Abstract

I - Is the existence of a fundamental reference frame compatible with a group structure?

Poincaré has brought an important contribution to the conventional relativity theory in demonstrating that the relativity principle implies, necessarily, a group structure for the space-time transformations. The question asked in this paper is the following: is the existence of a privileged reference frame compatible with the exact applicability of the relativity principle in the physical world (i.e. with a group structure)?

Instead of using the properties of group theory which suppose the application of the relativity principle in all generality, we start from the Galilean transformations including the Galilean composition of velocities law which applies to real speeds, and we subject them to the measurement distortions due to length contraction, and clock retardation and to the usual synchronization procedures (with light signals or by slow clock transport). We obtain a set of space-time transformations which assume a different mathematical form than the Lorentz-Poincaré transformations. Obviously these transformations do not constitute a group. In fact as we will see in ref (1b), replacing real speeds by apparent speeds, these transformations can be converted into a set of equations which assume the same mathematical form as the Lorentz-Poincaré transformations but which fundamentally differ from them because they connect distorted co-ordinates whose distortion varies as a function of the absolute speed of the frames where the measurements are made. Although the relativity principle seems to apply with these transformations, it cannot be regarded as a fundamental principle of physics since it depends on measurement distortions.

Poincaré, who acknowledged the Lorentz postulates, wanted to reconcile them with the relativity principle. To this end, using group theory, he was compelled to postulate that the speeds obey the relativist composition of velocities law (see his demonstration below) but as we demonstrate in the 2° paragraph of this text, the application of this law generates apparent speeds rather than real speeds a fact that Poincaré's approach does not show. The space-time transformations refered to as the Lorentz-Poincaré transformations are a set of transformations which are supposed to connect real co-ordinats although this is not the case.

II - The law of composition of velocities in Lorentz’s aether theory and in Einstein's relativity.

We demonstrate that the law of composition of velocities in Lorentz’s theory is obtained from the Galilean law . When we try to determine v, we commit a double error: one regarding the evaluation of the distances (resulting from the impossibility of bringing the contraction of moving lengths to the fore ), and the other regarding the evaluation of the time (resulting from the slowing down of moving clocks, and from the impossibility of synchronizing the clocks exactly by the usual methods).

As a result of these two errors, we find the law in place of . But the real law according to Lorentz's postulates is .

III - Some comments about the transformations of Tangherlini.

These are the Galilean relationships disguised by a false appreciation of lengths and of time. But here the synchronization of clocks is supposed to be carried out exactly.

( The present article comments on some aspects of our previous paper: "Relativity and cosmic substratum" updated in October 2002 and November 2005 in the web site www.levynewphysics.com (precirculated proceedings P.I.R.T., 1996, p. 231).

 

I. Is the existence of a privileged reference frame compatible with a group structure ?

Poincaré has brought an essential contribution to the understanding of the conventional relativity theory, in demonstrating that the relativity principle implies necessarily a group structure for the space-time transformations.

Indeed, if one takes for granted the equivalence of all "inertial" frames for the description of the physical laws, then the space-time transformations connecting any pair of such reference frames must take an identical mathematical form.

The exact demonstration of this will be recalled in appendix, at the end of the present chapter. Nevertheless, the following question is worth asking: are all the frames in the physical world really inertial (and therefore equivalent for the description of the physical laws?) According to Einstein, there is no doubt about this, but for Lorentz, this is not so obvious. Indeed for him:

1.  There is a privileged reference frame supporting the ether (fundamental inertial frame).

2.  The contraction of moving lengths is real and not reciprocal.

3.  The speed of light is isotropic and equal to C exclusively in the fundamental inertial frame (Cosmic Substratum).

4. Real speeds obey the Galilean composition of velocities law. This law was applied by Lorentz to explain Michelson's experiment.

Starting from these hypotheses, we have studied the example of a light signal traveling along the x axis, and we have demonstrated that the transformations regarding space between any two inertial systems S₁ and S₂ take, in this case, the following form¹ (figure 1)

                                                                                           (1)

(x₁ designates the distance O'B - In the same manner, x = OB and x₂ = O"B).

Figure 1

S₀ is the fundamental frame, S₁and S₂ are two "inertial" frames receding from S₀
along the x axis, is a rod of arbitrary length.

Important remarks

1/ Since the length of the rod is arbitrary, we can write = x₂. But the apparent length of the rod is:

                                                                                                                                   (2)

Indeed, using a contracted standard to measure the rod AB, observer S₂ finds in place of . So expression (1) can be written as follows:

                                                                        (3)

2/ In order to measure O'B observer S₁ also uses a contracted standard, so that the result of the measurement will be multiplied by . Therefore, the apparent distance O'B will be:

                                                                            (4)

This space transformation regarding a light signal differs from the conventional one in that it applies the Galilean compositon of velocities law to real speeds. Only apparent speeds obey the relativist composition of velocities law.

We see that when S₁is at rest in the cosmic substratum S₀, then v₀₁ = 0 and expression (4) is reduced to:

                                                                                   (5)
and since v₀₂ = v₁₂

                                                                                                    (6)
replacing x_2app by its value in (6) yields:

                                                                                                              (7)

 

Although his expression takes the same mathematical form as the conventional transformation, its meaning is quite different since it connects the real co-ordinates of the preferred frame to the apparent co-ordinates of the moving frame. This fact that characterizes aether theory, highlights its difference with relativity theory which regards all the co-ordinates as real.

— Transformation (4) is different from Einstein's relativistic transformation. Indeed, it depends on v₀₁ and v₀₂. But v₀₁ and v₀₂ are the speeds of reference frames S₁ and S₂ with respect to the fundamental frame S₀ which, as we can see, is omnipresent when one takes for granted the Lorentz assumptions.

Einstein's relativistic approach is completely different. Here v₀₁ and v₀₂ do not mean anything. x_2E= is the real length of AB, and C the real speed of light in all "inertial" frames. Einstein's transformation between frames S₁ and S₂ is:

                                                                                                           (8)

(E = Einstein)

It is different from (4) (but takes a similar form to (7)). According to Einstein, it is valid between any pair of "inertial" frames.

The same remark applies to the transformation regarding the time²

                   (9)
but                                                               (10)

Expressions (9) and (10) assume the same mathematical form only when the first frame is at rest with respect to the aether frame.

In any cases the meaning of the equations differ because according to relativity theory equation (10) is supposed to result from exact measurements while equation (9) connects distorted co-ordinates.

Notice that in the present example of a light signal, = C. But contrary to Einstein's special relativity, C is not the real speed of light in S₁.

We note that:

assumes the mathematical form of a Lorentz-Poincaré transformation

assumes the mathematical form of a Lorentz-poincaré transformation
but does not
and therefore T₀₁ T₁₂ T_02.                                                                                                           (11)

 

This seems paradoxical, but can be easily explained when we know that the measurements in frame S₀ are exact, but those carried out in the other frames are fictitious. (Moreover, in S₀ the speed of light is isotropic, but not in S₁ and S₂)

Poincaré was persuaded of the interest represented by a relativity principle and the group structure associated with it, but, at the same time, he believed in the postulates of Lorentz's theory and, although these concepts looked conflicting, he thought that they could be reconciled.. He did not call into question the existence of a privileged inertial frame, the real and non reciprocal contraction of moving lengths, and the necessity of an ether to convey the electromagnetic waves.

Indeed, in his article "Sur la dynamique de l'électron"³, page 1, he expresses the relativity principle as follows:

"It appears that the impossibility of observing the absolute motion of the Earth is a general law of nature. We are naturally led to assume this law which we will refer to as the Relativity postulate."

End of the 7° chapter, speaking of the Fitzgerald-Lorentz contraction (real and not reciprocal), he affirms:

"... Therefore, the hypothesis of Lorentz (contraction) is the only one which is compatible with the impossibility of bringing the absolute inertial frame to the fore".

So, it is clear that for Poincaré absolute motion exists. Besides, the Lorentz contraction implies this absolute motion. Indeed, if a rod really contracts when it passes from one inertial frame to another, it is because there is a hierarchy between the different inertial frames (and not an equivalence). And therefore, the theory of Poincaré tries to reconcile two incompatible notions: the relativity principle on the one hand, and the existence of a preferred inertial frame on the other hand. In one sense, Poincaré was right since, as we will see in ref (1b), when one replaces the real speeds by the apparent speeds, which obey the relativist composition of velocities law, the relativity principle seems to apply. But this result is not essential since it depends on these measurement distortions, a fact that the Lorentz-Poincaré transformations do not highlight. After correction of the distortions, the relativity principle no longer applies.

Enstein's relativity theory presents some flaws, but not this one, since, from the beginning, Einstein discarded the notion of a privileged reference frame.

So, the group structure applies to Einstein's transformations, but not to those we have derived from the Lorentz postulates (including expressions 4 and 9) which assume the Galilean composition of velocities law.

Appendix

The Lorentz group according to Poincaré

In his article "Sur la dynamique de l'électron"³, chapter 4, Poincaré demonstrates that the equivalence of all inertial frames (relativity principle), implies a group structure. We cannot see anything wrong with that if we suppose that all the frames are actually equivalent but, as we have seen, this is not the case if we assume the Lorentz postulates (see formulas 4 and 9). The demonstration only applies to Einstein's relativity where there is no privileged inertial frame, where the speed of light is supposed to be really constant in any inertial frame, and where the contraction of moving rods is only observational.

This can be seen in the following comments of the demonstration briefly given by Poincaré and derived here in more detail.

Let us consider three inertial frames S", S' and S (see figure). If one supposes that they are equivalent, we have (see figure 2):

                                (12)                                    (13)

                              (14)                                      (15)

(Here we have taken the same notations as Poincaré did for S, S' and S"):

 

Figure 2

 

replacing x' and t' by their value in x" we obtain:

                                                                      (16)

The numerator N can be written as follows:

                                               (17)

Let us define                                                                                             (18)
we obtain:                                 (19)

So                                                                             (20)

We easily demonstrate that expression (20) is reduced to:

                                                                                                               (21)

So, the equivalence of all inertial frames implies a group structure; but the existence of an aether drift makes sure that even if bodies are not subjected to any external forces, the frames associated to them are not perfectly inertial, moreover they are not equivalent since the magnitude of the aether drift depends on their absolute speed.

In fact the relativity principle would be a fundamental principle if v" in formula 18 was the real speed. But it is an apparent speed resulting from measurement distortions a fact that the Lorentz-Poincaré transformations do not highlight.

(Note that although they are mathematically consistent, the assumptions of Einstein can be challenged).

II. The law of composition of velocities in Lorentz's theory and in Einstein's relativity

In Einstein's special relativity, the speed of light is isotropic in any inertial frame. In Lorentz's theory the speed of light is isotropic in the ether frame; in all other frames, it is not. More precisely, C is neither the real speed of light, nor the real average two way speed of light; it is the apparent (fictitious) average velocity obtained as a result of the systematic errors made when we try to measure it. These are three in number:

1 - The clocks used in order to carry out the measurement are synchronized by Poincaré-Einstein's method (or by slow clock transport). These methods are equivalent to the measurement of the average two way speed of light.

2 - The standard used to measure a rod is also contracted; so, we cannot observe the contraction, and therefore we make a systematic error in measuring the rod.

3 - The clocks of any inertial frame, moving with respect to the Cosmic Substratum, are slowed down.

In order to derive the space-time transformations in Lorentz's theory, we have assumed that the speed of light in the Earth frame is

C - v₀ in the + x direction of motion of the Earth with respect to the Cosmic Substratum and C + v₀ in the - x direction.

First of all, we have obtained for the real transit time of light in the Cosmic Substratum⁴

                                                         (22)
and                                                                                                                (23)

In an exact (ideal) measurement we should also have obtained for t'

                                                                                           (24)
so that t = t' and                                                                                        (25)

But as a result of the anisotropy and of the systematic errors mentioned above, we have found:

                and                                                                                        (26)
replacing and L by their value in (22), we find the following expression whose mathematical form is identical to the Lorentz-Poincaré transformation (Despite a quite different meaning).

                                                                                                                  (27)

So, we are compelled to realize that the experimental space-time transformations, are a set of fictitious transformations obtained as a result of the systematic errors made at the time of the measurement.

— Now, it is easy to demonstrate that similar errors affect the speeds; indeed, in order to derive the law of composition of velocities in Lorentz's theory, we must start from the Galilean law:

                                                                                                                              (28)

For this purpose, let us consider two inertial frames S₀ and S₁. S₀ is at rest in the Cosmic Substratum, and S₁ moves with respect to S₀ at speed v₀ along the common x axis. At time t₀, S₀ and S₁ overlap. At this very instant, a body M passes by point OO' and travels toward point B (see figure 3).

 

Figure 3

v_r is the real speed of the body with respect to O' and V is its real speed with respect to S₀. We propose measuring the apparent speed of the body relative to O' with two clocks, firmly attached to frame S₁, placed at A and B, and synchronized by Einstein-Poincaré's method.

The real length of the moving rod, according to Lorentz, is , where L is the length at rest (in frame S₀). The real time needed by the body to cover the distance OB in S₀ and O'B in S₁ is :

                                                                                                                      (29)

Taking account of the slowing down of the clocks in S₁, and supposing that the clocks are exactly synchronized in this frame we should find:

                                                                                                                       (30)

The measured (fictitious) length of the moving rod AB is L (since the standard used to measure it, is contracted).

The measured (fictitious) time is obtained using the two clocks A and B synchronized by means of Einstein-Poincaré's method ⁵ or by slow clock transport.

The error of synchronization is equal to the difference between the real time of reflection of a light signal in B, and the measured time (half two way transit time ⁵). That is:

           — =                              (31)

Taking account of the slowing down of the clocks, we find in fact:

                                                                                                                                             (32)

So the apparent time needed by the body to cover the distance AB will be the difference between (30) and (32)

                                                                                          (33)
and the apparent speed of the body will be:

                                                                                            (34)
and conversely:

                                                                                                              (35)

Hence, this law of composition of velocities in Lorentz's aether theory is a fictitious law obtained as a result of the systematic errors made when we try to measure the real speed . And therefore, Lorentz's aether theory is nothing else than Galileo's theory disguised by false measurements. (This is different from Einstein's relativity which does not recognizes the existence of apparent speeds).).

Another consequence is that all the measurements of the speeds, carried out in frames moving with respect to the cosmic substratum, are false.

For example, let us try to measure the speed of a body moving on the surface of a celestial object receding from the cosmic substratum at speed km/sec. The apparent (measured) speed of the body is found equal to 10⁵ km/sec. We have:

        (36)

Now                                                                                                                      (37)

So that                                              (38)

So, the error commited in measuring the speed of the body is:

                                                        (39)

III - Some comments about the transformations of Tangherlini

The equations of Tangherlini assume the form

                          (40)                                    (41)

They are proposed as an alternative to the Lorentz transformations. They assume the existence of a privileged inertial frame and the contraction (real and non reciprocal) of moving lengths.

Now it is important to know what other assumptions are concealed in such transformations. To this end, let us return to figure 3. At the initial instant, body M starts from 00'. When it has covered the distance in frame S₀, frame S₁ has covered the distance When body M covers this distance in turn, S₁ has moved away from S₀ a distance of , and so on. So that the distance covered by the body when it reaches point B is:

                                              (42)

                                                                                                    (43)

So that                                                             (44)
from x = Vt we obtain

                                                                                                                  (45)

We notice that equation (45) is identical to equation (40). Since L is arbitrary we can write:

                                                   x'=L                                                                                             (46)

L is not the real length of AB. Indeed, due to length contraction, the real length is

Now the transformation for the time in Tangherlini's approach is:

                                                                                                                 (47)

Here the sychronization of clocks is supposed to be carried out exactly, so that the term of the Lorentz equation disappears.

We note two points:

1) x' = L and not , so x' is not an exact estimate of the distance covered by the body.

2) t' is not an exact estimate of the time, the real time being t ; so that the apparent speed of the body relative to S₁ is :

                                                                                                      (48)

(Notice that this expression is different from Lorentz's and Einstein's law)

But the real speed is:

                                                                                        (49)

So, we can conclude that the transformations of Tangherlini are the Galilean transformations disguised by a false estimate of the lengths and of the time. But here, contrary to Lorentz-Poincaré's transformations, the synchronization of clocks is carried out exactly.