Abstract
Starting from the experimental measurement
of the two way velocity of light which, apparently, equals
C in any inertial frame, and assuming the existence of an ether
frame in a state of absolute rest, the isotropy and homogneity of space and time, "time dilation" and some other simplifying
constraints, F. Selleri derives a set of general space-time
transformations based on an absolute synchronization procedure. These transformations distinguish from the conventional tansformations obtained with the usual measurement procedures in that they are not affected by the synchronism discrepancy effect.Yet they take their specific mathematical form because the measurement of the lengths and the time are made with contracted meter sticks and clocks slowed down by motion.
Using this very enlightening derivation and,
replacing the experimental data (which are affected by systematic measurement distortions),
by the corrected values, we demonstrate that
the same assumptions lead to the Galilan transformations. So, these
two concurrent viewpoints, apparently contradictory, are in
fact compatible and complementary.
1 - Introduction
Einstein's relativity is based on two assumptions:
the relativity principle and the isotropy of the one way velocity
of light.
Now, measuring the one way velocity of light
is a really difficult task, because the clocks used to make
this measurement have been generally synchronized beforehand
by means of light rays going to and fro, from one clock to another
which is equipped with a mirror (Einstein-Poincaré Method
[1, 2]). But in order to carry out the synchronization exactly,
we must know the one way velocity of light. "So, the situation
becomes circular [3]".
In fact, as demonstrated by Prokhovnik [4],
the apparent one way velocity of light, measured with
clocks slowed down by motion and synchronized by means of light signals, and with contracted meter sticks, is nothing else than the apparent average two way velocity
of light. So, all the measurements using light signals, lead
to this average round trip velocity.
Another method of synchronization available
is the method of slow clock transport. Now, according to Eddington
and Prokhovnik [4, 5], this method is equivalent to the Einstein-Poincaré
procedure. This point of view is not shared by Mansouri and
Sexl [6]. In any case, the method is not reliable because the
clock retardation introduced during the transport, is not, a
priori, known. Indeed, this retardation will be different
according to whether the assumptions of special relativity are
exact or not. In the first case, the retardation will be very
small; but if we take for granted the assumptions of Lorentz,
it would not be negligible.
Since the method needs the knowledge of the
true assumptions, it cannot be used to verify them, and therefore
to measure the one-way speed of light.
So, there is no evident proof that the one
way velocity of light is really isotropic, and several physicists,
today, consider that Lorentz's theory can be defended as much
as that of Einstein. On the contrary, we have good reasons today
to consider that the one way velocity of light is not isotropic.
See on that subject ref [14]
Lorentz's theory does not assume that the one
way velocity of light is identical in any inertial frame. The
explanation of Michelson's experiment by Lorentz, considers
that this speed is C - v in the direction of motion of the Earth
and C + v in the opposite direction. The meaning of C in Lorentz's
theory is quite different from C of Einstein's relativity. It
means the value of the one way velocity of light in the ether
frame.
It is easy to verify that the experimental
apparent two way velocity of light in any inertial frame,
also equals C, this has been done several times (Fizeau, Foucault
Shoran, Aslakson Bergstrand and others).
Starting from this result, and assuming the
existence of an ether frame in a state of absolute rest, the
isotropy and homogeneity of space and time, time dilation, and
some other simplifying constraints (see later), Selleri derives
a set of equations based on an absolute synchronization procedure.These transformations distinguish from the conventional transformations obtained with the usual measurement procedures in that they are not affected by the synchronism discrepancy effect.Yet they take their specific mathematical form because the measurement of the lengths and the time are made with contracted meter sticks and clocks slowed down by motion.When the synchronization with light signals is used, they reduce to the classical Lorentz transformations.
Now, the purpose of the present paper, is an
alternative approach in which we become aware that the apparent
isotropy of the two way velocity of light results from unavoidable
(or considered so) systematic measurement distortions. After correction
of these distortions, we realize that the Lorentz assumptions lead to the
Galilean transformations.
We finally demonstrate that assuming the existence of an aether frame in a state of absolute rest, the exerimental space-time
transformations are nothing else than the Galilean transformations,
disguised by the altered measurements.
To this end, we start from a demonstration
of Franco Selleri [7] which is in conformity with the experimental
observation. Then, we diverge in order to bring the hidden variables
of the problem to the fore.
2 - Transformations of space and time and the two way velocity
of light
A very enlightening paper, by Franco Selleri,
has been published in the Chinese Journal of systems,
engineering and electronics [7]. Let us give a brief summary
of the main points of the paper useful for our demonstration.
The author starts from a very general set of
transformations from one inertial reference frame 
to
another 
.
is
considered at rest with respect to the ether frame (absolute
rest), and S moves with respect to
with velocity v ; i.e:
(1)
Taking the homogeneity and the isotropy of
space and time for granted, the author assumes that expressions
(1) are linear and take the form:
(2)
Now, assuming that at time 
for
,
and t = 0 for S, the origins of
and S coïncide, we must have:
(3)
and consequently:
(4)
The condition that plane 
coincides with plane (x, y) at all times implies:
(5)
Therefore, for the third equation (2) we have:
(6)
The same condition applied to planes 
and (x, z) gives:
(7)
Now, the condition 
leads to:
(8)
(this means that at time 
the planes 
and (y, z) coincide). Assuming
that, the origin of S moves with respect to with
velocity v parallel to the + 
axis,
we have :
(9)
whence
(10)
Taking account of the previous results, equations (2) reduce to:
(11)
If we suppose that the cartesian coordinates are orthogonal, then 
and 
become completely equivalent.
Therefore:
(12)
Inverting (11), we easily obtain:
(13)
with R = 
(14)
Now, let us assume that x, y, z and t are the
coordinates of a light signal, and let us give a little increment
to them. We easily demonstrate that:
(15)
with
(16)
In 
the
speed of light is isotropic and equal to C. So, we must have:
(17)
Replacing 
and 
in
expression (17) by their values given in (15), we obtain a second
degree equation which, when solved, gives:
(18)
where 
from (16) and (18) we easily obtain 
(19)
With a suitable choice of polar coordinates
we finally obtain:
(20)
Let us designate the forward velocity of light as 
, and the backward one as 
. We have:
(21)
the two way velocity of light 
is given by:
(22)
From (20) and (22), F. Selleri obtains the
following general expression of this two way velocity of light:
(23)
3 - Experimental and real data
According to Einstein, the one way velocity
of light is isotropic and equal to C. But, as we have seen,
measuring the one way velocity of light by means of the usual
methods, (Einstein-Poincaré procedure or slow clock transport)
is a really difficult objective, and all the exact measurements lead to the apparent average round trip velocity . So, several authors consider today
that the isotropy is a property of the two way velocity of light
(and not of the one way), a fact that we shall check.
Probably an anisotropy of the two way transit time of light exists, in gas mode as demonstrated by Miller,
Esclangon and Allais [8,
9, 10]. Yet, recent experiments have significantly confirmed the null result of Michelson experiment in vacuo.
(Note that the measurement of the one way speed
of light is not in principle impossible, but we need for that
synchronization procedures different from the usual methods. See ref [13] and [15]).
Now, the questions asked here are the following:
1/ Do the hypotheses of Lorentz really lead to an isotropy of the two way speed of light in any inertial
frame ? In other words, are the usual measurements of this velocity really reliable ?
2/ What is the real meaning of the Lorentz-Poincaré transformations.
Are there some hidden variables behind them ?
Let us recall the hypotheses of Lorentz:
- existence of an ether reference frame 
in a state of absolute rest
- the one-way speed of light is constant and isotropic, exclusively with respect to the ether frame
- Real speeds obey the Galilean composition of velocities law
- rods moving with speed v with respect to
parallel to the direction of motion contract according to the law
(24)
where L is the length of the rod when it is at rest in the ether frame, and 
its
contracted length.
- clocks moving with respect to the aether frame
are subjected to a clock retardation whose amplitude is given by
(25)
Now, for Lorentz, only t0is the
real time, t is considered as fictitious. There is no relativity
of time but only a physical effect affecting the clocks.
- the mass of a body moving with respect to
the ether frame increases according to the law
(26)
This hypothesis has been introduced
finally in order to justify that all the experiments designed
to verify the contraction of moving rods gave a null result.
In fact, when we consider the explanation given
by Lorentz of Michelson's experiment, we can see that:
1/ In the direction of motion the two way transit
time of light is:
(27)
(28)
(
means T real in the direction of motion).
Taking account of clock retardation, we obtain:
=
(29)
(
means T experimental in the direction of motion ).
Indeed,
is not the real two way transit time of light, but it is the
clock display found experimentally with the clocks attached to reference frame S.).
Now, considering that the contraction of L
is not perceived by the observer of S due to the fact that his
standard also contracts, the apparent
average two way speed of light with respect to S is:
(30)
This is what we experimentally obtain. But in fact, the real average
two way speed of light along the direction of motion is:
(31)
2/ In the direction perpendicular to the motion,
the two way transit time 
is of course identical to 
(32)
Now, if we take account of the clock retardation, we obtain:
(33)
But here there is no length contraction along the directions y or z, so that:
(34)
and 
, obtained with the retarded clocks of frame S is:
(35)
We can see that 
. In agrement with the experiment. Conversely
(36)
which means that the real value of the average two way speed
of light is not isotropic, and is different from the speed of
light in the aether frame.
4 - Two viewpoints
4.1 - The point of view adopted
by Franco Selleri is the experimental one, according to which
the two way velocity of light appears isotropic and equal to
C, to an accuracy better than 
(37)
(
in the notation of Selleri means : average C forth and back)
As a result of the formula (23), this identity
leads to:
for 
, 
(38)
for 
, 
(39)
Taking account of (14), equation (38) can be written as follows:
(40)
Replacing R and 
by their values (given by expressions (38) and (39)) in expression
(11)) and replacing 
by its value, we obtain:
(41)
The author also demonstrates that 
are parameters independent of r, 
.
Due to our hypotheses we must have 
,
so 
(42)
Therefore, we are led to a set of transformations
equivalent to the Lorentz-Poincaré transformations, but whose mathematical
form depends on the way of synchronizing clocks. Using Einstein-Poincaré's
synchronization procedure, they take the form of the classical
equations of Lorentz-Poincaré. Others methods of synchronization
can change the form of the transformation of t (see ref. [7],
p. 37, section 7), but regarding x, provided that the transformations connect the aether frame with any other "inertial" frame and that the measurements are made with contracted meter sticks, and clocks slowed down by motion, all the procedures lead
to the expression:
(43)
This result is completely in accordance with
the experimental measurement.
4.2 - Now, there is another
point of view which takes into account and corrects the systematic
experimental distortions.
As we have seen, the assumptions of Lorentz
lead to an average two way speed of light equal to 
in
the direction of motion (see formula (31)), and to 
in
any direction perpendicular to the movement (see formula (34)).
Replacing 
in formula (23) by its value given by (31), we obtain:
(44)
Replacing 
in
formula (23) by its value given by (34), we obtain:
(45)
Therefore, f1= g2 (46)
Substituting these expressions in (13) we obtain:
(47)
Now, due to our hypotheses, we must have y0=
y and z0= z. This fixes the constant f1
which must be equal to 1. According to (44), we also have R = 1.
Now, it is well known that in Lorentz's approach,
the dependance of t0 on x, y and z, is not essential,
and can be eliminated by a suitable choice of synchronization
procedure [6, 7 p. 37, chap. 7], known as absolute synchronization.
In consequence, expressions (47) can be reduced
to the galilean relationships. These can be considered as the
hidden variables of Lorentz’s theory (and not of Einstein's
relativity).
(48)
Important note
The time in question here, is the real time,
which is the time of the ether frame. Now, if we had taken clock
retardation into account, we would have obtained for the transformation
regarding the clock display:
(49)
5 - Discussion
Contrary to Einstein's relativity, in Lorentz's
theory, by virtue of equations (48), real speeds are simply
additivel. This has also been directly demonstrated in ref.
[5] and [11].
On the other hand, the fact that the space-time
transformations conceal hiddenvariables asks delicate questions.
Indeed,, Einstein’s theory is based on the relativity
principle, so the laws of nature must be identical in any inertial
frame. And, as a consequence, the relativistic mass must be
introduced in order that the total quantity of motion (momentum),
during a collision, is conserved under a change of inertial
frame.
If the space-time transformations
are nothing else than the Galilean transformations disguised by unavoidable
(or considered so) errors, then, the question of the conservation
of the total relativistic momentum, in any inertial frame, arises.
Now, we have to be aware that the Lorentz postulates
are not compatible with the relativity principle. (But this
does not invalidate them since, today, weighty arguments
call into question the relativity principle).So there is no necessity for the laws of nature
to be strictly invariant. On the contrary, they should slowly
vary when one passes from one inertial frame to another.
And, as a consequence, if in a collision carried
out in the fundamental inertial frame, the total relativistic
momentum is conserved, we must not strictly expect the same
result if an identical experiment is performed in another
frame. At high speeds the action of the ether drift should not
be negligible and the laws of physics
should not appear strictly invariant.
So, since the law of conservation of the total
relativistic momentum does not exactly
apply, it cannot be used to demonstrate
the law 
.
But, at the same time, it can no longer be used to assert that
the law
is incompatible with the Lorentz assumptions.
This viewpoint was not ours in the initial
(non uptaded) version of reference [12] because at that time
we thought that the relativity principle was an unquestionable
law of physics.
But since that time we have realized that, insofar as an ether drift exists, real frames are not perfectly inertial. Therefore, the relativity principle does not exactly apply in the physical world [13,14]. For some authors, its
character is essentially conventional, depending on the synchronization
procedure and, using an exact synchronization method, it
should not be maintained [7]. (It nevertheless remains
approximately true for bodies moving at low speeds with respect
to one another and with respect to the ether frame 
. For this reason the conclusions of Galilei can be retained
as a good approximation).
The fact remains that real speeds, in Lorentz's
theory, are simply additive, and this seems, a priori, difficult
to reconcile with the law .
But, as we will see in ref [16] this is not impossible. Of course the law has not the same meaning than in conventional relativity..
Important remarks
If one takes the postulates of Lorentz for
granted, (including the fact that the Galilean composition of velocities law applies to real speeds,) one must know that the resulting space-time transformations
take the (usual) form of those of Lorentz-Poincaré exclusively
when one of the frames involved is the fundamental inertial
frame. In all other cases they take a different form (This is
explained in detail in ref [14]).
(N.B - In a previous paper [13], we have proposed
another method of synchronization which needs neither light
signals nor transported clocks. It is a question of a thought
experiment which could be applicable as it is, or with some
slight modifications).
- This paper has been published in its initial
form in the volume of supplementary papers of the 1998 P.I.R.T
meeting, p 86. The present version contains some clarifications
and some footnotes which were not included in the previous one.
Acknowledgements
- I would like to express my thanks to Pr.
F. Selleri, for interesting exchanges of views and for the support
given to this work.
- I am grateful to Dr. M.C. Duffy for supervising,
with authority, but also with devotion and kindness, a forum
open to new ideas, which permits scientists to fight against routine
and stagnation in the field of physics.
- I would also like to thank Prs. P. Marmet
and T. E. Phipps Jr. for the valuable alternative studies in
the field of relativity they sent to me, and for the interest
shown in my views.
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