. The principle of relativity states the following: if, using all their equipment, these two physicists were to study all the laws of nature, one in his stationary laboratory and the other in his laboratory on the train, they would discover exactly the same laws of nature, provided that the train is not shaking and is travelling in uniform motion. We can say: according to the principle of relativity, the laws of nature are independent of the translational motion of the reference system, and are identical in all acceleration-free systems.

The fact that Einstein has misunderstood the point Galileo was making, and how proper referencing is applied, is demonstrated by: provided that the train is not shaking and is travelling in uniform motion. Both Poincaré and Einstein thought that sensing had an effect on what they regarded as the actuality, and that this was negated so long as the reference was in a certain mode of motion. Both notions are invalid. This falsehood was repeated in General Relativity (1916) with two examples, a man on a train, and a man in a chest.

3.4.3 What Poincaré wrote
The most ingenious idea was that of local time. Imagine two observers who wish to adjust their timepieces by optical signals. When station B perceives the signal from station A, its clock should not mark the same hour as that of station A at the moment of sending the signal, but this hour augmented by a constant representing the duration of the transmission. Suppose, for example, that station A sends its signal when its clock marks the hour 0, and that station B perceives it when its clock marks the hour t. The clocks are adjusted if the slowness equal to t represents the duration of the transmission, and to verify it, station B sends in its turn a signal when its clock marks 0; then station A should perceive it when its clock marks t. The timepieces are then adjusted.

They mark the same hour at the same physical instant, but on the one condition, that the two stations are fixed. Otherwise the duration of the transmission will not be the same in the two senses, since the station A, for example, moves forward to meet the optical perturbation emanating from B, whereas the station B flees before the perturbation emanating from A. The watches adjusted in that way will not mark, therefore, the true time; they will mark what may be called the local time, so that one of them will be slow of the other. It matters little, since we have no means of perceiving it. All the phenomena which happen at A, for example, will be late, but all will be equally so, and the observer will not perceive it, since his watch is slow; so, as the principle of relativity requires, he will have no means of knowing whether he is at rest or in absolute motion.

This is invalid, because it involves de-synchronisation of timing devices. See Section 4.5. Einstein repeated this methodology in Part 1, paras 1 & 2 in 1905. His conclusion being: So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system. This error is similar to the above. On this occasion, the improper use of referencing has resulted in reality being ‘altered’ in respect of when the events occurred, which was actually at the same time.

Neither does it represent Lorentz’s concept of local time anyway, as explained in Section 3.3. His supposed contraction in shape meant that, for the duration of the effect, the spatial position occupied, and its time of occurrence, was different from what would have otherwise happened. Lorentz: The variable t’ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this variable the local time of this point, in contrast to the general time t. The transition from one time to another is provided by equation t’ = t – (p_x/V²)x – (p_y/V²)y – (p_z/V²)z. [Where p = translational velocity of ponderable matter, and V = velocity of light in the ether].

Finally, in 1905, Poincaré wrote: An explanation was proposed by Lorentz and FitzGerald, who introduced the hypothesis of a contraction by all bodies in the direction of the motion of earth, proportional to the square of aberration; this would give an account of the experiment of Michelson and all those which were carried out up to now. The hypothesis would become insufficient, however, if one were to assume the postulate of relativity in all its generality.

If one wants to preserve Lorentz’s theory, it is necessary to suppose a special force which explains at the same time the contraction, and the constancy of two of the axes. I sought to determine this force, I found that it can be compared to a constant external pressure, acting on the deformable and compressible electron, and whose work is proportional to the variations of the volume of the electron.

Having criticised Lorentz for deploying an additional hypothesis when some new feature arose, Poincaré did just that. In doing so, he demonstrated that he considered Lorentz’s explanation, ie contraction, of the (non-existent) issue with the Michelson experiments to be fundamentally correct, but just in need of some further refinement.

Hall of Mirrors: Section 3.5: Einstein

3.5.1 Einstein quotes.
The importance of Lorentz in Einstein’s own words, is:
The development [of the Special Theory of Relativity] starts off from the ideas of Faraday and Maxwell, according to which all physical processes involve a continuity of action, or, in the language of mathematics, they are expressed by partial differential equations. Maxwell succeeded in doing this for electro-magnetic processes in bodies at rest. The extension of electro-dynamics to the case of moving bodies fell to the lot of Maxwell’s successors.

From Fizeau’s experiment one had to conclude that the ether is not carried along completely by matter in motion, but instead there occurs a relative displacement. The earth being a body that rotates around its axis, and revolves around the sun, with velocities that change their directions, one should expect a spatial anisotropy in optical phenomena. Thus, in vacuum or in the atmosphere, light should propagate faster in the direction of the earth’s motion than in the opposite direction.

Experimental verification of this consequence was unthinkable, because the order of magnitude of the term considered is that of the ratio of the velocity of the earth to the velocity of light. Also, and this is a most important point, all terrestrial methods for measuring the velocity of light employ light rays that travel along a closed (back and forth) rather than a simple path, due to the fact that the times of departure and arrival of the rays must be determined with the help of one and the same device.

At this point, Lorentz came to the rescue. He achieved this by taking from ether its mechanical, and from matter its electromagnetic qualities. As in empty space, so too in the interior of material bodies, the ether was exclusively the seat of electromagnetic fields. Assuming that ether is completely immobile, Lorentz conceived a theory of electromagnetic phenomena. The special theory of relativity does not compel us to deny ether. We may assume [its] existence; only we must give up ascribing a definite state of motion to it.

The [Lorentz] theory appeared to be unsatisfactory only in one point. It appeared to give preference to one system of coordinates of a particular state of motion (at rest relative to the ether) as against all other systems of co-ordinates in motion with respect to this one. In this point the theory seemed to stand in direct opposition to classical mechanics, in which all inertial systems which are in uniform motion with respect to each other are equally justifiable as systems of co-ordinates. The Special Theory of Relativity owes its origin to this difficulty.

Lorentz and FitzGerald resorted to a strange hypothesis: they assumed that each body in motion, with respect to the ether, contracts in the direction of motion by a fraction equal to ½(v/c)², or, which amounts to the same if only terms of second order are considered, that the length of the body is diminished in that direction in the ratio: 1 to Ö(1-v²/c²). This hypothesis supplies us then with that particular law of motion.

The [Special] theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. It turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty. One had to realize that an auxiliary quantity, “local time”, could be defined as “time” in general. If one adheres to this definition of time, the basic equations of Lorentz’s theory correspond to the principle of relativity, provided that the [Galilean] transformation equations are replaced by ones that correspond to this new conception of time.

On the basis of the [Special] theory of relativity the method of interpretation is more satisfactory. There is no such thing as a “specially favoured” co-ordinate system. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point.

3.5.2 Comment

In respect of the first paragraph, the author raises the issue of what constitutes movement, as everything is moving? Earth movements are varied, and considerable, but the underlying ones are omnipresent. So, a differentiation between what is ‘at rest’, and what is ‘moving’ and how, can be misconceived. Only a proper system of referencing can identify what is actually happening. See Section 2.1.

Misinterpretation of Galileo, see Section 3.4.2, lead to the presumption that, so long as something was in the mode of uniform translational motion irrespective of velocity, then when something else, with a constant movement, was referenced to it, the outcome must always be that constant. For this notion to be true, everything involved would have to be in a fixed spatial relationship, not just uniform translational motion. In which case the idea is meaningless. This resulted in the false notion of inertial frames of reference.

When contrasting Lorentz’s hypothesis with classical mechanics, this misinterpretation becomes apparent: In this point the theory seemed to stand in direct opposition to classical mechanics, in which all inertial systems which are in uniform motion with respect to each other are equally justifiable as systems of co-ordinates. The Special Theory of Relativity owes its origin to this difficulty. This depiction of classical mechanics is incorrect, and the Special Theory of Relativity involves the fundamental flaw based on the misinterpretation of Galileo.

What became known as The Special Theory of Relativity was contained in the first part, the Kinematical Part, of the 1905 paper entitled: On the electrodynamics of moving bodies. According to Einstein this hypothesis: is based on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters. The issue with this being that there is no physical relationship between the electrodynamics of bodies (rigid or otherwise) and clocks, ie time, and systems of co-ordinates, ie space. The author assumes that after over 100 years, the actual effect of any given motion on any given ‘body’, in respect of electrodynamics, etc, is understood.

Although it only becomes clear in later papers, what became known as Relativity in special circumstances was an alternative explanation to Lorentz. This [Lorentz] hypothesis supplies us then with that particular law of motion [length of the body is diminished in that direction in the ratio: 1 to Ö(1-v²/c²)]. The [Special] theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. It turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty. One had to realize that an auxiliary quantity, “local time”, could be defined as “time” in general. And: Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point.

Lorentz’s hypothesis of contraction attempted to explain the apparent failure of Michelson’s experiments, which needed no explanation since those experiments were never going to manifest the outcomes that were expected, though the underlying notion was correct. Hence, there was no necessity to find some previously unknown factor.

For Lorentz, the contraction, albeit limited to one dimension and a particular direction, was not the function of referencing, but real: the motion of a rigid body, eg, a brass rod or the stone plate used in later experiments, would have an influence on the dimensions which, depending on the orientation of the body with respect to the direction of motion, is different, and thus cause a contraction in the direction of motion in the ratio of 1 to Ö(1 – p²/v²). Or: A change in length of the arms in the first experiment, and the stone plate in the second, is not inconceivable. Or: The displacement would cause a contraction in the direction of motion in the ratio of 1 to √(1-p²/V²).

Whilst it was implicit, Lorentz’s hypothesis presumed the existence of a force which was responsible for the variation in movement. In Einstein’s alternative hypothesis no such force was presumed, and the ‘contraction’ was not real, but explainable by other means: not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Yet again, what was thought to be reality, was deemed alterable with referencing. Any given motion can be described with respect to any given reference, but that is pointless, as it mostly does not define the actual.

As previously stated, there could be a force and a consequent alteration in shape, though not as an explanation for the discrepancy between expected outcome and experimental results. In comparing velocities, and different processes, both of them employed the same mathematics, ie use of the factor √(1 – v²/V²), as explained in Section 3.2.1.

Comparing his solution with Lorentz’s, Einstein thought: The method of interpretation is more satisfactory. Here the contraction of moving bodies follows from the two fundamental principles of the theory [Special] without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point.

There are four key phrases in the above quotes: First, with respect to the ether. Second: [Lorentz] assuming that ether is completely immobile. Third: one had only to realize that an auxiliary quantity introduced by Lorentz, “local time”, could be defined as “time” in general. Fourth: without the introduction of particular hypotheses.

1 With respect to the ether

As explained in Sections 3.1 and 3.3, even when Lorentz did refer to ‘with respect to the ether’, the mathematics did not reflect that, neither was the ether relevant to his argument. Subsequently, Lorentz explicitly stated that the reference was: Instead of the co-ordinate system used above [coordinate system that is at rest in the ether], we introduce another one which is rigidly connected with ponderable matter and therefore shares any displacement. By a fixed point, we now understand one point that has a steady position with respect to the new axis. From now on it will be assumed that the bodies are moving at a steady velocity of translation p, under which we will have to understand in almost all applications, the speed of the Earth in its motion around the sun. Real velocity is thus: the velocity of the previously mentioned relative motion plus p.

2 Ether is immobile

Einstein falsely attributed how Lorentz involved the ether, both in respect of its use as a reference, see above, and its immobility. Lorentz: That we cannot speak about an absolute rest of the ether is self-evident. When I say the ether is at rest, this only means that one part of this medium does not move against the other.

Einstein’s concern with the ether was its potential, according to him, to provide a reference which would contravene his own thinking, which is characterised by: There is no such thing as a “specially favoured” co-ordinate system. The corollary being: ‘everything is relative’.

Is it really impossible to reconcile the essential foundations of Lorentz’s theory with the principle of relativity? If we wish to attempt such a reconciliation, the first step we must take is to give up the ether. For, on the one hand, we have been obliged to admit that the ether is stationary, whereas, on the other hand, the principle of relativity demands that the laws of natural phenomena referred to a uniformly moving coordinate system S’, be identical with the laws of these same phenomena referred to a system S, at rest with respect to the ether. But there is no reason to assume the immobility of ether any more with respect to the system S’, than with respect to the system S; these two systems cannot be distinguished from each other, and it is therefore improper to make one of them play a special role by saying that it is at rest with respect to the ether. It follows that the only way to arrive at a satisfactory theory is to give up the notion of a medium filling all of space.

Leaving aside the fact that there had been no obligation to admit that the ether is stationary, see above, the argument that followed was false anyway. The principle of relativity does not demand that the laws of natural phenomena referred to a uniformly moving coordinate system S’, be identical with the laws of these same phenomena referred to a system S, at rest with respect to the ether. Or another way of expressing that is that this interpretation of a principle of relativity is invalid. This is the consequence of the Galileo misinterpretation.

Furthermore, it is irrelevant what the motion of the ether is, because while the motion of S and S’ can be depicted in many different forms depending upon which reference is used, that proves nothing. What needs to be known is their actual motion in reality, using a referencing system which accords with the nature of our closed system. Finally, giving up the notion of a medium filling all of space is not the way to arrive at a satisfactory theory, but the way to fulfil Einstein’s flawed ideas. Giving up the ether does not lead to a theory, satisfactory or otherwise. Ether was given up because it was irrelevant to the circumstance being considered, ie light and bodies in states of motion.

Further, it is easy to see that we must also abandon the idea of introducing a luminiferous ether into the theory. Because, if every light ray in vacuum is supposed to propagate with the velocity c, with respect to K, then we must conceive of this luminiferous ether as being everywhere at rest with respect to K. However, if the laws of propagation of light with respect to the system K’ (in motion relative to K), are the same as those with respect to K, then we would have to assume, with as much right, the existence of a luminiferous ether that is at rest with respect to K’. Since it is absurd to assume that the luminiferous ether is simultaneously at rest with respect to both systems, and since it would be hardly less absurd for the theory to privilege one of the two (or of infinitely many) physically equivalent systems over the others, one must dispense with that concept, which was, anyway, just a useless accessory of the theory ever since the mechanical interpretation of light had been abandoned.

This is another false argument, involving one amongst several characteristics that Einstein incorrectly attributed to Lorentz, which he then used to justify his alternative hypothesis. What Lorentz said about light is misrepresented, which was that it always started, not continued, at c, and that the previous movement of that light, and what it reacted with, were irrelevant. Moreover, the transmitted speed could not be c, as the bodies were not in a vacuum. Then there is a repeat of the flawed argument with k, based on the Galileo misinterpretation, by which Einstein attempts to justify ‘giving up the ether’.

Einstein claimed: It may be added that the whole change in the conception of the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility. This is incorrect. For both Lorentz and Einstein, given what was being considered, the ether was irrelevant. Einstein admitted this: the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it.

3 ‘local time’ could be defined as ‘time’ in general

As explained Section 3.3.2, the statement: one had only to realize that an auxiliary quantity introduced by Lorentz, “local time”, could be defined as “time” in general, is incorrect. Lorentz’s concept of local time (and local spatial position) was valid, given his view that under certain conditions the shape of bodies altered, and therefore the time and position of any such affected entity would be different from what it would otherwise have been.

As stated in Section 3.4.3, Poincaré gave an example on how to synchronise timing devices based on his perspective that: There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention. This methodology actually de-synchronised them. Einstein used this in Part 1 of the 1905 paper to justify the supposed relativity of time and space. See Sections 4.5 and 4.6.

4 Without the introduction of particular hypotheses

Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point.

Einstein implies that his alternative explanation, Relativity in special circumstances, must be correct as it does not involve the introduction of particular hypotheses. This is untrue, as it involved the misinterpretation of two fundamental principles, which led to the abandonment of a third valid principle, the invalid use of c as a universal reference point and instead of an ‘atmospheric’ speed of light, introduction of an incorrect methodology for calibrating time, space, and motion, and no gravitational effect. That is, Part 1 1905, implied an entirely new construct of physical existence, as noted by Minkowski.

Hall of Mirrors: Section 4: The Reconciliation Argument

Einstein did not fully explain the underlying argument that substantiated his new approach in the paper On the electrodynamics of moving bodies (1905). It is available in papers written between 1905 and 1916, and those written after that which review the entire development of Relativity. Quotes from those are provided in the next two sub sections, followed by commentary on the argument revealed therein.

Hall of Mirrors: Section 4.1: Introduction

In the Introduction to On the electrodynamics of moving bodies (1905) Einstein proposed two postulates which would suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies. The two postulates were:
1 The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. Principle of Relativity.
2 Light is always propagated in empty space with a definite velocity c, which is independent of the state of motion of the emitting body. Principle of constancy of the velocity of light.

Einstein stated that these two postulates were only apparently irreconcilable. But no proper explanation was given in that paper as to why these two principles could be considered irreconcilable in the first place, or why this was not so.

Einstein: The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place. The theory to be developed is based, like all electrodynamics, on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

The Lorentz explanation to the perceived issue, for which Einstein proposed this alternative hypothesis, was not concerned with electrodynamics. Neither did it involve the notion that the laws of optics will be valid for all frames of reference for which the equations of mechanics hold good, a reference with respect to the ether, an absolutely stationary space, or assign a velocity-vector to a point of the empty space.

Neither did Lorentz consider anything to be absolutely stationary. For him, stationary was the existent state of equilibrium that normally prevailed, before something temporarily caused an alteration therein. It still involved movement, but that was omnipresent. In respect of the constancy of light, his point was that only the start speed was c, since the interaction always involved the same process. After that, transmission speed depended on conditions.

The conclusion to the Kinematical Part, Part 1, 1905 was: We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.
Einstein then proceeded to apply this to electrodynamics, ie using the new transformation equations on any so-called ‘moving’ body: If we apply to these equations the transformation developed in para 3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations…And: Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations…

The problem with this is that there is no direct physical relationship between the electrodynamics of bodies (rigid or otherwise), and clocks, ie time, and systems of co-ordinates, ie space. Electrodynamics, and similar effects, are not a function of references.

Hall of Mirrors: Section 4.4: The Argument

4.4.1: Introduction
The principle of constancy of the velocity of light, and the relativity principle, are incompatible with each other only as long as one holds to the postulate of absolute time. It turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty. [Then] proceeding from these two principles one arrives at the theory designated as the “theory of relativity”.

Every reference-body (co-ordinate system) has its own particular time. Unless we are told the body to which the statement of time refers, there is no meaning in a statement of the time of an event. Before the advent of the theory of relativity it had always been assumed that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference.

One had only to realize that an auxiliary quantity introduced by Lorentz and named by him “local time” could be defined as “time” in general. If one adheres to this definition of time, the basic equations of Lorentz’s theory correspond to the principle of relativity, and the [Galilean] transformation equations are replaced by [those].

We call this “The Special Relativity Principle”. By the word special, it is signified that the principle is limited to the case when K′ has uniform translatory motion with reference to K, the equivalence of K and K′ does not extend to non-uniform motion.

The Special Relativity Theory does not differ from classical mechanics through the assumption of this [relativity] postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate gives, in a well-known way, the relativity of synchronism, as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

Einstein sometimes started with a comment about the classical approach, in order to distinguish it from his proposed ‘relativistic’ one, which he thought should replace it. The proponents of classism included Newton, Euclid, Descartes, and Galileo. Poincaré thought they invoked a ‘convention’, ie a construct imposed upon reality. Poincaré (1902): There is no absolute space. We only conceive of relative motion; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred. There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention. We have not even direct intuition of the simultaneity of two events occurring in two different places. Is not our Euclidean geometry in itself only a kind of convention of language? Thus, absolute space, absolute time, and geometry are not conditions imposed on mechanics.

As explained in Section 2.1 this is invalid. This flawed perspective underpinned the Kinematic Part, Part 1 of Einstein’s 1905 paper, subsequently known as Special Relativity. Einstein incorrectly depicted the theories of Newton, et al, as inadequate since they only involved one particular circumstance, which they did, but it was reality. The issue, for him, being that any such singularity ‘must’ be a ‘privileged’ reference, and hence not allowable, as ‘everything is relative’. While the classicists reflected the invalid ‘everyday’ perspective that physical existence comprises ‘things’ and ‘change’ to them, their modelling imposed a ‘freeze’ on that. So, albeit inadvertently, the classicists depiction was correct.

Newtonian laws of motion transform to laws of the same form when one passes from one co-ordinate system to another because they reflected reality, as per the closed system, ie not a ‘reality’ with respect to any reference that might be chosen arbitrarily. There is no physical relationship between reality and any such reference. Reality cannot be influenced by whatever it is referred to. Relativity is not an inherent characteristic of physical existence. So, Einstein’s concept cannot be depicting reality, but something else, ie the reality of light-based representations of reality. Einstein not only invoked an invalid referencing system to determine reality, but conflated reality with physically existent representations thereof. His ‘new kinematics’ was actually concerned with observation, not reality.

Einstein’s Special Relativity involved:
-only uniform rectilinear, and non-rotary, motion
-only fixed shaped bodies
-only light in a vacuum which travelled in straight lines at a constant speed (c)
-no gravitational force (or if there was, then it was of a type that had no effect, ie the equivalent of there being none).

That is, it was not the concept of Relativity that was ‘special’, but the circumstances within which it prevailed, which were subsequently abandoned in order to invoke general Relativity.

These quotes sum up the concept of Relativity:
There is no such thing as a “specially favoured” co-ordinate system (ie everything is relative)
Every reference-body (co-ordinate system) has its own particular time. Unless we are told the body to which the statement of time refers, there is no meaning in a statement of the time of an event. Before the advent of the theory of relativity it had always been assumed that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. (ie Relativity of simultaneity)

On the basis of the [Special] theory of relativity the method of interpretation is more satisfactory. Here the contraction of moving bodies follows from the two fundamental principles of the theory; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. (ie Relativity of distance)

The core of Einstein’s reconciliation argument was:
Strictly speaking, however, we have only learned that the following three things are incompatible with one another:
(a) the relativity principle
(b) the principle of the constancy of the velocity of light
(c) the transformation equations or the law of the parallelogram of velocities or the rule of addition of velocities
One arrives at the theory that is now called “the theory of relativity” by keeping
(a) and (b) but rejecting (c).
So that the theory based on these two principles should not lead to contradictory results, one must renounce the rule of addition of velocities. Well founded as this rule may seem to be at first glance, it conceals no less than two arbitrary hypotheses, which hold sway over all of kinematics.

A subset of this argument reflects the misinterpretation of Galileo. See Section 3.4.2. Whereby, anything moving in the mode of uniform translational motion, irrespective of individual velocity, was deemed to be equivalent as a reference. That led to the false notion of inertial frames of reference. This is exemplified by: The Special Relativity Theory does not differ from classical mechanics through the assumption of this [relativity] postulate. Or: Newtonian laws of motion transform to laws of the same form when one passes from one coordinate system to another one that is in uniform translational motion with respect to the first. This is the property we express when we say that the principle of relativity is satisfied in classical mechanics. Or: In addition to K, all bodies of reference K’ should be given preference in this sense, they should be equivalent to K for the formulation of natural laws, provided that they are in a state of uniform rectilinear and non-rotary motion with respect to K. The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity.

When comparing two movements in reality, the time of occurrence for both must be the same, as a non-existent entity cannot have a spatial position, ie t must equal t’. Having misinterpreted Galileo, Einstein then had to resolve the consequence, ie certain references being deemed equivalent when they were not. This prompted a revision of the measurement methodology for time, and hence space, which compensated for the original error, since the same mathematics was involved, ie √(1 – v²/V²).

More searching analysis of the physical significance of space and time rendered it evident that the Galileo transformation is founded on arbitrary assumptions, in particular the assumption that the statement of simultaneity has a meaning which is independent of the state of motion of the system of co-ordinates used.

These arbitrary assumptions were, supposedly, in respect of time:
t’ = t. The last of these equations expresses the assumption that temporal determinations have a meaning independent of the state of motion (assumption of “absolute time”).

We have no right to assume, a priori, that the clocks of the two groups can be set in a manner that the two time coordinates of the elementary event would be the same, or in other words, in a way that t would be equal to t’. To assume this would mean to introduce an arbitrary hypothesis which has been introduced into kinematics up to the present time.

In respect of space these arbitrary assumptions were, supposedly:
What should we understand by the “length of the bar”? One is at first inclined to believe that this concept does not require any special definition. However, we will immediately see that nothing of the sort is true, if we consider the following two methods of determining the length of the bar… All that one can deduct from the principle of relativity is that the two methods lead to the same numerical value for the length only when the bar AB is at rest relative to the system S. But in no way is it possible to assert that the second method yields a value for the length independently of the velocity v of the bar.

Or: If the configuration of a body in uniform translational motion with respect to S, is determined by ordinary geometric methods, by means of measuring rods, or other solid bodies, moving in exactly the same way, the measurements turn out to be independent of the velocity v of the translation: these results give us what we will call the geometric configuration of the body. By contrast, if one marks in the system S the positions of various points of the body at a given instant, and determines the configuration formed by these points by geometric measurements using measuring rods at rest with respect to S, one obtains what we will call the kinematic configuration of the body with respect to S. The second hypothesis used unconsciously in kinematics is: The kinematic and the geometric configuration are identical. [Whereas] the kinematic configuration of a body in uniform translational motion with respect to a co-ordinate system depends on the velocity v of the translation. [ In Einstein’s language, the geometric configuration is reality, while the kinematic configuration is the reality of light-based representations of reality].

Hence, Einstein’s new alternative presumptions were based on:
For time: Every reference-body (co-ordinate system) has its own particular time. Unless we are told the body to which the statement of time refers, there is no meaning in a statement of the time of an event. Before the advent of the theory of relativity it had always been assumed that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference.

For space: On the basis of the [Special] theory of relativity the method of interpretation is more satisfactory. Here the contraction of moving bodies follows from the two fundamental principles of the theory; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point.

Which resulted in:
This shows that one cannot reject, a priori, the possibility that the concept of spatial distance might also possess only a relative meaning. Thus, in addition to the “relativity of time”, we must also admit a “relativity of lengths”. This shatters the foundation on which the original transformation equations for spatial coordinates, and time values, are based. In the theory of relativity, the place of those equations is taken by equations that simultaneously satisfy the principle of relativity and the principle of constancy of the velocity of light. One finds the new equations by formulating mathematically the requirement that every light ray should propagate with the same velocity c in both systems K and K’. In this way one arrives at: x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, t’ = (t-(v/c²)x)/√(1 – v²/c²)

This is invalid. While the fact that reality is only manifest via representations of it is a statement of the obvious, it needs to be overtly stated, the consequences determined, and a theory formulated. The differentiation between reality, and the reality of representations of it, must be maintained. Furthermore, the underlying flaws in Einstein’s hypothesis were reified when Minkowski alighted on the supposed implications of the Kinematic Part 1, and developed a new model of physical existence, which is invalid. See Section 5.

Einstein’s misconceived notion of Relativity led to what might be termed the ‘localisation’ of time, which is why Einstein and Poincaré noted, and then misrepresented, Lorentz’s concept of local time. See Section 3.3. Einstein: It turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty. One had to realize that an auxiliary quantity [by Lorentz], “local time”, could be defined as “time” in general. If one adheres to this definition of time, the basic equations of Lorentz’s theory correspond to the principle of relativity, provided that the [Galilean] transformation equations are replaced. Poincaré: The most ingenious idea [of Lorentz] was that of local time.

Einstein’s new approach to timing needed a reference. He justified his choice as follows:
The theory of relativity is often criticized for giving, without justification, a central theoretical role to the propagation of light, in that it founds the concept of time upon the law of propagation of light. The situation, however, is somewhat as follows. In order to give physical significance to the concept of time, processes of some kind are required which enable relations to be established between different places. It is immaterial what kind of processes one chooses. It is advantageous, however, for the theory to choose only those processes concerning which we know something certain. This holds for the propagation of light in vacuo in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and Lorentz.

Or: For these signals we can use, for example, sound waves that propagate between A and B, through a medium that is at rest with respect to these points. We can use light rays propagating through the vacuum, or through a homogeneous medium at rest with respect to A and B. It does not make any difference whether we choose this or that kind of signals. Still, of all the signals that can be used, we are going to prefer those that make use of light rays propagating in the vacuum, because the synchronization requires that the path out, and the path back, be equivalent, and in our case this equivalence is satisfied, by definition, by virtue of the principle of the constancy of the velocity of light, as light in the vacuum always propagates with the velocity c.

Einstein’s falsehood about timing is encapsulated in: Before the advent of the theory of relativity it had always been assumed that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. A proper timing system does not assume time had an absolute significance, it just invokes a universal reference against which events can be registered. The notion of absolute significance is a function of the nature of reality, which independently occurs at a singular time, and not with respect to an arbitrary reference. In order to measure that, a constant, high frequency, numeric sequence of ‘ticks’ is required as the reference.

It has no relationship with any body of reference, let alone its state of motion. There was no need to give physical significance to the concept of time in the first place, which having done so then required processes of some kind which enable relations to be established between different places. The theory of relativity relied on a relationship which does not exist, resulting in a new methodology for timing which is invalid.

Einstein’s system of timing, first stated by Poincaré in 1900, involved:
Einstein: By an elementary event, we understand an event that is concentrated in one point, and is of infinitely short duration. By the time coordinate of an elementary event, we understand the indication, at the instant of the event’s occurrence, of a clock that is situated infinitely close to the point where the event takes place. An elementary event is defined by four coordinates, the time coordinate, and the three spatial coordinates.

The first hypothesis we wish to discuss concerns the physical notion of time measurement. The meaning of this definition is perfectly clear as long as the clock is sufficiently close to the place at which the event occurs. If the event is taking place in some corner far away from the clock, then it will no longer be possible to establish immediately a correspondence between the different phases of the event, and the different positions of the clock’s hands. If we choose a sufficiently large number of clocks, so that we can ascribe to each of them a sufficiently small domain, we will be able to fix any instant whatsoever, at any location in space, to any degree of accuracy desired. But we cannot obtain in this manner a definition of time useful to a physicist, because we did not say what the position of the clock hands should be at a given instant of time at different spatial points. We forgot to synchronize our clocks. To get a complete physical definition of time, we have to say in what manner all of the clocks have been set at the start. We proceed as follows:…

There is only one way of regulating the clock at B against the clock at A, in such a manner that the signal travelling from A to B would take the same amount of time as the signal travelling from B to A. That is: The reading of the clock at B at the moment the signal AB arrives at B (ie t(b)) minus the reading of the clock at A at the moment the signal AB leaves A (ie t(a)) equals the reading of the clock at A at the moment the signal AB arrives at A (ie t‘(a)) minus the reading of the clock at B at the moment the signal BA leaves B (ie t‘(b)), ie t(b) – t(a) = t’(a) – t’(b). Once the synchronization has been done, the totality of the readings in phase with one another is what we will call the physical time.

Poincaré: Imagine two observers who wish to adjust their timepieces by optical signals. When station B perceives the signal from station A, its clock should not mark the same hour as that of station A at the moment of sending the signal, but this hour augmented by a constant representing the duration of the transmission. Suppose, for example, that station A sends its signal when its clock marks the hour 0, and that station B perceives it when its clock marks the hour t. The clocks are adjusted if the slowness equal to t represents the duration of the transmission, and to verify it, station B sends in its turn a signal when its clock marks 0; then station A should perceive it when its clock marks t. The timepieces are then adjusted.

They mark the same hour at the same physical instant, but on the one condition, that the two stations are fixed. The watches adjusted in that way will not mark, therefore, the true time; they will mark what may be called the local time, so that one of them will be slow of the other. It matters little, since we have no means of perceiving it. All the phenomena which happen at A, for example, will be late, but all will be equally so, and the observer will not perceive it, since his watch is slow; so, as the principle of relativity requires, he will have no means of knowing whether he is at rest or in absolute motion.

This synchronisation, and hence the timing system based on it, meant that the given time for any given entity became a function of relative spatial position at that time. In other words, instead of t = t’ which represents reality, t = t’ with a delay equal to the transmission time in-built. Whilst invalid anyway as a system, this also required what was involved to maintain their spatial relationship during transmission, otherwise it was useless. Poincaré: but on the one condition, that the two stations are fixed.

This new timing methodology led to the following conclusions:
Evaluated from K, the time between two strokes of the clock is equal to: t = 1/√(1 – v²/c²), and thus longer than one second. A clock traveling with the velocity v, runs slower, judged from a non-comoving system, than the same clock when it does not travel. Generalizing, one can conclude: Every event in a physical system slows down if the system is set into translational motion. But this occurs only from the standpoint of a non-comoving coordinate system.

The kinematic configuration of a body in uniform translational motion with respect to a co-ordinate system depends on the velocity v of the translation. The kinematic configuration differs from the geometric configuration solely by a contraction in the direction of the motion, a contraction which is in the ratio 1:√(1-v²/c²).

Once it is understood that Einstein did not maintain a distinction between reality, and representations of it, then the notions he expressed make sense, but are still invalid:
We suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows, as viewed from K, that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light, will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time, he will interpret his observation by saying that the clock on the periphery “actually” goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position.

Clock H’ will indicate the end of a period at the times in the form: tn’ = n/P₀. Since we seek the time with respect to S, the first of the transformation equations will have to be written: t = β[t’ – vx’/c²]. Since clock H’ is at rest at the origin of S’, we must always have x’ = 0, which yields: tn = βtn’ = (β/P₀)n. Observed from S clock H’ indicates: P = P₀/β = P₀√(1-v²/c²) periods in a unit time. In other words, a clock moving uniformly with velocity v with respect to a reference system runs, as observed from this system, 1:√(1-v²/c²) times slower than an identical clock that is at rest with respect to this system.

How great is the length of a rod, as observed from the system K, that is at rest with respect to K’, is oriented parallel to the x’ axis, and possesses the length l’ with respect to K’? The x coordinates of the rod ends, satisfy the equation: x₁= (x₁ -vt)/√(1 – v²/c²), x₂’ = (x₂ -vt)/√(1 – v²/c²), so: x2’ – x1’ = (x₂ –x₁)/√(1 – v²/c²) or: l = l’[√(1 – v²/c²)]. This means, if a rod possesses the length l’ when measured at rest, then, if it moves with velocity v along its axis, it will possess the smaller length l = l’[√(1 – v²/c²)] for a non-comoving observer, whereas for a comoving observer, it will always, have the length l’. The greater the velocity v of the moving rod, the smaller the length.

4.4.2 The Principle of Relativity

1 The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. (Intro 1905)

2 The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translational motion. (Section 1 1905)

3 This independence from the state of motion of the system of coordinates used, which we call “the principle of relativity”. (Vol 2 Doc 47)

4 The laws of nature are independent of the translational motion of the reference system, and are identical in all acceleration-free systems. (Vol 3 Doc 17)

5 But this equivalence of the systems K and K’, that are moving uniformly relative to each other, is not limited to mechanics. So far as our experience extends, this equivalence holds generally. The assumption of the equivalence of all such systems K, K’, which rules out the privileging of one state of motion over all others, we will designate as the “relativity principle”. (Vol 4 Doc 21)

6 The principle which was the pivot of all our previous considerations was the special principle of relativity, ie the principle of the physical relativity of all uniform motion. It was at all times clear that every motion must be considered only as a relative motion. If it is simply a question of detecting, or of describing, the motion involved it is immaterial to what reference-body we refer the motion. But that must not be confused with the much more comprehensive statement called “the principle of relativity”. The principle we have made use of not only maintains that we may equally well choose either K or K’ for the description of any event, it also asserts what follows: if we formulate the general laws of nature as they are obtained from experience, by making use of either reference-body, then these general laws of nature (eg the laws of mechanics, or the law of the propagation of light in vacuo) have exactly the same form in both cases. (Special and General Theory 1916)

7 There exists a reference-body K, whose condition of motion is such that the Galileian law holds with respect to it, ie a particle left to itself, and sufficiently far removed from all other particles, moves uniformly in a straight line. All bodies of reference K’ should be exactly equivalent to K for the formulation of natural laws, provided that they are in a state of uniform rectilinear and non-rotary motion with respect to K. All these bodies are Galileian reference-bodies. The validity of the principle of relativity was assumed only for these, but not for others (eg those possessing motion of a different kind). In this sense we speak of the special principle of relativity. (Special and General Theory 1916)

8 The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics. If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to a uniform translational motion relative to K. We call this postulate “The Special Relativity Principle”. By the word special, it is signified that the principle is limited to the case when K′ has uniform translatory motion with reference to K. The equivalence of K and K′ does not extend to the case of non-uniform motion of K’ relative to K. (Foundation of GR 1916)

9 Now in order that the special principle of relativity may hold, it is necessary that all the equations of physics do not alter their form in the transition from one inertial system to another when we make use of the Lorentz transformation for the calculation of this change. In the language of mathematics, all systems of equations that express physical laws must be co-variant with the Lorentz transformation. (Brief Outline 1921)

As explained in Section 3.4, the principle of relativity as espoused by Einstein, and Poincaré, is invalid. It is based on a misinterpretation of a point made by Galileo. Let us take Galileo’s familiar law of inertia: a material point not acted upon by external forces moves uniformly in a straight line. It is clear that this law cannot hold true if the motion is referred to an arbitrarily moving coordinate system. This is incorrect. The law is a function of reality, not a reference. This need for a reference is driven by Einstein’s underlying belief that ‘everything is relative’. The situation becoming more confused when he conflates reality with representations thereof.

Generally, the issue is that while the movement of any given entity can be described with respect to any other movement, that is somewhat pointless, as its actual movement is only definable with respect to an appropriate reference. This was explained in Section 3.2.2. The ‘laws of nature’, and ‘nature’ itself (eg physical existence), are entirely independent, ie what occurs is not determined by references. There was therefore no need to reformulate Galileo’s fundamental law in the following way: It is possible to choose a coordinate system K, that is in such a state of motion that every freely moving material point moves rectilinearly and uniformly relative to it. Naturally, the law also holds then for all other coordinate systems at rest with respect to K.

This assertion was also incorrect: There exists a whole class of coordinate systems moving uniformly, relative to one another, that are strictly equivalent when it comes to formulating the laws of mechanics. But this equivalence of the systems K and K’ is not limited to mechanics. So far as our experience extends, this equivalence holds generally. The assumption of the equivalence of all such systems K, K’, which rules out the privileging of one state of motion over all others, we will designate as the “relativity principle”. [It is not clear what generally referred to].

We will state the principle of relativity in the following way: The laws governing natural phenomena are independent of the state of motion of the coordinate system with respect to which the phenomena are observed, provided that this system is not in accelerated motion. This is invalid. It is not true in respect of any motion. The more explicit indication that Relativity actually concerns the relativity of observation, not reality, is noted.

4.4.3 The Principle of the constancy of the velocity of light
To go a step further, we must reconcile the principle of relativity with an essential consequence of Lorentz’s theory, because giving up this consequence would amount to giving up the most fundamental formal properties of the theory. And here is the consequence in question: A ray of light in vacuum always propagates with the same velocity c, which velocity is independent of the motion of the body that emits the ray. In Lorentz’s theory this principle holds only for a system in a special state of motion: In effect, the system must be at rest relative to the ether. (Vol 2, Doc 47).

The most important content of this hypothesis can be expressed as: There exists a reference system (called in Lorentz’s theory “a system at rest relative to the ether”) with respect to which every light ray propagates in a vacuum with the universal velocity c. This ought to hold independently of whether the light-emitting body is in motion or at rest. We designate this proposition the principle of constancy of the velocity of light. (Vol 3, Doc 17).

Hence in accordance with Lorentz’s theory we can proclaim the following, which we call “the principle of the constancy of the velocity of light”: “There exists a coordinate system with respect to which every light ray propagates in vacuum with the velocity c.” This principle contains a far-reaching assertion. It asserts that the propagation velocity of light depends neither on the state of motion of the light source nor on the states of motion of the bodies surrounding the propagation space. The question as to what extent this principle can be considered certain is of fundamental significance for the theory of relativity. For the time being we will content ourselves with the realization that this principle is demanded by Lorentz’s theory. (Vol 4, Doc 1).

The fundamental assertion of Lorentz’s theory, that every light ray in a vacuum always propagates (at least with respect to a certain coordinate system K) with the definite constant velocity c, we will call the principle of the constancy of the velocity of light. (Vol 4, Doc 21)

As explained previously, Lorentz did not reference what Einstein thought was a special state of motion, ie at rest relative to the ether. Neither did his hypothesis involve the existence of a reference system with respect to which every light ray propagates in a vacuum with the universal velocity c. The conclusion Lorentz did come to was that the propagation velocity of light depends neither on the state of motion of the light source nor on the states of motion of the bodies surrounding the propagation space, to quote Einstein.

Lorentz was referring to the start speed, as the creation of any given light results from the same process, which is why the movement of anything involved had no effect. It is not a speed with respect to any given reference, but a constant (actual) within our existentially closed system. Put another way, it is that speed with respect to any reference, so long as that remains stationary. That is not the same as: There exists a coordinate system with respect to which every light ray propagates in vacuum with the velocity c. That rendition reflects Einstein’s misinterpretation of Galileo, which results in the belief that there is a certain mode of motion in which everything has equivalence, apart from the fact that it cannot be c.

What must be a ‘real’ light speed has been replaced with a hypothetical one. That is, there is a contradiction between a ray of light in vacuum always propagates with the same velocity c, and which is independent of the motion of the body that emits the ray. These are two different circumstances, ie bodies do not exist in a vacuum. Though in most practical circumstances, the actual transmission speed of any given light, on earth, can be presumed to be constant, but not c, and lights depicting the same occurrence transmit at that speed.

Apart from the incorrect choice of c as a universal reference, it could neither be c, or an equivalent ‘real’ constant, if the equations were supposed to be applicable to reality, rather than observation. Selecting one overarching reference is incorrect, since whilst any reference must be universal, that is only with respect to the characteristic being considered. Finally, from a practical standpoint, it is impossible to relate any given physically existent state of any given thing, at any given time, to the relevant existent state of a ray of light at that time.

Einstein justified his choice as follows:
The theory of relativity is often criticized for giving, without justification, a central theoretical role to the propagation of light, in that it founds the concept of time upon the law of propagation of light. The situation, however, is somewhat as follows. In order to give physical significance to the concept of time, processes of some kind are required which enable relations to be established between different places. It is immaterial what kind of processes one chooses. It is advantageous, however, for the theory, to choose only those processes concerning which we know something certain. This holds for the propagation of light in vacuo in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and Lorentz.

And: For these signals we can use, for example, sound waves that propagate between A and B, through a medium that is at rest with respect to these points. We can use light rays propagating through the vacuum, or through a homogeneous medium at rest with respect to A and B. It does not make any difference whether we choose this or that kind of signals. Still, of all the signals that can be used, we are going to prefer those that make use of light rays propagating in the vacuum, because the synchronization requires that the path out, and the path back, be equivalent, and in our case this equivalence is satisfied, by definition, by virtue of the principle of the constancy of the velocity of light, as light in the vacuum always propagates with the velocity c.

The conundrum that Einstein had created for himself was that having adopted the belief that ‘everything is relative’, he then needed some reference which was apparently constant, in order to devise transformation equations in what was actually the reality of light-based representations of reality. Moreover, had Einstein properly understood the concept of timing, then he would have known that it was unnecessary to give physical significance to the concept of time, which required processes of some which enable relations to be established between different places. It is immaterial what kind of processes one chooses. It is advantageous, however, for the theory, to choose only those processes concerning which we know something certain.

Reality occurs at a unique time and spatial position. The issue is to determine what that was on the basis of representations received, which is not limited to those conveyed by light, or humans, or direct experience. Finally, as explained in Section 3.1, the path out, and the path back are not equivalent unless the spatial relationship is fixed, which renders the supposed requirement meaningless, and the system for adjusting timing devices implied here, originally proposed by Poincaré, is an invalid method for synchronising timing devices.

The conundrum is manifest as: If we want to preserve the principle of relativity, we must assume that the principle of the constancy of the velocity of light holds for any arbitrary system not in accelerated motion. At first glance this seems impossible. For let us consider a light ray that propagates with velocity c with respect to the system S, and suppose that we seek to determine the velocity of propagation relative to a system S’, that is in uniform translational motion with respect to the first system. Applying the rule of addition of velocities, we will find a velocity different from c; in other words, the principle of the constancy of the velocity of light that is valid with respect to S is not valid with respect to S’.

This stems from the misinterpretation of Galileo. Einstein indicated that he had a solution. But, that involved more mistakes with his new conceptualisation of time and space, resulting in his repudiation of the rule addition of velocities. Obviously, if S and S’ have different speeds/directions, then a constant velocity will be different when calibrated with respect to them. That they are in uniform translational motion, per se, is irrelevant. The constant velocity has not changed, but when calculated with respect to a reference that has a movement of its own the outcome must be different.

Since there are two entities there needs to be two rays of light. Therefore, if it is presumed that the two rays are ‘identical’, and not affected differently during transmission, the velocity of those rays with respect to S and S’ will depend on their velocities. That determines any alteration in spatial position with respect to the light source whilst it is travelling. Because, the distance travelled, and the duration thereby incurred, by the light is a function of the spatial position of the occurrence with respect to the position of what became the ‘receiving’ entity when the representation was received. Not the distance between the event and that entity when it happened, or when the representation was received.

While an event occurred at a specific time and place, due to differing spatial positions, when the representation of it was received by various entities will likely vary. This could involve the original differences in spatial position, with respect to the occurrence, not altering during the delay. Or, during that delay there is some alteration in spatial position, either with respect to each other, and/or with respect to what was the position of the occurrence.

The path of a representation, at any given t, would be defined by a distance starting from a point on the path of the ‘emitting’ entity at that t (which is also the start of the distance (x) between the entities at that t), to a point on the path of the ‘receiving’ entity, which would be beyond the end point of distance x. The relationship between ‘distance x’ and ‘distance transmission’, at any given t, is governed by the factor √(1 – v²/V²). With a light-based representation, V would be the actual speed of light in that particular circumstance, ie not c.

The Doppler effect was explained in the final paragraph of Section 2.1: Assuming a regularity in the sequence, and a constancy of velocity and direction of travel in the ‘emitting’, and ‘receiving’, entities, then the frequency of the sequence, when assessed with respect to what is receiving the representations, will be constant. However, if the relationship of distance, and/or direction, varies from occasion to occasion, then their relative spatial position could vary on each such occasion, and the frequency appear to differ from that emitted to that received. Because, as the spatial relationship alters, successive representations have to travel distances which alter at a varying rate. In this circumstance, a sequence measured with respect to the ‘receiving’ entity, either increases or decreases at a rate dependent upon the rate at which relative spatial position is altering whilst each representation is being transmitted. That is, it appears, with respect to the ‘receiving’ entity, to be speeding up/slowing down, even though the actual emitted rate did not change.

Einstein then had to resolve this self-inflicted issue, described by his S and S’ example. Why does E=mc²: Cox and Forshaw. Da Capo Press © 2009 addresses the same conundrum:
By page 74, the next step is seen as calculating the ‘distance’ between what are actually existent and non-existent (ie a previously existent) entities. That is, their supposed spatial relationship, given that there is a time difference in their occurrence. The notion of distance continuing over time being a contradiction, by definition, since something cannot be a distance from something else when they do not exist at the same time.

This is ‘resolved’ by using x = vt. The problems with reifying t and x, and representing x as a duration, have been stated many times in this paper. The authors state: …because they are not of the same type of quantity. We can, however, convert distances into times and vice versa if we use the equation…v = xt…distance and time can be interchanged using something that has the currency of a speed. Let us therefore introduce a calibrating speed; call it c. We can then measure time in metres provided we take any time interval and multiply it by our calibrating speed. At this point in our reasoning c really can be any old speed.

So, the ‘distance’ in terms of a combination of space and time, known as s, was deemed to be the hypotenuse of an imaginary triangle relating the two entities, with distance and elapsed time representing the other sides. It was either: s² = (ct)² + x², or s² = (ct)² – x².

The argument then invoked to substantiate the choice between them is self-fulfilling. On page 89 it is asserted in relation to an examination of causality that: we have finally managed to interpret the speed c: It is the cosmic speed limit. Nothing can travel faster than c because if it did it could be used to transmit information that could violate the principle of cause and effect. Which is incorrect, because that presumes that the other entity must be a conveyor of a representation, ie as per light, which is not a necessary condition. So, what started as supposedly ‘just any old calibrating speed’ was interpreted, on the basis of its functionality, as a conveyor of representations, and therefore had a speed limit. Also, it was not the actual light which enables sight, but a hypothetical entity in a vacuum. Whether anything can travel faster than light is determined by whether there is an entity in physical existence that can do so. It is not determined by the constraints of a mathematical construct, as per Lorentz 1904.

Having proved the presumption, the argument continues on page 89 that: No matter how two different observers are moving they must always measure c to be the same. This is incorrect, since by definition, if the observers (entities) are moving at different speeds and directions, albeit in a uniform translational motion, then the speed of light with respect to each observer is likely to be different. As it would be for any entity, ie light does not have some ‘special’ characteristic. This is a function of changes in spatial position occurring during the delay whilst a representation travels.

These invalid assumptions lead in the next sentence to the conclusion that: The speed c is beginning to look like another special speed we have encountered in this book: the speed of light, but we haven’t proved the connection yet. Their final rejoinder being: Certainly the existence of a universal speed limit offers promise, especially if we can interpret it as the speed of light. The authors then go on to see: if we can obtain the slowing down of moving clocks…an invalid notion arising from the conflation of occurrence and representation.

The author acknowledges that this was a description of Einstein’s argument, and far more readable and understandable than the original.

4.4.4 The Reconciliation
On the basis of his flawed interpretations, Einstein then had to reconcile the two principles. This relates to the opening statement in the 1905 paper, when they were asserted to be only apparently irreconcilable.

To go a step further, [from concerns about the ether] we must reconcile the principle of relativity with an essential consequence of Lorentz’s theory. A ray of light in vacuum always propagates with the same velocity c, which velocity is independent of the motion of the body that emits the ray.

At first glance this seems impossible. For let us consider a light ray that propagates with velocity c with respect to the system S, and suppose that we seek to determine the velocity of propagation relative to a system S’, that is in uniform translational motion with respect to the first system. Applying the rule of addition of velocities, we will find a velocity different from c; in other words, the principle of the constancy of the velocity of light that is valid with respect to S is not valid with respect to S’.

So that the theory based on these two principles should not lead to contradictory results, one must renounce the customary rule of addition of velocities. How have we to modify the theorem of the addition of velocities? This question leads to a general one. How are we to find the place and time of an event in relation to S, when we know the place and time of the event with respect to S’? Is there a thinkable answer to this question of such a nature that of light in vacuo does not contradict the principle of relativity? In other words: can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to S and S’?

Having correctly, albeit coincidentally as it was on the basis of a flawed argument, rendered the ether an irrelevant concern, Einstein then noted a new problem, illustrated by S and S’. His assertion that Lorentz relied on a particular reference was incorrect. However, he was correct in stating that we must assume that the principle of the constancy of the velocity of light holds for any arbitrary system not in accelerated motion, except for the latter caveat, as maintaining the condition of not in accelerated motion, still reflected the misinterpretation of Galileo. The notion that this was needed to preserve the principle of relativity was incorrect.

This outcome, ie that c would vary, was contrary to his usual stance, which was a function of the false equivalence of references in a certain mode of motion. This was written in 1907, Vol2 Doc 47. Normally, S and S’, like K and K’, were, albeit wrongly, considered equivalent. But that might be because it concerned any arbitrary system, so when he applied the rule of the addition of velocities, he revealed the truth, ie the outcome is a velocity different from c. Or, Einstein revealed the difference, as this, in his thinking, proved it was wrong, as the outcome ‘must’ always be c in accordance with his valid, as he saw it, hypothesis, and therefore the rule of the addition of velocities needed amendment. As indicated by at first glance this seems impossible, Einstein thought he had a solution.

One must renounce the customary rule of addition of velocities. How have we to modify the theorem of the addition of velocities? Since that theorem is self-evidently valid, nobody would consider trying to renounce it. Or put the other way round, whatever argument was used in an attempt to do so, must be invalid. As will be seen in Section 6.2, when Einstein tried to illustrate this with examples, rather than maths, he had to confuse the references.

This led to: How are we to find the place and time of an event in relation to S, when we know the place and time of the event with respect to S’? By definition, the time being considered is the same (t = t’), as the quest is to compare the existence of the two entities, not the existence of one with whatever existed of the other at some other time. An existent entity cannot be compared to a non-existent entity. The latter cannot have a spatial position if it is non-existent at that time. Leaving aside how it was achieved, whatever was done to ‘know the place’ of S’ could have been repeated in order to know the place of S. A more generic answer is that the time and place is derived from the times when the two light-based representations of S & S’, over a sequence of such receipts, were received. Which, when taking into account the known movement of the recipient entity over that duration, reveals what must be the movement, ie sequence of successive spatial positions, of S & S’.

In other words, Einstein asked a false question based on his flawed conception of time and place, which he wrongly considered to be relative instead of independently unique. In order to resolve the predicament, he rephrased the issue: In other words: can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to S and S’? This is not an accurate portrayal of the circumstance, but one that Einstein can address with his invalid concepts. The two events occur at the same time, it is incorrect to conceive of a different time, which then enables the constant to be c when referenced to both S and S’.

Einstein: we have only learned that the following three things are incompatible:
(a) the relativity principle
(b) the principle of the constancy of the velocity of light
(c) the transformation equations or the law of the parallelogram of velocities or the rule of addition of velocities
One arrives at the theory that is now called “the theory of relativity” by keeping
(a) and (b) but rejecting (c).

Which is justified by: So that the theory based on these two principles should not lead to contradictory results, one must renounce the rule of addition of velocities. Well founded as this rule may seem to be at first glance, it conceals no less than two arbitrary hypotheses, which hold sway over all of kinematics.

The first unjustified hypothesis we wish to discuss concerns the physical notion of time measurement… But we have no right to assume, a priori, that the clocks of the two groups [S & S’] can be set in a way that t would be equal to t’. To assume this would mean to introduce an arbitrary hypothesis which has been introduced into kinematics up to the present time. [There is then an argument on how to tell the time ‘properly’] with the conclusion that: A clock moving uniformly with velocity v with respect to a reference system runs, as observed from this system, 1:√(1-v²/c²) times slower than an identical clock that is at rest with respect to this system. See Section 4.5

The second unjustified hypothesis concerns the configuration of a body in motion…What should we understand by the “length of the bar”? One is at first inclined to believe that this concept does not require any special definition. However, we will immediately see that nothing of the sort is true, if we consider the following two methods of determining the length of the bar… The second hypothesis used unconsciously in kinematics can thus be expressed as follows: The kinematic configuration and the geometric configuration are identical. [There is an argument on how to measure a length ‘properly’] with the conclusion that: The kinematic configuration differs from the geometric configuration solely by a contraction in the direction of the motion which is in the ratio 1:√(1-v²/c²). See Section 4.6

One recognizes in this the hypothesis of Lorentz and FitzGerald. This is the hypothesis that looked so strange, and had to be introduced to explain the negative results of the experiment of Michelson. Here this hypothesis appears naturally as an immediate consequence of the principles assumed.

In summary: The rule of addition of velocities, which made one think that Lorentz’s theory cannot be reconciled with the theory of relativity, is based on unacceptable hypotheses which lead to the following transformation equations: t’ = t, x’ = x – vt, y’ = y, z’ = z. The first of these equations, expresses an ill-founded hypothesis about the time coordinates of an elementary event taken with respect to two systems, S and S’, that are in uniform translational motion with respect to each other. The other three equations, express the hypothesis that the kinematic configuration of the system S’, with respect to the system S, is identical with the geometric configuration of the system S’. [For Einstein, the geometric configuration is reality, while the kinematic configuration is the reality of light-based representations of reality].

Another flawed argument to justify the abandonment of the rule of addition of velocities is: It turns out that the result of the Fizeau experiment can be derived quantitatively from (a) and (b). [Because] If we take this standpoint [keeping (a) and (c)], then Fizeau’s experiment forces us to conclude that in a transparent medium, various velocities of light must be possible. For if the velocity of light in the medium, as assessed by an observer moving with the medium, were always equal to Vo, then, according to the rule of addition of velocities, the velocity of light as assessed by an observer not moving with the medium would be Vo + nl, We are thus forced to assume that, in Fizeau’s experiment, the velocity of light relative to the moving medium is, in this medium, different from what it would be in the same medium if the latter were at rest.

Einstein explored the logic of his hypothesis ad absurdum (Vol 3, Doc 17):
For it turns out that precisely those two seemingly incompatible axioms which were imposed on us by experience, the principle of relativity and the principle of constancy of the velocity of light, lead us to a perfectly definite solution of the space-time transformation problem.

If we have a solid body that is moving uniformly with respect to the coordinate system k, which we take as the basis for our analysis, then this body appears contracted by a perfectly definite ratio in the direction of its motion, as compared with the shape it has when it is in a state of rest, with respect to this system. If we denote the velocity of motion of the body by v, and the velocity of light by c, then each length measured in the direction of motion, and equal to l in a motionless state, will be diminished because of the body’s motion relative to the non-comoving observer to the length l.√(1 – v2/c2).

If the body has a spherical shape in the state of rest, it will have the shape of a flattened ellipsoid if we move it in a certain direction. When its velocity reaches the velocity of light, it will collapse to a plane. This result loses very much of its oddness, if one considers that this assertion about the shape of a moving body has a complicated meaning, since this shape can be ascertained only with the aid of determinations of time. The feeling that this concept, “the shape of the moving body,” has an immediately obvious meaning is due to the fact that in our day-to-day experience we are accustomed to encountering only such velocities of motion that are practically infinitely small compared with the velocity of light.

And now a second purely kinematic consequence of the theory that strikes us as even more peculiar. We imagine that there is given a clock capable of indicating the time of a reference system k, provided that it is arranged at rest relative to this system. It can be proved that this same clock, when set into uniform motion relative to the reference system k, runs slower, as judged from the system k, in such a way that when the time reading of the clock has increased by 1, the clocks of the system k indicate that, with respect to the system k, there has elapsed the time 1/√(1-v²/c²). Were we to succeed in making the clock move with the velocity of light, the hands of the clock, as judged from k, would move forward infinitely slowly.

The thing is at its funniest when one imagines that the following is being done: One imparts to this clock a very great velocity (almost equal to c), then lets it fly on in uniform motion, and after the clock has covered a long stretch, one imparts to it a momentum in the opposite direction, so that it returns to the point from which it has been launched. It then turns out that the positions of the clock’s hands have hardly changed during the clock’s entire trip, while an identically constituted clock, that remained at rest at the launching point during the entire time, changed the setting of its hands quite substantially. It should be added that whatever holds for this clock, which we introduced as a simple representation of all physical phenomena, holds also for closed physical systems of any other constitution.

This ‘back to the future’ notion is alone sufficient to indicate the something is wrong with the concept of Relativity, as expressed. That is the purpose of the ad absurdum test.

4.4.5: Conclusion
The two principles, Relativty and Constancy of the velocity of light, as interpreted by Einstein, are interrelated, and both are invalid. Neither can they be reconciled, logically. Einstein’s principle of Relativity deemed that everything in a mode of movement known as uniform translational motion, irrespective of individual velocity, is equivalent as a reference, for calibrating the laws of nature, with anything else which is also in that mode of motion, ie this referencing always results in the same outcome, which is impossible.

Einstein’s principle of the Constancy of light, deemed that everything in uniform translational motion, irrespective of individual velocity, is equivalent as a reference for calibrating a constant. That is, when the constant is referenced to anything in that mode of motion, then the outcome will always be the value of the constant, which is impossible.

Considerations about the relative motion of the conductor and the magnet, etc, in the context of electrodynamics, is not the same as considerations about the relative movement of ‘bodies’ in general. Part 1 proposed a new definition of simultaneity based on a new method of timing, a new definition of measuring length using vt, a new transformation equation for relating ‘stationary’ and ‘moving’ bodies, and a repudiation of the rule of addition of velocities. The concluding sentence of Part 1 was: We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics. Whereas, what Einstein actually did was conflate reality with the reality of existent light-based representations thereof, ie observation.

While the subject matter was ostensibly electrodynamics, Part 1 was, in effect, based on a different model of physical existence than what had gone before. That was recognised by Minkowski who formulated a construct which became known as space-time. This meant that Einstein’s ideas became inherent characteristics of a model supposedly depicting our existentially closed system, ie it created a self-fulfilling circumstance.

The only way to reconcile Einstein’s two principles, where reconcile entails doing so within their own rationale instead of on the basis of logic, was: 1) abandon the rule of addition of velocities which rendered Einstein’s interpretation of both principles invalid, even though it is obviously correct, 2) compensate for the effect of that by invoking the hypothesis of Relativity in a circumstance which involves relativity, ie conflate reality with representations thereof. The specific sensory system involved being irrelevant, since in terms of generic physical form, no one sensory system has a precedence, or ‘special’ characteristic.

The whole argument is circular, based on a sequence of flawed assumptions.

Hall of Mirrors: Section 6.1: The Special and General Theory

[Note: In the introduction, the author explained the necessity of using quotes in an unorthodox manner. In this section in particular, it is often best to let Einstein ‘speak for himself’, albeit whilst ‘curating’ the narrative, rather than paraphrase what he said].

6.1.1 Special Relativity Theory: Quotes
If one abandons the ordinary kinematics and builds a new kinematics based on the new foundations, one arrives at [different] transformation equations. We are going to show that based on:
1. The principle of relativity
2. The principle of the constancy of the velocity of light
we arrive at transformation equations that allow us to see that Lorentz’s theory is compatible with the principle of relativity. The theory based on these principles we shall call the theory of relativity. (Vol 3, Doc 2)

The theory of relativity requires that systems of equations in physics turn into systems of equations of the same form if one transforms them by means of the Lorentz transformation. (Vol 4, Doc 1)

It turns out that the principle of constancy of the velocity of light, and the relativity principle, are incompatible with each other only as long as one holds to the postulate of absolute time, ie to the absolute meaning of simultaneity. If one admits the relativity of time, then the two principles turn out to be compatible with each other; proceeding from these two principles, one arrives at the theory designated as the “theory of relativity”. (Vol 4, Doc 21)

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate [relativity], but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks. (Foundation 1916)

The theory [Lorentz] appeared to be unsatisfactory only in one point. It appeared to give preference to one system of coordinates of a particular state of motion (at rest relative to the ether) as against all other systems of co-ordinates in motion with respect to this one. In this point the theory seemed to stand in direct opposition to classical mechanics, in which all inertial systems which are in uniform motion with respect to each other are equally justifiable as systems of co-ordinates. (Brief Outline 1921)

The whole change in the conception of the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility. The special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it. (Ether and Relativity 1920)

Development of the special theory of relativity consists of two steps, the adaptation of the space-time “metrics” to Maxwell’s electro-dynamics, and an adaptation of the rest of physics to that altered space-time “metrics”. The first of these processes yields the relativity of simultaneity, the influence of motion on measuring-rods and clocks, a modification of kinematics, and in particular a new theorem of addition of velocities. (Brief Outline 1921)

This theory [Special Relativity] originated as the answer to the question: Is the special principle of relativity really contradictory to the field equations of Maxwell for empty space? The answer to this question appeared to be in the affirmative. [But] if those [Galileo transformation] equations are valid with reference to a system of co-ordinates K, and we introduce a new system of co-ordinates K’ in conformity with them, then Maxwell’s field equations are no longer valid in the new co-ordinates (x’, y’, z’, t’). (Brief Outline 1921)

A more searching analysis of the physical significance of space and time rendered it evident that the Galileo transformation, x’ = x – vt & t’ = t, is founded on arbitrary assumptions, and in particular on the assumption that the statement of simultaneity has a meaning which is independent of the state of motion of the system of co-ordinates used. It was shown that the field equations for vacuo satisfy the special principle of relativity, provided we use the [Lorentz equation]: x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, t’ = (t-vx/c²) / √(1 – v²/c²). (Brief Outline 1921)

In these equations x, y, z represent the co-ordinates measured with measuring-rods which are at rest with reference to the system of co-ordinates, and t represents the time measured with suitably adjusted clocks of identical construction, which are in a state of rest. Now in order that the special principle of relativity may hold, it is necessary that all the equations of physics do not alter their form in the transition from one inertial system to another, when we make use of the Lorentz transformation for the calculation of this change. In the language of mathematics, all systems of equations that express physical laws must be co-variant with respect to the Lorentz transformation. (Brief Outline 1921)

6.1.2 Special Relativity: Comment
What became known as the Special Theory of Relativity was contained in the first part, the Kinematical Part, of the 1905 paper entitled On the electrodynamics of moving bodies. According to Einstein this hypothesis: is based on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

As previously noted, what Einstein referred to as the ordinary kinematics represented reality, whereas the new kinematics based on the new foundations concerned the reality of light-based representations of reality. So, the hypothesis in Part One is not based on the kinematics of the rigid body, but the relativity of observation. This is why one arrives at [different] transformation equations, based on: 1. The principle of relativity, and 2. The principle of the constancy of the velocity of light, both of which, as interpreted by Einstein, are invalid. There is no direct physical relationship between the electrodynamics (or any other such feature) of bodies, rigid or otherwise, and clocks, ie time, systems of co-ordinates, ie space, and whatever sensed that by whatever sensory system in any form of movement.

Einstein 1905, Part 1, the Kinematical Part

Para 1, Definition of Simultaneity: using a methodology first proposed by Poincaré, explains how to carry out timing ‘properly’. This is invalid. See Section 4.5.

Para 2, On the Relativity of Lengths and Times: is based on the principle of relativity and on the principle of the constancy of the velocity of light… where the time interval is to be taken in the sense of the definition in para 1. The interpretation of both these principles is invalid, as explained in Section 4.4. Para 2 explains how length (space) varies depending on what referencing system is chosen to carry out the measurement. One methodology, in effect, times it. Length does not actually alter with varying references. See Section 4.6.

Para 3, Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former: explains the new transformation laws, which inevitably involve √(1 – v²/V²), since the same mathematics is always used. c replaced V for invalid reasons, as explained in Section 4.4. As a function of the mathematics, c cannot exceed v. The new transformation equations were: t’ = β[t – (v/c²)x], x’ = β(x – vt), y’ = y, z’ = z, or t’ = [(t-(v/c²)x]/√(1 – v²/c²)], x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, where β = 1/√(1-v²/c²).

Para 4, Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks: It follows that the time marked by the clock (viewed in the stationary system) is slow by 1- √(1 – v2/c2) seconds per second, and whereas the y and z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the x dimension appears shortened in the ratio 1: √(1 – v2/c2).

Para 5, The Composition of Velocities: negates the rule of addition of velocities with a self-fulfilling mathematical proof based on the above.

Conclusion to Part 1: We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.

Having derived a new transformation rule based on the relativity of observation, Einstein then proceeded to apply this to electrodynamics. This involved using the new transformation equations for any so-called ‘moving’ body. If we apply to these equations the transformation developed in para 3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations…And: Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations…

There are two issues with this:
1 Specific electrodynamic, etc, effects relating to differential movement, are not the equivalent of relativity effects in the receipt of light-based representations of reality.

2 What constitutes ‘movement’, given that everything is moving? On earth the movements are varied, and considerable. It is just that, at least in respect of the underlying motions, they are omnipresent. So, a differentiation between what is ‘at rest’, and what is ‘moving’, needs validation. Only a proper system of referencing, ie one which takes account of the nature of our existentially closed system, can identify actuals, albeit only from within the system.

Einstein derived the Special Theory of Relativity, with its new transformation equations, by forcing a reconciliation between his invalid interpretations of the principle of relativity and the principle of the constancy of light. That entailed abandoning the rule of addition of velocities, which he achieved with his new conceptualisation of ‘time’. The equations are based on the factor √(1 – v2/V2). That became known as the Lorentz transformation, but is not a necessary characteristic of Lorentz’s hypothesis, just a simple mathematical expression of the generic relationship between two movements. See Section 3.2.

These conditions were what was ‘special’, a hypothetical state with no relationship to reality, not Relativity. This hypothesis was asserted to be an explanation to a supposed issue, ie the discrepancy between expectations and the outcomes of the Michelson experiments. It was proposed as an alternative to Lorentz’s explanation that matter contracted in a particular way when subjected to a force, causing movement which was different to that which normally prevailed, and reformulated several of his ideas, along with those of Poincaré.

Einstein thought contraction did not occur, and it was explained with his new understanding of time, and space. Whereas, he actually explained it with the inherent relativity in the receipt of light-based representations of reality. Since both of them used the same mathematics to compare circumstances, they both ‘proved’ that their explanation compensated for their supposed effect, ie it was just self-fulfilling. Also, they both had the same outcome, because the same factor, √(1 – v2/V2), was used.

Hints that what was originally presented needed amendment were expressed by Einstein in his reply to a comment by Abraham (1912): The theory presently designated as “the theory of relativity” rests on two principles that are totally independent of one another, namely 1 The principle of relativity (with respect to uniform translation) 2 The principle of the constancy of the velocity of light From these two principles it is possible to develop the theory currently known as “the theory of relativity”.

It is common knowledge that it is impossible to base a theory of the transformation laws of space and time on the principle of relativity alone. As we know, this is connected with the relativity of the concepts of “simultaneity” and “shape of moving bodies.” To fill this gap, I introduced the principle of the constancy of the velocity of light, which I borrowed from Lorentz’s theory of the stationary luminiferous ether.

But what about the limits of the validity of the two principles? We have not the slightest reason to doubt the general validity of the principle of relativity. On the other hand, I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of gravitational potential. This is where, in my opinion, the limit of the validity of the principle of the constancy of the velocity of light, though not the principle of relativity, and therewith the limit of the validity of our current theory lies. In my opinion, the present theory of relativity will always retain its significance as the simplest theory for the important limiting case of spatio-temporal events in the presence of a constant gravitational potential.

Leaving aside what the actual faults are with this hypothesis, and accepting Einstein’s rationale, per se, there are still a number of issues:

The notion of the limit of the validity. A theory is either valid, or it is not. Gravity was not new knowledge. The Lorentz hypothesis of actual contraction inherently presumed a force, as some such effect was necessary to cause an actual alteration in movement from that which normally prevailed. Whereas, Einstein’s hypothesis did not provide such an opportunity. Apparently, this new, alternative, idea was based on a better understanding of the relativity of the concepts of “simultaneity” and “shape of moving bodies”, coupled with the principle of the constancy of the velocity of light borrowed from Lorentz’s theory. But that has nothing to do with reality.

Einstein asserted that: It is common knowledge that it is impossible to base a theory of the transformation laws of space and time on the principle of relativity alone. This is incorrect, as there is nothing else a transformation equation can be based on, assuming it concerns reality, and not something else. This is why Galileo’s transformation equation t’ = t, x’ = x – vt, y’ = y, z’ = z is correct, as it reflects reality. That is, the equation concerns the relationship of two motions which must occur at the same time, ie t = t’, because a non-existent entity cannot be compared to an existent one. There is then a difference in one of the three spatial dimensions from the original, which is expressed in terms of vt, where that t is a duration, and not at a given time, which is what t and t’ refer to.

The notion that it was ‘common knowledge’ was a false assertion, and incorrect, per se. This is a device Einstein often used, ie an assertion was described as ‘obvious/well known/etc’ and therefore no explanation was required. Or, in reverse, anyone who thought otherwise was put on the defensive, the implication being they ‘should have known better’.

However, the main point here is that having stated that a theory of the transformation laws cannot be developed on just one of the two principles, a circumstance which was apparently common knowledge, he then proceeded to do just that.

That is, he declared that the hypothesis, which had become known as Special Relativity, still had validity, albeit ‘limited’. Having admitted that gravity was not accounted for in his alternative explanation (unlike Lorentz), which meant that one of the two principles was apparently rendered obsolete, when gravity was introduced [though this was not the actual reason for the flaw in his interpretation of the principle of the constancy of light]. This is summarised in the somewhat arcane language: the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of gravitational potential and the present theory of relativity will always retain its significance as the simplest theory for the important limiting case of spatio-temporal events in the presence of a constant gravitational potential.

In other words, the special circumstance in which the concept of Relativity still supposedly had some validity was a hypothetical circumstance that had no relationship with reality. Scientifically, it is not acceptable to pick and choose what conditions are to be considered, from amongst those that must have relevancy, in order to formulate a theory. Otherwise, it is just a self-fulfilling belief of no scientific value.

His understanding of what could constitute a principle of relativity was invalid, as it was based on a misinterpretation of Galileo’s ship analogy. His understanding of what could constitute a principle of the constancy of light was invalid, as it is only the start speed of light that is always c. Any constant is an actual, ie not a function of a reference. However, when referenced to something that has a movement of its own then the outcome will vary, while the actual remains constant. That reflects the valid rule of addition of velocities.

All this resulted in the false conclusion that when a constant (be it c, or for practical purposes an ‘atmospheric’ constant for light) was calculated with reference to anything in uniform translational motion, it would always be that constant, ie the Galileo misinterpretation, or the invalid concept of inertial frames of reference. This is impossible, unless everything remained in the same spatial relationship (a solid state), in which case the point is meaningless. The mode of uniform translational motion does not preclude variations in velocity, ie speed and divergence. That presumption contradicted the rule of addition of velocities, which was abandoned. The final compensatory error involved the conflation of reality, with observation.

The circularity of the argument was enhanced when Minkowski reformulated Part 1 of the 1905 paper into a construct which became known as space-time. That is, the errors in Part 1 were then inherent in what became a new, but invalid, configuration of physical existence. Relativity is a hypothesis about the relativity of the receipt of light-based representations, as occurs in either ‘special’ or ‘general’ conditions. Relativity is not a characteristic of reality. As such, the ‘special’ version is invalid, because c is used as a universal reference, and instead of an ‘atmospheric’ c, and references which, just because they are in a certain mode of motion, are incorrectly deemed equivalent.

All this resulted in conclusions such as:
Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous, while observers in the stationary system would declare the clocks to be synchronous. So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

Evaluated from K, the time between two strokes is equal to: t = 1/√(1 – v²/c²), longer than one second. A clock traveling with the velocity v, runs slower, judged from a non-comoving system, than the same clock when it does not travel. Generalizing, one can conclude: Every event in a physical system slows down if the system is set into translational motion. But this slowing occurs only from the standpoint of a non-comoving coordinate system.

If a rod possesses the length l’ when measured at rest, then, if it moves with velocity v along its axis, it will possess the smaller length l = l’[√(1 – v²/c²)] for a non-comoving observer, whereas for a comoving observer, it will always, have the length l’. The greater the velocity v of the moving rod, the smaller the length. If v approaches the velocity of light c, then the length of the rod approaches the value zero. For values of v that exceed the velocity of light, our result becomes meaningless.

6.1.3 Special Relativity: The replacement transformation laws were derived as follows:
[S and S’] each possess a group of clocks that run-in synchrony [as determined by Einstein] when the two systems are at relative rest with respect to each other. Thus the velocity of light in a vacuum [used as it is constant and not because it is the speed of observational light] must be expressed by the same number in the two systems. Let t, x, y, z be the coordinates of an elementary event with respect to S, and t’, x’, y’, z’ the coordinates of the same event with respect to S’. We seek to find the relations that link these two groups.

These relations must be linear, because of the homogeneity of time and space, and time t is therefore linked with time t’ by a formula of the form: t’ = At + Bx + Cy + Dz. We will count the time from the instant when the origins of the two systems coincide, ie when: x’ = 0 and x – vt = 0, y’ = 0 and y = 0, z’ = 0 and z = 0, are equivalent ie: x, y, z, x’, y’, z’ are linked in the form: x’ = E(x – vt), y’ =Fy, z’ = Gz.

[Footnote: it is easy to understand what we mean by the homogeneity of time and space, or why we assumed a priori that the transformation equations must be linear. For if a rate of a clock at rest with respect to S’, is observed from S, this rate does not have to depend on the location of the clock in S’, nor on the value of the time of S’ in the vicinity of the clock. An analogous remark applies to the orientation and length of a bar linked with S’, and observed from S. Only when the transformation equations are linear are these conditions satisfied].

To determine the constants A, B, C, D, E, F, G, we assert that, according to the principle of the constancy of the velocity of light, it has the same value, c, with respect to the two systems, ie the two equations: x² + y² + z² = c²t² & x’² + y’² + z’² = c²t’² are equivalent. Replacing in the second of these equations t’, x’, y’, z’ by their values obtained from above [the two equations involving A – G], and equating it with the first of these equations, the transformation equations sought are of the form: t’ = ϕ(v)β[t – (v/c²)x], x’ = ϕ(v)β(x – vt), y’ = ϕ(v)y, z’ = ϕ(v)z where β = 1/√(1-v²/c²), and ϕ(v) is a function of v that is to be determined.

We can find ϕ(v) by introducing a third coordinate system S”, which is equivalent to the first two systems, is moving relatively to S’ with a uniform velocity -v, and is oriented with respect to S’, as S’ is oriented with respect to S. Applying the above set of equations twice, we obtain: t” = ϕ(v)ϕ(-v)t, x” = ϕ(v)ϕ(-v)x, y” = ϕ(v)ϕ(-v)y, z” = ϕ(v)ϕ(-v)z

Since the origins of S and S’ are permanently coincident, the axes have the same orientation, and the systems are equivalent, ϕ(v)ϕ(-v) = 1. Moreover, as the relation between y and y’, (as also that between z and z’), does not depend on the sign of v, we have: ϕ(v) = ϕ(-v). It follows that ϕ(v) = 1, as ϕ(v) = -1 is inappropriate.

The transformation equations, which we call the Lorentz transformations, are: t’ = β[t – (v/c²)x], x’ = β(x – vt), y’ = y, z’ = z, or t’ = [(t-(v/c²)x]/√(1 – v²/c²)], x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, where β = 1/√(1-v²/c²)

Consider a body attached to S’. Let x’1, y’1, z’1, and x’2, y’2, z’2, be co-ordinates of two points of the body. At any instant t of the system S, we have the following relations between these two: x₂– x₁= [(√(1 – v²/c²)(x’2 – x’1)], y2 – y1 = y’2 – y’1, z2 – z1 = z’2 – z’1. The kinematic configuration of a body in uniform translational motion with respect to a co-ordinate system depends on the velocity v of the translation. Furthermore, the kinematic configuration differs from the geometric configuration solely by a contraction in the direction of the motion, a contraction which is in the ratio 1:√(1-v²/c²).

Clock H’ will indicate the end of a period at the times in the form: tn’ = n/P₀. Since we seek the time with respect to S, the first of the transformation equations will have to be written: t = β[t’ – vx’/c²]. And since clock H’ is at rest at the origin of S’, we must always have x’ = 0, which yields: tn = βtn’ = (β/P₀)n. Observed from S, clock H’: P = P₀/β = P₀√(1-v²/c²) periods in a unit time. In other words, a clock moving uniformly with velocity v with respect to a reference system runs, as observed from this system, 1:√(1-v²/c²) times slower than an identical clock that is at rest with respect to this system.

Let us consider the equations of motion of a point moving in uniform translation with velocity u’ with respect to S’: x’ = uₓ’t’, y’ = uᵧ’t’, z’ = uz’t’. If one replaces x’, y’, z’, t’ by their values as functions of x, y, z, t, one obtains x, y, z as functions of t and, hence, the components ux, uy, uz of the velocity u of the point with respect to the system S. In this way it is possible to obtain the formula that expresses the theorem of the addition of velocities in its general form, and one can immediately see that the law is valid only in first approximation. In the special case when the velocity u’ has the same direction as the velocity v of the translation of S’ with respect to S, one easily obtains: u = (v + u’)/(1 + vu’/c²). This equation shows that if one adds two velocities, each smaller than the velocity of light in a vacuum, one always obtains a resultant velocity that is smaller than the velocity of light. If one sets v = c – λ, u’ = c – μ, where λ and μ are positive and smaller than c, one gets: u = c[(2c – λ – μ)/(2c – λ – μ + λμ/c)] <c. When one adds the velocity of light c, and a velocity smaller than c, one always obtains the velocity of light.

Let us now consider the most general co-ordinate transformation compatible with the theory of relativity. This transformation is characterized by the fact that x’, y’, z’, t’ must be linear functions of x, y, z, t, such that the condition: x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² will be satisfied identically. The transformations compatible with Newtonian mechanics can be obtained by setting c = ∞. We arrive at the equations of ordinary kinematics if, instead of the principle of the constancy of the velocity of light, we assume the existence of signals whose propagation does not require any time [ie reality rather than representations of reality].

6.1.4 General Relativity
The next step was to ‘generalise’ the conditions within which Relativity supposedly prevailed. The following notions were used by Einstein to support this:

1 Because it was ‘obvious’.
Without any proper evidence, Einstein asserted that Relativity should be universally applicable. Ironically, this is true, but that is because Relativity is actually a function of the relativity of the receipt of light-based representations, ie an inherent feature of the observation of reality, not reality.

The modification which the theory of space and time has undergone through the special relativity theory is a profound one, but a weightier point remains. According to the special relativity theory, the theorems of geometry are the laws about any possible relative positions of solid bodies at rest, and more generally the theorems which describe the relation between measurable bodies and clocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.

Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system; to these positions there always corresponds a time-interval of a definite length, independent of time and place. It would be shown that the general relativity theory cannot hold fast to this simple physical significance of space and time.

Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another, and from all other masses, that only that sort of gravitational forces are to be taken into account which any of these bodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. The bodies (S₁ and S₂) are measured with the help of measuring rods, relatively at rest; it is then found that the surface of S₁ is a sphere and the surface of the other is an ellipsoid of rotation. Why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory from the epistemological standpoint, when the thing adduced as the cause is an observable fact of experience.

The weightier point was actually that, in reality, entities can have different movements and shapes, and that is what needed an explanation. These variations do not occur as a function of referencing. If it is assumed that they do, then a compensatory factor is required, ie conflation of reality and representations thereof. He then continued the argument:

The Newtonian mechanics does not give any satisfactory answer. It says: the laws of mechanics hold true for a space R₁ relative to which the body S₁ is at rest, not however for a space relative to which S₂ is at rest. The Galilean space introduced, is a purely imaginary cause, not an observable thing. Thus Newtonian mechanics does not, in the case treated here, actually fulfil the requirements of causality.

A satisfactory explanation can only be thus given: that the physical system composed of S₁ and S₂ shows for itself alone no conceivable cause to which the different behaviour of S₁ and S₂ can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specifically the forms of S₁ and S₂ must be of such a kind that the mechanical behaviour of S₁ and S₂ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration.

Among all the conceivable spaces R₁ and R’₁ etc. moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised, from the standpoint of the theory of knowledge, cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate. (Foundation 1916)

But no person can rest satisfied with this condition of things. He asks: “How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion)? What is the reason for this preference? I seek in vain for a real something in classical mechanics (or in the special theory of relativity) to which I can attribute the different behaviour of bodies considered with respect to the reference systems K and K’. It can only be got rid of by means of a physics which is conformable to the general principle of relativity, since the equations of such a theory hold for every body of reference, whatever may be its state of motion.(General Relativity 1916)

So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law: That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations. This condition of general covariance which takes away the last remnants of physical objectivity from space and time, is a natural requirement. (Foundation 1916)

Having failed to allow for some cause, ie a force, which accounts for alteration in movement and shape, and having conflated reality with observation, Einstein was in a self-inflicted predicament. The above arguments about the need for explanations of differences, and privileged references, are all false since the start point is invalid. This is illustrated by: The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. And: How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion)? These arguments resulted in the incorrect concept that: Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations. This is only valid in the context of the receipt of representations of reality, light-based or otherwise, not reality.

2 Because classical theory is inadequate.
Since, according to Einstein, classical theory was inadequate, something else was needed.

In the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-like co-ordinate difference with the aid of a normal clock. (Foundation 1916)

As explained previously, Einstein’s depictions of the inadequacies of classical theory are based on his failure to understand that they are referring to what actually occurs, whereas Einstein is referring to the reality of representations of what actually occurred. Whether their specific explanations were correct is irrelevant, since that is a different issue.

3 Because there was a problem with the special circumstances.
Einstein’s dysfunctional logic led him to deduce that: The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specifically the forms of S₁ and S₂ must be of such a kind that the mechanical behaviour of S₁ and S₂ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration.

The fact that he had not taken gravity into account in his special circumstances, and an admission that it could be relevant, was referred to in his response to Abraham (1912). But Relativity had apparently been able to occur in the special circumstances, even though one condition of this was no gravitational effect.

In the absence of a feature which could be the causal factor for the variations, when considering the ‘real world’, ie general circumstance, Einstein had to alight on gravity. This was ‘justified’ because it was an ‘extrinsic’ factor, and had certain characteristics that supposedly meant it did not invalidate his concept of Relativity. Indeed, Einstein tried to argue that gravitational theory could be derived from Relativity, and conversely, gravitational theory proved Relativity. These assertions are invalid, since Relativity concerns a characteristic in the sensing process of reality. This does not necessarily mean that whatever Einstein subsequently wrote about gravity is faulty.

Some of this thinking is encapsulated in the following (General Relativity 1916):
In contrast to electric and magnetic fields, the gravitational field exhibits a most remarkable property. Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not depend either on the material, or on the physical state, of the body. A piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo), when they start off from rest, or with the same initial velocity.

Let us suppose that we know the space-time “course” for any natural process as regards the manner in which it takes place in the Galileian domain, relative to a Galileian body of reference K. By means of purely theoretical operations (ie simply by calculation) we are then able to find how this known natural process appears, as seen from a reference-body K’, which is accelerated relatively to K. Since a gravitational field exists with respect to this new body of reference K’, our consideration also teaches us how the gravitational field influences the process studied. For example, we learn that a body which is in a state of uniform rectilinear motion with respect to K, is executing an accelerated, and in general curvilinear motion, with respect to K’. This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to K’.

We obtain a result of fundamental importance when we carry out the consideration for a ray of light. With respect to the Galileian reference-body K, such a ray of light is transmitted rectilinearly with the velocity c. It can be shown that the path of the same ray of light is no longer a straight line when we consider it with reference to K’. From this we conclude, that, in general, rays of light are propagated curvilinearly in gravitational fields.

According to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on light.

If we now refer such a domain to a reference-body K’ possessing any kind of motion, then relative to K’ there exists a gravitational field which is variable with respect to space and time. The character of this field will of course depend on the motion chosen for K’. According to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way.

Another part of the argument is (General Relativity 1916):
Let us refer this domain to a random Gauss coordinate system, or to a “mollusc”, as reference-body K′. With respect to K′ there is a gravitational field G. We learn the behaviour of measuring-rods and clocks, and also of freely-moving material points, with reference to K′ simply by mathematical transformation. We interpret this behaviour as the behaviour of measuring-rods, clocks and material points under the influence of the gravitational field G. We introduce a hypothesis: the influence of the gravitational field on measuring-rods, clocks and freely-moving material points, continues to take place according to the same laws, even in the case where the prevailing gravitational field is not derivable from the Galileian special case, simply by means of a transformation of co-ordinates.

The next step is to investigate the space-time behaviour of the gravitational field G, which was derived from the Galileian special case simply by transformation of the coordinates. This behaviour is formulated in a law, which is always valid, no matter how the reference-body (mollusc) used in the description may be chosen.

This law is not yet the general law of the gravitational field, since the gravitational field under consideration is of a special kind. In order to find out the general law-of-field of gravitation we still require to obtain a generalisation of the law. This can be obtained by taking into consideration the following demands: (a) The required generalisation must likewise satisfy the general postulate of relativity, (b) If there is any matter in the domain under consideration, only its inertial mass, and thus only its energy is of importance for its effect in exciting a field, (c) Gravitational field and matter together must satisfy the law of the conservation of energy (and of impulse).

Finally, the general principle of relativity permits us to determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent i.e. which have already been fitted into the frame of the special theory of relativity. In this connection we proceed in principle according to the method which has already been explained for measuring-rods, clocks and freely moving material points. The theory of gravitation derived in this way from the general postulate of relativity excels not only in its beauty; nor in removing the defect attaching to classical mechanics; nor in interpreting the empirical law of the equality of inertial and gravitational mass; but it has also already explained a result of observation in astronomy, against which classical mechanics is powerless.

If we confine the application of the theory to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the coordinate system with velocities which are small compared with the velocity of light, we then obtain as a first approximation the Newtonian theory. Thus, the latter theory is obtained here without any particular assumption, whereas Newton had to introduce the hypothesis that the force of attraction between mutually attracting material points is inversely proportional to the square of the distance between them. If we increase the accuracy of the calculation, deviations from the theory of Newton make their appearance, practically all of which must nevertheless escape the test of observation owing to their smallness.

An alternative form of argument from Foundation 1916 is:
There is also a well-known physical fact which favours an extension of the relativity theory. Let there be a Galilean co-ordinate system K relative to which, at least in the four-dimensional region considered, a mass at a sufficient distance from other masses moves uniformly in a line. Let K’ be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K’ any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction of acceleration is independent of its material composition and its physical conditions.

Can any observer, at rest relative to K’ conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to K’ can be explained in as good a manner in the following way. The reference-system K’ has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K’. This conception is feasible, because to us the experience of the existence of a field of force, namely the gravitation field, has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The mechanical behaviour of the bodies relative to K’ is the same as experience would expect with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K’ can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.

It may be that gravitational waves cause alteration in speed and divergence in direction. But that is not the same as attributing this effect to the affect on a sequence of light-based representations, when conflating reality with representations thereof. Neither can it be assumed that such a detected effect is necessarily a function of a gravitational wave. Varying the reference alters what is perceived, not reality. Since a gravitational field exists with respect to this new body of reference K’, our consideration also teaches us how the gravitational field influences the process studied. For example, we learn that a body which is in a state of uniform rectilinear motion with respect to K, is executing an accelerated, and in general curvilinear motion, with respect to K’. This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to K’.

How something ‘appears’, when compared to any given reference, is not the same as what actually happens. In an, albeit brief, attempt to discern how gravitational waves might actually affect light, the author only found jargon about ‘time slowing down’/’space expanding’/etc. As has been stated previously, any given light is a physically existent entity, per se, and therefore potentially subject to alteration during transmission. Since changes to its velocity as it travels in different mediums are known, the author presumes that the actual effect on light of gravitational waves is now known. That is, while Einstein’s notions of acceleration and curvilinearly are functions of his referencing, they may be, but only coincidentally, what actually occurs.

Another pertinent example of this fallacy is: We see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easily that the path of a ray of light with reference to K’ must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K. Any given occurrence can be compared to any given reference, which will result in different outcomes. The resulting statements are correct, but only given the rules invoked. However, it is a fundamentally pointless exercise, as the requirement is to establish what actually happened, and there is a possibility of reifying concepts, and conflating them with reality, in this approach.

The character of this field will depend on the motion chosen for K’ only within the constraint of the specified comparison. The effect on whatever is affected by any given gravitational field, at any given time, is the result of the physical relationship between the field and the entity involved.

Neither can any kind of motion, whatever that actually refers to, just be attributed to gravity. The general law of the gravitational field has no relationship with the general form of relativity, hence, gravitational fields are not obtainable in this way. The observed effects can be used to discern what actually occurred, as they can with anything else. Neither was the intial form of Relativity a function of the law of the constancy of the velocity of light. The velocity of light does not depend on the space-time co-ordinates (ie references), but on what physical effects it encounters during transmission, which could include a specific gravitational force, ie not some hypothetical generality. The presence of a gravitational field does not invalidate the definition of the coordinates and the time, which led us to our objective in the special theory of relativity. Those, as defined by Einstein, were invalid, per se, because they were based on an invalid methodology for timing, and measuring distance.

The following assertions are invalid:
If we now refer such a domain to a reference-body K’ possessing any kind of motion, then relative to K’ there exists a gravitational field which is variable with respect to space and time. The character of this field will of course depend on the motion chosen for K’. According to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. (General Relativity 1916)

We see that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can “create” a gravitational field by a simple variation of the co-ordinate system. (Foundation 1916)

The pysical relationship implied does not exist. Put simply, Einstein identified a relativity in observation which was conflated with reality. Then, when he had to involve gravity, the effects thereof were seen to be similar. So he deduced an interchangeability, on the basis of an apparent equivalence, between gravitational, and relativity, theory.

Ultimately, this led to the following (General Relativity 1916):
In the first part we were able to make use of space-time co-ordinates which allowed of a simple and direct physical interpretation, and can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity of light. But the general theory of relativity cannot retain this law. We arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. We found that the presence of a gravitational field invalidates the definition of the coordinates and the time, which led us to our objective in the special theory of relativity.

In view of the results of these considerations we are led to the conviction that, according to the general principle of relativity, the space-time continuum cannot be regarded as a Euclidean one. Here it is impossible to build up a system (reference-body) from rigid bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect to one another, shall indicate position and time directly.

If we desire to adhere to our “old-time” three-dimensional view of things, then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains, ie to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, ie a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively to it. Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies.

In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity. For this reason non-rigid reference-bodies are used, which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion. Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time.

Or (Foundation 1916): In the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-like co-ordinate difference with the aid of a normal clock.

Contrary to what Einstein asserts, the space-time continuum, or more precisely physical existence, is a Euclidean one. That is, as explained in Section 2.1.3, any given reality only occurs for as long as there is no alteration in it. The extent to which the co-ordinates, ie the shape, as opposed to the co-ordinates defining spatial position, of any given body alter would need to be proven. In practical terms, it is necessary to assume that any given body retains its shape over a duration of more than one, until there is evidence to the contrary, and particularly not to assume there is some all pervading alteration in shape which is constant and applies to all entity types in all physical conditions.

That is, what Einstein refers to as our “old-time” three-dimensional view of things/ Galileian domains/ a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively to it, is reality, and not the reality of light-based representations. Furthermore, according to Einstein a gravitational field of a special kind is then present, if referenced in a certain way: Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies. A notion, like any other of this type, which is invalid, as the reality of any given gravitational wave is not a function of its reference.

Summary
Relativity first supposedly existed in a special circumstance involving only uniform rectilinear and non-rotary motion, fixed shaped bodies, light in a vacuum which travelled in straight lines at a constant speed, and no effective gravitational force. With the general circumstance there were gravitational fields, no such things as rigid bodies, the idea of a straight line also loses its meaning, and bodies which possessed any kind of motion. So now: the fictitious rigid body of reference is of no avail in the general theory of relativity. Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity. For this reason non-rigid reference-bodies are used, which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form, ad lib, during their motion.

In other words, all the conditions which prevailed originally, and were asserted to be important, were abandoned.

6.1.5 The mathematical extension of the transformation equations
On the basis of the above, Einstein evolved the following:

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as “the theory of Relativity;” this latter theory I differentiate from the former “Special Relativity theory,” and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the “Absolute Differential Calculus”, which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics.

The method of Cartesian coordinates must then be discarded, and replaced by another which does not assume the validity of Euclidean geometry for rigid bodies. According to Gauss…

For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and attach numbers to them, we have: ds² = du² + dv². Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. It is clear that Gauss co-ordinates are an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points “in space”. So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied also to a continuum of three, four or more dimensions.

If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum, we associate arbitrarily four numbers, x_1,x_2,x_3,x₄, which are known as “co-ordinates.” Adjacent points correspond to adjacent values of the coordinates. Only when the continuum is a Euclidean one is it possible to associate the co-ordinates x_1,x_2,x_3,x₄, with the points of the continuum so that we have simply ds² = dx₁² + dx₂² + dx₃² + dx₄² In this case relations hold in the four-dimensional continuum which are analogous to those holding in our three-dimensional measurements.

However, the Gauss treatment for ds² which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which “size-relations” (“distances” between neighbouring points) are defined. To every point of a continuum are assigned as many numbers (Gaussian coordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the assignment, and that numbers (Gaussian coordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian coordinate system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined “size” or “distance,” small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

We are now in a position to formulate more exactly the idea of Minkowski. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these “Galileian co-ordinate systems.” For these systems, the four co-ordinates x, y, z, t, which determine an event, or in other words, a point of the four-dimensional continuum, are defined physically in a simple manner. For the transition from one Galileian system to another, which is moving uniformly with reference to the first, the equations of the Lorentz transformation are valid. These last form the basis for the derivation of deductions from the special theory of relativity, and in themselves they are nothing more than the expression of the universal validity of the law of transmission of light for all Galileian systems of reference.

Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body K, by the space co-ordinate differences, dx, dy, dz, and the time-difference dt. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are dx’, dy’, dz’, dt’. Then these magnitudes always fulfil the condition: dx² + dy² + dz² – c²dt² = dx’² + dy’² + dz’² – c²dt’²

The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude ds² = dx² + dy² + dz² – c²dt² which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x, y, z, √(-1ct), by x_1,x_2,x_3,x₄, we also obtain the result that ds² = dx₁² + dx₂² + dx₃² + dx₄² is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or four-dimensional points. Thus, if we choose the imaginary variable √(-1ct) instead of t, we can regard the space-time continuum, in accordance with the special theory of relativity, as a “Euclidean” four-dimensional continuum.

We refer the four-dimensional space-time continuum in an arbitrary manner to Gauss co-ordinates. We assign to every point of the continuum (event) four numbers, x_1,x_2,x_3,x₄, (co-ordinates), which have not the least direct physical significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement does not even need to be of such a kind that we must regard x_1,x_2,x_3, as “space” co-ordinates and x4, as a “time” co-ordinate.

Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary existence without duration, then it would be described in space-time by a single system of values x_1,x_2,x_3,x₄. Thus its permanent existence must be characterised by an infinitely large number of such systems of values, the co-ordinate values of which are so close together as to give continuity; corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional continuum. In the same way, any such lines in our continuum correspond to many points in motion. The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of co-ordinate values x_1,x_2,x_3,x₄, in common.

When we were describing the motion of a material point relative to a body of reference, we stated nothing more than the encounters of this point with particular points of the reference-body. We can also determine the corresponding values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials. It is just the same in the case of space-measurements by means of measuring-rods, as a little consideration will show.

The following statements hold generally: Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x_1,x_2,x_3,x₄. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.

We are now in a position to replace the provisional formulation of the general principle of relativity by an exact formulation. The form [above] used, “All bodies of reference K, K‘, etc, are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature”.

We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time variables, x, y, z, t, of a Galilean reference body K, by the space-time variables, x’, y’, z’, t’ of a reference body K’. According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables, x_1,x_2,x_3,x₄, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusc”, is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusc” a certain comprehensibility as compared with the Gauss co-ordinate system is the (really unjustified) formal retention of the separate existence of the space co-ordinates as opposed to the time co-ordinate. Every point on the mollusc is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusc is considered as reference-body. The general principle of relativity requires that all these molluscs can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusc.

Einstein went from an existent state of only uniform rectilinear, and non-rotary, motion, and no gravitational force which meant there were only fixed shaped bodies, to one of complete ‘fluidity’. As a representation of physical existence this is invalid, because any given reality involves no variation. This supposed alteration in the nature of existence, and time being treated as a fourth dimension, resulted in Einstein’s suggestion that certain mathematical constructs should be used to relate physically existent entities in terms of space and time.

6.1.6 General Relativity Theory: Progression
The first manifestation of the “general principle of relativity” was: All bodies of reference K, K′, etc, are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion.

2 This then became: Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems which are covariant for all possible transformations.

Since, this condition takes away the last remnants of physical objectivity from space and time. In mathematics, all systems of equations that express physical laws must be co-variant with the Lorentz transformation. But, this cannot be maintained because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. So the Gauss co-ordinate system has to take the place of the body of reference. Therefore, it became:

3 All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.

Which can be stated in another form which renders it more intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by use of the Lorentz transformation, we replace the space-time variables, x, y, z, t, of a Galilean reference body K, by x’, y’, z’, t’ of a reference body K’. According to the general theory of relativity, on the other hand:

4 By application of arbitrary substitutions of the Gauss variables, x_1,x_2,x_3,x₄, the equations must pass over into equations of the same form, for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

However, in gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity. For this reason, non-rigid reference-bodies are used, which are as a whole not only moving in any way, but also suffer alterations in form, ad lib, during their motion. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only one condition, that the “readings” observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount.

This non-rigid reference-body, which might be termed a “reference-mollusc”, is equivalent to a Gaussian four-dimensional co-ordinate system. What gives the “mollusc” a certain comprehensibility, compared with the Gauss co-ordinate system, is the, really unjustified, retention of the separate existence of the space co-ordinates as opposed to the time co-ordinate. So:

5 All molluscs can be used as reference-bodies with equal right, and success, in the formulation of the general laws of nature; the laws themselves must be independent of the choice of mollusc. Every point on the mollusc is treated as a space-point, and every material point which is at rest relatively to it is at rest, so long as the mollusc is considered as a reference-body.

Hall of Mirrors: Section 6.2: General Theory examples

When Einstein provided examples in General Relativity 1916 to substantiate his ideas, the invalidity of the concept of Relativity is more obvious, as mathematics can obscure flaws.

6.2.1 Section 3: Space and Time in Classical Mechanics. Droppping a stone.
The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. This is incorrect. Once the stone was let loose it was no longer travelling with the carriage and the person who was holding it. With respect to the pedestrian on the embankment, the stone was seen to drop vertically, while the person who dropped the stone saw it falling in a ‘curve’ as they moved away from the falling stone.

Leaving that aside, the underlying point is what can be deemed to be the ‘actuality’? Einstein: Do the “positions” traversed by the stone lie “in reality” on a straight line, or on a parabola? First, observation, or any form of representation receipt by any type of sensory system, is not reality. Second, as everything is moving, then all movement is relative. The movement of any given entity could be defined by comparing it to any other given entity. But, proper measurement systems use of a reference that has validity, and consistency, in order to ensure comparability of results, and identify actuals.

Given the circumstance that Einstein defines, the best reference is the spatial position occupied by the pedestrian. This is a fixed co-ordinate on the earth, which is not susceptible to variation, and is a practical reference from which to compare outcomes. The point about the spatial position is that it is only deemed to be stationary as a reference, ie it is known to be moving, but it provides a valid, practical, and consistent reference from which to compare all other movements. For now, the additional complication that knowledge is only available through the receipt of representations, in this case light-based representations, can be ignored.

Einstein then asked another fundamental question: Moreover, what is meant here by motion “in space”? Distance is the spatial difference between any pair of entities/representations at any given time. It is a spatial quantity reflecting a unique occurrence in terms of when it happened, and what was involved. For there to be a distance, whatever is under consideration must exist at the same time, since something that is non-existent cannot have a spatial position. Movement is alteration in distance. It is revealed when discrete distances in sequences are compared retrospectively. Since calculating movement, as opposed to distance, involves sequence, any given entity/representation has to be deemed to be the ‘same’ over the duration of the measurement. This comparison enables the calculation of velocity, which is speed, the rate of alteration in distance, and divergence, the rate of alteration in direction.

A distance can only properly be expressed in terms of spatial units with respect to a specific time of occurrence. Alternatively, distance can be expressed in terms of duration, ie x = vt, a hypothetical expression which is defined as the duration that would have been incurred had any given entity, at a known velocity, travelled that distance in either direction. The potential issue with this approach is that t and x can become reified, ie disassociated from reality.

With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory, but only a trajectory relative to a particular body of reference. What can be clearly seen is that, apart from being invalid, this is a statement of the obvious and of no added value. The assertion that there is no such thing as an independently existing trajectory is invalid. Another implication of this notion is that nothing exists other than what is identified via any sensory system, which is incorrect.

In respect of the idea that a complete description of the motion should have referred to the processes described above: Only one timing device is required, though that is not a necessity, and it is assumed that the process of timing was properly understood. That device should have been provided to the pedestrian, as the purpose of the exercise was to define the path of the fall of the stone with respect to the earth. Describing its path, having allowed for the delay whilst each representation travels especially in a circumstance where spatial position varies, to any passenger, or pedestrian elsewhere, or anything else, is pointless.

However, Einstein then invokes what seemed to be unnecessary conditions: These data must be supplemented by such a definition of time that these time-values can be regarded essentially as magnitudes capable of observation. If we take our stand on the ground of classical mechanics…Einstein proposed providing both those involved receive a timing device. Explanation of the example then stops, with the assertion that there are, apparently, issues: In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this, and with a second difficulty prevailing here, we shall have to deal in detail later. That is Sections 8, 9 & 10, but no direct reference is made therein as to how this particular example was supposedly resolvable.

6.2.2 Section 5: The Principle of Relativity in the Restricted Sense. The raven.
Let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation (“uniform” because it is of constant velocity and direction, “translation” because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line.

Again, the issue is not that any given alternative perspective must be of a certain type of motion which is then the same as the original. The point is to understand the dynamics of any given perspective, and thereby deduce what ‘actually’ happened using a proper process. Assuming a constancy in the speed of travel of the representation, which would otherwise just be an unnecessary additional complexity. If the entity and observer are both in an identical uniform motion, then the sequence of actuality will be the same as the sequence of representation receipt. If the entity has a different speed and/or direction, then the sequence of representation receipt will vary at a constant rate, and what happened could be discerned.

To put it simply, given that the same principle applies, then the observer could be on a fairground carousel, the calculations would just become much more complex. Another way of illustrating the point is to assume that the raven croaks at a regular frequency. The same process would be applied in order to discern the raven’s flight path, it would just involve a different form of sensory system. Varying perceptions do not alter reality, not least because it has already occurred. Perceptions, whether sight based or whatever, are not reality.

Although this Section is concerned with addressing the examples Einstein gave in his 1916 paper, it is worth digressing in order to emphasise, again, an underlying issue. Einstein follows his comment about the raven as follows: Expressed in an abstract manner we may say: If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K′, provided that the latter is executing a uniform translatory motion with respect to K. It follows that: if K is a Galileian co-ordinate system, then every other co-ordinate system K′ is a Galileian one when, in relation to K, it is in a condition of uniform motion of translation. Relative to K′ the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.

We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, K’ is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K’ according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).

If the principle of relativity does not hold, then the Galileian co-ordinate systems K, K′, K″, etc, which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. We should be constrained to believe that natural laws are capable of being formulated in a particularly simple manner, and only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K₀) particular state of motion as our body of reference. We should then be justified in calling this system “absolutely at rest,” and all other Galileian systems K “in motion.”

If the principle of relativity were not valid, we should expect that the direction of motion of the earth would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth. Owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year.

The key sentences are;

1 Relative to K′ the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K. [Or the next sentence which is fundamentally a repeat: If, relative to K, K’ is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K’ according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense)].

2 If the principle of relativity does not hold, then the Galileian co-ordinate systems K, K′, K″, etc, will not be equivalent for the description of natural phenomena. We should be constrained to believe that natural laws are capable of being formulated in a particularly simple manner.

3 and on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one particular state of motion as our body of reference. We should then be justified in calling this system “absolutely at rest,” and all other Galileian systems K “in motion.”

4 If the principle of relativity were not valid, we should expect that the direction of motion of the earth would enter into the laws of nature. For owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year.

This encapsulates why the concept of Relativity is invalid. Einstein’s numerous allusions to Galileo stems from his misinterpretation of him. He understood it to mean that if in a certain mode of motion, then mechanical laws/general laws/laws of nature, or any other descriptor Einstein chose to use, are the same with respect to any entity involved in such a motion, irrespective of their velocity, in the context of representation receipt. This is incorrect. Neither was it the point Galileo was illustrating, as explained in Section 3.4.

The corollary being that in a different mode of motion this is no longer so, hence his allusion to particularly simple manner. This is true. Except that what is true is that the original idea was incorrect, and therefore the corollary of Einstein’s idea is incorrect. Whereas, from Einstein’s flawed perspective, that was asserted to be true (see last paragraph below).

In that circumstance, he argued that this would necessitate choosing one particular state of motion as our reference. That was true. Although, Einstein often pointed out that a privileged set of coordinates/body of reference is unacceptable, ie this statement was a negative as far as Einstein was concerned. And that it needed to be from amongst all possible Galileian co-ordinate systems, which it did not, as it could be anything.

This, according to Einstein, then meant that the chosen reference could be deemed to be absolutely at rest. The correctness of that depended on what he meant by ‘absolute’. Given his comments made elsewhere, and adoption of Poincaré’s declarations, ie There is no absolute space. There is no absolute time. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places, it can be assumed Einstein was referring to some state which is extrinsic to our existentially closed system, rather than the ‘actuality’. Finally, other than the reference, he asserted all other Galileian systems “in motion.”. Whereas, everything is in motion, including the reference. Which, by definition, as the reference it is deemed to be ‘stationary’in so far as all other motion is referred to it, not that it is ‘actually’ stationary.

His last point concerns what Galileo’s ship analogy was intended to prove. That is, the old idea that earth is stationary was incorrect, but because its movement is omnipresent for everything ‘of’ the earth, the effect of its movement was not necessarily always apparent. Since K0 is of the earth, then it is always at rest relative to the earth, contrary to what Einstein thought. In summary, this demonstrates that Einstein had a flawed understanding of how measuring systems operate properly, conflated representation of reality with reality, and had no understanding of the fundamental nature of our physical existence.

6.2.3 Section 6: The Theorem of the Addition of Velocities Employed in Classical Mechanics. Man walking in moving carriage.
The actuality must be that that the man was walking at a speed of v + w with respect to the embankment (ie earth). However, according to Einstein this may not be true: We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained. Again, later refers to Sections 8, 9 and 10, though this time a reference was at least made to this example. But it involved the two strikes of lightening example, where light from two separate events moved in opposite drections. That is, not only was the lightening example invalid, but it was not comparable anyway with the walking man on the carriage, or the dropping stone, or the raven, examples, and thus what was referred to did not provide an justification for Einstein’s assertions.

Einstein stated the following about the man walking example in Section 9: We were led to that conflict by the considerations of Section 6, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each second. But, according to the foregoing considerations, [in Section 9 with man observing lightening strikes from the train] the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body).

Einstein Section 10: This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section 6. Namely, if the man in the carriage covers the distance w in a unit of time, measured from the train, then this distance, as measured from the embankment is not necessarily also equal to w.

That is, no real explanation was provided, just a reaffirmation of the original assertion. In the example Einstein stated: the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. But, apparently this was not so anymore. Indeed, in another example Einstein thought he was able to conclude that the same train was of different lengths (Section 10).

Section 10: On the Relativity of the Conception of Distance. Man walking.
At first, in order to measure a length on the train, Einstein describes the correct approach. Then, he described an alternative approach which is incorrect. This amounts to, in effect, ‘timing’ the distance. Einstein does not go into detail in this Section, that is covered in his earlier papers. The alternative methodology for timing and measuring space is explained in Sections 4.5 and 4.6 using that material.

Not only are rods used to measure the distance in the first approach, but the reference is the train. The fact that it is moving is irrelevant, and that should be so, because the objective is to measure a distance on the train. Whereas, in the second method, the distance is measured with respect to the embankment, which is ‘stationary’ (ie with respect to the earth), whilst the train is moving with respect to the earth. Einstein is therefore trying to measure a distance on the train which is moving away from the point of reference, via sight. As has been explained many times, since there is a delay between creation of a representation and its receipt, then relative spatial position may vary. This is why the two methods of measuring difference can result in different outcomes.

Einstein’s assertion: A priori it is by no means certain that this last measurement will supply us with the same result as the first, is true, but that is because the first method was a valid way of measuring distance, while the second method was invalid, which is not what Einstein meant. As can be seen by his next statement: Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself. This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section 6. Namely, if the man in the carriage covers the distance w in a unit of time, measured from the train, then this distance, as measured from the embankment is not necessarily also equal to w.

The actuality is that the distance on the train was a fixed length, it does not alter with differing ways of perceiving it through sensory systems. Similarly the man on the train moved at the speed of the train, plus the speed of his walking, with respect to the earth.

6.2.4 Section 7: The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity. Ray of light.
The actuality is that the speed of the ray of light with respect to the carriage will be c – v. The justification of using c, rather than the actual speed, is noted: We shall imagine the air above it to have been removed. The following statement is incorrect: The velocity W of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. Einstein specified three sentences previously that: the tip of the ray will be transmitted with the velocity c relative to the embankment. The man was on a carriage that was moving at v with respect to the embankment. He then walked forward at w with respect to the carriage, which meant that overall his speed was v + w with respect to the embankment. The ray of light had no relationship with the carriage, and had been specified by Einstein as c with respect to the embankment anyway.

So, as Einstein states: The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c. That is, c – v. But, because of Einstein’s flawed understanding of the principle of relativity, he had a self-inflicted problem with that obvious outcome which accords with reality. This is encapsulated in: For, like every other general law of nature, the law of the transmission of light in vacuum must, according to the principle of relativity, be the same for the railway carriage as a reference-body, as when the rails are the body of reference. But, from our above consideration, this would appear to be impossible. This is the other principle which Einstein misconceived, since an entity cannot be the same speed with reference to two entities which are travelling at different velocities.

As noted above, the detail of light speed in a vacuum can be ignored. The point here is that there was one entity (a ray of light) which had an actual velocity with respect to the earth, which Einstein himself defined. Einstein resolved the apparent issue with his new methodology for timing, and measuring space. Which for this example made the speed of light the same with respect to the carriage and the embankment, contrary to reality.

6.2.5 Section 8: On the Idea of Time in Physics, Section 9: The Relativity of Simultaneity. Section 10: On the Relativity of the Conception of Distance. Strikes of lightening.
Lightening struck the embankment at two places simultaneously. So, yet again, despite the fact that this was a statement of the actuality, Einstein tried to develop a methodology to validate simultaneity, and thereby arrive at a different outcome. He rejected a sensible solution on the basis that: your definition would certainly be right, if only I knew that the light by means of which the observer at M perceives the lightning flashes travels along the length A to M with the same velocity as along the length B to M. But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. This was an acceptable rephrasing of the criterion for proof. Though it ignored the facts that the point M was equidistant from the two strikes at A and B, and the light from both these points was travelling at the same speed, ie there was nothing to prove in the first place.

Einstein then placed a timing device at three places, ie the locations of the lightening strikes (A & B), and the mid-point (M), assuring that they were of similar construction, would work in the same way, and that ‘telling the time’ involved watching the position of the pointers on them. But on this occasion, his flawed methodology of timing which he usually deployed to demonstrate that an alternative existed to the obvious solution, was irrelevant. Careful examination of what he actually said reveals that Einstein used another method.

Section 9 is concerned with Eistein’s attempt to define simultaneity. Poincaré asserted in 1902: There is no absolute time. When we say that two periods are equal, the statement has no meaning. We have no direct intuition of the simultaneity of two events occurring in two different places. A sentiment which Einstein followed. Einstein starts the Section: Up to now our considerations have been referred to a particular body of reference, a “railway embankment”. This is correct, since the example has been about two simultaneous lightening strikes on the embankment, and the possibility of placing an observer there.

However, Einstein alters the scenario: We suppose a very long train travelling along the rails with the constant velocity v. People travelling in this train regard all events in reference to the train. But originally there was no train, whether long or short, or whether moving or stationary. Then, in order to address Einstein’s irrelevant concerns about simultaneity, he suggested that an observer be at the midpoint on the train in order to prove simultaneity. A similar solution having already been rejected by him when he wanted proof that the light travelled at the same speed in both directions. Another irrelevant concern, since the light speed has been deemed to be constant by everybody, including Einstein.

The issue now was that there were two different reference points, train and embankment, and that the observers were on the train which was moving, whilst the lightening struck the embankment. Einstein: the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. This is incorrect as there is a confusion of references. That is proven by Einstein’s next statement: are two events which are simultaneous with reference to the railway embankment, also simultaneous relatively to the train? We shall show that the answer must be in the negative.

The notion that the answer to the question posed might not be true, once any adjustment is made to allow for how that knowledge was obtained by observers, again starkly demonstrates the invalidity of the concept of Relativity. Einstein tried to extricate himself from this position by: When we say that the lightning strikes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B meet each other at the mid-point M of the length A to B of the embankment. But the events A and B also correspond to positions A and B on the train. This is incorrect, and not because originally the lightening struck the embankment and there was no train. It is incorrect because the only reference point was the embankment, which was stationary (in the sense that it was ‘of the earth’), whilst the train was moving with respect to the embankment.

Einstein then makes the most fundamental of mistakes: Let M’ be the mid-point of the distance A to B on the train. Just when the lightning occurs, M’ coincides with M, but it moves with the velocity v of the train. If an observer sitting in the position M′ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, ie they would meet just where he is situated. [This is equivalent to the original proposition to prove simultaneity, except that the observer was to be on the embankment midway between the two stikes, as there was no train].

Now in reality, ie considered with reference to the railway embankment, he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result: Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).

Until this point, for the purposes of this example, the fact that Einstein had conflated a light- based representation of the event with the event itself, could have been ignored. Since, the observable light, though not c, was constant, distances were equidistant, and any movement was uniform rectilinear and non-rotary. Therefore, representation of actuality could be equivalenced with actuality.

If the recipent of the representations is moving towards one representation (light B), and hence away from the other (light A), then obviously the timings of the receipt are going to be different. Whether the observer comes to the conclusion, or not, that the lightning flash B took place earlier than the lightning flash A is irrelevant. Observation is not reality.

The difference was simply the function of an alteration in spatial position, with respect to the two lights, which occurred whilst the representations were travelling. A circumstance that was not included in the original definition of the example. It did not prove that: Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). What it did prove was that Einstein made a fundamental mistake with referencing. The presumption by Einstein that this could have been in any way correct can be traced back to his misinterpretation of Galileo’s ship analogy. This erroneous solution was used, either explicitly or implicitly, to ‘explain’ other previously described examples (eg stone dropping, man walking).

6.2.6 Section 10: On the Relativity of the Conception of Distance
Einstein attempts to explain how to measure space. In effect he ‘times’ it, which leads to the same flawed conclusion arrived at in the context of timing. For example: Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself.

6.2.7 Section 18: Special and General Principle of Relativity. In the train carriage
The flaw is in the caveat as long as it is moving uniformly. Einstein asserted that if the motion was such that the occupant could not feel it, then it is impossible to discern the relative motion. Although, all other sensory systems which could detect the movement would have to be ‘negated’ as well. The requirement of moving uniformly, in conjunction with other restrictions not explicitly stated, is yet another statement of the obvious which is meaningless, but presented as if it reveals a significant fact.

Einstein then stated that: according to the special principle of relativity, this interpretation is quite justified also from a physical point of view. But that is because, as explained previously, the special circumstances within which Relativity was supposed to prevail, were based on a misinterpretation of Galileo, not because they were inherently valid. The notion of how the carriage moved, etc, is the equivalent of Galileo’s smooth seas and no view out of the cabin. But the smooth seas were necessary so that the motion experiments would have the same outcome even though the ship was sailing, and not moored in the harbour. Galileo was proving that the earth could be moving since the motions were the same. That is, a notion that could be argued to demonstrate that the earth was not moving, was really a function of omnipresent pysical effects of being on the earth.

Einstein then introduced a different form of motion: a powerful application of the brakes, then the occupant of the carriage experiences a correspondingly powerful jerk forwards a powerful application of the brakes, then the occupant of the carriage experiences a correspondingly powerful jerk forwards. Given his flawed understanding, this is then considered by him in terms of referencing (usually K & K’, & S & S’), with the conclusion that certain laws no longer appear to hold: The mechanical behaviour is different from that of the case previously considered, and for this reason it would appear to be impossible that the same mechanical laws hold relatively to the non-uniformly moving carriage, as hold with reference to the carriage when at rest or in uniform motion. It is clear that the Galileian law does not hold with respect to the non-uniformly moving carriage.

The point being made is true, if it is understood that the circumstance involves sensory perception, and a misinterpretation of Galileo. On the basis of these errors, Einstein then had a problem: So, we feel compelled at present to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relativity. That is, the underlying notion of the general circumstances cannot hold, given his interpretation. However, that is because the circumstance of his special version is invalid. The underlying notion of general relativity, in the context of sensory perception, is valid.

Einstein then needed to resolve this self-induced conundrum: But in what follows we shall soon see that this conclusion cannot be maintained. In Section 18 (General Relativity 1916) Einstein was striving for: In contrast to this we wish to understand by the “general principle of relativity” the following statement: All bodies of reference K, K′, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion. Although he immediately explained that this version will be superseded by another: But before proceeding farther, it ought to be pointed out that this formulation must be replaced later by a more abstract one, for reasons which will become evident at a later stage.

The supposed proof that the conclusion reached at the end of that Section, ie we feel compelled at present to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relativity, could be discounted, was given in Section 20. Section 19 introduced the gravitational field, with one ultimate objective being to prove an interdependence between Relativity and Gravitational Theory, which was invalid.

6.2.8 Section 20: The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity. Man in chest
This contains all the fallacies Einstein has previously used to substantiate his argument, and therefore will not be commented on again. The result of this dysfunctional logic is supposedly: We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity. Expressed another way: We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity [above]. That is, it is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises, in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.

The actual argument Einstein proposed, as opposed to what he claimed it was, is as follows. Relativity was at all times concerned with the relativity of the receipt of representations of reality, not reality. These two existent states were conflated, both in the ‘special’ and ‘general’ circumstances. The special circumstance, within which Relativity was first described, was invalid. Since it involved ascribing a false equivalence to a certain mode of movement due to a misinterpretation of Galileo, a non-existent hypothetical state with no gravitational effect, and the invalid use of light speed in a vacuum, c, as the universal reference for movement. In order to invoke a general circumstance within which Relativity could be deemed to exist, albeit actually, as opposed to what Einstein thought, only in respect of the receipt of representations, Einstein had to discount the special circumstance on the basis of similar spurious arguments which had originally been used to justify it, while trying to claim that the original hypothesis still had some validity.

So, there was a man on a train who experienced braking, and hence, according to Einstein, but not really, this appeared to invalidate a general form of Relativity. However, it did not, according to Einstein, but not really, with the introduction of the experience of gravity, which has certain characteristics which mimic the misinterpretation of Galileo. Meanwhile, while all this variation in perception is happening, reality actually occurs in a definitive spatial position, at a definitive time. The nature of any given gravitational field, as with any given physically existent entity, is not a function of whatever it is referenced to. Whereas, the representation of any given reality by any given sensory system, in terms of time and space, is a function of spatial relationships.

Hall of Mirrors: Section 6.3: Einstein’s attempt to define physical existence

In the last section of General Relativity Einstein attempted to define our existentially closed system, ie physical existence. He started by referring to an earlier section, where he wrote: Classical mechanics starts out from the following law: Material particles sufficiently far removed from other material particles continue to move uniformly in a straight line or continue in a state of rest. We have also repeatedly emphasised that this fundamental law can only be valid for bodies of reference K which possess certain unique states of motion, and which are in uniform translational motion relative to each other. Relative to other reference-bodies K the law is not valid. Both in classical mechanics and in the special theory of relativity we therefore differentiate between reference-bodies K relative to which the recognised “laws of nature” can be said to hold, and reference-bodies K’ relative to which these laws do not hold.

This rendition of the law can be written more precisely by eliminating the misleading distinction between ‘moving’ and ‘at rest’, since everything is moving. That then illuminates the fact that while the statement is true, it is also a statement of the obvious, and as such, meaningless. By definition, if any given existent entity does not encounter any given physical effect which could alter its movement, then it will continue to have that movement.

Einstein took issue with this: This fundamental law can only be valid for bodies of reference K which possess certain unique states of motion, and which are in uniform translational motion relative to each other. Relative to other reference-bodies K’ the law is not valid.

That it is true, so long as it is understood that this is an assertion about observation, not reality. The classicists were referring to reality. That is, the law is a statement of an actual, albeit within a closed system, not a function of any given reference. Any given physically existent attribute can be compared to any other, and within that specified confine the outcome is valid, but that does not mean it is the actuality. Such a process is fundamentally pointless. More importantly, it becomes invalid when reality is conflated with a representation of it.

Einstein continued: If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. This is invalid, see Section 2.1. The author: We cannot transcend our own existence, and can never have knowledge, as opposed to belief, of any other possibility. Which means we are confined to an existentially closed system, and the issue becomes how to determine validity from within that, given its inherent nature.

We cannot know anything that is extrinsic to our existentially closed system. The notion of infinity is a belief, ie a possible alternative, since physical existence, as manifest to us, must be finite. The issue is that finite must ultimately finish, otherwise it is not finite. For us, from within that confine, the fact that finite must finish results in a relentless sequence of possibilities which we cannot know and has no end. Section 2.1: There is always the possibility of a mutually exclusive alternative, ie if A, there must be the possibility of not-A. For each subsequent ‘A’, there is the possibility of another ‘not-A’. In other words, we are unable to resolve this conundrum objectively, and (Section 2.1) must accept, when it is the case, that being unknowable is the proper scientific decision, rather than invoking a belief system in an attempt to solve the unsolvable.

Einstein asserted: The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience. This is invalid, since the concept which Einstein refers to (see below) is not infinity, and the notion of non-Euclidean geometry, or any other similar idea, being a characteristic of reality is incorrect.

As explained in Section 2.1: To be physically existent must entail physical unambiguity. That involves a unique physically existent state of any given ‘entity’, or representation, because different physical versions of them cannot co-exist. Neither can more than one physically existent state occupy the same spatial position at the same time. Thus, at any given time, unique physically existent states of some ‘substance’ exist independently of sensory detection, and have altered from the preceding existent version thereof. Therefore, physical existence is an existential sequence, of which there can only ever be one occurrence at a time, ie a reality. The frequency of alteration being determined by the quickest type(s) of attribute change, and the overall rate of alteration is one unit of duration, assuming constancy.

Similarly, that underlying physical unambiguity means that the sequence itself must progress incrementally. Every physically existent state, and its spatial position, must comprise whatever constitutes the next in the sequence, and that must involve only one degree of difference (a change) in any given attribute of that state. When considering any given existent state, and comparing it to its previous existence, change will have occurred in at least one of its attributes, but not necessarily all of them.

The differentiation of realities is a function of one degree of change in any given attribute, not some arbitrary percentage, and/or type. Any other circumstance presupposes that a reality both exists and is altering concurrently, which is an impossibility. This means, any reality is purely a spatial phenomenon. The notion of duration arises from physical existence being a sequence, which is actually a succession of discrete realities.

Hence, the usual, understandable, view that our existence involves ‘entities’ which persist whilst also undergoing alteration, is incorrect. This also applies to representations. The misconception that entities or representations exist in the same state over many durations is a reflection of the continued manifestation of certain superficial physical attributes. We tend to only deem them to have changed when the extent of the alteration reaches an arbitrary point. This perspective represents a higher-level conceptualisation of physical existence.

That is, nothing alters in any given manifestation of reality. The difference is between the manifestations of reality as the sequence progresses. The reason it supposedly does not come into conflict with the laws of thought or with experience, is because the hypothesis of Relativity concerns observation, not reality.

Einstein: Let us consider a second two-dimensional existence on a spherical surface instead of a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods as the realisation of “distance”? They cannot do this. For if they attempt to realise a straight line, they will obtain a curve, which we “three-dimensional beings” designate as a great circle, i.e. a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area that can be compared with the area of a square constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits.

This is incorrect, as it does not have no limits, as in the proper definition of infinite, it is just a circle, or a sphere. Einstein was at liberty to invoke whatever beliefs he wanted, but not to assert them under the guise of fact. A straight line, in any given direction, cannot, by definition, be drawn on a sphere. Again, by definition, progressing in a ‘straight line’ on a sphere will delineate a circle eventually, as such a line must return to its start point. These simple facts are not necessarily transferable to the notion that the universe (physical existence) must be a sphere. The actual shape of the universe cannot be known, as that involves being extrinsic to our own existentially closed system.

Einstein: On their spherical surface our flat beings would find this ratio smaller than π, the difference being the more considerable, the greater the radius of the circle in comparison with the radius of the “world-sphere.” By means of this relation the spherical beings can determine the radius of their universe (“world”), even when only a relatively small part of their world-sphere is available for their measurements. But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

Einstein: For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value π, which is independent of the diameter of the circle. This is an actual, the caveats of the same rod, Euclidean geometry, and independent are irrelevant. Either a measurement is carried out properly, or it is not. As happened earlier, when a flawed methodology was deployed to measure distance with the result that Einstein derived a different outcome, even though the distance remained unaltered, now Einstein proposed that by virtue of these caveats, the ratio is less than π.

Thus if the spherical surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

One key word is practically. In reality, it is either flat, or it is not. This is a general flaw in Einstein’s thinking. For example, when considering clocks, he considered being sufficiently close as equivalent to occupying the same spatial position, which it is not. This fallacy is exposed when a definition is required as to what constitutes being practically/sufficiently close/about the same as/etc, as there must be a point at which this is no longer valid. Then an explanation is needed as to why that extra instance of differentiation justified the notion that there has been change, instead of it being at some preceding instance.

Whether or not there is an ability to detect the nature of the surface is irrelevant. This, and the latter part of this statement, is similar in form to Einstein’s misrepresentation of Galileo, and his attempt to argue that Relativity was still valid in some way, even when a critical condition had been abandoned. Einstein imbues humans with the ability to alter reality by their perception, which is obviously an invalid concept, since physical existence occurs independently of its sensory detection, and has already happened by then.

Neither does it follow that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. The actual shape of the universe is unknowable.

Einstein then admitted to his presumption: Perhaps the reader will wonder why we have placed our “beings” on a sphere rather than on another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio of the circumference c of a circle to its radius r depends on r, but for a given value of r it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”

His argument is circular, and therefore meaningless. The fact that the sphere is unique in possessing the property that all points on it are equivalent is irrelevant. Obviously, the ratio of the circumference c of a circle to its radius r depends on r, in that the ratio of a circle’s circumference to its diameter is a constant, ie π. That is, this notion of a dependency is false, as the two dimensions are in a fixed spatial relationship, by definition. Which means, as the radius alters the circumference alters accordingly, otherwise it would no longer be a circle/sphere. This is admitted by Einstein: but for a given value of r it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”. There is no but, it is a fact. Einstein was trying to make a statement of the obvious appear meaningful, and invoke the possibility of an alternative which does not exist.

Einstein then asserted: According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.

We already know that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be nil. Thus such a universe could not be inhabited by matter everywhere.

This statement is invalid. It has similarities with Einstein’s attempts to link Gravitational theory and Relativity, and thereby present Relativity with a justification it does not possess. The false start is: the geometrical properties of space are not independent, but they are determined by matter. This is stated to be a function of the general theory of relativity. Which it is, once it is realised what that notion actually means according to Einstein. Because Relativity, whether as manifest in the so-called special circumstances, or generally, is a hypothesis about human observation, not reality. This is confirmed in a subsequent sentence: We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. The conclusion asserted by Einstein is invalid: Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. From within our existentially closed system we can conclude, on the basis of properly established knowledge, the nature of that system, but nothing else.

The next false statement was: We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. Whether any given timing device, or distance measuring device, is affected in some way by any given gravitational field is irrelevant. These devices are assumed, in order to carry out their function, to be unaffected, whether that is virtue of construction, or compensation for know affects. Otherwise, the measuring system is useless. This is how, in effect, all such devices for any given measuring system act as if they were one device, with the measurement being compared to a generic conceptualisation of the attribute being considered. That alone is why the following statement is incorrect: This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. More importantly, it is invalid because what we can know about our existentially closed system is that it is a sequence of manifestations, each of which involves no alteration, ie Euclidean geometry is relevant.

Einstein then caveated that with the flawed notion of slightly. Again, either ‘it’ ‘is’, or not. There is no equivalent state as ‘nearly’. This is based on a presumption anyway, about how the universe differs, indicated by the word conceivable, which needs to be proven first, otherwise, yet again, the argument is self-fulfilling.

Hall of Mirrors: Section 7: Conclusions

1 We are confined to an existentially closed system. This means there is a ‘boundary’ beyond which nothing is knowable, and there are fundamental ‘rules’ concerning how it occurs, both of which must be taken into account with any scientific endeavour.

2 Our closed system (physical existence), at any given time, comprises whatever is physically existent, and physically existent representations of what previously existed. It is an existential sequence, of which there can only ever be one occurrence at a time, ie a reality. Any given reality is purely a spatial phenomenon as it only exists for one duration. The notion of a greater duration arises from physical existence appearing to have a continuity, whereas it is actually a sequence of discrete occurrences.

3 We can only become aware of what existed from what is manifested to us via certain physical processes (light, sound, etc). The resultant representations are independently physically existent. The time of occurrence, and spatial position at that time, can only be determined from when, and where, the representations were received. As different entities cannot occupy the same spatial position, there is always a distance between occurrence and recipient entity, so there is always a delay between occurrence and the representation being created, and when it was received. Hence, the distance travelled, and the duration thereby incurred, is a function of the spatial position of the occurrence with respect to the position of what became the recipient entity when the representation was received. Not the distance between the event and that entity when it happened or when the representation was received. Therefore, alterations in spatial position, whilst any representation is travelling, are critical. This is Relativity. Relativity is not a characteristic of the physically existent occurrence.

4 Relativity is an inherent feature of the reality of any given type of physically existent representations, which are associated with any given valid sensory system.

5 Einstein’s hypothesis of Relativity concerned the relativity in the reality of the receipt of light-based representations of reality, ie observation. It is not a valid hypothesis of existent occurrences. Neither is it in its original (special) circumstance a valid hypothesis of observation, since it involved conflation of reality with observation, ascribing a false equivalence to a certain mode of movement, no gravitational effect, the invalid use of light speed in a vacuum as the universal reference for movement, the use of that instead of an ‘atmospheric’ light speed, the abandonment of the valid rule of addition of velocities, and an invalid timing methodology. Nor is it a valid hypothesis of observation in the general circumstance, since it still involved conflation of reality with observation.

6 There is no interdependence between Relativity Theory and Gravitational Theory, as asserted by Einstein. There is no direct physical relationship between whatever sensed the occurrence, irrespective of sensory system deployed or form of movement it was in, and the physically existent reality of gravitational waves (also electrodynamics, etc) in relation to bodies, rigid or otherwise, and clocks (ie time). This does not necessarily mean that whatever Einstein subsequently wrote about gravity was faulty.

References

Einstein (1905): On the electrodynamics of moving bodies
Einstein (1907): Volume 2: The Swiss Years: Writings 1900-1909, Doc 47: On the relativity principle and the conclusions drawn from it
Einstein (1910): Volume 3: The Swiss Years: Writings 1909-1911, Doc 2: The principle of relativity and its consequences in modern physics
Einstein (1911), Volume 3: The Swiss Years: Writings 1909-1911, Doc 17: The Theory of Relativity
Einstein (1912): Volume 4: The Swiss Years: Writings 1912-1914, Doc 1: Manuscript of Special Relativity
Einstein (1912): Volume 4: The Swiss Years: Writings 1912-1914, Doc 8: Relativity and Gravitaion: Reply to a comment by M. Abraham
Einstein (1913): Volume 4: The Swiss Years: Writings 1912-1914: Document 21: Theory of Relativity,
Einstein (1916): Relativity: The Special and General Theory,
Einstein (1916): The Foundation of the Generalised Theory of Relativity
Einstein (1920): Ether and The Theory of Relativity
Einstein (1921): A Brief Outline of the Development of the Theory of Relativity
Einstein (1922): The meaning of Relativity

Michelson (1881): The Relative Motion of the Earth and the Luminiferous Ether
Michelson and Morley (1886): Influence of Motion of the Medium on the Velocity of Light
Michelson and Morley (1887): On the Relative Motion of the Earth and the Luminiferous Ether

Lorentz (1892): The Relative Motion of the Earth and the Aether
Lorentz (1895): Attempt at a Theory of Electrical and Opical Phenomena in Moving Bodies
Lorentz (1904): Electromagnetic phenomena in a system moving with any velocity smaller than that of light

Poincaré (1900): The Theory of Lorentz and the principle of reaction
Poincaré (1902): Science and Hypothesis
Poincaré (1904): The Principles of Mathematical Physics
Poincaré (1905): On the Dynamics of the Electron

Minkowski (1908): Space and time

Section 3.1: Quotes

Michelson (1881): The Relative Motion of the Earth and the Luminiferous Ether

Paras 1-2
Assuming the ether is at rest, the earth moving through it. Suppose the direction of a line joining the two points to coincide with the direction of earth’s motion. Then the time required for light to pass from one point to another on the earth’s surface, would depend on the direction in which it travels. Let:

V = velocity of light

v = speed of the earth with respect to the ether

D = distance between the two points

d = distance earth moves while light travels between points [away from each other]

d’ = distance earth moves while light travels between points [towards each other]

T = time for light to pass between points when travelling away from each other

T’= time for light to pass between points when travelling towards each other

To = time for light to pass between points when earth is at rest.

Then: T = (D + d)/V = d/v and T’ = (D – d’)/V = d’/v

Therefore: d = D[v/(V – v)] and d’ = D[v/(V + v)]

Therefore: T = D/(V – v) and T’ = D/(V + v)

T + T’ = 2D[V/( V²– v²)], T – T’ = 2To(v/V) nearly, and v = V[(T – T’)/2To] nearly If the light had travelled in a direction at right angles to the earth’s motion it would be entirely unaffected and the time of going and returning is 2To = 2D/V. Difference between T+T’ and 2To is 2DV[v²/(V²(V²– v²)]nearly or 2Tov²/ V² nearly. The distance light travels in the first case [T+T’] is greater than in the second [2To], by the quantity 2D(v²/V²).

Start of Comment

Correct expressions are:
Total time when earth moving is: T+T’ = 2DV/(V²- v²)]

Total time when earth is stationary is: 2To = 2D/V

Difference between these is: 2Dv²/[V( V²- v²)]

Example:
Let V = 10, v = 2, D = 6

When the earth and the light are moving in the same direction:

T = 0.75, the earth travels 1.5, the light travels 7.5

When the earth and the light are moving towards each other:

T’ = 0.5, the earth travels 1, light travels 5

When the earth does not move:

To = 0.6, the light travels 6.

Total time is 1.25 when earth moves, and 1.2 when stationary.

Total distance the light travels is 12.5 when the earth moves, and 12 when stationary.

Total distance the earth moves is 2.5.

End of Comment

Para 3
If, therefore, an apparatus is so constructed as to permit two pencils of light, which have traveled over paths at right angles to each other, to interfere, the pencil which has traveled in the direction of the earth’s motion, will in reality travel 4/100 of a wave-length farther than it would have done, were the earth at rest. The other pencil being at right angles to the motion would not be affected. If, now, the apparatus be revolved through 90° so that the second pencil is brought into the direction of the earth’s motion, its path will have lengthened 4/100 wave-lengths. The total change in the position of the interference bands would be 8/100 of the distance between the bands, a quantity easily measurable.

Conclusion (after experimentation)
The interpretation of these results is that there is no displacement of the interference bands. The result of the hypothesis of a stationary ether is thus shown to be incorrect. This conclusion directly contradicts the explanation of the phenomenon of aberration which has been hitherto generally accepted, and which presupposes that the earth moves through the ether, the latter remaining at rest.

Michelson and Morley (1886): Influence of Motion of the Medium on the Velocity of Light

Para 1
The only work of any consequence, on the influence upon the velocity of light of the motion of the medium through which it passes, is the experiment of Fizeau. He announced the remarkable result that the increment of velocity which the light experienced was not equal to the velocity of the medium, but was a fraction of this velocity which depended on the index of refraction of the medium.

Para 2
But the ratio of the velocity of light in the external ether to that within the prism is n, the index of refraction, and is equal to the inverse ratio of the square root of the densities, (n²-1)/n², which is Fresnel’s formula. Fresnel’s statement amounts then to saying that the ether within a moving body remains stationary with the exception of the portions which are condensed around the particles. If this condensed atmosphere be insisted upon, every particle with its atmosphere may be regarded as a single body, and then the statement is, simply, that the ether is entirely unaffected by the motion of the matter which it permeates.

Conclusion (after experimentation)
The result of this work is therefore that the result announced by Fizeau is essentially correct; and that the luminiferous ether is entirely unaffected by the motion of the matter which it permeates.

Michelson and Morley (1887): On the Relative Motion of the Earth and the Luminiferous Ether

Para 2
According to Fresnel, first, the ether is supposed to be at rest except in the interior of transparent media, in which secondly, it is supposed to move with a velocity less than the velocity of the medium in the ratio (n²-1)/n², where n is the index of refraction. These two hypotheses give a complete and satisfactory explanation of aberration. The second hypothesis, notwithstanding its seeming improbability, must be considered as fully proved, first, by the celebrated experiment of Fizeau and secondly, by the ample confirmation of our own work. The experimental trial of the first hypothesis forms the subject of the present paper.

If the earth were a transparent body, it might perhaps be conceded that the inter-molecular ether was at rest in space, notwithstanding the motion of the earth in its orbit; but we have no right to extend the conclusion from these experiments to opaque bodies. There can hardly be question that the ether can and does pass through metals, but again we have no right to assume that it makes its escape with perfect freedom, and if there be any resistance, however slight, we certainly could not assume an opaque body such as the whole earth to offer free passage through its entire mass.

Conclusion (after experimentation)

The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part. But since the displacement is proportional to the square of the velocity, the relative velocity of the earth and the ether is probably less than one sixth the earth’s orbital velocity, and certainly less than one-fourth. It appears, from all that precedes, reasonably certain that if there be any relative motion between the earth and the luminiferous ether, it must be small.

Lorentz (1892): The Relative Motion of the Earth and the Ether

Paras 1, 2 & 3
To explain the aberration of light it was assumed by Fresnel that the ether does not share the annual motion of earth which requires that our planet is completely permeable for that medium. Later, Stokes sought an explanation on the assumption that the ether is dragged by the earth and thus to each point at the earth’s surface the speed of the ether is the same as that of the earth. Of the two extreme views I thought it was necessary to reject that of Stokes. It was possible to explain nearly all considered phenomena by Fresnel’s theory, if we assume the “dragging coefficient” for transparent ponderable substances given by Fresnel.

Paras 4 & 5
A great difficulty was posed by an interference experiment executed by Michelson, in order to decide between the two theories. It was noted by Maxwell, that if the ether remains at rest, then the motion of earth must have an influence on the time required by light to travel forth and back between two points regarded as fixed to earth. If L is the distance of the points, V the velocity of light, and p that of earth, then: the time (if line of points parallel to direction of motion) is: 2(L/V)[1 + p²/V²] and if it is perpendicular to it: 2(L/V)[1 + p²/2V²], making a difference of Lp²/V³.

Para 6
Michelson used a device with two equally long horizontal arms perpendicular to each other, with mirrors at the ends and perpendicular to their direction. An interference phenomenon occurred, when (from the radius of the intersection) a ray travelled forth and back along one arm, and another along the other arm. The device, including the light source and observation telescope, could be rotated around a vertical axis, and the observation time was chosen so that one can bring one arm or the other arm into the direction of motion of earth. Assuming that this is the case, then, if Fresnel’s theory is correct, due to the earth’s motion the rays that travel forth and back into the earth’s direction, must suffer a certain delay by Lp²/V³ in respect to the other [perpendicular] ray. When rotated by 90°, all phase shifts must be altered by an amount, which, expressed in unit time, can be given by the double of magnitude Lp²/V³. But a displacement of the interference fringes could not be observed.

Para 7
Against this experiment one can argue, that the length of the arms was too small to obtain any observable displacement of the fringes. Michelson and Morley repeated the experiment on a larger scale. The light rays were travelling forth and back in mutually normal directions several times, because they were reflected every time by mirrors; the latter as well as all other parts of the apparatus stood on a stone plate, that swam on mercury and which could be rotated in horizontal direction. However, the shift as required by Fresnel’s theory could not be observed again.

Para 8
I have sought a long time to explain this experiment without success, and eventually I found only one way to reconcile the result with Fresnel’s theory. It consists of the assumption that the line joining two points of a solid body doesn’t conserve its length, when it is once in motion parallel to the direction of motion of earth, and afterwards it is brought normal to it. The difference would be removed when a = p²/2v². A change in length of the arms in the first experiment, and the size of the stone plate in the second, is not inconceivable.

Paras 9 & 10
Indeed, what determines the size and shape of a solid body? Apparently, the intensity of molecular forces; any cause that could modify it, could modify the shape and size as well. We can assume that electric and magnetic forces act by intervention of the ether. It is not unnatural to assume the same for molecular forces, but then it can make a difference, whether the connecting line of two particles, which move together through the ether, is moving parallel to the direction of motion or perpendicular to it. Since we know nothing about the nature of molecular forces, it is impossible to verify the hypothesis. We can, by introducing more or less plausible assumptions, calculate the influence of the motion of ponderable matter on electric and magnetic forces.

Para 14 & 15
We cannot ascribe great importance to this result; the transfer to molecular forces of what we have found for electrical forces, may be too risky for some. Moreover, if we want to do this, it remains undecided whether earth’s motion shortens the dimensions in one direction, as it was supposed before, or elongates the length perpendicular to it, by which assumption we could reach the same result. Anyway, changes of the molecular forces, and consequently of the body’s size of order 1 – p²/2v² are possible.

Lorentz (1895): Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies

Introduction
That we cannot speak about an absolute rest of the ether, is self-evident; this expression would not even make sense. When I say for the sake of brevity, that the ether would be at rest, then this only means that one part of this medium does not move against the other one.

To come to the basic equations for the phenomena of electricity in moving bodies, I joined an opinion that has been represented in recent years by several physicists; I have indeed assumed that small electrically charged molecules exist in all bodies, and that all electric processes are based on the location and motion of these “ions”.

The periodically changing polarization, which forms a light ray according to Maxwell’s theory, become vibrations of the ions in this conception. It is well known that many researchers, who stood on the basis of the older theory of light, considered the resonance of ponderable matter as the cause of color dispersion, and this explanation can in the main also included into the electro-magnetic theory of light, for which it is only necessary to ascribe to the ions a certain mass. This I have shown in a previous paper, in which I derived the equations of motion from actions at a distance, and not from Maxwell’s expressions.

In general, the assumptions that I introduce represent in a certain sense a return to the earlier theories of electricity. The core of Maxwell’s views is therefore not lost, but it cannot be denied that with the adoption of ions we are not far away from the electric particles, which were used earlier. Since the essence of electric charge is seen by us in the accumulation of positive or negative charged particles, and since the basic formulas for stationary ions give Coulomb’s law, therefore, for example, the entire electrostatics can be brought into the earlier form.

Section 1: The fundamental equations for a system of ions located in the ether.

The equations for the ether
When forming the equations of motion, we express all magnitudes in electromagnetic measure, and use a coordinate system that is at rest in the ether. Now according to Maxwell, two kinds of deviations from the equilibrium state can exist in this medium. The deviation of first kind, which can be found in the vicinity of any charged body, we call the dielectric displacement. It is solenoidally distributed in “pure” ether, ie in the spaces between the ions. We now want to assume, that ether exists in the space where an ion is located, and that a dielectric displacement can happen here, ie dielectric displacement caused by a single ion is extended over the interior of the other ions.

The second deviation of the equilibrium state of the ether will be determined by the magnetic force. It depends on the instantaneous current distribution, and satisfies [a condition of magnetic force] whose applicability we also presuppose for the interior of ponderable matter. We also assume the relation for the interior of the ions as well as for the interspaces, by which, in Maxwell’s theory, the dielectric displacement is connected with the temporal variation of the magnetic force.

Section 2: Electric phenomena in ponderable bodies that are moving with constant velocity through the stationary ether.

Transformation of the fundamental equations.
From now on it will be assumed that the bodies to be considered are moving at a steady velocity of translation p, under which we will have to understand in almost all applications, the speed of the earth in its motion around the sun. It would be interesting at first to further develop the theory for stationary bodies, but let us immediately turn to the more general case.

The treatment of the problems that are now coming into play is most simple, when instead of the co-ordinate system used above, we introduce another one which is rigidly connected with ponderable matter and therefore shares its displacement. While the coordinates of a point with respect to the fixed system were called x, y, z, let those, which refer to the moving system and which I call the relative coordinates, denoted by (x), (y), (z) for the time being.

Under a fixed point, we now understand one point that has a steady position with respect to the new axis; in the same way, by rest or motion of a physical particle, we shall mean the relative rest or the relative motion in relation to ponderable matter. Real velocity is thus: the velocity of the previously mentioned relative motion plus p.

Application to electrostatics.
To clearly define the meaning of the above formulas, we will compare the considered system S₁ with a second one S₂. The latter should not be moved, and it arises from S₁ by increasing all the dimensions that have the direction of the x-axis (therefore the relevant dimensions of the ions as well), in the ratio √(V² – p²) to V, or: between the coordinates x, y, z of a point of S₁ and the coordinates x’, y’, z’ of the same corresponding point of S₂, we let remain the relations: x = x’√(1 – p²/V²), y = y’, z = z’. If we apply to all magnitudes, which are related to the second system, a prime so they can be distinguished, then: p’ = p√(1 – p²/V²).

Finally, it should be noted that by our formulas, the distribution of a charge over a given conductor, as well as the attraction or repulsion of charged bodies by the motion of the earth, must be changed. But this influence is limited to the second order, namely if the fraction p/V is called a magnitude of first order, and thus the fraction p²/V² is called a magnitude of second order.

Since p/V = 1/10000, we may not hope, neglecting some very special cases, to find with respect to electrical and optical phenomena an influence of earth’s motion that depends on p²/V². The only thing that could be observed in relation to bodies at rest on earth is the magnetic force. At first glance, we might expect a corresponding effect on the current elements.

Section 3: Investigation of oscillations excited by oscillating ions.

General formulas
Once the motion of the ions is given, known functions of x, y, z and t appear on the right-hand side of equations [dielectric displacement and magnetic force at spatial co-ordinates] with respect to the last variable [t], these are periodic functions, if the ions carry out oscillations with constant amplitude, and a common oscillation interval T. It is easy to see that in this case the equations are satisfied by values of [dielectric displacement and magnetic force at spatial co-ordinates] which also have the period T. Therefore, the important and almost self-evident theorem is given: If ion oscillations of period T take place in a light source, then the dielectricd displacement and the magnetic force indicate the same periodicity at each point that shares the translation of the source.

The variable t’ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this variable the local time of this point, in contrast to the general time t. The transition from one time to another is provided by equation t’ = t – (p_x/V²)x – (p_y/V²)y – (p_z/V²)z. [Where p = translational velocity of ponderable matter, and V = velocity of light in the ether. T = oscillation period, t = time, and t’ = local time].

Section 5: Application to optical phenomena.

Reduction to a resting system.
The specification of the influence that the motion of ponderable bodies exerts on the phenomena of light can be achieved in a very simple manner, if we neglect circular polarization. Namely we want, as we did it earlier already, by continuing omission of magnitudes of second order, to introduce (instead of t) the “local time” as an independent variable: t’ = t – 1/V²(p_xx +p_yy + p_zz).

Although we have given (in the previous consideration) to the coordinate axes the directions of the symmetry axis, the derived theorem applies to any right-angled coordinate system. We can recognize this, when we consider, that for local time t’, it can also be written: t – p_rr/V² where r is the line drawn from the coordinate origin to the point (x, y, z), and t′t’ is independent of the direction of the coordinate axes.

Different applications.
We want to call the two states of motion, the stationary and the moving system, of bodies, corresponding states. Now, they shall be mutually compared more precisely.

If in a stationary system the magnitudes [dielectric polarisation in a ponderable body, electric force, magnetic force] are periodic functions of t with the period T, then in the other system those magnitudes have the same period with respect to t’, thus also with respect to t, when we let x, y, z remain constant. When interpreting this result, we have to consider, that two periods must be distinguished in the case of translation, which we accordingly can call absolute and relative period. We are dealing with the absolute one, when we consider the temporal variations in a point that has a fixed position against the ether; but we are dealing with the relative one, when we consider a point that moves together with ponderable matter. The things found above can now be expressed as follows: If a state of oscillation in the moving system shall correspond to a state in the stationary system, then the relative oscillation period in the first mentioned case must be equal to the oscillation period in the second mentioned case.

In the stationary system, no motion of light may take place at an arbitrary location. From that it directly follows, that a surface that forms the border of a space filled with light within a stationary body can have the same meaning when the body is moving. In the moving system, relative light rays of relative oscillation period T were mirrored and refracted by the same laws, as rays of the oscillations period T in the stationary system.

Section 6: Experiments whose results cannot be explained without further ado.

The interference experiment of Michelson.
As it was first noticed by Maxwell, and which follows from a very simple calculation, the time required by a light ray to travel forth and back between two points A and B must change, as soon as these points are subject to a common displacement, without dragging the ether. Although the variation is a magnitude of second order, it is nevertheless big enough that it can be demonstrated by means of a sensitive interference method.

The experiment was executed by Michelson in the year 1881. His apparatus, a kind of interference-refractor, had two equally long, horizontal, mutually perpendicular arms P and Q, and from the two mutually interfering light beams, one went forth and back along arm P and the other one along arm Q. The whole instrument, including the light source and the observation device, could be rotated around a vertical axis, and especially the two locations come into consideration, at which arm P or arm Q had (so far as possible) the direction of earth’s motion. Now, during the rotation from one “main-position” into the other, a displacement of the interference fringes was expected on the basis of Fresnel’s theory.

However, the change in this displacement caused by the variation of the propagation times, we want to call it Maxwell’s displacement for sake of brevity, was [such that] Michelson thought that he is allowed to conclude that the ether wouldn’t remain at rest when the Earth is moving, a conclusion however, whose correctness was soon questioned. By inadvertence, Michelson has estimated the change of the phase differences, as expected by the theory, to double of the correct value; if we correct this error, we arrive at displacements, which could be hidden by the observational errors.

Together with Morley, Michelson has started again the investigation, where (to increase the sensitivity) he let reflect every light beam by some mirrors back and forth. This artifice gave the same advantage, as if the arms of the earlier apparatus would have been considerably extended. The mirror was carried by a heavy stone plate, that floated on mercury and thus was easily rotatable. Altogether, every beam had to traverse a path of 22 meters, and by Fresnel’s theory, when passing from one main-position to the other, a displacement of 0,4 of the fringe-distance was to be expected. Nevertheless, during the rotation only displacements of at most 0,02 of the fringe-distance were obtained; they probably might stem from observational errors.

Now, is it allowed to assume on the basis of this result, that the ether shares the motion of earth, and thus Stokes’ aberration theory is the correct one? The difficulties, with which this theory is confronted when explaining aberration, seem too great to me as for having that opinion, so I rather should try to remove the contradiction between Fresnel’s theory and Michelson’s result. Indeed this can be achieved by means of a hypothesis which I already have spoken out some time ago, and to which, as I found out later, also Fitzgerald arrived.

We want to assume, that we would work with an instrument as that during the first experiments, and that with respect to one main-position, the arm P coincides exactly with the direction of earth’s motion. Let p be the velocity of this motion, and L the length of every arm, thus 2L the path of the light rays. Then by the theory, the translation causes that the time in which one light-beam travels forth and back along P, is longer by: L.(p²/V³) than the time, in which the other beam completes its path. Exactly this difference would also exist, when without that the translation, arm P would be longer by: L.(p²/2V²) than arm Q. Similar things are true for the second main-position.

If we assume, that the arm lying in the direction of earth’s motion, is shorter by L.(p²/2V²) than the other one, and simultaneously the translation would have an influence which follows from Fresnel’s theory, then the result of Michelson’s experiment is fully explained. Consequently, we have to imagine that the motion of a rigid body, e.g. a brass rod or of the stone plate used in later experiments, would have an influence on the dimensions throughout the ether which, depending on the orientation of the body with respect to the direction of motion, is different, and thus it would cause a contraction in the direction of motion in the ratio of 1 to √(1 – p²/v²).

Consequently, if, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Σ’, if the velocity w is imparted to it, will of itself change into the system Σ. In other terms, the translation will produce the deformation.

As strange as this hypothesis would appear at first sight, it’s not so far off, as soon as we assume that the molecular forces, similarly as we now definitely can say it of the electrical and magnetic forces, are transmitted through the ether. If this is so, then the translation will change the action between two molecules or atoms, most likely in a similar way, as the attraction or repulsion between charged particles. Since the shape and the dimensions of a fixed body are, in the last instance, determined by the intensity of the molecular effects, then also a change of dimensions is inevitable.

From a theoretical perspective there is no objection to the hypothesis. As regards the experimental confirmation, it is to be noticed at first, that the relevant elongations and contractions are extremely small. If we would like to observe those magnitudes, then we probably can hope to succeed only by an interference method. We would have to work with two mutually perpendicular rods, and of two mutually interfering light beams, let one travel back and forth with respect to the first rod, and the other with respect to the second rod. By that we come again to Michelson’s experiment, and we wouldn’t observe any displacement of the fringes during the rotation. In reverse as we have expressed it earlier, we could say now, that the displacement stemming from the changes of length, is compensated by Maxwell’s displacement.

It is noteworthy, the we are led exactly to the above presupposed changes of dimensions, when we first (without consideration of the molecular motion) assume, that in a rigid body which remains at its own, the forces, attractions or repulsions which act on an arbitrary molecule, are mutually in equilibrium, and second, for which, however, there is no reason, when we apply to these molecular forces the law which we have derived in [para 23 Section 2: Application to electrostatics].

If we understand by S₁ and S₂ not two systems of charged particles as in that paragraph, but two systems of molecules, the second at rest and the first with the velocity p in the direction of the x axis, between whose dimensions the relation given early exists, and if we assume, that in both systems the x-components of the forces are the same, but the y- and z-components are mutually different by the factors given in [para 23 Section 2: Application to electrostatics], then it is clear, that the forces in S₁ will be mutually cancelled, as soon as this happens in S₂. Consequently, if S₂ is the state of equilibrium of a stationary, rigid body, then in S₁ the molecules have exactly those positions, in which they can remain under the influence of translation. The displacement would of course cause this configuration by itself, and thus it would cause a contraction in the direction of motion in the ratio of 1 to √(1-p²/V²).

Lorentz (1904): Electromagnetic phenomena in a system moving with any velocity smaller than that of light

Para 1
The problem of determining the influence exerted on electric and optical phenomena by a translation, such as all systems have in virtue of the earth’s annual motion, admits of a comparatively simple solution, so long as only those terms need be taken into account which are proportional to the first power of the ratio between the velocity of translation w and the velocity of light c. Cases in which quantities of the second order, i.e. of the order w²/ c² may be perceptible, present more difficulties.

The negative result of Michelson’s interference experiment led Fitzgerald and myself to the conclusion that the dimensions of solid bodies are slightly altered by their motion through the ether. Some new experiments in which a second order effect was sought have recently been published. Rayleigh and Brace have examined the question whether the earth’s motion may cause a body to become doubly refracting; at first sight this might be expected, if the just mentioned change of dimensions is admitted. Both physicists have however come to a negative result.

Para 2
The experiments of which I have spoken are not the only reason for which a new examination of the problems connected with the motion of the earth is desirable. Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelsons’s negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light.

Surely, this course of inventing special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory, if it were possible to show by means of certain fundamental assumptions, and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. I believe to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light.

Para 4
We shall further transform these formulae by a change of variables: c²(c² – w²) = k², and understanding by l another numerical quantity to be determined further on. I take as new independent variables x’ = klx, y’ = ly, z’ = lz and t’ = (l/k)t – kl(w/c²)x. As to the coefficient l, it is to be considered as a function of w, whose value is 1 for w=0 The variable t’ may be called the local time; indeed, for k=1, l=1 it becomes identical with what I have formerly understood by this name.

Para 8
Thus far we have only used the fundamental equations without any new assumptions. I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming kl times, and those in perpendicular direction l times, smaller. Our assumption amounts to saying that in an electrostatic system, moving with a velocity, all electrons are flattened ellipsoids with their smaller axes in the direction of motion. In this deformation each element of volume is understood to preserve its charge.

In the second place I suppose the forces between uncharged particles, as well as those between such particles and electrons, are influenced by a translation in quite the same way as the electric forces in an electrostatic system. In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do not move relatively to each other. we shall have between the forces acting in a system without, and the same system with, a translation, the same relation [of electric force for stationary ions].

We see by this that, as soon as the resulting force is 0 for a particle in Σ’ , the same must be true for the corresponding particle in Σ. Consequently, if neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Σ’ , if the velocity w is imparted to it, will of itself change into the system Σ. In other terms, the translation will produce the deformation.

It will easily be seen that the hypothesis, formerly made in connexion with Michelson’s experiment, is implied in what has now been said. However, the present hypothesis is more general because the only limitation imposed on the motion is that its velocity be smaller than that of light. We are now in a position to calculate the electromagnetic momentum of a single electron.

Para 10
I shall show that, if we start from any given state of motion in a system without translation, we may deduce from it a corresponding state that can exist in the same system after a translation has been imparted to it, the kind of correspondence being as specified in what follows

We are therefore led to suppose that the influence of a translation on the dimensions (of the separate electrons and of a ponderable body as a whole) is confined to those that have the direction of motion, these becoming k times smaller than they are in the state of rest. If this hypothesis is added to those we have already made, we may be sure that two states, the one in the moving system, the other in the same system while at rest, corresponding as stated above, may both be possible.

Moreover, this correspondence is not limited to the electric moments of the particles. If, in the system without translation, there is a state of motion in which, at a definite place, the components [charge of an ion, dielectric displacement, magnetic force] are certain functions of the time, then the same system after it has been put in motion (and thereby deformed) can be the seat of a state of motion in which, at the corresponding place, the [other form of those] components are the same functions of the local time.

Para 11
It is easily seen that the proposed theory can account for a large number of facts. Let us take in the first place the case of a system without translation, it appears that those parts which are dark while the system is at rest, will remain so after it has been put in motion. It will therefore be impossible to detect an influence of the earth’s motion on any optical experiment made with a terrestrial source of light, in which the geometrical distribution of light and darkness is observed. Many experiments on interference and diffraction belong to this class.

In the second place, if in two points of a system, rays of light of the same state of polarization are propagated in the same direction, the ratio between the amplitudes in these points may be shown not to be altered by a translation. The latter remark applies to those experiments in which the intensities in adjacent parts of the field of view are compared.

The above conclusions confirm the results I have formerly obtained by a similar train of reasoning, in which however the terms of the second order were neglected. They also contain an explanation of Michelson’s negative result, more general and of somewhat different form than the one previously given, and they show why Rayleigh and Brace could find no signs of double refraction produced by the motion of the earth. As to the experiments of Trouton and Noble, their negative result becomes at once clear, that the only effect of the translation must have been a contraction of the whole system of electrons and other particles constituting the charged condenser and the beam and thread of the torsion-balance. Such a contraction does not give rise to a sensible change of direction.

It need hardly be said that the present theory is put forward with all due reserve. Though it seems to me that it can account for all well established facts, it leads to some consequences that cannot as yet be put to the test of experiment. One of these is that the result of Michelson’s experiment must remain negative, if the interfering rays of light are made to travel through some ponderable transparent body.

Our assumption about the contraction of the electrons cannot in itself be pronounced to be either plausible or inadmissible. What we know about the nature of electrons is very little and the only means of pushing our way farther will be to test such hypotheses as I have here made. Of course, there will be difficulties, e.g. as soon as we come to consider the rotation of electrons.

Poincaré (1900): The Theory of Lorentz and The Principle of Reaction

The principle of reaction appears to us, therefore, as a consequence of the principle of energy and the principle of relativity of motion. The latter weighs heavily on our thoughts when we consider an isolated system. But in the case we’re considering, we’re not dealing with an isolated system, since we’re considering the ordinary matter, and in addition to that there is still an ether. If all material objects are carried along by a common translation, as, for example, the motion of the earth, phenomena could be different from those which we would observe in the absence of that translation since the ether could not be carried along by the translation.

It seems like the principle of relativity of motion should not just apply to ordinary matter; so experiments have been carried out to detect the motion of the earth. Those experiments have produced negative results. The theory of Lorentz explains that negative result. It appears that the principle of relativity of motion, which is not clearly true a priori, is verified aposteriori, and that the principle of reaction should follow. Yet the principle of reaction does not hold; how can that be?

It is the case that, in reality, that which we call the principle of relativity of motion has been verified only imperfectly, as shown by the theory of Lorentz. This is due to the compensation of multiple effects:

That compensation does not take place unless we neglect v²
For the compensation to work, we must relate the phenomena not to the true time t, but to a certain local time t’ defined in the following fashion:

Let us suppose that there are some observers placed at various points, and they synchronize their clocks using light signals. They attempt to adjust the measured transmission time of the signals, but they are not aware of their common motion, and consequently believe that the signals travel equally fast in both directions. They perform observations of crossing signals, one traveling from A to B, followed by another traveling from B to A. The local time t is the time indicated by the clocks which are so adjusted. If V = 1/√K₀ is the speed of light, and v is the speed of the Earth which we suppose is parallel to the x axis, and in the positive direction, then we have: t’ = t – vx/V².

Poincaré (1902): Science and Hypothesis

Para 90

There is no absolute space, and we only conceive of relative motion; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred.
There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention.
Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places.
Finally, is not our Euclidean geometry only a kind of convention of language? Mechanical facts might be enunciated with reference to a non-Euclidean space which would be less convenient but as legitimate as our ordinary space; the enunciation would become more complicated, but still would be possible. Thus, absolute space, absolute time, and geometry are not conditions which are imposed on mechanics.

Poincaré (1904): The Principles of Mathematical Physics

The Principle of Relativity
The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for an observer carried along in a uniform motion of translation; so that we have not, and cannot have, any means of discerning whether or not we are carried along in such a motion.

Consider two electrified bodies; though they seem to us at rest, they are both carried along by the motion of the earth; an electric charge in motion, Rowland has taught us, is equivalent to a current; these two charged bodies are, therefore, equivalent to two parallel currents of the same sense and these two currents should attract each other. In measuring this attraction, we shall measure the velocity of the earth; not its velocity in relation to the sun or the fixed stars, but its absolute velocity.

I well know what will be said: It is not its absolute velocity that is measured, it is its velocity in relation to the ether. Is it not evident that from the principle so understood we could no longer infer anything? If we succeed in measuring anything, we shall always be free to say that this is not the absolute velocity, and if it is not the velocity in relation to the ether, it might always be the velocity in relation to some new unknown fluid which might fill space.

Indeed, experiment has taken upon itself to ruin this interpretation of the principle of relativity; all attempts to measure the velocity of the earth in relation to the ether have led to negative results. Finally, Michelson pushed precision to its last limits; nothing came of it. It is precisely to explain this obstinacy that the mathematicians are forced today to employ all their ingenuity. Their task was not easy, and if Lorentz has got through it, it is only by accumulating hypotheses.

The most ingenious idea was that of local time. Imagine two observers who wish to adjust their timepieces by optical signals; they exchange signals, but as they know that the transmission of light is not instantaneous, they are careful to cross them. When station B perceives the signal from station A, its clock should not mark the same hour as that of station A at the moment of sending the signal, but this hour augmented by a constant representing the duration of the transmission. Suppose, for example, that station A sends its signal when its clock marks the hour 0, and that station B perceives it when its clock marks the hour t. The clocks are adjusted if the slowness equal to t represents the duration of the transmission, and to verify it, station B sends in its turn a signal when its clock marks 0; then station A should perceive it when its clock marks t. The timepieces are then adjusted.

And in fact they mark the same hour at the same physical instant, but on the one condition, that the two stations are fixed. Otherwise the duration of the transmission will not be the same in the two senses, since the station A, for example, moves forward to meet the optical perturbation emanating from B, whereas the station B flees before the perturbation emanating from A. The watches adjusted in that way will not mark, therefore, the true time; they will mark what may be called the local time, so that one of them will be slow of the other. It matters little, since we have no means of perceiving it. All the phenomena which happen at A, for example, will be late, but all will be equally so, and the observer will not perceive it, since his watch is slow; so, as the principle of relativity requires, he will have no means of knowing whether he is at rest or in absolute motion.

It is necessary to admit that bodies in motion undergo a uniform contraction in the sense of the motion. One of the diameters of the earth, for example, is shrunk by one two-hundred-millionth in consequence of our planet’s motion, while the other diameter retains its normal length. Thus the last little differences are compensated. And then, there is still the hypothesis about forces. Forces, whatever be their origin, gravity as well as elasticity, would be reduced in a certain proportion in a world animated by a uniform translation; or, rather, this would happen for the components perpendicular to the translation; the components parallel would not change.

Resume, then, our example of two electrified bodies; these bodies repel each other, but at the same time if all is carried along in a uniform translation, they are equivalent to two parallel currents of the same sense which attract each other. This electrodynamic attraction diminishes, therefore, the electrostatic repulsion, and the total repulsion is feebler than if the two bodies were at rest. But since to measure this repulsion we must balance it by another force, and all these other forces are reduced in the same proportion, we perceive nothing.

Thus, all seems arranged, but are all the doubts dissipated? What would happen if one could communicate by non-luminous signals whose velocity of propagation differed from that of light? If, after having adjusted the watches by the optical procedure, we wished to verify the adjustment by the aid of these new signals, we should observe discrepancies which would render evident the common translation of the two stations. Are such signals inconceivable, if we admit with Laplace that universal gravitation is transmitted a million times more rapidly than light? Thus, the principle of relativity has been valiantly defended in these latter times.

Poincaré (1905): On the Dynamics of the Electron

Introduction
It seems at first sight that the aberration of light, and the related optical and electrical phenomena, will provide us a means of determining the absolute motion of the Earth, or rather its motion, not in relation to the other stars, but in relation to the ether. Fresnel had already tried it, but he recognized soon that the motion of the earth does not alter the laws of refraction and reflection. Similar experiments, like that of a telescope filled with water and all those which take into consideration only terms of first order in respect to aberration, give no other but negative results; soon an explanation was discovered; but Michelson, having imagined an experiment where the terms depending on the square of the aberration became sensitive, failed as well.

It seems that this impossibility of demonstrating experimental evidence for absolute motion of the Earth is a general law of nature; we are naturally led to admit this law, which we will call the Postulate of Relativity and admit it without restriction.

An explanation was proposed by Lorentz and FitzGerald, who introduced the hypothesis of a contraction by all bodies in the direction of the motion of earth and proportional to the square of aberration; this would give an account of the experiment of Michelson and all those which were carried out up to now. The hypothesis would become insufficient, however, if one were to assume the postulate of relativity in all its generality. Lorentz sought to supplement and modify it in order to put it in perfect agreement with this postulate. He succeeded in doing so in his 1904 article. The idea of Lorentz can be summarized as follows: if we can bring the whole system to a common translation, without modification of any of the apparent phenomena, it is because the equations of the electromagnetic medium are not altered by certain transformations, which we will call Lorentz transformation; two systems, one motionless, the other in translation, thus become exact images of one another.

Section 4.2: Einstein Quotes

4.2.1 Volume 2: The Swiss Years: Writings 1900-1909, Doc 47: On the relativity principle and the conclusions drawn from it, 1907
Newton’s equations of motion retain their form when one transforms to a new system of coordinates that is in uniform translational motion relative to the system used originally. One could not doubt that the laws of nature are the same, without regard to which of the coordinate systems they are referred. This independence from the state of motion of the system of coordinates used we call “the principle of relativity”.

After the acceptance of that [Lorentz] theory one had to expect that one would succeed in demonstrating an effect of the terrestrial motion relative to the luminiferous ether on optical phenomena. Lorentz proved that in optical experiments an effect of that relative motion on the ray path is not to be expected, as long as the calculation is limited to terms in which the ratio v/c, the relative velocity to the velocity of light in a vacuum, appears in the first power. But the negative result of Michelson and Morley’s experiment showed that an effect of the second order, (proportional to v²/c²), was not present either.

This contradiction was formally removed by the postulate of Lorentz and FitzGerald, according to which moving bodies experience a certain contraction in the direction of their motion. However, this ad hoc postulate seemed to be only an artificial means of saving the theory. Michelson and Morley’s experiment had actually shown that phenomena agree with the principle of relativity.

It turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty. One had only to realize that an auxiliary quantity introduced by Lorentz and named by him “local time” could be defined as “time” in general. If one adheres to this definition of time, the basic equations of Lorentz’s theory correspond to the principle of relativity, and the above [Galilean] transformation equations are replaced by ones that correspond to the new conception of time. [Then] Lorentz’s and FitzGerald’s hypothesis [of length contraction] appears as a compelling consequence of the theory.

4.2.2 Volume 3: The Swiss Years: Writings 1909-1911, Doc 2: The principle of relativity and its consequences in modern physics, 1910

Part 2: The optics of moving bodies and the ether
Assuming that ether is completely immobile, Lorentz conceived in 1895 a very satisfactory theory of electromagnetic phenomena, a theory which not only permitted a quantitative prediction of Fizeau’s experiment, but also provided a simple explanation of almost all the experiments that one can imagine in this sphere.

Part 3: Experiments and consequences not reconcilable with the theory
From Fizeau’s experiment, one had to conclude that the ether is not carried along completely by matter in motion, but that, instead, there occurs a relative displacement. The earth being a body that rotates around its axis, and revolves around the sun, with velocities that change their directions, one was bound to believe that, in our laboratories, the ether would take a slight part in the motion of the earth.

From this, it would seem to follow that the relative velocity of the ether, with respect to our equipment, must vary with time, and that one therefore should expect that an apparent spatial anisotropy be observed in optical phenomena, ie these phenomena should depend on the orientation of the equipment. Thus, in vacuum, or in the atmosphere, light should propagate faster in the direction of the earth’s motion than in the opposite direction.

Experimental verification of this consequence of the theory was unthinkable, because the order of magnitude of the term considered is that of the ratio of the velocity of the earth to the velocity of light, and one could not hope to attain such precision in the direct determination of the velocity of light. Also, and this is a most important point, all terrestrial methods for measuring the velocity of light employ light rays that travel along a closed (back and forth) rather than a simple path, this due to the fact that the times of departure and arrival of the rays must be determined with the help of one and the same device.

The negative result of one of these experiments presented a real headache for the theoreticians, and as a result the foundation of Lorentz’s theory seemed shaky. Lorentz and FitzGerald resorted to a strange hypothesis: they assumed that each body in motion with respect to the ether contracts in the direction of motion by a fraction equal to ½(v/c)², or if only terms of second order are considered, that the length of the body is diminished in that direction in the ratio: 1 to √(1-v²/c²).

Part 4: The principle of relativity and the ether
What is the source of the difficulties we have just seen? Lorentz’s theory contradicts the purely mechanical models to which physicists hoped to reduce all phenomena. For while mechanics admits no absolute motion, but only the motions of bodies relative to each other, there is a particular state in Lorentz’s theory that corresponds physically to the state of absolute rest: that is the state of a body which is not in motion with respect to the ether.

If the equations of Newtonian mechanics of a coordinate system that is not undergoing accelerated motion are referred, by means of the relations t’ = t, x’ = x -vt, y’ = y, z’ = z, to a new coordinate system which is in uniform translational motion with respect to the first, one obtains equations in t’, x’, y’, z’ that are identical to the original equations in t, x, y, z. In other words, the Newtonian laws of motion transform to laws of the same form when one passes from one coordinate system to another one that is in uniform translational motion with respect to the first.

This is the property we express when we say that the principle of relativity is satisfied in classical mechanics. More generally, we will state the principle of relativity in the following way: The laws governing natural phenomena are independent of the state of motion of the coordinate system with respect to which the phenomena are observed, provided that this system is not in accelerated motion.

Part 5: On the arbitrary hypotheses contained implicitly in the customary notions of time and space
The question arises: Is it really impossible to reconcile the essential foundations of Lorentz’s theory with the principle of relativity? If we wish to attempt such a reconciliation, the first step we must take is to give up the ether. For, on the one hand, we have been obliged to admit that the ether is stationary, whereas, on the other hand, the principle of relativity demands that the laws of natural phenomena referred to a uniformly moving coordinate system S’, be identical with the laws of these same phenomena referred to a system S, at rest with respect to the ether. But there is no reason to assume the immobility of ether any more with respect to the system S’, than with respect to the system S; these two systems cannot be distinguished from each other, and it is therefore improper to make one of them play a special role by saying that it is at rest with respect to the ether. It follows that the only way to arrive at a satisfactory theory is to give up the notion of a medium filling all of space.

To go a step further, we must reconcile the principle of relativity with an essential consequence of Lorentz’s theory, because giving up this consequence would amount to giving up the most fundamental formal properties of the theory. And here is the consequence in question: A ray of light in vacuum always propagates with the same velocity c, which velocity is independent of the motion of the body that emits the ray.

In Lorentz’s theory this principle holds only for a system in a special state of motion: In effect, the system must be at rest relative to the ether. If we want to preserve the principle of relativity, we must assume that the principle of the constancy of the velocity of light holds for any arbitrary system not in accelerated motion. At first glance this seems impossible. For let us consider a light ray that propagates with velocity c with respect to the system S, and suppose that we seek to determine the velocity of propagation relative to a system S’, that is in uniform translational motion with respect to the first system. Applying the rule of addition of velocities, we will find a velocity different from c; in other words, the principle of the constancy of the velocity of light that is valid with respect to S is not valid with respect to S’.

So that the theory based on these two principles should not lead to contradictory results, one must renounce the customary rule of addition of velocities. Well founded as this rule may seem to be at first glance, it conceals no less than two arbitrary hypotheses, which hold sway over all of kinematics.

The first hypothesis we wish to discuss concerns the physical notion of time measurement. The meaning of this definition is perfectly clear as long as the clock is sufficiently close to the place at which the event occurs, so that the clock and the event can be observed simultaneously. If, on the contrary, the event is taking place in some corner far away from the clock, then it will no longer be possible to establish immediately a correspondence between the different phases of the event, and the different positions of the clock’s hands.

To determine the time at each point in space, we can imagine it populated with a very great number of clocks of identical construction. Suppose, for example, that we are studying the motion of a material point whose trajectory passes through the points A, B C. Let us consider the points A, B, C, each of which is furnished with a clock, and is referred to a system in non-accelerated motion with the aid of time-independent coordinates. We will now be able to know the time at any of these locations.

If we choose a sufficiently large number of clocks, so that we can ascribe to each of them a sufficiently small domain, we will be able to fix any instant whatsoever, at any location in space, to any degree of accuracy desired. But we cannot obtain in this manner a definition of time useful to a physicist, because we did not say what the position of the clock hands should be at a given instant of time at different spatial points. We forgot to synchronize our clocks.

To get a complete physical definition of time, we have to say in what manner all of the clocks have been set at the start. We proceed as follows: First, we furnish ourselves with a means of sending signals, be it from A to B, or from B to A. This means should be such that we have no reason whatsoever to believe that the phenomena of signal transmission in the direction AB will differ in any way whatsoever from the signal transmission in the direction BA.

In that case there is, obviously, only one way of regulating the clock at B against the clock at A, in such a manner that the signal travelling from A to B would take the same amount of time, measured with the clocks described above, as the signal travelling from B to A. That is: The reading of the clock at B at the moment the signal AB arrives at B (ie t(b)) minus the reading of the clock at A at the moment the signal AB leaves A (ie t(a)) equals the reading of the clock at A at the moment the signal AB arrives at A (ie t‘(a)) minus the reading of the clock at B at the moment the signal BA leaves B (ie t‘(b)), ie t(b) – t(a) = t’(a) – t’(b)

For these signals we can use, for example, sound waves that propagate between A and B, through a medium that is at rest with respect to these points. We can use light rays propagating through the vacuum, or through a homogeneous medium at rest with respect to A and B. It does not make any difference whether we choose this or that kind of signals. Still, of all the signals that can be used, we are going to prefer those that make use of light rays propagating in the vacuum, because the synchronization requires that the path out, and the path back, be equivalent, and in our case this equivalence is satisfied, by definition, by virtue of the principle of the constancy of the velocity of light, as light in the vacuum always propagates with the velocity c.

We have to synchronize our clocks in such a way that the time spent by a signal travelling from A to B, be equal to the time spent by an identical signal travelling from B to A. Now we possess a well-defined method by which to synchronize two clocks, with respect to each other. Once the synchronization has been done, we will say that the two clocks are in phase. The totality of the readings of all of these clocks in phase with one another, is what we will call the physical time.

By an elementary event, we will understand an event that is concentrated in one point, and is of infinitely short duration. By the time coordinate of an elementary event, we understand the indication, at the instant of the event’s occurrence, of a clock that is situated infinitely close to the point where the event takes place. An elementary event is thus defined by four coordinates, the time coordinate, and the three coordinates that define the spatial position. Thanks to this, we can give a perfectly defined meaning to the concepts of simultaneity, and non-simultaneity, of two events occurring at locations removed from one another.

In order to define the physical time with respect to a coordinate system, we used a group of clocks in a state of rest relative to that system. According to this definition, the time readings, or the establishment of the simultaneity of two events, have meaning only if the motion of the group of clocks, or that of the coordinate system, is known.

Consider two non-accelerated coordinate systems, S and S’, in uniform translational motion with respect to one another. Suppose that each of these systems is provided with a group of clocks attached to it, and that all clocks belonging to the same system are in phase. Under these conditions, the readings of the group attached to S will define the physical time with respect to S; the readings of the group attached to S’ define the physical time with respect to S’. Each elementary event will have a time coordinate t with respect to S, and a time coordinate t’ with respect to S’.

But we have no right to assume, a priori, that the clocks of the two groups can be set in a manner that the two time coordinates of the elementary event would be the same, or in other words, in a way that t would be equal to t’. To assume this would mean to introduce an arbitrary hypothesis which has been introduced into kinematics up to the present time.

The second arbitrary hypothesis introduced in kinematics concerns the configuration of a body in motion. Consider a bar AB moving in the direction of its axis with velocity v, with respect to a coordinate system S not in accelerated motion. What should we understand by the “length of the bar”? One is at first inclined to believe that this concept does not require any special definition. However, we will immediately see that nothing of the sort is true, if we consider the following two methods of determining the length of the bar:

One accelerates the motion of an observer furnished with a measuring rod until he attains the velocity v, ie until he is at relative rest with respect to the bar. The observer then measures the length AB by successively applying the measuring rod along the bar.
Using a group of clocks in phase with each other, and at rest with respect to the system S, one determines the two points P₁ and P₂ of S where one finds the two ends of the bar, at the instant t; after that, one determines the length of the straight line connecting the two points P₁ and P₂ by successively applying the measuring rod along the line P₁P₂, which is assumed to be a material line.

As one can see, it is with some justification that the results obtained in the first, and in the second, case are designated as the “length of the bar.” But in no way does this mean, a priori, that these two operations must necessarily lead to the same numerical value for the length of the bar. All that one can deduct from the principle of relativity, and this is easy to demonstrate, is that the two methods lead to the same numerical value for the length, only when the bar AB is at rest relative to the system S. But in no way is it possible to assert that the second method yields a value for the length, independently of the velocity v of the bar.

More generally, if the configuration of a body in uniform translational motion, with respect to S, is determined by ordinary geometric methods, by means of measuring rods, or other solid bodies, moving in exactly the same way, the measurements turn out to be independent of the velocity v of the translation: these results give us what we will call the geometric configuration of the body. By contrast, if one marks in the system S the positions of various points of the body at a given instant, and determines the configuration formed by these points by geometric measurements using measuring rods at rest with respect to S, one obtains as a result what we will call the kinematic configuration of the body, with respect to S. The second hypothesis used unconsciously in kinematics can thus be expressed as follows: The kinematic configuration and the geometric configuration are identical.

Part 6: The new transformation equations (the Lorentz transformation) and their physical meaning
The rule of the parallelogram of velocities, which made one think that Lorentz’s theory cannot be reconciled with the theory of relativity, is based on unacceptable hypotheses which lead to the following transformation equations: t’ = t, x’ = x – vt, y’ = y, z’ = z. The first of these equations, expresses an ill-founded hypothesis about the time coordinates of an elementary event taken with respect to two systems, S and S’, that are in uniform translational motion with respect to each other. The other three equations, express the hypothesis that the kinematic configuration of the system S’, with respect to the system S, is identical with the geometric configuration of the system S’.

If one abandons the ordinary kinematics and builds a new kinematics based on the new foundations, one arrives at transformation equations different from those given above. And now, we are going to show that based on:

1. The principle of relativity
2. The principle of the constancy of the velocity of light

we arrive at transformation equations that allow us to see that Lorentz’s theory is compatible with the principle of relativity. The theory based on these principles we shall call the theory of relativity.

[S and S’] each possess a group of clocks that run-in synchrony when the two systems are at relative rest with respect to each other. Thus [in that circumstance] the velocity of light in a vacuum must be expressed by the same number in the two systems. Let t, x, y, z be the coordinates of an elementary event with respect to S, and t’, x’, y’, z’ the coordinates of the same event with respect to S’. We seek to find the relations that link these two groups.

x’ = 0 and x – vt = 0, y’ = 0 and y = 0, z’ = 0 and z = 0, are equivalent

ie: x, y, z, x’, y’, z’ are linked in the form: x’ = E(x – vt), y’ =Fy, z’ = Gz.

t’ = ϕ(v)β[t – (v/c²)x], x’ = ϕ(v)β(x – vt), y’ = ϕ(v)y, z’ = ϕ(v)z

where β = 1/√(1-v²/c²), and ϕ(v) is a function of v that is to be determined.

t” = ϕ(v)ϕ(-v)t, x” = ϕ(v)ϕ(-v)x, y” = ϕ(v)ϕ(-v)y, z” = ϕ(v)ϕ(-v)z

t’ = β[t – (v/c²)x], x’ = β(x – vt), y’ = y, z’ = z, or

t’ = [(t-(v/c²)x]/√(1 – v²/c²)], x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z,

where β = 1/√(1-v²/c²)

We call them the Lorentz transformations.

Part 7: Physical interpretations of the transformation equations

1 Consider a body attached to S’. Let x’₁, y’₁, z’₁, and x’₂, y’₂, z’₂, be co-ordinates of two points of the body. At any instant t of the system S, we have the following relations between these two co-ordinates: x₂- x₁= [(√(1 – v²/c²)(x’₂ – x’₁)], y₂ – y₁ = y’₂ – y’₁, z₂ – z₁ = z’₂ – z’₁.

The kinematic configuration of a body in uniform translational motion with respect to a co-ordinate system depends on the velocity v of the translation.

Furthermore, the kinematic configuration differs from the geometric configuration solely by a contraction in the direction of the motion, a contraction which is in the ratio 1:√(1-v²/c²). A relative motion of two reference systems with a velocity v that exceeds the velocity of light in vacuum is incompatible with the principle here assumed. One recognizes in this the hypothesis of Lorentz and FitzGerald. This is the hypothesis that looked so strange, and had to be introduced to explain the negative results of the experiment of Michelson. Here this hypothesis appears naturally as an immediate consequence of the principles assumed.

2. Let us consider a clock H’ which is at rest at the origin of S’, and runs P₀ times faster than one of the clocks used for the determination of physical time in the systems S or S’, ie when the two clocks are compared while at relative rest, clock H’ will indicate P₀ periods during the unit time indicated by the other clock. How many periods will clock H’ indicate during unit time if observed from the system S?

Clock H’ will indicate the end of a period at the times in the form: tn’ = n/P₀. Since we seek the time with respect to S, the first of the transformation equations will have to be written: t = β[t’ – vx’/c²]. And since clock H’ is at rest at the origin of S’, we must always have x’ = 0, which yields: tn = βtn’ = (β/P0)n. Observed from S, clock H’ thus indicates: P = P0/β = P0√(1-v²/c²) periods in a unit time. In other words, a clock moving uniformly with velocity v with respect to a reference system runs, as observed from this system, 1:√(1-v²/c²) times slower than an identical clock that is at rest with respect to this system.

3. Let us consider the equations of motion of a point moving in uniform translation with velocity u’ with respect to S’: x’ = uₓ’t’, y’ = uᵧ’t’, z’ = uz’t’. If one replaces x’, y’, z’, t’ by their values as functions of x, y, z, t, one obtains x, y, z as functions of t and, hence, the components ux, uy, uz of the velocity u of the point with respect to the system S. In this way it is possible to obtain the formula that expresses the theorem of the addition of velocities in its general form, and one can immediately see that the law of the parallelogram of velocities is valid only in first approximation. In the special case when the velocity u’ has the same direction as the velocity v of the translation of S’ with respect to S, one easily obtains: u = (v + u’)/(1 + vu’/c²). This equation shows that if one adds two velocities, each smaller than the velocity of light in a vacuum, one always obtains a resultant velocity that is smaller than the velocity of light. For if one sets v = c – λ, u’ = c – μ, where λ and μ are positive and smaller than c, one gets: u = c[(2c – λ – μ)/(2c – λ – μ + λμ/c)] <c. When one adds the velocity of light c, and a velocity smaller than c, one always obtains the velocity of light.

Part 8: Remarks about Some Formal Properties of the Transformation Equations
Let us consider two coordinate systems S and S’ the origins of which coincide and which have the same orientation. There are two kinds of co-ordinate transformations in Newtonian mechanics that do not alter the laws of motion. These are: 1 A change in orientation of the system S’, with respect to the system S, about the common origin. This first transformation is characterized by equations linear in x’, y’, z’ and x, y, z, between the coefficients of which there exist relations, such that the condition: x’² + y’² + z’² = x² + y² + z² is identically satisfied.

2 Uniform motion (translation) of the system S’, with respect to the system S. This second transformation is characterised by the equations: x’ = x + αt, y’ = y + βt, z’ = z + γt, where α, β, γ, are constants. For these two kinds of transformation, the condition t’ = t must be satisfied. In other words, time is an invariant under these two transformations.

Combining these two transformations we obtain the most general transformation, by means of which one can transform the equations of mechanics without altering them. This transformation is characterised by the equation t’ = t, and by three equations that express x’, y’, z’ as linear functions of x, y, z. The coefficients of these three equations are connected by relations that, for t = 0, satisfy condition: x’² + y’² + z’² = x² + y² + z² identically.

Let us now consider the most general co-ordinate transformation compatible with the theory of relativity. From what we have seen, this transformation is characterized by the fact that x’, y’, z’, t’ must be linear functions of x, y, z, t, such that the condition: x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² will be satisfied identically. It should be noted that the transformations compatible with Newtonian mechanics can be obtained by setting c = ∞, in condition x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t². Thus, if we take the same route as before, we arrive at the equations of ordinary kinematics if, instead of the principle of the constancy of the velocity of light, we assume the existence of signals whose propagation does not require any time. The group characterized by equation x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² contains the transformations that correspond to a change in the orientation of the system. These are the transformations compatible with the condition t = t’.

The simplest transformations compatible with x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² are those for which two of the four coordinates of an elementary event remain invariant. Let us consider, for example, the transformations under which x and t do not change. Instead of the general condition x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t², we will have the special condition: t’ = t, x’ = x, y’² + z’² = y² + z². To this condition corresponds a rotation of the system about the x-axis. If, on the other hand, we consider transformations under which two of the spatial coordinates, for example y and z, remain invariant, we will have instead of the general condition x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² the special condition: y’ = y, z’ = z, x’² – c²t’² = x² – c²t².

These are what we encountered in the preceding section while investigating a system in uniform motion parallel to the x-axis of an identically oriented system at rest. The two systems of equations differ only by a change of sign in the third condition. But even this difference can be made to disappear if one chooses, with Minkowski, to take ict instead of t as a variable, where i is the imaginary unit. In that case this imaginary temporal coordinate plays the same role in the transformation equations as the spatial coordinates. If we set x = x₁, y = x₂, z = x₃, ict = x₄, and consider x₁ x₂ x₃ x₄ as the coordinates of a point in a four-dimensional space such that to each elementary event there corresponds a point in this space, we reduce everything that happens in the physical world to something static in the four-dimensional space. In that case the condition x’² + y’² + z’² – c²t’² = x² + y² + z² – c²t² will be written as: x’₁² + y’₂² + x’₃² + x’₄² = x₁² + x₂² + x₃² + x₄².

This is the condition that corresponds to a rotation without relative translation of a four-dimensional coordinate system. The principle of relativity demands that the laws of physics not be altered by the rotation of the four-dimensional coordinate system to which they are referred. The four coordinates x₁ x₂ x₃ x₄ must appear in the laws symmetrically. To express the different physical states, one can use four-dimensional vectors which behave in the calculations in a manner analogous to ordinary vectors in three-dimensional space.

4.2.3 Volume 3: The Swiss Years: Writings 1909-1911, Doc 17: The Theory of Relativity, 1911
How to put kinematics back on its feet? The answer is the very same circumstances that led us into so many difficulties in the past, lead us to a negotiable path by putting aside the arbitrary assumptions mentioned above. For it turns out that precisely those two seemingly incompatible axioms, which were imposed on us by experience, namely the principle of relativity and the principle of constancy of the velocity of light, lead us to a perfectly definite solution of the space-time transformation problem.

One arrives at results that, in part, run very much counter to customary conceptions. First, things purely kinematic. Since we defined the coordinates, and the time, in a definite way in physical terms, all relationships between these will have a perfectly definite physical content. We obtain the following: If we have a solid body that is moving uniformly with respect to the coordinate system k, which we take as the basis for our analysis, then this body appears contracted by a perfectly definite ratio in the direction of its motion, as compared with the shape it has when it is in a state of rest, with respect to this system. If we denote the velocity of motion of the body by v, and the velocity of light by c, then each length measured in the direction of motion, and equal to l in a motionless state, will be diminished because of the body’s motion relative to the non-comoving observer to the length l.√(1 – v2/c2).

If the body has a spherical shape in the state of rest, it will have the shape of a flattened ellipsoid if we move it in a certain direction. When its velocity reaches the velocity of light, it will collapse to a plane. However, judged by a comoving observer, the body retains, before and after, its spherical shape; on the other hand, to the observer moving with the body, all non-comoving objects appear, in exactly the same way, contracted in the direction of the relative motion. This result loses very much of its oddness, if one considers that this assertion about the shape of a moving body has a complicated meaning, since this shape can be ascertained only with the aid of determinations of time. The feeling that this concept, “the shape of the moving body,” has an immediately obvious meaning is due to the fact that in our day-to-day experience we are accustomed to encountering only such velocities of motion that are practically infinitely small compared with the velocity of light.

4.2.4 Volume 4: The Swiss Years: Writings 1912-1914, Doc 1: Manuscript of the Special theory of Relativity, 1912

Part 5: Principle of the Constancy of the Velocity of Light
By processing a number of equations, some of which include the propagation of light in empty space, Einstein asserts that: Hence in accordance with Lorentz’s theory we can proclaim the following, which we call “the principle of the constancy of the velocity of light”: “There exists a coordinate system with respect to which every light ray propagates in vacuum with the velocity c.” This principle contains a far-reaching assertion. It asserts that the propagation velocity of light depends neither on the state of motion of the light source nor on the states of motion of the bodies surrounding the propagation space. The question as to what extent this principle can be considered certain is of fundamental significance for the theory of relativity. For the time being we will content ourselves with the realization that this principle is demanded by Lorentz’s theory.

Part 7: Apparent Incompatibility of the Principle of the Constancy of the Speed of Light with the Relativity Principle
Lorentz’s theory seemed incompatible with the relativity principle. For we have seen that, according to Lorentz’s theory, there exists a reference system K in which every light ray propagates in vacuum with the velocity c. It would appear that a system in uniform translational motion with respect to K could not have this property. For if again we choose K’ such that its origin glides along the x axis of K with velocity v, and if we choose a ray of light that propagates along the x axis of K with the velocity c with respect to K, then according to the law of the parallelogram of velocities this same light ray propagates along the x’ axis with velocity c-v with respect to K’. It would follow that a law of nature, namely the principle of the constancy of the velocity of light, is indeed valid with respect to K but not with respect to K’. Thus the principle of the constancy of light (in vacuum) seems to be incompatible with the relativity principle.

Strictly speaking, however, we have only learned that the following three things are incompatible with one another:
(a) the relativity principle
(b) the principle of the constancy of the velocity of light (Lorentz’s theory)
(c) the transformation equations or the law of the parallelogram of velocities

One arrives at the theory that is now called “the theory of relativity” by keeping
(a) and (b) but rejecting (c). But whoever has not yet studied the electrodynamics
of moving bodies thoroughly will be inclined to keep (a) and (c) but give up (b).

Therefore, we will first draw several consequences starting out from the latter standpoint in order to justify in that way the theory of relativity.”

[There then follows commentary on Fizeau’s experiment with the conclusion that:] On the other hand, since the Maxwell-Lorentz theory requires the principle of the constancy of the (vacuum) velocity of light, there is every reason to stick with it. It turns out that the result of the Fizeau experiment can be derived quantitatively from (a) and (b). [Because] If we take this standpoint [keeping (a) and (c)], then Fizeau’s experiment forces us to conclude that in a transparent medium, various velocities of light must be possible. For if the velocity of light in the medium, as assessed by an observer moving with the medium, were always equal to Vo, then, according to the parallelogram law of velocities, the velocity of light as assessed by an observer not moving with the medium would be Vo + nl, We are thus forced to assume that, in Fizeau’s experiment, the velocity of light relative to the moving medium is, in this medium, different from what it would be in the same medium if the latter were at rest.

Part 8: The Physical Meaning of Spatial and Temporal Determinations
We now seek to solve the dilemma that consists in the incompatibility of postulates (a), (b), and (c) by relinquishing (c) but keeping (a) and (b). We realize that the postulate (c) can be given up, by investigating the meaning that the spatial and temporal determinations have in physics.

Physical meaning of spatial determinations:
The propositions of Euclidean geometry acquire a physical content through our assumption that there exist objects that possess the properties of the basic structures of Euclidean geometry. We assume that appropriately constructed edges of solid bodies not subjected to external influences have the definitional properties that straight lines have (material straight line), and that the part of a material straight line between two marked material points has the properties of a segment. Then the propositions of geometry turn into propositions concerning the arrangements of material straight lines and segments that are possible when these structures are at relative rest. By virtue of its properties, every material straight line can be extended. By repeatedly laying a shorter segment on a longer one and counting these operations, one can measure the latter by means of the former; ie the length of the latter can be expressed by a number if the former (the measuring rod) is regarded as being given once and for all. The position of a material point on a material straight line on which a fundamental point is given once and for all can be defined by its distance from the latter, that is, by a number (coordinate).

From among all the material straight lines at rest relative to one another, we think of three that are perpendicular to one another and intersect in one point as being distinguished, and from among all the material straight line segments we likewise imagine a (transportable) segment, which we will think of as being used for measuring all lengths (measuring rod). We call these structures, taken together, “the coordinate system.” Geometry teaches how we can express the position of every material point with respect to such a coordinate system K by means of three numbers (coordinates). The shape and position of a material structure at rest relative to K are given by the totality of the positions (coordinates) of all of its material points with respect to K.

The physical meaning of temporal determinations:
If we consider what happens physically with respect to the coordinate system K, then we know that spatial coordinates x, y, z do not suffice for the determination of the physical variables. The specification of a fourth basic variable, the temporal coordinate, is also needed. This coordinate as well we will define in such a way that each temporal determination shall appear as the result of a precisely defined measurement procedure. We imagine a completely isolated physical system that repeatedly assumes a specific state Z. Then the state Z is always followed by the states Z’, Z”, etc., until the state Z is reached again. The system changes its state periodically. We can then count how often the system, which we shall call a “clock,” assumes the state Z; we will call this number the “temporal determination” of the clock.

We imagine that such a clock is permanently arranged at the point of origin O of K, and that the spatial dimensions of the clock are so small that, from a geometrical point of view, it may be treated as a “point.” By means of the determination of this clock, every event that is spatially infinitely close to the coordinate origin can be assigned a temporal determination, the “time coordinate,” or, in brief, the “time” of the event, if we have arbitrarily fixed the zeroth temporal determination of the clock. We cannot evaluate directly the time of an event taking place at another point A(x, y, z) of K by means of the clock set up at the origin of K; for we possess no means, at first, for deciding which temporal determination of the clock set up at the origin of K is “simultaneous” with the event, ie is to be assigned to it.

Therefore, in order to evaluate the events occurring at A, we imagine a clock of exactly the same constitution as that at O set up at A, with the zeroth temporal determination of this clock chosen arbitrarily. By this means we arrive at a temporal valuation of all events occurring at A. But with a procedure of this kind there is, as yet, no connection between the temporal determinations concerning the places O and A; for these temporal determinations to fulfill the demands we impose on temporal determinations in physics, we still have to indicate a procedure by which the clock at A can be “regulated according to the one at O.” To that end we imagine some physical arrangement that makes it possible to send “signals” both from O to A and from A to O in such a way that the signal O – A and the signal A – O are to be conceived, for reasons of symmetry, as completely identically constituted processes.

Suppose a signal that was sent from O at the determination to of the clock at O (O-time) arrives at A at the A-time tₐ. Let a signal sent from A at the A-time tₐ‘ arrive at O at the O-time to‘. We stipulate that the clock at A should be regulated in such a manner that we shall always have tₐ – t₀ = t‘₀– tA‘ . If this is possible and is accomplished, then we say that the clock at A is synchronised with the clock at O. Using the same procedure, clocks arranged at rest at other points B, C, etc., and of identical constitution as the above-mentioned clocks, can also be synchronized with the clock at O. If, upon the completion of this procedure, two other arbitrary clocks, eg those at points A and B, are synchronised in accordance with the same definition, then it is finally no longer of consequence that we privileged the clock at O by regulating all of the other clocks according to it. Each clock is synchronized with each of the others.

We define the time coordinate of an event taking place at an arbitrary point of K (point event) as the simultaneous reading of the clock set up at this point and regulated according to the given procedure. Two point events (occurring at different points) are simultaneous if their time coordinates are equal.

It is of importance that the determination of a time coordinate, and thereby also the determination of simultaneity, has meaning only if the coordinate system K to which that determination refers is indicated. Because for the measurement of a time coordinate, or the verification of simultaneity, we need a system of identically constructed clocks that are set up at rest relative to K. We will designate the coordinate system K, together with the measuring rod and the regulated system of clocks at rest relative to K, as the “reference system Σ.”

We have now completely defined the physical meaning of spatial and temporal determinations with respect to the reference system Σ. Let us now introduce a second reference system, Σ‘, which is in uniform translational motion with respect to the first one. Spatial and temporal determinations with regard to this second reference system Σ‘ can be interpreted in exactly the same way as with regard to the original system Σ, and we will stipulate that the measuring rod and the clocks of Σ‘ are constituted in the same way as the measuring rod and clocks of Σ ,when these objects are compared in a state of relative rest.

The space-time coordinates of a specific event with respect to Σ(x, y, z, t), and those of the same event with respect to Σ‘(x’, y’, z’, t’) are totally independent of one another by the terms of their definition, even if the state of motion of Σ‘ is given relative to Σ. Hence it does not follow from these definitions that two point events that are simultaneous with respect to Σ must also be simultaneous with respect to Σ‘.

Let us consider, further, a body in arbitrary motion with respect to Σ. Obviously, its shape and position with respect to Σ is determined by the totality of the spatial coordinates of the material points of the body at a specific time t (of Σ). A corresponding remark can be made about the shape and position of the same body with respect to Σ‘ at a specific time t’ (of Σ‘). The fact that the definitions of the space-time coordinates with respect to Σ and Σ‘ are independent of each other entails that no relationships between the shape of the body with respect to Σ and that with respect to Σ‘ can be given on the basis of these definitions.

If one takes into account these two consequences, one finds oneself unable to derive the [standard] transformation equations. If we replace these equations with transformation equations that are consistent not only with the relativity principle but also with the principle of the constancy of the velocity of light, we arrive at the theory presently designated as the “theory of relativity.” Only in this way can we reconcile Lorentz’s eminently fruitful extension of Maxwell’s theory with the relativity principle.

Part 9: Derivation of the Lorentz Transformation
The principle of the constancy of the velocity of light demands the existence of a reference system Σ relative to which every light ray propagates in vacuum with velocity c. According to the relativity principle, all reference systems Σ‘ in uniform translational motion relative to Σ must possess the same property.

What kind of transformation equations must obtain between the space-time coordinates x, y, z, t (with respect to Σ) and the space-time coordinates x’, y’, z’, t’ (with respect to Σ‘) of the same point event so that the principle of the constancy of the velocity of light would hold with respect to both systems?

The functions that are being sought must be completely linear functions of these variables. We demand this in order to preserve the homogeneity properties of physical space. If one did not make this assumption, then bodies that are at rest, congruent, and identically located with respect to Σ‘ would be differently shaped or located when referred to Σ; or clocks that are at rest and identically constructed with respect to Σ‘ would have different or time-dependent rates when referred to Σ.

We can further demand that the transformation equations be homogeneous, because all that is needed for this is that the path described by the origin of Σ‘ with respect to Σ pass through the origin of Σ, and that the origin of the time scales in Σ and Σ‘ be chosen in such a way that the clocks located at the origins of the systems Σ and Σ‘ both read zero at the moment when the two points coincide.

According to the principle of the constancy of the velocity of light, then the spatial points that are just reached by the signal at times t and t’ with respect to Σ and Σ‘, respectively, will be determined by the equations: √(x² + y² + z²) = ct & √(x’² + y’² + z’²) = ct’ This means that the equations: √(x² + y² + z²) -ct = 0 & √(x’² + y’² + z’²) – ct’ = 0 must be equivalent.

The transformations must therefore make the equation: λ²(x² + y² + z² -c²t²) = (x’² + y’² + z’² – c²t’²) into an identity, where all we know about the factor λ² for the time being is that it must not vanish. But one can see that λ² must be independent of x, y, z, t, for otherwise the right hand side divided by λ² could not be an homogeneous, complete function of second order in x, y, z, t, after the substitution is carried out. We then have: x² + y² + z² -c²t² = x’² + y’² + z’² – c²t’²

If one introduces the variable u = ict’ or u’ = ict’ in place of the time variables, where i denotes the imaginary unit, one obtains: x² + y² + z² + u² = x’² + y’² + z’² + u’². As is well known this choice of time variables derives from Minkowski. By means of it, the equation: x² + y² + z² -c²t² = x’² + y’² + z’² – c²t‘², is brought into a form into which the spatial co-ordinates and the temporal co-ordinate enter the same manner.

The transformations we seek are exactly the same as those we have to apply to the spatial coordinates when passing from an orthogonal coordinate system to another one with the same origin, the only difference being that here one deals with a four-dimensional manifold rather than with a three-dimensional manifold.

The simplest Lorentz transformations are those in which, in addition to the time coordinate, only one of the spatial coordinates (e.g., the x-coordinate) undergoes a transformation, then the transformation equations assume the form: x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, u’ = (t-(v/c²)x)/√(1 – v²/c²)

If one performs several Lorentz transformations in succession, one obtains again a Lorentz transformation; i.e the set of the Lorentz transformations forms a group. We can now also formulate the relativity principle in the following way: The theory of relativity requires that systems of equations in physics turn into systems of equations of the same form if one transforms them by means of the Lorentz transformation. Simple calculation shows that the fundamental equations of classical mechanics do not have this property. Thus, they are not compatible with the theory of relativity.

Part 10: The Physical Content of the Lorentz Transformation
For x’ and t’ to be real in the Lorentz transformation equations then v must be <c. The translational velocity of Σ‘ relative to Σ must be smaller than c. Thus, according to the theory of relativity, it is impossible in principle for a body (coordinate system) to move with superluminal velocity. If the coefficients in the Lorentz transformation were also dependent on x, etc, and t, then it would be not only differences of the spatial and temporal coordinates that entered into propositions about moving bodies and clocks, but their values as well. This is out of the question from a physical standpoint.

Further, in λ²(x² + y² + z² -c²t²) = (x’² + y’² + z’² – c²t‘²) we arbitrarily set λ² = 1. Had we not done this, we would have obtained: x’=λ(v)[(x -vt)/√(1 – (v/c²)x], y’=λ(v)y, z’=λ(v)z, t’=λ(v)[(t – v²/c²)/ )/√(1 – v²/c²)] instead of: x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, u’ = (t-vx/c²)/√(1 – v²/c²)

For, since λ is independent of x and t while the special Lorentz transformation is defined by v alone, λ can only depend on v alone. If we introduce a third system E” that possesses the velocity (-v) in the X direction relative to E’, we obtain transformation equations between E and E” which means the coordinate axes of E and E” are permanently coincident. Since the coordinates are measured with identical measuring rods in both systems, we must have x” = x, etc. Hence λ(v)λ(-v) = 1. Furthermore, transformation shows that 1/λ(v) is the length, with respect to E, of a rod of length 1, that moves with E’ and is oriented parallel to the y’ axis. From a physical standpoint the length of a rod moving perpendicularly to its extension must be independent of the orientation of the motion, which means that we must have λ(v) = λ(-v). From these two equations, it follows that λ = ±1. The special case v = 0 shows that the choice of the positive sign is the only one that is possible.

4.2.5 Volume 4: The Swiss Years: Writings 1912-1914: Document 8: Relativity and Gravitation, Reply to comment by M Abraham, 1912
The theory presently designated as “the theory of relativity” rests on two principles that are totally independent of one another, namely

1 The principle of relativity (with respect to uniform translation)

2 The principle of the constancy of the velocity of light

From these two principles it is possible to develop the theory currently known as “the theory of relativity”.

But what about the limits of the validity of the two principles? As I have already emphasised, we have not the slightest reason to doubt the general validity of the principle of relativity. On the other hand, I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of gravitational potential. This is where, in my opinion, the limit of the validity of the principle of the constancy of the velocity of light, though not the principle of relativity, and therewith the limit of the validity of our current theory lies.

In my opinion, the present theory of relativity will always retain its significance as the simplest theory for the important limiting case of spatio-temporal events in the presence of a constant gravitational potential.

4.2.6 Volume 4: The Swiss Years: Writings 1912-1914, Doc 21: Theory of Relativity, 1913.
The phenomena of interference and refraction of light compelled physicists to view light as a wave-like process. Until the end of the last century, it was thought that light consisted in mechanical oscillations of a hypothetical medium, the ether. When the electromagnetic theory of light gained the upper hand, this conception changed, only insubstantially, in that light was no longer conceived as a motion of the ether, but rather as an electromagnetic process in the ether. Still, the conviction persisted that, in addition to ponderable matter, there must exist a second matter, the ether, considered the carrier of light. This conception led to the question of how this ether behaves, mechanically, with respect to matter.

Does ether take part in the motions of ponderable matter? Fizeau demonstrated that the hypothesis, according to which the luminiferous ether simply takes part in the motions of matter, is untenable. The next simplest hypothesis is that the luminiferous ether does not participate at all in the motions of matter. According to Lorentz’s theory, the electromagnetic laws of the ether are independent of the state of motion of matter. Matter interacts with the ether, only in the sense that matter is to be conceived as the carrier of electric masses, the motions of which are produced and influenced by electromagnetic processes in the ether.

Physical phenomena depend only on the motions of bodies relative to each other, ie from the physical standpoint, absolute motion does not exist. More precisely, wherever spatial determinations play a role in physics, they always represent determinations concerning the relative position of some object, or marker, relative to a solid body. In a theory, the coordinate system is the representative of this solid body.

Let us take some simple empirical law in which spatial determinations appear, eg Galileo’s familiar law of inertia: a material point not acted upon by external forces moves uniformly in a straight line. It is clear that this law cannot hold true, if the motion is referred to an arbitrarily moving coordinate system (eg one undergoing arbitrary rotation). We must therefore formulate Galileo’s fundamental law in the following way: It is possible to choose a coordinate system K, that is in such a state of motion that every freely moving material point moves rectilinearly and uniformly relative to it. Naturally, the law also holds then for all other coordinate systems at rest with respect to K.

If Galileo’s fundamental law were not valid for any coordinate system that is in motion relative to K, then the state of motion of K would be privileged, with respect to all other states of motion. We could then designate this state of motion as that of absolute rest. However, a simple argument shows that every freely moving material point satisfies Galileo’s fundamental law, not only with respect to K, but also with respect to every coordinate system K’, in uniform translational motion relative to K. The laws of mechanics are just as valid relative to such systems K’, as they are relative to K. There exists a whole class of coordinate systems moving uniformly, relative to one another, that are strictly equivalent when it comes to formulating the laws of mechanics. But this equivalence of the systems K and K’, that are moving uniformly relative to each other, is not limited to mechanics. So far as our experience extends, this equivalence holds generally. The assumption of the equivalence of all such systems K, K’, which rules out the privileging of one state of motion over all others, we will designate as the “relativity principle”.

According to Lorentz’s theory, the motion of matter does not result in any motion of the luminiferous ether. Instead, the parts of the latter are at rest relative to each other. If we choose a coordinate system K, that is at rest relative to the ether, then this coordinate system is privileged, with respect to all coordinate systems K’, that are in motion relative to K. Thus, the theory does not agree with the relativity principle. We can also carry out this argument without making use of the concept of the luminiferous ether. According to Lorentz’s theory, there exists a coordinate system K, relative to which every light ray propagates in a vacuum with the definite constant velocity c. If we refer such a light ray to a coordinate system K’, that is in motion relative to K, moving, say, in the direction of the propagation of the light, then we feel intuitively compelled to assume that this same light ray has a different propagation velocity relative to K’.

Thus, one would have to conclude that, in contradiction to the relativity principle, the coordinate system K is privileged, with respect to all coordinate systems K’, that are in motion relative to it. The fundamental assertion of Lorentz’s theory, that every light ray in a vacuum always propagates (at least with respect to a certain coordinate system K) with the definite constant velocity c, we will call the principle of the constancy of the velocity of light. The previously indicated difficulty with Lorentz’s theory consists in the fact that the principle of the constancy of the velocity of light seems to be incompatible with the relativity principle.

If one maintains, in accordance with Lorentz’s theory, that there exists a privileged coordinate system K, in which the velocity of light in a vacuum equals c, then one cannot assume that the earth is at rest relative to this coordinate system. Because one cannot then assume that the (stationary) ether participates in the motion of the earth around the sun. A great number of experiments were carried out to demonstrate this relative motion. But it proved wholly impossible to demonstrate experimentally such a privileged direction. However, most of these negative findings did not prove anything against the theory. In a very ingenious theoretical investigation, Lorentz showed that, to a first degree of approximation, the relative motion is without effect on the way rays proceed in arbitrary optical experiments. Only one optical experiment [Michelson], employing a method so extremely sensitive that its negative outcome could not be accounted for, even by Lorentz’s theoretical analysis.

To reconcile the negative result of this experiment with the theory, Lorentz and FitzGerald set forth the hypothesis that the stone plate, together with all the objects mounted on it, experience a minuscule contraction in the direction of the earth’s motion, the magnitude of which is such that the expected effect is compensated by an opposite effect resulting from this contraction. This manner of providing theoretical justification for experiments with a negative outcome, by means of hypotheses invented ad hoc, is very unsatisfactory.

It becomes hard to avoid the idea that no reality attaches to this relative motion of the earth with respect to the system K, ie that it is, in principle, impossible to demonstrate such a relative motion. In other words: we arrive at the conviction that the relativity principle is universally, and strictly, valid. On the other hand, the foundation of Lorentz’s theory, and therewith, the principle of the constancy of the velocity of light, seem to be incompatible with the relativity principle. But anyone who has made an attempt to replace Lorentz’s theory with another one, that would do justice to the experimental facts, would have to admit that, given the current state of our knowledge, this undertaking seems quite hopeless.

Given this state of affairs, one has to ask oneself, again, whether Lorentz’s theory, and the principle of the constancy of the velocity of light, are really incompatible with the relativity principle. Close examination shows that the two principles are compatible with each other, and that Lorentz’s theory does not contradict the relativity principle. However, our conception of space and time must be subjected to a fundamental revision.

As far as its interpretation in theoretical physics is concerned, the coordinate system is nothing but a rigid measuring framework, on which to mark off the values of the spatial coordinates, with the help of rigid rods. We now have to pose the question of the physical meaning of the temporal determinations that usually occur in physics, in association with spatial determinations.

We designate as a clock a system that automatically repeats the same process. The number of processes of this kind that have already taken place, counting from an arbitrarily chosen one, is the temporal determination of the clock. The temporal determination of the clock that is simultaneous with an event, is called the time of the event, as measured by the clock. Suppose that a clock is set up at the origin of our coordinate system, and that some event takes place at a location immediately adjacent to this origin. Then we are in a position to specify the clock time that is simultaneous with the event, ie the time of the event (referred to our clock).

But, if the location of the event is far away from the location at which the clock is set up, then we cannot ascertain directly the clock reading that is simultaneous with the event. For an observer standing next to the clock cannot perceive the event directly, but only via some intermediary process (signal) induced by the event, and propagated to the observer (eg via light rays). The observer determines only the time of arrival of the signal, but not the time of the event. He could ascertain the latter only if he knew the length of time during which the signal was en route. But it is impossible in principle to ascertain this length of time by means of the clock set up at the origin of K. Only the times of events occurring in the immediate vicinity of the clock can be ascertained directly by means of that clock.

If we also had a clock at the location at which the [distant] event took place (we will assume that this clock is of exactly the same constitution as the other one), and if an observer stood there who determined the time of the event from this clock, even this would not yet do us any good. For we are, at present, unable to determine the temporal reading of the first clock, that is simultaneous with the temporal determination read off the other clock. From this we see that, for a definition of time, a physical definition of simultaneity is also needed. If the latter is given, then the physical definition of time that we are seeking is complete. In other words, we also need a rule by which to regulate the other clock according to the first clock.

Let some means be given for sending signals from the origin of the system K to the other location, and vice versa, such that the signals either way are completely equivalent physical processes. Then, the other clock should be adjusted such that the time taken for the signal to travel in each direction is the same. The process can then be repeated with other clocks.

As regards the given definition, one has to be careful about one thing in particular. We use for the definition of time, a system of clocks that are at rest relative to the system K. Hence this definition has meaning only with respect to a coordinate system K, in a specific state of motion. If in addition to the coordinate system K, we introduce another system K’ that is in uniform translational motion relative to K, then we can define a time with respect to K’ just as well as we have done previously with respect to K. But it is not, a priori, evident that agreement can be produced between the readings of these two systems of clocks. There is no reason a priori, why two events that are simultaneous with respect to K, must also be simultaneous with respect to K’. This is what one understands by the “relativity of time”.

Thus, it turns out that the principle of constancy of the velocity of light, and the relativity principle, are incompatible with each other only as long as one holds to the postulate of absolute time, ie to the absolute meaning of simultaneity. But if one admits the relativity of time, then the two principles turn out to be compatible with each other; proceeding from these two principles, one arrives at the theory designated as the “theory of relativity”.

The fundamental problem connected with this conception is the following: We are given two coordinate systems K and K’. Let K’ be in a state of uniform translation with respect to K, and let v be the velocity of this motion. We are given the location and time of an arbitrary event (ie the coordinates x, y, z, and the time t) with respect to K. One has to determine location and time (x’ y’, z’, t’) with respect to K’. Conventional kinematics solves this problem by means of the following equations: x’ = x – vt, y’ = y, z’ = z, t’ = t. The last of these equations expresses the assumption that temporal determinations have a meaning independent of the state of motion (assumption of “absolute time”).

But these equations also have hidden in them another implicit assumption. The state of motion, of the two systems K and K’, as these appear when observed from K. Let us now consider a point P’ on the x’ axis, whose distance from O’ is equal to l’. That means: an observer moving with K’, must lay his meter stick l’ times along the x’ axis, in order to get from O’ to P’. But observers who are at rest in system K, will have to proceed differently in order to evaluate the distance O’P’. They determine those spatial points in the system K, at which O’ and P’ are located, at a specified time (of the system K). The distance l between these two points, subsequently determined by laying the meter stick along the x axis of K, is the length we are seeking.

One sees that the two procedures are fundamentally different, so that it is, a priori, possible that their numerical results, l and l’, differ from each other. This shows that one cannot reject, a priori, the possibility that the concept of spatial distance might also possess only a relative meaning. Thus, in addition to the “relativity of time”, we must also admit a “relativity of lengths”. This shatters the foundation on which the original transformation equations for spatial coordinates, and time values, are based. In the theory of relativity, the place of those equations is taken by equations that simultaneously satisfy the principle of relativity and the principle of constancy of the velocity of light. One finds the new equations by formulating mathematically the requirement that every light ray should propagate with the same velocity c in both systems K and K’. In this way one arrives at the transformation equations: x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, t’ = (t-(v/c²)x)/√(1 – v²/c²)

The last of these equations shows that, in general, the equality of time values (simultaneity) of two events with respect to K, does not result in the equality of time values (simultaneity) of the same events with respect to K’. Thus, the absolute meaning of simultaneity is lost.

x₁ = (x₁ -vt)/√(1 – v²/c²), x₂’ = (x₂ -vt)/√(1 – v²/c²), so: x₂’ – x₁’ = (x₂ – x₁)/√(1 – v²/c²)

or: l = l’[√(1 – v²/c²)]

This means the following. If a rod possesses the length l’ when measured at rest, then, if it moves with velocity v along its axis, it will possess the smaller length l = l’[√(1 – v²/c²)] for a non-comoving observer, whereas for a comoving observer, it will always, have the length l’. The greater the velocity v of the moving rod, the smaller the length. If v approaches the velocity of light c, then the length of the rod approaches the value zero. For values of v that exceed the velocity of light, our result becomes meaningless; such velocities of motion are not possible according to the theory of relativity. One sees that the above mentioned hypothesis of Lorentz and FitzGerald, which was advanced in order to explain Michelson’s experiment, follows as a consequence of the theory of relativity. According to the latter, bodies at rest relative to K, evaluated from K’, will display exactly the same contraction as bodies at rest in K’, if these are evaluated from K.

Let a clock with a second hand be located at the origin of K’. For this clock we always have x’ = 0, and the clock strikes the seconds at times t’ = 0, 1, 2, 3, etc. The first and the fourth of our equations yield the following values for the times t of these strokes: t = 0/√(1 – v²/c²), 1/√(1 – v²/c²), 2/√(1 – v²/c²), 3/√(1 – v²/c²), etc

Thus, evaluated from K, the time between two strokes of the clock is equal to: t = 1/√(1 – v²/c²), and thus longer than one second. A clock traveling with the velocity v, runs slower, judged from a non-comoving system, than the same clock when it does not travel. Generalizing, one can conclude: Every event in a physical system slows down if the system is set into translational motion. But this slowing occurs only from the standpoint of a non-comoving coordinate system.

Section 4.3: Other Einstein Quotes

4.3.1 The Foundation of the Generalised Theory of Relativity (1916)
The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as “the theory of Relativity;” this latter theory I differentiate from the former “Special Relativity theory,” and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the “Absolute Differential Calculus”, which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics.

Section A: Principal considerations about the Postulate of Relativity
Part 1: Remarks on the Special Relativity Theory
The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics. If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to a uniform translational motion relative to K. We call this postulate “The Special Relativity Principle”. By the word special, it is signified that the principle is limited to the case when K′ has uniform translatory motion with reference to K, but the equivalence of K and K′ does not extend to the case of non-uniform motion of K’ relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate [relativity principle], but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

Part 2: About the reasons which explain the extension of the relativity-postulate
Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another, and from all other masses, that only that sort of gravitational forces are to be taken into account which any of these bodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. Each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity, ie definite relative motion for both the masses.

The bodies (S₁ and S₂) are measured with the help of measuring rods, relatively at rest; it is then found that the surface of S₁ is a sphere and the surface of the other is an ellipsoid of rotation. Why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory, from the epistemological standpoint, when the thing adduced as the cause is an observable fact of experience.

A satisfactory explanation to the question put forward above can only be thus given: that the physical system composed of S₁ and S₂ shows for itself alone no conceivable cause to which the different behaviour of S₁ and S₂ can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specifically the forms of S1 and S₂ must be of such a kind that the mechanical behaviour of S₁ and S₂ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration.

They take the place of the imaginary cause R₁. Among all the conceivable spaces R₁ and R’₁ etc. moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised, from the standpoint of the theory of knowledge, cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.

Beside this epistemological argument, there is also a well-known physical fact which favours an extension of the relativity theory. Let there be a Galilean co-ordinate system K relative to which, at least in the four-dimensional region considered, a mass at a sufficient distance from other masses moves uniformly in a line. Let K’ be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K’ any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction of acceleration is independent of its material composition and its physical conditions.

Can any observer, at rest relative to K’, then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to K’ can be explained in as good a manner in the following way. The reference-system K’ has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K’.

This conception is feasible, because to us the experience of the existence of a field of force, namely the gravitation field, has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The mechanical behaviour of the bodies relative to K’ is the same as experience would expect with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K’ can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.

From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can “create” a gravitational field by a simple variation of the co-ordinate system. Also we see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easily that the path of a ray of light with reference to K’ must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K.

Part 3. The time-space continuum. Requirements of the general Co-variance for the equations expressing the laws of Nature in general
In classical mechanics as well as in the special relativity theory, the co-ordinates of time and space have an immediate physical significance; when we say that any arbitrary point has x₁ as its x₁ co-ordinate, it signifies that the projection of the point-event on the x₁ axis ascertained by means of a solid rod according to the rules of Euclidean Geometry is reached when a definite measuring rod, the unit rod, can be carried x1 times from the origin of co-ordinates along the x₁ axis. A point having x₄ = t as the x₁ co-ordinate signifies that a unit clock which is adjusted to be at rest relative to the system of co-ordinates, and coinciding in its spatial position, with the point-event and set according to some definite standard has gone over x₄ = t periods before the occurrence of the point-event.

We introduce in a space which is free from a gravitation-field, a Galiliean co-ordinate System K(x, y, z, t), and another system K'(x’, y’, z’, t’) rotating uniformly relative to K. The origin of both the systems as well as their Z-axes might continue to coincide. We will show that for a space-time measurement in the system K’, the above established rules for the physical significance of time and space cannot be maintained. On grounds of symmetry, it is clear that a circle round the origin in the X-Y plane of K, can also be looked upon as a circle in the X’ – Y’ plane of K’.

Let us now think of measuring the circumference, and the diameter of these circles, with a unit measuring rod (infinitely small compared with the radius) and take the quotient of both the results of measurement. If this experiment be carried out with a measuring rod at rest relatively to the Galiliean system K we would get π, as the quotient. The measurement with a rod relatively at rest as regards K’ would be a number which is greater than π. This can be seen when we regard the whole measurement-process from the system K, and remember that the rod placed on the periphery suffers a Lorentz-contraction, not however when the rod is placed along the radius. Euclidean Geometry therefore does not hold for the system K’; the above fixed conceptions of co-ordinates which assume the validity of Euclidean Geometry fail with regard to the system K’.

We cannot similarly introduce in K’ a time corresponding to physical requirements, which will be shown by all similarly prepared clocks at rest relative to the system K’. In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows, as viewed from K, that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light, will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time, in the way under consideration, he will interpret his observation by saying that the clock on the periphery “actually” goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position.

The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature. So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law: That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations. This condition of general covariance which takes away the last remnants of physical objectivity from space and time.

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. We fit to the world our space-time variables x₁, x₂, x₃, x₄ such that to any and every point-event corresponds a system of values of x₁…x₄. Two coincident point-events correspond to the same value of the variables x₁…x₄, ie the coincidence is characterised by the equality of the co-ordinates. If we now introduce any four functions: x₁, x₂, x₃, x₄ as coordinates, so that there is a unique correspondence between them, the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points. As the purpose of all physical laws is to allow us to remember such coincidences, there is a priori no reason present, to prefer a certain co-ordinate system to another; i.e. we get the condition of general covariance.

4.3.2 Ether and The Theory of Relativity (1920)
Lorentz brought theory into harmony with experience by means of a wonderful simplification of theoretical principles. He achieved this, the most important advance in the theory of electricity since Maxwell, by taking from ether its mechanical, and from matter its electromagnetic qualities. As in empty space, so too in the interior of material bodies, the ether, and not matter viewed atomistically, was exclusively the seat of electromagnetic fields. According to Lorentz the elementary particles of matter alone are capable of carrying out movements; their electromagnetic activity is entirely confined to the carrying of electric charges. Thus Lorentz succeeded in reducing all electromagnetic happenings to Maxwell’s equations for free space.

As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobility is the only mechanical property of which it has not been deprived by Lorentz. It may be added that the whole change in the conception of the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility.

The space-time theory and the kinematics of the special theory of relativity were modelled on the Maxwell-Lorentz theory of the electromagnetic field. This theory therefore satisfies the conditions of the special theory of relativity, but when viewed from the latter it acquires a novel aspect. For if K be a system of co-ordinates relatively to which the Lorentzian ether is at rest, the Maxwell-Lorentz equations are valid primarily with reference to K. But by the special theory of relativity the same equations without any change of meaning also hold in relation to any new system of co-ordinates K’ which is moving in uniform translation relatively to K. Now comes the anxious question: Why must I in the theory distinguish the K system above all K’ systems, which are physically equivalent to it in all respects, by assuming that the ether is at rest relatively to the K system? For the theoretician such an asymmetry in the theoretical structure, with no corresponding asymmetry in the system of experience, is intolerable. If we assume the ether to be at rest relatively to K, but in motion relatively to K’, the physical equivalence of K and K’ seems to me from the logical standpoint, not indeed downright incorrect, but nevertheless unacceptable.

The next position which it was possible to take up in face of this state of things appeared to be the following. The ether does not exist at all. The electromagnetic fields are not states of a medium, but they are independent realities which are not reducible to anything else, exactly like the atoms of ponderable matter. This conception suggests itself the more readily as, according to Lorentz’s theory, electromagnetic radiation, like ponderable matter, brings impulse and energy with it. More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it.

4.3.3 A Brief Outline of the Development of the Theory of Relativity (1921)
The entire development [of the theory of Relativity] starts off from, and is dominated by, the idea of Faraday and Maxwell, according to which all physical processes involve a continuity of action (as opposed to action at a distance), or, in the language of mathematics, they are expressed by partial differential equations. Maxwell succeeded in doing this for electro-magnetic processes in bodies at rest by means of the conception of the magnetic effect of the vacuum-displacement-current, together with the postulate of the identity of the nature of electro-dynamic fields produced by induction, and the electro-static field.

At this point, Lorentz came to the rescue. In view of his unqualified adherence to the atomic theory of matter, Lorentz felt unable to regard the latter as the seat of continuous electromagnetic fields. He thus conceived of these fields as being conditions of the ether, which was regarded as continuous. Lorentz considered the ether to be intrinsically independent of matter, both from a mechanical and a physical point of view. The ether did not take part in the motions of matter, and a reciprocity could be assumed only in so far as the latter was considered to be the carrier of attached electrical charges. The great value of the theory of Lorentz lay in the fact that the entire electrodynamics of bodies at rest and of bodies in motion was led back to Maxwell’s equations of empty space.

The theory appeared to be unsatisfactory only in one point. It appeared to give preference to one system of coordinates of a particular state of motion (at rest relative to the ether) as against all other systems of co-ordinates in motion with respect to this one. In this point the theory seemed to stand in direct opposition to classical mechanics, in which all inertial systems which are in uniform motion with respect to each other are equally justifiable as systems of co-ordinates.

The Special Theory of Relativity owes its origin to this difficulty, which, because of its fundamental nature, was felt to be intolerable. This theory originated as the answer to the question: Is the special principle of relativity really contradictory to the field equations of Maxwell for empty space? The answer to this question appeared to be in the affirmative. For if those equations are valid with reference to a system of co-ordinates K, and we introduce a new system of co-ordinates K’ in conformity with the equations of transformation: x’ = x-vt, y’ = y, z’ = z, t’ = t, the Galileo transformation, then Maxwell’s field equations are no longer valid in the new co-ordinates (x’, y’, z’, t’).

A more searching analysis of the physical significance of space and time rendered it evident that the Galileo transformation is founded on arbitrary assumptions, and in particular on the assumption that the statement of simultaneity has a meaning which is independent of the state of motion of the system of co-ordinates used. It was shown that the field equations for vacuo satisfy the special principle of relativity, provided we make use of the equations of transformation stated below (ie the Lorentz transformations): x’ = (x -vt)/√(1 – v²/c²), y’ = y, z’ = z, t’ = (t-vx/c²) / √(1 – v²/c²) Galilaean transformation: x’ = x – vt & t’ = t.

Later, H. Minkowski found an expression for this condition of co-variance:

Euclidean Geometry of Three Dimensions

Euclidean Geometry of
Three Dimensions Special Theory of
Relativity

Corresponding to two neighbouring points in space, there exists a numerical measure (distance ds) which conforms to the equation:
ds² = dx1² + dx2² + dx3²

Corresponding to two neighbouring points in space-time (point events), there exists a numerical measure (distance ds) which conforms to the equation:
ds² = dx1² + dx2² + dx3² + dx4²

It is independent of the system of co-ordinates chosen, and can be measured with the unit measuring-rod

It is independent of the system of co-ordinates chosen, and can be measured with the unit measuring-rod It is independent of the inertial system chosen, and can be measured with the unit measuring-rod and a standard clock. x1, x2, x3 are are here rectangular co-ordinates, whilst x₄= √(-1ct) is the time multiplied by the imaginary unit and by the velocity of light

The permissible transformations are of such a character that the expression for ds² is invariant, i.e. the linear orthogonal transformations are permissible

The permissible transformations are of such a character that the expression for ds² is invariant, i.e. the linear orthogonal transformations are permissible The permissible transformations are of such a character that the expression for ds² is invariant, i.e. those linear orthogonal substitutions are permissible which maintain the semblance of reality of x1, x2, x3, x4. These substitutions are the Lorentz transformations

With respect to these transformations, the laws of Euclidean geometry are invariant

With respect to these transformations, the laws of physics are invariant

This reveals a formal relationship between Euclidean geometry of three dimensions and the space-time continuum of physics.

The development of the special theory of relativity consists of two main steps, namely, the adaptation of the space-time “metrics” to Maxwell’s electro-dynamics, and an adaptation of the rest of physics to that altered space-time “metrics”. The first of these processes yields the relativity of simultaneity, the influence of motion on measuring-rods and clocks, a modification of kinematics, and in particular a new theorem of addition of velocities. The second process supplies us with a modification of Newton’s law of motion for large velocities, together with information of fundamental importance on the nature of inertial mass.

4.3.4 Einstein 1922: The Meaning of Relativity. Lecture 2
The theory of relativity is often criticized for giving, without justification, a central theoretical role to the propagation of light, in that it founds the concept of time upon the law of propagation of light. The situation, however, is somewhat as follows. In order to give physical significance to the concept of time, processes of some kind are required which enable relations to be established between different places. It is immaterial what kind of processes one chooses for such a definition of time. It is advantageous, however, for the theory, to choose only those processes concerning which we know something certain. This holds for the propagation of light in vacuo in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and Lorentz.