Paper for peer-reviewed scientific journal

  

Manuscript for submission to a peer-reviewed scientific journal

 

 

 

Paradox of the invariance

of the speed of light

 

 

Abstract

 

 The author of this paper explains why the speed of light cannot be invariant in all possible cases, given one of the logical implications of Einstein’s Light Speed invariance postulate. The results of this analysis could lead to a major change in our approach to the concept of space-time.

 

Introduction

 

        The author describes here how the invariance of the speed of light necessarily entails that the relativity of simultaneity at the physical level holds in the case of two different inertial reference frames. The relativity of simultaneity at the physical level is a metaphysical principle which is implicitly used in the theory of special relativity, a theory which gave rise to a particular conception of space-time.[1] This principle will be presented below, along with the reasons why it has to be differentiated from the simple relativity of the simultaneity of events perceived by two different observers. We will go on to see, however, that when an observer is accelerating, this principle can lead us to say one thing and its very opposite at the same time. Although this situation is nothing new, the formulae used so far to account for the Theory of Relativity do not express this contradiction. This is an important finding because it leads to the conclusion that the speed of light cannot be physically invariant[2] in all possible inertial situations, and it is therefore necessary to rethink how to present space-time.  It should also be possible to define those situations in which a difference in the speed of light can be actually measured, although this would first require further theoretical and technical investigations.

 

 

 Is there some inaccurate reasoning at the root of special relativity?

 

        It was once assumed that the speed of light was invariant with respect to the aether, and that it could therefore not be invariant with respect to the Earth. Although the aim of the Michelson-Morley experiment was to confirm this assumption, the results obtained in the latter experiment seem to show on the contrary that the speed of light is invariant with respect to the Earth. It was this finding that probably led Albert Einstein to adopt the following reasoning: if the speed of light is invariant with respect to the Earth, then, since the Earth is a moving body, the speed of light must be invariant with respect to any body in a state of inertia. If we apply this reasoning to Einstein’s train thought experiment, this means that if the speed of light is invariant with respect to the station, then it must also be invariant with respect to the train, which is in constant motion with respect to the station. However, as I propose to establish here, this reasoning is not entirely accurate. As we will see, it is probably far more reasonable to expect the speed of light to constantly adapt to the current spatial configuration – although no proof of this hypothesis has been definitely established so far.

 

 

Invariance of the speed of light entails the principle of the relativity of simultaneity at the physical level

 

        It is proposed in this section to comment on Einstein’s train thought experiment[3], which was originally used to illustrate the concept of space-time according to the theory of special relativity, whereas Einstein’s elevator thought experiment leads rather to the principle of equivalence he used to illustrate his theory of general relativity. Einstein’s train thought experiment  showed what effect the hypothetical invariance of the speed of light would have from the point of view of all inertial observers.  The train thought experiment focuses on what two observers perceive while a train is moving through a station without stopping: the one observer (a passenger) is sitting in the middle of the train, which is travelling at a constant speed through the station, and the second observer (the stationmaster) is standing motionless on the platform. At the instant when the two observers are crossing each other, two light sources are visible at equal distances from the stationmaster: one ahead of the train and the other, behind the train. In this situation, the two beams of light reach the stationmaster at the same time. Since they are emitted at equal distances from the stationmaster, he will conclude that both beams were emitted at the same instant because he assumes the speed of light to be invariant with respect to him. It is quite possible, however, that these two events did not actually occur simultaneously, but if the speed of light is assumed to be invariant with respect to the stationmaster and the distances between the stationmaster and the two beams of light are known, the two events will be taken by this observer to be simultaneous. In this context, we cannot therefore say that there is no simultaneity whatsoever, since the simultaneity of the two events perceived by the stationmaster is a logical conclusion based on the given premises.

 

         Generally speaking, two distant events that appear to be simultaneous from the point of view of a stationary observer will not be perceived as being simultaneous by a second observer who is constantly moving past the first observer unless both of them are in exactly the same place when observing or measuring these events. This could be called the relativity of the simultaneity of events perceived by two different observers. As we will see below, it is necessary to differentiate between this relativity of the simultaneity of events perceived by different observers at the time of measuring the events (that is to say during a measurement procedure) from the relativity of the simultaneity of the emission of beams of light[4], as well as from the relativity of  simultaneity at the physical level. One particular feature of this analysis of Einstein’s train thought experiment is that it shows that this supposed relativity of the simultaneity of light beam emission necessarily entails the relativity of simultaneity at the physical level. Let us now consider two different scenarios. First scenario: the two observers were both in the same place when the two beams reached the stationmaster (as in the version of the train thought experiment presented by Yann le Roux); second scenario: the two observers were both in the same place when the two beams were emitted, which corresponds to what was perceived by the stationmaster, assuming the speed of light to be invariant with respect to him (as in the version proposed by Albert Einstein).

 

        In the first scenario, if we assume that the two rays were emitted simultaneously from the point of view of both observers (i.e. absolute simultaneity) and that the speed of light was invariant with respect to the stationmaster, then it follows that the speed of light cannot be invariant with respect to the observer in the train because in this scenario, at the instant when the two rays were emitted, the distances from the two light sources were not perceived as being the same by the passenger in the train travelling through the station. However, in the first scenario, the two rays both reach the passenger at the same time. Therefore, in this scenario, for the speed of light to also be invariant with respect to the passenger in the train, the two rays cannot have been taken by the latter  to have been emitted simultaneously, although this was what the stationmaster perceived. 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 takes the two rays of light to have been emitted simultaneously, assuming the speed of light to be invariant with respect to him. However, as we saw in the previous paragraph, in order for the observer in the train to be able to conclude that the speed of light is invariant with respect to himself, then from his point of view, the two rays cannot have been emitted simultaneously: what has been said about the first scenario should presumably also be applicable to the second.  The two scenarios are indeed fairly similar, except that the timing of the train is different. The passenger in the train will assume the beam of light occurring in front of the train to have been emitted before the two observers were aligned, and the second ray of light to have been emitted from behind the train after the two observers were aligned. We can therefore now define the metaphysical principle underlying the invariance of c between the point of origin of the beam of light and the point where the beam is perceived by the two inertial observers.

 

        In the second scenario, at the instant when the two observers pass each other, the ray of light originating from the rear of the train is taken by the stationmaster, but not by the passenger in the train, to have been already emitted – i.e. to actually still exist at its point of origin. The passenger in the train will take the ray to exist only a short time later when he is a little further away from the stationmaster.  For a body to be moving with respect to an observer, that body must be taken by that observer to exist. I propose to use  the expression “to exist for” from now on because according to the latter principle, even though the two observers are in the same physical position, the beam of light originating from the rear of the train exists for the observer on the platform, but not for the one in the train.[5] This is what I have referred to here as the principle of the relativity of simultaneity at the physical level, which is 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 was strongly implied. One must remember that the latter events occurred in a space-like interval, and that the beam of light was not actually observed in the second scenario by either of the two observers at that particular instant. Here we are simply presenting theoretically what is implicitly stated in this particular case. The time lag, i.e. the additional time it takes for the ray to exist for the observer sitting in the train, 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 longer the time lag will be.[6] 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 check the simultaneity of two distant events with complete certainty. This having been said, it is possible to explore this subject theoretically. That is what it is now proposed to do.

 

 

The principle of the relativity of simultaneity at the physical level is self-contradictory

 

Still in the framework of  the scenario presented 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 in the train may have travelled 300 feet after passing the observer on the platform. If the stationmaster were to get into the moving train to join the passenger before the train had covered the whole distance of 300 feet, then the stationmaster, by whom the beam of light is assumed to have already existed in the past, would be joining the passenger in 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 in the train, we therefore now have a beam of light – which could be replaced by any physical body – which existed in the stationmaster’s past before he accelerated upon getting onto the moving train, but will not yet have existed for him after he accelerated. 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 adopted the same viewpoint as the observer sitting in the train, he will end up with two contradictory calculations: one in which the body in question has already travelled a certain distance, and another in which the body in question has not yet started to move.[8]

 

 

Objection involving a shuttle and a missile

 

The stationmaster could be replaced here by a shuttle accelerating at high speed in outer space, and the beam of light could be replaced by a missile that is assembled only just before being launched, and fired at the shuttle. Based on the above analysis of Einstein’s train thought experiment, one objection to the invariance of the speed of light can be presented in terms of this shuttle and missile scenario, whereby the space shuttle would end up with two contradictory assessments of the missile’s position. According to special relativity, the temporal order between events can change when an object is accelerating, providing these events are separated by a space-like interval. However, this no longer holds if we take the existence of bodies into account. The only contribution I will make here is to show that the principle of the relativity of simultaneity at the physical level, which results from the invariance of c, requires us to consider the existence of the bodies usually featuring in space-time diagrams. This is why the principle of relativity at the physical level resulting from the invariance of c is so important: it helps us to understand that the speed of light cannot actually be invariant in every possible situation. The framework it provides can only be either that of absolute simultaneity or the relativity of simultaneity at the physical level; it does not allow for any third possibility.[9] Once it has been established that the relativity of simultaneity at the physical level results in a contradiction, it follows that there necessarily exists an absolute simultaneity at the physical level.  However, as we saw earlier in the context of absolute simultaneity, the speed of light cannot be invariant in all possible situations.[10] I will explain the logic underlying my reasoning in greater detail in Appendices 1 and 2, as well as providing some relevant equations in Appendix 3.

 

 

Response to a major criticism

 

Some people may feel it is fallacious 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, which has often been presented in space-time diagrams, gives a useful picture of the relativity of simultaneity. If we take the framework in which Einstein’s train thought experiment was set and compare the two observers’ perception of events, we can see that it is a question of the relativity of simultaneity as regards the emission of the beams of light. This relativity of simultaneity was already implied by the postulate of the invariance of the speed of light.

 

 The assumed chronology of events separated by a light-like interval would in turn have implications as far as an assumed chronology of events separated by a space-like interval is concerned.  If the distance from the light source and the speed of the beam of light are assumed to be known at the instant when the beam arrives (i.e. after the light-like time lag has elapsed between the moment when the beam was emitted and that when it was received), it will in fact be possible to say at what moment it was emitted in the past from the point of view of  the observer in question,  (i.e. to specify the time  elapsing between the space-like event when the beam of light was emitted and the occurrence of the event when it was received by the waiting observer). This means that by using a supposed chronology of events separated by a light-like interval, we can, by a process of theoretical reasoning, obtain a supposed chronology of events separated by a space-like interval. This is why Einstein in his train thought experiment stated that there existed a relativity of simultaneity as regards the emission of the two beams of light.

 

By stating that a body is moving with respect to an observer, Einstein implicitly stated that that body exists for that observer even if the latter has not yet consciously perceived its existence. If the initial statement is true, then that body must seem to exist according to the observer. The relativity of the simultaneity of events involving the emission of beams of light, which is often presented in space-time diagrams, therefore involves the assumption that the relativity of simultaneity at the physical level holds true. This means that we must take into account the existence of the bodies featuring in space-time diagrams. If this reasoning is fallacious, the fallacy is actually introduced  by the postulate that the speed of light is invariant: this postulate not only implicitly states that a body exists with respect to a given observer at given times because that body is assumed to be moving with respect to that observer; but it also suggests that this body is not perceived to exist by the same observer because it cannot yet be detected by that observer at that specific moment (due to the existence of a space-like interval between events).  The verb "to exist" is being used here in two different ways with two different meanings, and it is necessary to distinguish between the two to be able to grasp the fact that the invariance of c leads to the principle of the relativity of simultaneity at the physical level.

 

The relativity of simultaneity, which is presented in space-time diagrams and is based on the postulate of the invariance of the speed of light, means in some cases, when the two observers are opposite each other, that this body exists from the  point of view of one observer but not from that of the other. This is what I call the principle of the relativity of simultaneity at the physical level. Based on this conclusion, which is naturally not the only one possible, the principle of causality in the Theory of Relativity can be challenged from the perspective of realist philosophy. In the Theory of Relativity, there is a possible causal relationship between events when either a time-like or a light-like interval is involved. But ultimately, because of the postulate of the invariance of the speed of light, this principle of causality necessarily involves the principle of the relativity of simultaneity at the physical level – which does not fully take the existence of bodies into account.[12] In terms of natural philosophy, the causal principle can be understood as what accounts for the existence of bodies, their structure and their behavior.  For a relationship to exist 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 reasons I have explained above, with the principle of causality in Relativity.

 

Some people have noted that the term "simultaneity" in the expression "relativity of simultaneity", would give rise, if we take simultaneity to be something physical, to the problem that I am highlighting here. It has therefore been suggested that it might be replaced by the expression “same time coordinate”. It has been rightly pointed out that with any pair of spatially separate events, it is possible to establish a coordinate system in which the two events have the same temporal coordinates. However, we cannot totally overlook 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 speeds of the light signals are known. These clocks synchronized via light signals are assumed to run simultaneously at the same rate. We thus obtain a hypothetical simultaneity which has a physical sense and is based on the postulate of the invariance of the speed of light. Bearing in mind what has been said above, this means that the relativity of simultaneity, in its physical sense, does indeed result from 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 other authors or not. When we bring together the block universe theory of time and special relativity, or when we explore the possibility of  semi-closed time loops[14] in general relativity, things are often said indirectly. We need only consider the argument put forward by Roger Penrose, which he called the Andromeda paradox[15]. But the fact remains that it may be difficult, at least for some people, to come to any definite conclusions on this question. By showing that the relativity of simultaneity in special relativity entails the principle of the relativity of simultaneity at the physical level, I have at least shown where the origin of the problem lies. When we examine the subject closely enough, we see that something is not quite right. This is an interesting situation, as it leads us to question the picture of space-time that physicists have adhered to for over a hundred years. It is therefore a subject of crucial importance to the field of physics as a whole. This question will have to be solved first from the theoretical point of view, however exasperating this situation may be for those who favor a purely operational approach.

 

 

Towards a new concept of space-time

 

 Based on all that has been said 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 current spatial configuration[16], which we will predictably 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 near the Earth, but it would be variable relative to any moving body located near the Earth. In addition, the Earth’s rotation around its own axis can also complicate matters. The Sagnac effect has to be interpreted differently depending on whether we are dealing with absolute simultaneity or the relativity of simultaneity at the physical level. It might therefore be worth taking these measurements in outer space, away from any large masses.

 

 

Conclusion

 

        The notion of time in special relativity and the ensuing paradoxes  result from the postulate that the speed of light is invariant. By showing that the invariance of the speed of light entails the principle of the relativity of simultaneity at the physical level, I have shown where the origin of the problem lies. This problem has not been sufficiently thoroughly investigated by any previous author, to my knowledge. Since the principle of the relativity of simultaneity at the physical level gives rise to contradictions and since there is no third possibility, it follows that absolute simultaneity must exist at the physical level. This means that there is a present instant in the Universe – which is the thesis underlying Presentism, a theory of time where only the present instant 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 one. From the operational point of view, we cannot be absolutely certain that two distant events are simultaneous, but this does not mean that simultaneity does not exist in the physical world. If we want to succeed in developing a general theory of the Universe, we cannot totally dismiss the metaphysical aspects. From the metaphysical point of view, the principle of the relativity of simultaneity at the physical level, which is implied by the invariance of c, does not hold because something real cannot exist and not exist at the same time. The relativity of simultaneity is nonetheless still a useful tool for making some calculations, because it can be used, however approximately, to replace a difference in the speed of light.[17]

 

        It was special relativity that moved us away from the concept of absolute time and space, which, although it constituted an important conceptual advance, probably did not occur for the right reasons, as we have just seen.[18] Some physicists are now proposing to adopt a completely relational approach to time and space.[19] The theory of relativity is not fully compatible with an approach of this kind for at least two reasons. First, a completely relational view of space would include  the existence of a present instant in the Universe.  With this approach, bodies actually have to be set in relation to each other in order for space-time to exist.[20] Secondly, a completely relational approach to space would necessarily include a relational approach to motion, which would change our approach to the terms inertia and impulse and would certainly require further investigations.[21] The idea of a present instant in the Universe does not necessarily mean a return to Newtonian time or absolute time. Two similar clocks placed in different spatial contexts may very well work simultaneously at different rates, for example on two different floors in the same building.

 

        The postulate that the speed of light is invariant does in many ways have perfectly logical consequences, which can be found in the conventional equations relating to special relativity and space-time diagrams (see the Mathematical Appendix). However, if we assume the relativity of simultaneity to apply at the physical level, the latter postulate will lead to contradictory conclusions. Which is to say that this postulate, if one were tacitly to accept the concepts to which it leads, would generate some inconsistency in the theory[22]: the principle of the relativity of simultaneity at the physical level, to which the invariance of the speed of light leads implicitly, has metaphysical implications. And if we take the metaphysical aspects into account (i.e. the existence of a body such as those featuring in space-time diagrams), we end up with two contradictory results (as to the position of the missile in my shuttle and missile objection, for example). This illustrates the fact that – even mathematically speaking – this relativity of simultaneity is impossible. The speed of light cannot be physically invariant in every possible situation, which calls into question the second postulate of special relativity. This conclusion may understandably lead the field of physics to make a significant change of paradigm in its approach to the concept of space-time. The ideas presented in this paper could be the beginning of a radical change in the field of physics.

 


Philippe de Bellescize

 

English translation by EasyTranslate, edited by Dr. Jessica Blanc.

 

 

Appendix 1

 

Summary of my reasoning:

I am attempting here to make the following important points:

a) that the physical invariance of the speed of light implies belief in the principle of relativity of simultaneity at the physical level;

b) that the principle of relativity of simultaneity at the physical level is  self-contradictory, which shown by the shuttle and missile objection presented here;

c) that absolute simultaneity at the physical level must exist, since there is no third possibility in addition to 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 instant in the Universe;

e) that the speed of light cannot, in every possible situation, be physically invariant with respect to different inertial frames of reference, which we will predictably be able to measure at some point in the future.

 

Appendix 2 

 

According to  Dr. Gilles Plante, a Doctor of Philosophy specialized in the art of critical commentary:

 

Proposals A, B and C are the premises of a conditional syllogism based on the modus tollens form of argument, the conclusions of which can be found in lines D and E.

An example of a conditional syllogism based on the modus tollens form of argument:

If A is true, then B is true

However, B is not true.

Therefore A is not true.

A conditional syllogism using the modus tollens form of argument:

If the "invariance of the speed of light at the physical level " holds true, then the "principle of relativity of simultaneity at the physical level" also holds true.

However, since the  "principle of relativity of simultaneity at the physical level", has been disproved by the shuttle and missile objection,

 the "invariance of the speed of light at the physical level" does not hold true.

 It can therefore be concluded:

D) that there has to be a present instant in the Universe;

E) that the speed of light cannot in every possible situation be physically invariant in different inertial frames of reference, which we will predictably be able to measure at some point in the future.

I have only taken into account the form of your proposals, since I am unfamiliar with the content.

 

 Appendix 3 

 

Mathematical reasoning:

 

Bibliography

 

Philippe de Bellescize, Et si Einstein s'était trompé sur un point capital dans son analyse aboutissant à la relativité restreinte ? Vers une approche relationnelle de l'espace-temps Chapitre.com, 2017

Philippe de Bellescize, Et il survolait les eaux vers une nouvelle vision du monde physique? Chapitre.com, 2019

Albert Einstein, La théorie de la relativité restreinte et générale, Dunod, October 2012

Albert Einstein Relativity   the special and general theory, Translated by Robert W. Lawson, New York: Henry Holt, 1920

Etienne Klein, Le facteur temps ne sonne jamais deux fois, Flammarion, 2007

Marc Lachièze-Rey, 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 : Au-delà du quantique, Quai des Sciences,  Dunod,  September 2019


[1] In the theory of special relativity, space and time are assumed to be closely related, but as we will see, the theory of restricted relativity involves a misunderstanding about this relationship, which means that we cannot completely rely on special relativity in our picture of space-time.

[2] Physically invariant: this also refers to light travelling from one point to another at a constant speed, and not only to the invariance of the speed of light expressed in equations. The invariance of c defined with respect to an inertial reference frame refers to light travelling from one point to another at a constant velocity c in this same frame of reference, as long as no significant gravitational field is present.

[3]

 

Experience pensee train

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 reference-body (co-ordinate 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 mid-point 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 mid-point 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 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 (in line with the relativity of simultaneity). Every reference-body (every co-ordinate system) has its own particular time; unless we are told to which reference-body the statement of time refers, there is no meaning in a statement of the time of an event.

From "Relativity, the special and general theory", translated by Robert W. Lawson,  Henry Holt, New York 1920

[4]  Although both observers are in the same place when the beams are emitted, they are taken to be simultaneous by the stationmaster, but not by the passenger in the train.

[5]  This metaphysical position is admittedly very odd because if a body exists with respect to another body, it follows that the latter body must exist with respect to all other bodies in the physical world at that particular instant in time. The fact that a body exists is therefore a confirmation in itself that there is a present instant in the universe. However, it is not necessarily possible to determine whether 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 theory, if, at the instant when the two observers are aligned, the ray is taken by the stationmaster, but not yet by the passenger in the train, to have been emitted. This would amount to saying that time – as far as this ray is concerned – is already written for the observer in the train.

[7] Absolute simultaneity: if we accept the idea that the stationmaster takes the two rays to have been emitted at the instant when the two observers were aligned, this should also presumably be so in the context of absolute simultaneity,  in the case of the passenger in the train.

[8] In the case of this particular objection, there is no need to take into account the period of time during which the stationmaster is accelerating. We only have 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 when the two observers are aligned, the ray of light is taken to exist by the observer on the platform, then it either also exists or not for the observer in the train. Once the possibility of there being no simultaneity of events whatsoever has been ruled out, there is no third possibility. However, we cannot uphold the idea that there is 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 gave a certain time; there is simultaneity between the two events - even if it is not always easy to be entirely sure when relying on a purely theoretical account 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 both the train station and the train speeding through the station. This statement is therefore not invalidated by the findings of the Michelson-Morley experiment. The question arising here is whether the model presented by special relativity truly reflects reality. It seems to account accurately for the physical world in some situations, but not in others. To understand this point more clearly, we can look at what this model implies. As mentioned above, the principle of relativity at the physical level results from the postulate that the speed of light is invariant. Since this principle leads to contradictory conclusions, it cannot truly reflect what exists in reality. This means that the speed of light cannot be physically invariant with respect to some inertial reference frames.

[11] Hypersurface of simultaneity: what is assumed to be simultaneous by a given observer. The reference to an observer in this definition is important, as what is taken to be simultaneous by one observer is not necessarily taken to be simultaneous by another one.

[12] According to the principle of the relativity of simultaneity at the physical level, a body can exist from one point of view, but not from another (see also notes 5 & 14 on this subject). In the Theory of Relativity, when we are outside the causal sphere (in a space-like interval), the temporal order between events is not fixed, as the past can become the future. But this no longer applies when we start taking into account the existence of bodies (for example, in the objection involving the shuttle and the missile, which focuses on the existence of the missile).

[13] From the logical and mathematical point of view, the invariance of the speed of light entails the principle of the relativity of simultaneity at the physical level. At this level, the shuttle and missile objection presented above shows that this principle is self-contradictory. In the first place, the present theoretical considerations have shown that the speed of light cannot be physically invariant in all possible cases; and in the second place, it will predictably become possible one day to confirm this finding experimentally. The reason why the invariance of light is used to synchronize clocks is that this seems to be an appropriate method. Actually the synchronization of clocks does not depend directly and physically on the invariance of the speed of light, but it is rather an indirect operational effect. It is primarily at the theoretical (logical and mathematical) level that the invariance of the speed of light entails the relativity of simultaneity at the physical level. This statement is clearly of a general nature, even from the operational point of view, because it is intended to apply to any inertial reference frame, providing no significant gravitational field is present.

[14]On this subject, see for example Marc Lachièze-Rey's excellent 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 semi-closed time loops theoretically possible If there truly exists an absolute simultaneity at the physical level, and thus a present instant in the Universe, it will be impossible to travel through time. In this very instructive book, Marc Lachièze-Rey studies the consequences of the way in which general relativity has been formulated. He shows that one such consequence would be that semi-closed time loops would be theoretically possible: for example, a billiard ball could theoretically hit its own double self in its own past. It would be something 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 realm of science fiction due to the implicit presence in the principle of the relativity of simultaneity at the physical level at any given moment. Here we have a doubling of reality originating from the principle of the relativity of simultaneity at the physical level.

[15] First of all, I would like to point out that I had never heard about the Andromeda paradox when I was writing my shuttle and missile objection. I believe the shuttle and missile paradox is actually the more relevant of the two since it is the same observer who is accelerating, and the contradiction involves a single observer and not two different observers, as in the case of the Andromeda paradox. In fact, the Andromeda paradox merely repeats what Einstein said in his train thought experiment, except that greater distances are involved.  It does highlight a very important point, however, namely what is inferred by two observers who are not subjected to the same movement, but are both in the same place. This is how I went about highlighting the principle of the 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), Chapitre.com, page 43.

Two people pass each other on the street; and according to one of them, an Andromedan space fleet has already set off on its journey, while according 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 either person believes that 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 two people can know yet whether the launching of the space fleet has taken place. They can know only later, when telescopic observations from Earth have established that the fleet is actually 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 indefinite future, while according to the other, it lay in the definite 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 instant in the Universe because the relativity of simultaneity at the physical level gives rise to a contradiction. On this basis, if the speed of light, leaving aside the Sagnac effect, is invariant with respect to the Earth near the Earth, the same must apply to any other planet with the same mass as the Earth – which  amounts to saying that the speed of light constantly adapts to the current 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 Einstein’s train thought experiment, the time lag, i.e. the additional time required for the light to strike the rear of the train from the point of view of the observer in the train, replaces the difference in the speed of light with respect to the train.

[18] The relativity of simultaneity, which has been upheld since the beginnings of the theory of special relativity, has led to time being regarded 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 instant in the Universe, however, it is easy to understand that the various instants in time do not coexist, while this may be so in the case of two different physical bodies. Etienne Klein has addressed this subject in several of his lectures. A completely relational concept of space-time would also have to take into account – but differently – the spatial aspect of the progression of time.

[19] See for example the works of Lee Smolin  (see the present Bibliography).

[20] The existence of bodies suggests the existence of a present instant in the universe. The same can be said about the relationships between bodies in the present. If this approach did not take present-instant relationships into account, 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] The Larousse Dictionary defines the term "Inconsistance" as follows: "Propriété d’une théorie logique dans laquelle une même formule est à la fois démontrable et réfutable". (A property of a logical theory whereby the same formula is both demonstrable and refutable).