Paper for peerreviewed scientific journal
Paradox of the invariance of
the speed of light
Abstract
Reflections on the impossibility of the invariance of the speed of light in certain cases, with regard to one of the logical implications of this postulate: acknowledgement of this analysis could lead to an major paradigm shift concerning our concept of spacetime.
Introduction
In this paper we will show that the invariance of the speed of light with respect to different inertial frames of reference necessarily implies “relativity of simultaneity at the physical level” when two distant points are taken into account. “Relativity of simultaneity at the physical level” is an implicit metaphysical principle implied by special relativity, a theory which gave rise to a particular conception of spacetime.[1] This principle will be explained later, and we will understand that we need to differentiate it from the simple relativity of simultaneity of events as perceived by two different observers. However, we will go on to see that, when the observer is accelerating, this principle can lead us to say one thing and its very opposite at the same time. While this fact is nothing new, we see that the framework of interpretation in the theory of relativity appears to erase this aspect. Such a finding is important, because it lets us conclude that the speed of light cannot be physically invariant[2] in every situation, and that we consequently need to rethink how we represent spacetime. Moreover, it should also be possible to define situations in which a speed difference for light is measurable, although this would require further developments.
Is partially inaccurate reasoning at the origins of special relativity?
It was once thought that the speed of light was invariant with respect to the aether, and therefore could not be invariant with respect to the Earth. The MichelsonMorley experiment was an attempt to confirm this. Surprisingly, the experiment rather seemed to demonstrate that the speed of light is invariant with respect to the Earth. It was this finding that probably led Albert Einstein to the following reasoning: if the speed of light is invariant with respect to the Earth, given that the Earth is a moving body, then the speed of light must be invariant with respect to any body in a state of inertia. If we apply this to the thought experience of Einstein’s train, this means that if the speed of light is invariant with respect to the station, then it must be also invariant with respect to the train that is in constant motion with respect to the station. However, such a reasoning, as may be shown in this article, is partly inaccurate. We will see that it is probably far more reasonable to think that the speed of light constantly adapts to the spatial configuration – even if we do not immediately ascertain the cause.
Invariance of the speed of light implies the principle of relativity of simultaneity at the physical level
We will comment on Einstein’s train thought experiment[3] that was originally used to illustrate the conception of spacetime according to special relativity, and then on Einstein’s elevator thought experiment and its equivalence principle he used to illustrate general relativity. Einstein’s train thought experiment is used to demonstrate the effect of the supposed invariance of the speed of light for all the inertial observers. In this thought experiment, we consider the observations of two people while a train is moving through the station without stopping: one observer is sitting midway inside the train, with constant speed with respect to the station (which we shall refer to as “the observer inside the train car”) and a second observer standing immobile on the platform (“the stationmaster”). At the instant the two observers pass each other, the train is struck by two light sources, one from behind the train car and one from in front of the train car, at equal distance from the stationmaster. In this situation, the two rays of light reach the stationmaster at the same time. Since the two light sources are at equal distance from the stationmaster, and given that the speed of light is invariant with respect to him, he therefore concludes that the two rays of light were emitted at the same instant for him. It is however quite possible that these two events did not actually occur simultaneously for the stationmaster, but if the speed of light is assumed to be invariant with respect to him, and the distances between the stationmaster and the two light sources are known, it is inferred that the two events are simultaneous. In such a context, we cannot therefore say that there is no simultaneity whatsoever, since the simultaneity of these two events, as measured by the stationmaster, is a logical conclusion based on the premises.
Generally speaking, two distant events that appear to be simultaneous for a stationary observer will not be perceived as being simultaneous by a second observer who is constantly moving with respect to the first observer unless both observers are at the same position at the time of observing or measuring the events. This can be called a relativity of the simultaneity of events as perceived by two different observers. A little further on, we will see that it is important to differentiate this relativity of the simultaneity of events as perceived by different observers at the time of measuring the events (that is to say “during a measurement procedure”) from a “relativity of simultaneity involving rays of light[4]”, as well as from a “relativity of simultaneity at the physical level”. A particular feature of our analysis of Einstein’s train thought experiment is that it shows that this supposed relativity of simultaneity involving rays of light necessarily implies “a relativity of simultaneity at the physical level”. Let us now consider two different scenarios. First scenario: the two observers are at the same position when the two rays reach the stationmaster (variant of the train thought experiment proposed by Yann le Roux); second scenario: the two observers were at the same position when the two rays were emitted, according to the viewpoint of the stationmaster, assuming that the speed of light is invariant with regard to him (variant proposed by Albert Einstein).
In the first scenario, if we are to consider that the two rays were emitted simultaneously for both observers (i.e. absolute simultaneity) and that the speed of light is invariant with respect to the stationmaster, then it follows that the speed of light cannot be invariant with respect to the observer inside the train car. Indeed, in this scenario, at the instant the two rays were emitted, the distances from the two light sources were not the same for the passenger inside the train car who was moving with respect to the station. However, in the scenario we just mentioned, the two rays reach this observer at the same time. Therefore, in this scenario, for the speed of light to also be invariant with respect to the observer inside the train car, the two rays cannot have been emitted simultaneously for the observer inside the train car, although this was the case for the stationmaster. Bearing this first conclusion in mind, we can now move on to the second scenario.
In the second scenario, we already know that the observer on the platform considers the two rays of light to have been emitted simultaneously, assuming that the speed of light is invariant with respect to him. However, we saw in the previous paragraph that, in order for the observer inside the train car to be able to conclude that the speed of light is invariant with respect to himself, then according to him, the two rays cannot have been emitted simultaneously – as what has been said for the first scenario should also be applicable to the second. Indeed, the two scenarios are comparable – only the timing of the train is different. For the observer inside the train car, the ray of light from in front of the train must be emitted before the event where the two observers align, and the second ray of light has to be emitted from behind the train after the two observers align. From this we can define the implicit metaphysical principle implied by the invariance of C between the point of origin of the ray, and the point where the ray reaches the two inertial observers.
In the second scenario, at the instant the two observers pass each other, it is admitted that the ray of light from the rear of the train has already been emitted according to the stationmaster – i.e. that it does indeed exist at its point of origin for this observer – but that it has not yet been emitted according to the observer inside the train car – i.e. that it does not yet exist at its point of origin for the second observer. It will only exist for the observer inside the train car a short time later when he is a little further away from the stationmaster. Indeed, in order for a body to be moving with respect to an observer, that body must exist for that observer. I am using the expression “to exist for”, because according to this principle, even though the two observers are at the same physical position, the ray of light from the rear of the train exists for the observer on the platform, but it does not exist for the observer inside the train car.[5] This is what can be referred to as the “principle of relativity of simultaneity at the physical level”, which is implicitly implied by the postulate of the invariance of the speed of light. This metaphysical principle was not explicitly stated by Albert Einstein in his train thought experiment, but it is present implicitly. It is important to remember that these events are occurring in a spacelike interval, and that the ray was not actually observed in the second scenario by either of the two observers at that particular instant. We are merely theoretically reconstructing what we believe would happen in such a scenario. The time discrepancy, i.e. the additional length of time it takes for the ray to exist for the observer inside the train car, depends on the observer’s distance from the light source and the speed of the train: the faster the train is moving, the greater the distance from the light source, and the greater the discrepancy will be.[6] However, as far as the light source is concerned, any distance can be used. In actual fact, it is not possible, in terms of what is being measured, to ascertain whether there is indeed a relativity of simultaneity at the physical level, or absolute simultaneity[7], because it is not possible to be entirely certain of the simultaneity of two distant events. That said, it is possible to explore this subject theoretically. That is what we will attempt to do now.
The principle of relativity of simultaneity at the physical level is in contradiction with itself
Staying with the scenario proposed by Albert Einstein, if we were to imagine a situation in outer space where the light source is very far away, the difference in simultaneity might be, say, 10 seconds, during which time the observer inside the train car may have traveled 300 feet after having passed by the observer on the platform. If the observer on the platform were to get into the moving train to join the observer inside the train before the train had covered the whole distance of 300 feet, then the observer on the platform, for whom the ray is supposed to have already existed in his past, would be joining the observer inside the train for whom the ray does not yet exist at that precise moment in time. And, as the observer who was previously on the platform will at that precise moment in time have the same viewpoint as the observer inside the train, we therefore now have a ray of light – which can be replaced by any body – which existed in the past of the observer on the platform before his acceleration upon getting into the moving train, and which should not yet have existed for him after his acceleration. If the observer on the platform makes an initial calculation of the trajectory of the body in question before accelerating, and subsequently makes a second calculation when he has the same viewpoint of the observer inside the train, he would end up with two contradictory calculations: one calculation where the body in question had already traveled a certain distance, and another where the body in question had not yet moved.[8]
Objection of the shuttle and of the missile
For this, we could replace the stationmaster with a shuttle that is accelerating quickly in outer space, and we could replace the ray of light with a missile that is fired at the shuttle at the very moment the shuttle is launched. In other words, based on our analysis of Einstein’s train thought experiment, one objection to the invariance of the speed of light can be raised using the example of the shuttle and missile scenario, whereby the space shuttle would end up having two contradictory calculations regarding the missile’s position. According to special relativity, the temporal order between events may change when an object is accelerating, provided those events are separated by a spacelike interval. However, this is no longer correct if we take account of the existence of bodies. The only contribution I will make here is to show that the principle of relativity of simultaneity at the physical level, implied by the invariance of c, requires us to consider the existence of bodies according to what is shown in spacetime diagrams. This is why the principle of relativity at the physical level, implied by the invariance of c, is so important. It indeed helps us to understand that the speed of light cannot actually be invariant in every situation. The framework it provides only allows for either absolute simultaneity or for relativity of simultaneity at the physical level; it does not offer any third possibility.[9] Once it has been demonstrated that the relativity of simultaneity at the physical level results in a contradiction, it follows that its framework would have to also allow for absolute simultaneity at the physical level. However, we saw earlier that, in the context of absolute simultaneity, the speed of light cannot be invariant in every situation.[10] I will explain the logic of my reasoning further in appendices 1 and 2, as well as provide some relevant equations in appendix 3.
Response to a major critique
Some may consider it a fallacy to want to link the hypersurface of simultaneity[11] to the existence of bodies. This is indeed the main reason why the conclusion of this paper may be rejected by many scientists specialized in special relativity. But the hypersurface of simultaneity, as presented in spacetime diagrams, is a representation of the relativity of simultaneity. When we consider the framework proposed in Einstein’s train thought experiment, and when we compare the two observers’ perception of events, we see that it is a matter of relativity of simultaneity with respect to the emission of the rays of light. This relativity of simultaneity is implied by the postulate of the invariance of the speed of light.
This is based on an assumed chronology with regard to events separated by a lightlike interval, which would in turn have implications on an assumed chronology with regard to events separated by a spacelike interval. Indeed, if the distance from the light source and the speed of the ray of light are considered to be known at the moment the ray arrives (i.e. the time interval – by definition lightlike – between the moment the ray is emitted and the moment it is received), it is therefore indeed possible to say at what moment it was emitted, for the observer concerned, in his past (i.e. the interval – which at this moment is spacelike – between the event where the ray of light is emitted and the event where it is received by the observer). This means that by using a supposed chronology for events separated by a lightlike interval, we can, through theoretical reconstruction, obtain a supposed chronology for events separated by a spacelike interval. This is why Einstein, in his train thought experiment, stated that there was a relativity of simultaneity with regard to the emission of the rays of light.
By asserting that a body is moving with respect to an observer, Einstein implicitly accepts that that body exists for that observer even if that observer has not yet perceived it. If the initial statement is deemed to be correct, then that body has to exist for the observer. The relativity of simultaneity of events involving the emission of the rays of light, which is presented in spacetime diagrams, therefore assumes a relativity of simultaneity at the physical level. This means that we must take into account the existence of bodies according to what the spacetime diagrams show us. If there is a fallacy in taking such a position, that fallacy is actually implied by the postulate of the invariance of the speed of light. Indeed, it implicitly states that at certain times, a body does exist with respect to a given observer – because that body is supposed to be moving with respect to that observer; and at other times, that body is supposed not to exist for the same observer – because it cannot be detected by that observer at that specific moment (spacelike interval between events). Here, the verb "to exist" is being used in two different ways with two different meanings, and it is necessary to distinguish between the two to be able to understand that the invariance of c leads to the principle of relativity of simultaneity at the physical level.
The relativity of simultaneity, which is presented in spacetime diagrams and which is a consequence of the postulate of the invariance of the speed of light, implies – in certain cases and although the two observers are facing each other – that the same body exists for one observer, but not for the other. This is what I call the principle of relativity of simultaneity at the physical level. Based on this observation, which of course is not the only one possible, we can question the principle of causality in the theory of relativity from a perspective of realist philosophy. Indeed, in relativity, there is a possible causal relationship between events when dealing with a timelike or lightlike interval. But, ultimately, because of the postulate of the invariance of the speed of light, this principle of causality of relativity necessarily implies the principle of relativity of simultaneity at the physical level – which does not fully respect the existence of bodies.[12] In natural philosophy, the causal principle can be understood as what accounts for the existence of bodies, their structure and their behavior. Indeed, for there to be a relationship between antecedent and consequent, something real has to behave in a particular way. In realist philosophy, the causal principle therefore acknowledges the existence of bodies, which is not however altogether the case, for the reason I have explained above, with the principle of causality in relativity.
Certain people have seen that the term “simultaneity”, in the expression “relativity of simultaneity”, would give rise, if we consider simultaneity to be something physical, to the problem that I am highlighting. They therefore suggest it be replaced by the expression “same time coordinate”. They rightly point out that, for any pair of spatially separated events, it is possible to establish a coordinate system in which the two events have the same temporal coordinate. However, we cannot totally dismiss the question of the relationship between these time coordinates and the rates of clocks. With the Poincaré–Einstein synchronization method, clocks are synchronized by means of light exchanges, assuming that the distance from the light sources and the speed of the light signals are known. These clocks, synchronized by means of light signals, are supposed to run simultaneously at the same rate. We thus find a supposed simultaneity which has a physical sense and which is based on the postulate of the invariance of the speed of light. Bearing in mind what we have already said, this means that the relativity of simultaneity, in its physical sense, is indeed a result of the postulate of the invariance of the speed of light.[13] Therefore, questioning the relativity of simultaneity in its physical sense is tantamount to questioning the postulate of the invariance of the speed of light.
We could spend much time discussing whether things have already been said by others or not. When we bring together the block universe theory of time and special relativity, or when we explore the possibility of semiclosed time loops[14] in general relativity, things are said indirectly. We need only consider the argument advanced by Roger Penrose, which he called the Andromeda paradox[15]. But the fact remains that it may be difficult, at least for some, to come to any real conclusions on this issue. By showing that the relativity of simultaneity in special relativity implies the principle of relativity of simultaneity at the physical level, I have at least shown where the origin of the problem lies. When we develop the subject clearly enough, we see that something is not right. This is interesting, as it leads us to question the conception of spacetime that physicists have adhered to for over a hundred years. It is therefore a subject of crucial importance for physics. This question must first be resolved from a theoretical point of view, however exasperating that may be for those who favor a purely operational approach.
Towards a new concept of spacetime
Bearing in mind our considerations up to this point, if we were to take a completely relational approach to space and motion, we could hypothesize that the speed of light constantly adapts to the spatial configuration[16] – which we should be capable of measuring at some point in the future. In this case, the speed of light would be locally invariant with respect to the Earth or near the Earth, but it would have a variable speed relative to any moving bodies located near the Earth. Moreover, the Earth’s rotation on its own axis can also complicate matters. For example, the Sagnac effect is to be interpreted differently depending on whether we are dealing with absolute simultaneity or a relativity of simultaneity at the physical level. Consequently, it might be worth taking such measurements in outer space, away from any large mass.
Conclusion
The notion of time in special relativity and the resulting paradoxes are a consequence of the postulate of the invariance of the speed of light. By showing that the invariance of the speed of light implies the principle of relativity of simultaneity at the physical level, I have shown where the origin of the problem lies. This, to my knowledge, had not been sufficiently demonstrated by anyone else before. Since the principle of relativity of simultaneity at the physical level gives rise to contradictions and since there is no third possibility, it follows that there has to be an absolute simultaneity at the physical level. This means that there is a present moment for the universe – which is the thesis behind Presentism, a theory of time where only the present moment and present things exist. This is not compatible with the concept of time associated with special relativity, where what exists in the present for one observer is not what exists in the present for another observer. From an operational point of view, we cannot be absolutely certain of the simultaneity of two distant events, but it is not for this reason that simultaneity does not exist in the physical world. If we want successfully to devise a general theory of the universe, we cannot totally disregard the metaphysical aspect. From a metaphysical point of view, the principle of relativity of simultaneity at the physical level, which is implied by the invariance of c, does not stand up. Indeed, something real cannot exist and not exist at the same time. The relativity of simultaneity is nonetheless still of interest for certain calculations, because it can be used, albeit approximately, to replace a speed difference for light.[17]
It was special relativity that moved us away from a concept of absolute time and space, which, albeit an important conceptual advance, probably did not happen for the right reason, as we have just seen.[18] Some physicists are now looking to adopt a completely relational approach to space and time.[19] The theory of relativity is not fully compatible with such an approach for at least two reasons. Firstly, a completely relational view of space embraces the notion of a present moment for the universe. Indeed, with such an approach, bodies actually need to be in relation to each other in order for spacetime to exist.[20] Secondly, a completely relational approach to space necessarily implies a relational approach to motion, which would consequently change our approach to inertia and impulse and would require further development.[21] The idea of a present moment for the universe does not necessarily imply a return to Newtonian time or absolute time. Two similar clocks that are placed in different spatial conditions may very well work simultaneously at different rates, for example on two different floors of the same building.
The postulate of the invariance of the speed of light does in many ways have perfectly logical consequences, which can be found in the equations relating to special relativity and in spacetime diagrams. However, if we assume relativity of simultaneity at the physical level, the postulate leads to contradictory assertions. Which is to say that this postulate, if one were tacitly to accept the concepts arising from it, causes a certain inconsistency in the theory[22]. Indeed, the principle of relativity of simultaneity at the physical level, implied by the invariance of the speed of light, has metaphysical implications. And if we take account of the implied metaphysical aspect (i.e. the existence of a body as shown in the spacetime diagram), we end up with two contradictory calculations, namely regarding the position of the missile as demonstrated in my objection involving the shuttle and the missile. This illustrates – even mathematically speaking – that such a relativity of simultaneity is impossible. Thus, the speed of light cannot be physically invariant in every situation, which calls into question the second postulate of special relativity. Such a conclusion may understandably lead physics to a significant paradigm shift in its conceptual system and its representation of spacetime. What is proposed in this paper could be the beginning of a radical change for physics.
Philippe de Bellescize
Appendix 1
Summary of my reasoning:
I am attempting to demonstrate the following important aspects:
a) that the physical invariance of the speed of light implies the principle of relativity of simultaneity at the physical level;
b) that the principle of relativity of simultaneity at the physical level is in contradiction with itself, which can be demonstrated by the objection involving the shuttle and the missile;
c) that there has to be an absolute simultaneity at the physical level, since there is no third possibility besides the relativity of simultaneity at the physical level and absolute simultaneity at the physical level;
From this we can conclude:
d) that there has to be a present moment for the Universe;
e) that the speed of light cannot, in every situation, be physically invariant with respect to different inertial frames of reference, which we will most likely be able to measure at some point in the future.
Appendix 2
Critical commentary by Gilles Plante, Doctor of Philosophy specializing in critique:
Proposals a, b, and c are the premises of a conditional syllogism using the argument form modus tollens, the conclusions of which can be found at lines d and e.
Example of a conditional syllogism using such a modus tollens argument form:
If A, then B
However, not B.
Therefore: not A.
Conditional syllogism using the modus tollens argument form:
If “invariance of the speed of light at the physical level”, then “principle of relativity of simultaneity at the physical level”.
However, not “principle of relativity of simultaneity at the physical level, as demonstrated by the objection involving the shuttle and the missile”.
Therefore not “invariance of the speed of light at the physical level”.
“From this [you] can conclude”:
d) that there has to be a present moment for the Universe;
e) that the speed of light cannot, in every situation, be physically invariant with respect to different inertial frames of reference, which we will most likely be able to measure at some point in the future.
I have only taken into account the form based on what you are proposing, since I am unfamiliar with the content; it is not obvious to me.
Appendix 3
Mathematical reasoning:
Bibliography
A. Einstein, La théorie de la relativité restreinte et générale, Dunod, October 2012
A. Einstein Relativity the special and general theory, Translated by Robert W. Lawson, New York: Henry Holt, 1920
E. Klein, Le facteur temps ne sonne jamais deux fois, Flammarion, 2007
Marc LachièzeRey, Voyager dans le temps : la physique moderne et la temporalité, Science ouverte, Seuil, October 2013
Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, 4 March 1999
Carlo Rovelli, Et si le temps n'existait pas ?, Quai des sciences, Dunod, September 2014
Lee Smolin Rien ne va plus en physique ! L'échec de la théorie des cordes, Quai des sciences, Dunod, 18 April 2007
Lee Smolin La renaissance du Temps : Pour en finir avec la crise de la physique, quai des sciences, Dunod, May 2014
Lee Smolin La révolution inachevée d'Einstein : Audelà du quantique, Quai des Sciences, Dunod, September 2019
[1] According to special relativity, space and time are supposed to be intimately connected, but this paper shows that this tends not to be the case, which means that we cannot completely rely on special relativity in our conception of spacetime.
[2] Physically invariant: this also refers to light going from one point to another at a constant speed and not only the invariance of the speed of light used in equations. The invariance of c with respect to an inertial reference frame refers to light going from one point to another at a constant velocity c with respect to this same frame of reference, provided there is no significant gravitational field.
UP to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid referencebody (coordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: 

Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. 
When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the midpoint M of the length A —> B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the midpoint of the distance A —> B on the travelling train. Just when the flashes 1 of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram 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, i.e. they would meet just where he is situated. Now in reality (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 referencebody 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). Every referencebody (coordinate system) has its own particular time; unless we are told the referencebody to which the statement of time refers, there is no meaning in a statement of the time of an event. 

Relativity the special and general theory, TRANSLATED BY ROBERT W. LAWSON, NEW YORK: HENRY HOLT, 1920
[4] Even though both observers are at the “same position” when the rays are emitted, the rays are considered to be simultaneous for the observer standing on the platform, but not so for the observer inside the train car.
[5] We can immediately acknowledge that this metaphysical position is very odd, because if a body exists with respect to another body, it follows that that body must exist with respect to all other bodies in the physical world at that particular moment in time. The fact that a body exists is therefore a confirmation in itself that there is a present moment for the universe. However, it is not necessarily possible to know if two distant clocks are truly synchronized.
[6] This is actually the basis of the block universe theory of time, according to which all events occurring at any time and at any place coexist, and time no longer evolves. Therefore, according to this view, if, at the instant the two observers align, the ray is considered to have been emitted for the stationmaster, but not yet for the observer inside the train car, this would amount to saying that time – as far as this ray is concerned – is already written for the observer inside the train car.
[7] Absolute simultaneity: if we accept that the stationmaster perceives the two rays to have been emitted at the instant the two observers align, this should, in the context of absolute simultaneity, also be valid for the observer inside the train car.
[8] For this particular objection, there is no need to take into account the period during which the stationmaster is accelerating. We merely need to consider what each of the observers concludes before the stationmaster's period of acceleration, and what the stationmaster concludes after his period of acceleration.
[9] If, at the instant the two observers align, the ray of light is considered to exist for the observer on the platform, then either it does also exist for the observer inside the train, or it does not. Once we have eliminated the possibility of there being no simultaneity of events whatsoever, there is no third possibility. However, we cannot uphold the idea of there being no simultaneity, because, for a ray of light – or any body for that matter – to be moving with respect to an observer, that body has to be at a certain distance from the observer at any given time. When the body was located at a certain distance from the observer, the observer’s clock displayed a certain time; there is simultaneity between the two events  even if it is not always easy to be entirely certain when relying on the theoretical reconstruction of events.
[10] I am not questioning the fact that the speed of light may be locally invariant with respect to the Earth. I am only saying that the speed of light cannot be both invariant with respect to the train station and to the speeding train driving through the station. This statement is consequently not invalidated by the findings of the MichelsonMorley experiment. It has to be questioned whether the model proposed by special relativity truly reflects reality. It appears to represent the physical world in certain situations, but not in others. To discern this more clearly, we can look at what this model implies. We have shown in this paper that the principle of relativity at the physical level is a result of the postulate of the invariance of the speed of light. Since this principle leads to contradictions, it cannot truly reflect what exists in reality. This means that the speed of light cannot be physically invariant with respect to certain inertial reference frames.
[11] Hypersurface of simultaneity: what is supposed to be simultaneous for a given observer. The reference to an observer in this definition is important, as what is supposed to be simultaneous for one observer is not necessarily so for another observer.
[12] According to the principle of relativity of simultaneity at the physical level, a body does exist from one point of view, but it does not exist from another point of view (see also notes 5 & 14 on this subject). For relativity, when we are outside the causal sphere (spacelike interval), the temporal order between events is not fixed, as the past can occur in the future. But this no longer works once we start taking into account the existence of bodies (for example, in the objection involving the shuttle and the missile where we take into account the existence of the missile).
[13] If the invariance of the speed of light is used to synchronize clocks, it is because there are people who consider that clocks can actually be synchronized in such a way. Of course, the synchronization of clocks is not the direct physical consequence of the invariance of the speed of light, but it is an indirect operational consequence. It is primarily from a theoretical (logical and mathematical) point of view that the invariance of the speed of light implies the relativity of simultaneity at the physical level. This assertion is clearly of a general nature, even from an operational point of view, because it is supposed to be valid for any inertial reference frame, provided there is no significant gravitational field.
[14]On this subject, see for example Marc LachièzeRey's very good book  “Voyager dans le temps. La physique moderne et la temporalité” (Traveling Through Time  Modern Physics and Temporality) Paperback  October 2013. Relativity at the physical level is also present in general relativity. It is one of the basic principles that makes semiclosed time loops theoretically possible. Indeed, if there is truly an absolute simultaneity at the physical level, and thus a present instant for the Universe, it is impossible to travel through time. In this very instructive book, Marc LachièzeRey studies the consequences associated with the formalism of general relativity. He shows that one such consequence would be that semiclosed time loops would be theoretically possible – for example, the billiard ball that could theoretically hit its double self in its own past. It would be a little bit of a stretch to say there is no paradox here, because we would end up with two billiard balls instead of one. The theory tends to lose its footing and enter the realms of science fiction due to the implicit presence in the principle of relativity of simultaneity at the physical level at any given moment. Indeed, we see a doubling of reality originating from the principle of relativity of simultaneity at the physical level.
[15] First of all, I would like to point out that I had not heard about this paradox when I was writing my objection involving the shuttle and the missile. I believe the shuttle and missile paradox to actually be more relevant because, since it is the same observer who is accelerating, the contradiction concerns the same observer, and not two different observers, as is the case with the Andromeda paradox. In reality, the Andromeda paradox merely repeats what Einstein said in his train thought experiment, but involving greater distances. Nevertheless, it does highlight a very important point, namely what is inferred by two observers who do not have the same movement, but who are in the “same position”. This is how I went about highlighting the principle of relativity of simultaneity at the physical level.
Philippe de Bellescize, “Et il survolait les eaux vers une nouvelle vision du monde physique” (“And he flew over the waters towards a new vision of the physical world”), Chaptre.com, page 43.
Two people pass each other on the street ; and according to one of the two people, an Andromedean space fleet has already set off on its journey, while to the other, the decision as to whether or not the journey will actually take place has not yet been made. How can there still be some uncertainty as to the outcome of that decision? If to either person the decision has already been made, then surely there cannot be any uncertainty. The launching of the space fleet is an inevitability. In fact neither of the people can yet know of the launching of the space fleet. They can know only later, when telescopic observations from Earth reveal that the fleet is indeed on its way. Then they can hark back to that chance encounter, and come to the conclusion that at that time, according to one of them, the decision lay in the uncertain future, while to the other, it lay in the certain past. Was there then any uncertainty about that future? Or was the future of both people already "fixed"?
Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics
[16] There has to be a present moment for the universe, because the relativity of simultaneity at the physical level gives rise to a contradiction. On that basis, if the speed of light, ignoring the Sagnac effect, is invariant with respect to the Earth or near the Earth, the same must apply for any other planet with the same mass as the Earth – which amounts to saying that the speed of light constantly adapts to the spatial configuration.
[17] If the speed of light is locally invariant with respect to the train station – which is not necessarily the case in every scenario – it cannot be so with respect to the train that is moving with respect to the station. In this case, when we consider the thought experiment of Einstein’s train, the time discrepancy, i.e. the additional time required for the light to strike the rear of the train for the observer inside the train, is actually the result of a difference in the speed of light with respect to the train.
[18] The relativity of simultaneity, present from the beginnings of the theory of special relativity, led to the representation of time as a dimension on the same footing as the other three dimensions – which is a way of spatializing time. With the notion of a present moment for the universe, however, we clearly realize that the various moments in time do not coexist, while this may be the case for two different physical bodies. Etienne Klein has addressed this subject in a number of his lectures. A completely relational concept of spacetime would also have to take into account – but differently – the spatial aspect of the progression of time.
[19] We will refer to the works of Lee Smolin for example (see bibliography).
[20] The existence of bodies in the present implies a present moment for the universe. The same can be said with regard to the relationships between bodies in the present. If such an approach did not take into account presentmoment relationships, space would no longer exist.
[21] I deal with this subject in my book Et il survolait les eaux vers une nouvelle vision du monde physique (“And He Flew Over the Waters Towards a New Vision of the Physical World”) Chapitre.com, 2019.
[22]Larousse definition of "Inconsistency": Property of a logical theory in which the same formula is both demonstrable and refutable.
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