Updated on November 20th 2005 and August 30 th 2008

Abstract

This paper aims at demonstrating that

1/assuming the equality of the two way transit time of light along the two arms of Michelson's interferometer (modern versions of Michelson’s experiment oparating in vacuo), and the anisotropy of the one way speed of light in the Earth frame, two facts more and more confirmed today, length contraction appears necessary and can be easily deduced.

2/taking account of the above mentioned assumptions and of clock retardation, the apparent (measured) two way speed of light is demonstrated to be always identical and equal to C.

 

It is greatly to the credit of Builder and Prokhovnik to have demonstrated that, assuming Lorentz-Fitzgerald contraction, the two way transit time of light along a rod is independent of the angle assumed by the rod.

We agree with the approach of these authors until a certain point, but, contrary to what is often believed, their theory failed to demonstrate that the one way speed of light, although anisotropic, is (paradoxically) always found equal to C.

The reasons of this paradoxical but important result, will be explained here. They represent a decisive argument in favour of Lorentz-Fitzgerald contraction.

I - Introduction

Since the first steps of relativity, Lorentz-FitzGerald contraction has been the subject of a debate which is not closed today, and divides the physicists in opposite camps.

Some of them consider length contraction (L.C) as a naïve opinion, for example Wesley ¹, Phipps ², Cornille ³, Galeczki ⁴. Some others consider it as a fundamental process which explains a lot of experimental facts. Among them Bell ⁵, Selleri ⁶, Builder, Prokhovnik ⁷, Dishington ⁸, Mansouri and Sexl ⁹, Wilhelm ^10b.

Length contraction has been proposed by Lorentz ¹¹ and FitzGerald ¹² in order to explain the null result of Michelson’s experiment.

(In fact the fringe shift which was observed was not completely null, but much smaller than the result expected).

L. C has never been observed. Of course it cannot be demonstrated directly by an observer of the moving frame, since the standard used to measure it, also contracts. But it could be verified indirectly. This was the objective of different renowned physicists who tried to observe the physical modifications brought about by motion: variation of the refractive index of a refringent solid (Rayleigh ¹³ and Brace ¹⁴), influence of the aether wind on a charged condenser the plates of which make a certain angle with the direction of translation (Trouton and Noble ¹⁵), experiments of Trouton and Rankine ¹⁶ and of Chase ¹⁷ and Tomashek ¹⁸ on the electrical resistance of moving objects, and finally of Wood Tomlison and Essex ¹⁹ on the frequency of the longitudinal vibration of a rod.

But the experimentsl proved all negative.

In order to justify the lack of experimental evidence, Lorentz was compelled to postulate the variation of mass with speed (see ref ²⁰, p 99 and 105). But, in ref ^21-23, we demonstrated that the space-time transformations derived from the Lorentz postulates, are reducible to the transformations of Galilei after correction of the systematic measurement distortions resulting from length contraction, clock retardation and unreliable clock synchronization (i.e Einstein-Poincaré procedure or slow clock transport). Therefore, at first sight, these transformations do not seem really compatible with the law of variation of mass with speed. (Indeed in order to demonstrate this law, we generally make use of Einstein’s relativity principle which supposes the conservation of the total relativistic momentum in any inertial frame, and does not assume the Galilean transformations).

And then, the other Lorentz assumptions seemed incompatible with one of them. For this reason we regarded them as questionable in some earlier publications (ref ²⁷).

But, since that time, we have become aware that the relativity principle does not strictly apply in the physical world (see ref ²⁴ ). So, starting from this point, the objections opposed to Lorentz’s justification can be overcome. Using other arguments we could demonstrate in Ref 25 that the law of variation of mass with speed is compatible with the other Lorentz assumptions.

(Note that if Lorentz’s theory is reducible to Galilei’s, this implies another assumption: the speed of a body with respect to the aether frame must be < C. So if a body A has the speed with respect to the aether frame and another body B has the speed with respect to the first, the two bodies and the aether frame being aligned, must be limited in such a way that

A more recent experiment by Sherwyn ²⁶ proved also negative : the author considered an elastic rod rotating about one end in the laboratory frame. At low rotation rates, the length of the rod adiabatically follows the length demanded by the equilibrium lengths of the molecular bonds, which obviously cannot be estimated by laboratory meter sticks, since they experience the same dependence of length on angle. However, according to the author, at high rotation rates, when the time required to rotate 90° becomes comparable to the period of vibration of the structure, the macroscopic length would not be able to exactly follow the "bond equilibrium" length.
To support his demonstration, the author assumed that "the relativist contraction is a physiccal process and proceeds with the speed of sound, and "it will occur relatively slowly in a time comparable to L/v where v is the speed of sound in the rod.

This statement is not based on experimental grounds and nothing proves that it corresponds to reality. In his book "Light in Einstein's universe", Prokhovnik objects that the contraction should occur in a time comparable to L/C. In any cases there is no certainty that under the conditions of the experiment the adiabatic process would not have occurred.

We must add that for an aether drift estimated at 300 Km/sec the variation of L due to length contraction would have been of the order of half a micron for a rod of 1 meter long, a contraction very difficult to highlight. Yet the spring used in Sherwyn's experiment measured 0.123 m.

Note that another argument seemed, at first sight, to go against Lorentz-Fitzgerald contraction: the compressibility of matter is limited, and length contraction seems difficult to justify at very high speeds.

For example at 0,9999C the ratio would be reduced to 1,4%.

But we can answer that the law has been proposed following an experiment performed at low speed (Michelson’s experiment). It would not exactly adopt the same form at very high speeds.

Today, the author of this text realizes that there exists some strong arguments lending support to Lorentz-Fitzgerald contraction (L.C.). One of these arguments is that L.C enables us to explain (in all directions of space and not only in two perpendicular directions) the isotropy of the apparent (measured) average two way speed of light.

II - Length contraction is no longer an ad hoc hypothesis today

We know that all the usual attempts to measuring the one way speed of light with clocks synchronized by the Einstein-Poincaré procedure are in fact equivalent to measurements of the apparent to and fro speed of light ^7,27 or ³⁵. This is also true when we use clocks synchronized by slow clock transport ³⁵ which is, in fact, approximately equivalent to the Einstein-Poincaré procedure.

According to Anderson, Vetharaniam and Stedman ²⁸, all the recent experiments purporting to illuminate the isotropy of the one way speed of light were based on erroneous ideas (because they considered that the slow clock transport procedure allows exact synchronization).

On the contrary, a number of arguments speak in favour of the anisotropy of the real one way speed of light. Although a direct measurement comes against major difficulties, it can be deduced from the measurement of the terrestrial aether velocity, based on the fact that light signals propagate isotropically in the aether frame.

A first estimation of the absolute velocity of the solar system was already made in 1968 by De Vaucouleurs and Peters, by measuring the anisotropy of the red shift relative to many distant galaxies. The experiment was made again by Rubin in 1976.

A more reliable estimate of the solar system velocity was obtained by measuring the anisotropy of the 2.7° K microwave background, uniformly distributed throughout the Universe. "An observer moving with velocity v relative to this microwave background can detect a larger microwave flux in the forward direction (+v) and a smaller microwave flux in the rearward direction (-v). He can observe a violet shift in the forward direction (+v) and a red shift in the rearward direction (-v) (Wilhelm)".

From this data, the absolute velocity of the solar system could be measured (Conklin (1969), Henry (1971), Smoot et al (1977), Gorenstein and Smoot (1981), Partridge (1988)).

Let us also quote the method of measurement based on the determination of the muon flux anisotropy (Monstein and Wesley (1996)). An assessment of all these experiments is given by Wesley ^10a and Wilhelm ^10b.
Marinov ²⁹ also attested having measured the absolute velocity of the solar system by means of different devices (coupled mirror experiment, toothed wheels experiment). The experiments are described in detail in the book of Wesley ¹, and are quoted by Wilhelm ^10b.

Some physicists were suspicious about Marinov and regret not having the proof that the experiments were actually performed. But Wesley, who knew him very well, attested that he had no reason to doubt his reported results. According to him, "Marinov was extremely reliable and scrupulously honest in all of his personal dealing with people. He had a firm grasp on reality" ³⁰.

The photograph of the toothed wheels apparatus has been published and is available.

The toothed wheels experiment estimated that the absolute velocity of the solar system v is of the order of 360 ± 40 km/sec, and that the speed of light is C-v in the direction of motion of the solar system and C + v in the opposite direction. (Notice that the orbital motion of the Earth around the sun is far weaker (about 30 km/sec) and that the rotational motion at the latitude of the experiment was of the order of 0,5 km/sec).

This result was in agreement with most of the experiments described above.

More recently (April 2003) Cahill and Kitto³⁶  reinterpreted Michelson's experiments. They asserted that Michelson's interferometers operating in gas mode are capable of revealing absolute motion. The authors have found that, after correcting for the refractive index of the air, the Miller experiment gives a speed of v = 335 Km/sec plus or minus 57 Km/sec.. A more recent evaluation yielded a value of the order of 400 Km/sec. If confirmed, these studies provide additional weighty arguments in favour of an absolute aether frame

These experiments proved to be in good agreement with the other methods designed to measuring the absolute velocity of the solar system (coupled mirrors experiment ^1,10,29, anisotropy of the red shift of many distant galaxies and anisotropy of the 2.7°K microwave background radiation¹⁰, experiment of Wesley (muon flux anisotropy) ¹⁰, De Witte experiment (cited by Cahill).

 

— Consider now a Michelson interferometer operating in vacuo whose longitudinal arm is aligned along the x₀ axis of a co-ordinate system S₀ (0, x₀, y₀, z₀) attached to the Cosmic Substratum. The arm is at rest in the Earth frame which moves along the x₀ axis at speed v.

It is easy to verify that, in reply to the statement that the speed of light is C-v in the + x₀ direction, and C+v in the opposite direction, the arm will be contracted in the ratio

                                                                                                      (1)
where is the length of the arm in motion, and the length at rest.

With the same premises, we will demonstrate that the apparent (measured) average two way speed of light along the x₀ axis, is equal to C independently of the speed v.

Let us demonstrate formula (1).

A priori, we do not know if or not. The two way transit time of light along the longitudinal arm will be :

                                                                         (2)

Now, in the arm perpendicular to the direction of motion, there is no length contraction. The speed of light is C exclusively in the aether frame. The signal starts from a point P in this frame towards a point at the extremity of the arm and then comes back towards point P’. During that time, the interferometer has covered the path vt₂ (see figure1).

Figure 1

We have:

                              
                             
therefore:                                                                                                     (3)

Neglecting the minute displacement of the fringes observed when we change the orientation of the apparatus** , which is really too small to explain the existence of an aether wind of the order of 300 km/sec, we can write , so:

                              

Hence                   

So, if we take the anisotropy of the real one way speed of light into account, length contraction must no longer be considered as an ad hoc hypothesis. On the contrary, it must be seen as a necessary cause of the Michelson result.

Now, on account of clock retardation, the apparent (measured) two way transit time of light will be (from (3)):

                              

Since the length of the longitudinal arm is determined with a contracted standard, it is found equal to and not to , so that the apparent (measured) average two way speed of light along the x₀ axis will be found equal to C. (It is in fact different from its real value, which according to formula (2) is .

NB - In the absence of length contraction, the average two way speed of light would not have been found equal to C, in contradiction with the experiment.

III - Length contraction explains the apparent speed of light invariance

 

- But this is not all. We will now demonstrate that L.C leads to the independence of the apparent average two way speed of light from any direction of space and from the speed v.

The demonstration is based on Builder and Prokhovnik’s ⁷ studies whose importance is indisputable but, as we will see, some of the conclusions of Prokhovnik were questionable and could not enable us to demonstrate that the apparent average two way speed of light is C in any direction of space.

Consider the frame of the substratum and the Earth frame mentioned above, and suppose that a rod A B, enclosed in a vacuum tube, makes an angle with the x₀, x axis. The rod is at rest with respect to frame S (see figure 2).

Figure 2

 

At the two ends of the rod, let us place two mirrors facing one another by their reflecting surface, which is perpendicular to the axis of the rod . At the initial instant, the two frames S₀ and S are coincident. At this very instant a light signal starts from the common origin and travels along the rod towards point B. After reflection the signal comes back to point A.

We do not suppose a priori that (where is the length of the rod when it is at rest in frame S₀).

We note that the path of the light signal along the rod is related to the speed C₁ by the relation.

                                                            (see figure 3)
where t is the time needed by the signal to cover the distance AB.

In addition, when the signal reaches point B, frame S has moved away from frame S₀ a distance:

                              
so that                             

Now, from the point of view of an observer in frame S₀, the signal goes from point A to point B’ (see figure 3)

Figure 3

C being the speed of light in frame S₀, we have

                              
and then, the projection of the speed of light with respect to frame S along the x axis will be:

                              
we remark that

                              

The three speeds, C, C₁ and v being proportional to the three lengths AB’, AB and AA’ with the same coefficient of proportionality, we have

                              
so                                                                     (4)
(we must emphasize that equation (4) implies that the three speeds C, C₁ and v have been measured with the same clock, which obviously is a clock of frame S₀)
resolving the second degree equation, we obtain:

                              

The condition C₁ = C when v = 0 compels us to only retain the + sign so:

                              

- Now, the return of light can be illustrated by the figure 4 below:

 

 

Figure 4

 

From the point of view of the observer of frame S, the light comes back to its initial position with the speed C₂.

So we can write:

                              

For the observer of frame S₀ the light comes from B’ to A’’ with the speed C, so that

                              

During the light transfer, frame S has moved from A’ to A’’ with the speed v so:

                              

The projection of the speed of light along the x axis will be

                              
we easily verify that:

                              
so

                              

The two way transit time of light along the rod AB (which could be measured with clocks attached to frame S₀) is:

                                                                                                              (5)

According to the experiment, T must be essentially independent of the angle . So 2T must be equal to

                              
which is the two way transit time along the y direction (previously calculated).

We can see that, in order that this condition be satisfied, the projection of the rod along the x axis must shrink in such a way that:

                                                                                  (see figure 5)
where was the angle separating the rod and the x₀ axis when the rod was at rest in frame S₀.

 

Figure 5

from:                         
and                           
we easily verify that:

                                                 
finally:                       
replacing by this expression in (5) we obtain, as expected:

                                                                                                    (6)

Finally, from the above mentioned experiments demonstrating the anisotropy of the real one way speed of light, we deduced that the projections of this speed along the x axis of frame S are when the light travels in the forward direction and when it travels in the rearward direction. These results, combined with the invariance of the two way transit time of light along a rod, whatever its direction in space, are enough to verify that the projection of the rod along the x axis shrinks in such a way that

                              

and conversely.

- But this is not all. The same conditions combined with clock retardation, allow us to demonstrate that the apparent (measured) two way speed of light is C in any direction of space.

Clock retardation is an experimental fact.

Let us designate as the two way transit clock display in frame S along the rod. We will have (from (6)):

                              

                              

Now, the length of the rod, measured with the contracted meter stick of frame S, is always found equal to , so that the average two way speed of light is found equal to C in any direction of space and independently of the speed v. (As we have seen this is also the case for the apparent one way speed of light measured by means of the Einstein-Poincaré synchronization procedure or by slow clock transport).

This result is highly meaningful and is a direct consequence of the facts deduced from the experiments of Michelson and Morley and the experiments demonstrating the anisotropy of the one way speed of light.

Note

In our demonstration, although we are indebted to Prokhovnik, we differ with his conclusion⁷; indeed, since and also , it is obvious that t and t’ are the real transit times of light along the rod (which could be measured with clocks attached to the aether frame).

Now, since and there is no doubt that C₁ and C₂ are also measured with clocks attached to frame S₀.

This is also the case for .

Nevertheless, according to Prokhovnik, in "The logic of special relativity" ⁷ chapter " The logic of absolute motion", the time is identified to the two way transit time of light along the rod, as measured with clocks attached to the moving frame, (see formula 5.2.4 of ref 22-1, French edition).

This cannot be true for the reason mentioned above.

(Note that in our notation the moving frame is designated as S, while in Prokhovnik’s notation, S designates the aether frame and A the moving frame. We will continue the demonstration with our own notation).

In addition, if Prokhovnik's approach were true, the apparent average two way speed of light, measured with a standard in frame S, would not be C. Indeed, since this standard is also contracted, observer S would find for the length of the rod.

Therefore, the apparent (measured) average two way speed of light in frame S would have been:

                              
which is not in agreement with the experimental facts.

The real two way transit time of light along the rod, as measured in frame S₀, is in fact , and the two way clock display, measured by an observer of frame S, is . This corresponds to the experimental facts, since, with these values, the apparent average two way speed of light in frame S is found equal to

                              

(Note also that the real average two way speed of light along the x₀, x axis is

                              
which, as expected tends towards 0 when ).

Note

The hypothesis of the aether dragged by the Earth has generally been rejected because it seemed in contradiction with the theory of aberration. But, as demonstrated by Beckmann ³¹, Mitsopoulos ³² and Makarov ³³, this is not the case.

But the dragged aether theory is contradicted by the experiment of Lodge ³⁴, who demonstrated that the speed of light is not modified in the neighbourhood of a rotating wheel; and by the experiments of Marinov ^29,1 (toothed wheels experiment and coupled mirrors experiment), or all other experiments demonstrating the anisotropy of the one way speed of light.

The present study was presented at the VIIIth International Conference, Physical Interpretations of Relativity Theory, M.C Duffy editor, sponsored by the British Society for the Philosophy of Science.

 

References

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