Abstract

I - New definition of simultaneity

According to Einstein, two instantaneous flashes of lightning striking at the two ends of a segment of straight line, are considered simultaneous, if the light issuing from the flashes reaches the middle of the segment at the same instant. We demonstrate that this definition is only true if the sources of light are at rest with respect to the segment. In all other cases, the definition is incorrect.

A new definition is given, according to which, the absolute character of simultaneity is rediscovered.

II - An experiment designed to verify the possible existence of a privileged reference frame, and to test the validity of the relativity principle

III - Theories of Lorentz and Poincaré and Group structure

The application of Group theory without precautions, could make people believe that the existence of a preferred aether frame is compatible with the Relativity principle. This attitude has misled several physicists. Indeed, the use of this mathematical tool without discernment, ignores the assumption made by Poincaré who for his demonstration considered that the speeds should obey the relativist composition of velocities law (see ref 2) . Only with this assumption, the space-time transformations should assume a group structure. But this relativist law is plagued by measurement distortions, it is not the true law after the distotions are corrected, in which case the space-time transformations do not constitute a group. Yet the transformations derived by Poincaré (i.e, the Lorentz-Poincaré transformations) do not highlight this fact.

 

I - New definition of simultaneity

According to Albert Einstein, simultaneity is defined as follows :
two instantaneous flashes of lightning, striking at the ends of a segment of straight line, AB, are considered simultaneous, if the photons issuing from A and from B and traveling along AB (in opposite directions), reach the middle of AB at the same instant.

Let us now consider a railway platform and a train travelling along this platform in such a way that the two flashes of lightning strike at the very instant when the middle of the train is in front of point O (middle of AB) (Einstein) (see fig 1).

Now suppose that the light signals reach point O at the same instant. Knowing that the train moves away from point A and moves towards point B, the photons issuing from B will reach point O' (middle of the train), before those issuing from A. An observer placed at O' will conclude that the two flashes are not simultaneous.

So, for Einstein, two events simultaneous for one observer, at rest in a given reference frame, are not simultaneous for another observer moving with respect to this frame.

Criticism of this approach

The flaws in Einstein's reasoning, come from the fact that he has made a confusion between the instantaneous flashes and the light issuing from them and that he has not taken care to define the initial conditions. Nevertheless, contrary to what this approach indicates, it is necessary to distinguish two different cases :

1st case

The sources of light are firmly fixed to the frame of the platform. Then, if we suppose that the speed of light is constant in the frame of the platform, the flashes of lightning will appear simultaneous to the observer standing on the platform, and not to the observer in the train.

2nd case

The sources of light are firmly fixed to the frame of the train. If we suppose that the speed of light is constant in the train's frame, , the two flashes will appear simultaneous to the observer in the train, and not to the observer standing on the platform.

In fact, Einstein's definition of simultaneity implies that the speed of light is isotropic. On the other hand,, it is only valid for the observer at rest in the reference frame on which the two sources of light are fixed. For the other, it will be necessary to define simultaneity differently. (We should add that according to aether theory the speed of light is isotrpic exclusively in the aether frame, in which case the definition is really valid only in this frame).

We will suppose that the two sources are fixed to the platform. (see fig. 1)

Figure 1

Let us define AB = L and AO = l

If we suppose that in the frame of the platform the speed of light is constant, we can say with good reason that, for an observer standing on the platform, the flashes are simultaneous if the photons issuing from A and B arrive at point O at the same instant.

Now, as regards the train, this definition of simultaneity is no longer valid. Indeed, when the light issuing from A arrives at O, the train has covered the distance:

                                                   

When the photons cover this distance in turn, the train will have covered the distance:

                                                   
and so on

Therefore, when the photons issuing from A reach point O' (middle of the train), they have covered the distance:

                                                   

The time needed to cover this distance will be:

                                                   

In order to reach O', the photons issuing from point B must cover a distance x such that :

(see figure 2)

Figure 2

Therefore:

                                                   

And the time they need for this will be:

                                                   

We notice that the photons issuing from A and B reach O' at different instants.

New definition of simultaneity

Taking account of the previous considerations, we can say that two instantaneous flashes issuing from the two sources, firmly fixed at two points A and B of the platform, will be simultaneous if the photons issuing from A and B and traveling along the line AB in opposite directions, reach O' (middle of the train), at two instants t_Aand t_Bsuch that:

                                                   

Taking notice of this correction the absolute simultaneity is found again.

Note that according to aether theory the speed of light is equal to C only in the aether frame. Therefore this formula is quite exact exclusively in this case. In the other frames, the speed of light is anisotropic.

Let us designate respectively as C_AB and C_BA the speed of light in the two reverse directions:

We can conclude that two instantaneous events occurring at A and B can be considered simultaneous if the light issuing from them reaches the middle of the train at two instants t _AO’and t _BO, such that
                                    

We should also take account of length contraction and of the anisotropy of the speed of light in the Earth frame assumed by Lorentz. But, whichever corrections needed, they should slightly change the formulas but not modify the basic criticisms we make. (The role of the speed of light anisotropy will be studied in a subsequent paper^*).

II - An experiment designed to verify the existence of a privileged reference frame, and to test the validity of the relativity principle

Let us consider two rockets moving with a rectilinear uniform motion in opposite directions, along a line AB (see fig. 3):

Figure 3

At the initial instant, the two rockets meet at O, the middle of AB. At this very instant, two observers inside the rockets, set their clocks to zero. The rockets move towards two points A and B, symmetrically, at speed v. When they reach A and B, the observers note the reading indicated by their clocks. These will be compared subsequently. (Notice that the argument that follows implies that the speed v of the rockets is determined with sufficient accuracy. A task difficult but not theoretically impossible.)

If the clocks indicate the same reading whatever "inertial" frame (O, x, y, z) we have a good argument in favour of the relativity principle.

If the clocks indicate a different reading, then the existence of a privileged reference frame becomes very likely. Confirmation of this will be obtained if the ratio is variable. More precisely, according to Lorentz, the hypothesis of a fundamental reference frame implies that, if rocket 1 is at rest with respect to this frame, then:

                                                   
conversely, if rocket 2 is fixed to the privileged frame, we will have:

                                                   

All the intermediary situations exist in which the privileged frame is neither at rest with respect to rocket 1, nor with respect to rocket 2.

If frame (O, x, y, z) is attached to the fundamental reference frame we will have:

                                                   

In all other situations, one of the clocks will be slow with respect to the other. In such cases, both ratios and will be different from (which proves that, if we take for granted the postulates of Lorentz, provided that the speeds of the rockets are exactly measured, one can determine by an experiment internal to a given "inertial" frame whether this frame is at rest or in motion relative to the aether frame in contadiction with the relativity principle.

 

Important note: One can demonstrate that if we use the Einstein-Poincaré synchronization procedure with light signals in order to measure the speed of the rockets, the measurement is affected by a systematic error. In this case, the readings of the clocks when the rockets reach points A and B are always found to be identical. But the relativity principle cannot be viewed as a fundamental principle if it depends on an erroneous measurement of the speeds (see Ref 4).

III - Theories of Lorentz and Poincaré and Group structure

In a previous publication (Ref 3) we have demonstrated that, assuming the Lorentz postulates (including the application of the Galilean composition of velocities law to real speeds) the space-time transformations take the form of the Lorentz-Poincaré transformations^** in two cases:

1/ When the basic reference frame (at rest with respect to the observer) is the fundamental frame (aether frame) (fig. 4, case 1).

2/ When the basic reference frame (the frame of the observer) is any" inertial" frame, while the frame of the moving body is the fundamental aether frame (fig. 4, case 2).

(We have also shown that the relativistic composition of velocities law applies to apparent speeds because of the systematic measurement distortions see Ref 2).

Figure 4

 

Combining these two results (cases 1 and 2) it is clear, that if group theory applied, the space-time transformations connecting two "inertial" frames receding in opposite directions with respect to the fundamental frame should take the form of the Lorentz-Poincaré transformations (see fig. 5).

But as can be seen in Ref 1, in order to derive the transformations mentioned above (Cases 1 and 2) we started from the Galilean transformations and submitted them to the systemetic measurement distortions due to length contraction, clock retardation and the synchronization procedure with light signals. Therefore the galilean addition of velocities law was assumed in the demonstration, and for the sake of logic the transformation connecting the frames 1 and 2 in figure 5 must be derived using the same addition of velocities law.

Yet in his demonstration to prove that the space-time transformations should constitute a group, Poincaré used the relativistic composition of velocities law (see ref 2). This law is a fictitious law that applies after the the space-time transformations have been submitted to the measurement distortions; it should not be used a priori for their derivation.

 

 

Figure 5

According to group theory, given that , and the transformatins connecting two "inertial" frames receding from one another in opposite directions should assume the mathematical form of Lorentz transformations such that:

 

 

We shall show that , using the Galilean composition of velocities law, the space-time transformations do not constitute a group.

(For an exhaustive demonstration see Ref 5 and 6)

Consider, to this end, two reference frames S₁and S₂ moving symmetrically at constant speed v in opposite directions with respect to the fundamental inertial frame S₀, along the x₀, x₁, x₂axis. At the initial instant the three frames overlap, at this instant their clocks are set to zero.

Let us now derive the space-time transformations between the frames 1 and 2 which move in opposie directions.

At the initial instant, the three reference frames are coincident. At this very instant, a light ray starts from the common origin, and runs along a rod AB of length L at rest in reference frame S₂ and aligned along the x axis of this frame (see fig. 6).

Figure 6

 

We assume that, in conformity with Lorentz's theory, the rod contracts along the direction of motion and assumes the length:

                                                    
the real speed of S₂, with respect to S₁, is 2v, and the speed of the signal, with respect to S₁, is C + v from A to B and C - v after reflection in a mirror firmly fixed to this frame which is in front of point B at the instant of the reflection. (Indeed according to the Lorentz assumptions, the light ray assumes the speed C exclusively with respect to S₀)

When the signal covers the distance l with respect to S₁, the rod covers the distance:

                                                   

When the signal has covered this distance in turn, the rod has moved away a further distance equal to:

                                                   
and so on.

So that in order to reach the end B of the rod, the signal must cover the distance:

                

However, the apparent distance found by observer S₁ will be different, since he carries out his measurements with a contracted standard. He will find:

                                                   

- Transit time of the signal.

If the time could be measured exactly, the result would be:

                                                   

Nevertheless, we must make allowance of the imperfection of our measurements. If the measurements were made correctly, the theory would reduce to Galilean's theory as we have previously shown.

In fact, the time measured experimentally is nothing else than an average round trip time.

When the signal reaches point B, the photons are reflected in a mirror firmly fixed to frame S₁.

The transit time of the signal from B to O₁, after reflection, is:

                                                   

The time that would be measured in absence of clock retardation, using the Einstein-Poincaré synchronization procedure, is in fact equal to half the round trip time. That is:

                                                   

This expression does not take account of the slowing down of the clocks of reference frame S₁. The apparent time measured will be equal to this expression multiplied by that is:

                                                   

                                                   

Therefore, with the assumptions we have made, the transformations obtained do not assume the form of Lorentz-Poincaré transformations, contrary to what seemed to indicate a hasty application of Group theory.

It is interesting and paradoxical to notice the ratio:

                                                   

So the apparent (fictitious) speed of light in frame S₁ is found equal to C in conformity with the experiment.

 

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