Updated on November
20th 2005 and August 30th 2008
Abstract
This
work (although self-sufficient) continues the studies previously presented;
but in the light of new data, certain commonly accepted views will be
re-examined. On the other hand, some notions considered before as
controversial, will now appear far better founded.
Indeed, in previous papers, the Lorentz assumptions appeared suspect to us
because they lead to space-time transformations reducible to
Galilei’s and, as a consequence, the law of conservation of the total
relativistic quantity of motion could not be used to demonstrate the law  . But
since that time we have become aware that the relativity principle is not
an unquestionable concept of physics since it does not strictly apply in
the physical world 13. This result, in conjunction with the
experiments verifying the anisotropy of the one way speed of light, and
some others considerations, demonstrates that all the arguments opposed to
Lorentz’s assumptions can be overcome. In particular the law can be derived using other arguments, yet it only strictly applies when the original location of the mass is the aether frame S0 (see ref 41).
On the
contrary, the said assumptions explain some fundamental experimental
results such as the apparent isotropy of the two-way speed of
light. So, it appears justified to carry on with our studies on the bases
of Lorentz’s postulates.
In ref 9 starting from the Galilean transformations and subjecting them to the systematic measurement distortions entailed by length contraction, clock retardation and the usual clock synchronization procedures, we have derived a set of transformations relative to a light signal connecting any pair of "inertial" frames. We extend here the study to any body moving uniformly.
These "extended space-time transformations", which are based on Lorentz postulates, lean on the Galilean composition of velocities law applied to real speeds. (This law was used by Lorentz in particular to explain Michelson's experiment). We demonstrate that the relativist composition of velocities law applies to velocities whose measurement is altered by the systematic measurement distortions.
The
transformations are demonstrated to be consistent. They
reduce to the Lorentz-Poincaré transformations when one of the inertial
frames they connect is the fundamental frame, and they explain why the
velocity of light, although anisotropic, is paradoxically always found to
be constant. This approach, which shows that the experimental transformations are nothing else than the Galilean transformations altered by measurement distortions, allows some obscure points of physics to be
explained.
The
main ideas of this manuscript were registered at the French Society of
authors.
Important
notes: 1/ We call "inertial" the frames in which a body at rest is
not submitted to perceptible external forces, a term sanctioned by use. Yet,
we must be aware that, insofar as an aether drift acts on them, the frames
cannot be perfectly inertial, and therefore, the relativity principle does
not strictly apply.
2/ In previous versions of this text we applied the term Lorentz-Poincaré transformations to transformations which assume the same mathematical form as the conventional transformations. In fact, as we shall see, the term should be reserved to transformations which connect any "inertial" frame to the aether frame in which the measurements are not altered by measurement distortions (see Ref 42). In all other cases, the term is not appropriate since the space-time transformations connect reference frames whose co-ordinates are altered by measurement distortions, a fact that the Lorentz-Poincaré transformations do not highlight
I - Introduction
The
theory of space-time transformations was the collective work of several
physicists. Among the best known we must quote Heaviside 1,
Fitzgerald 2, Voigt 3 and Larmor 4. The
principal actors being Lorentz 5, Poincaré 6,
Einstein 7 and Minkowski 8. Although the fact is
often ignored among scientists, there were two opposite currents of
thought. The relativist current prevailed for a long time, but today, weighty
experimental and theoretical arguments lend support to the alternative
viewpoint which assumes the existence of a fundamental aether frame..
The
purpose of the present paper is, in the light of this new data, to
reexamine the Lorentz assumptions, and to verify whether they are in
agreement with different well established concepts of physics. In that
case, it should be justified to search for a set of transformations
deriving from the Lorentz postulates, but which could permit to solve the difficulties and the paradoxes inherent in theconventional transformations, and which could be applied between any
pair of inertial frames.
(Let us
bear in mind that the Lorentz assumptions are: existence of a privileged
inertial frame, in which there is no aether drift, designated as Cosmic Substratum,
length contraction, clock retardation, speed of light isotropic and equal
to C exclusively in the aether frame, Galilean composition of velocities law)
Let us
briefly recapitulate the reflections to which we devoted our previous
papers. In ref. 9-12 we have derived a set of space-time transformations for a light signal emitted from an "inertial" frame different from the fundamental frame. Using the Galilean composition of velocities law, we demonstrated that the space-time transformations connecting this reference frame and another, take a different form than the Lorentz-Poincaré transformations. Conversely, the transformations which connect the aether frame to any other "inertial" frame assume a mathematical form identical to the Lorentz-Poincaré transformations. So, the whole of
these transformations do not constitute a group and, as a consequence, they
do not, obey the relativity principle. We also verified that the Galilean
composition of velocities law can be converted into the law  in a simple
manner, by introducing the systematic measurement distortions resulting
from length contraction, clock retardation and imperfect clock
synchronization 10 It should be noted that, taking into account the assumptions we have just mentioned , the law takes this form
exclusively when one of the frames under consideration is the fundamental
frame. In all other cases, as we will see later, it takes another form.)
From
these considerations and some others, we concluded that these space-time
transformations which are derived from the Lorentz postulates assume the following
character:
1. They
are reducible to the Galilean transformations after correction of the
measurement distortions just mentioned.
2. They
take their usual form (Lorentz-Poincaré transformations, i-e L.P tr)
exclusively when one of the frame they connect is the privileged
inertial frame (aether frame).
3. They
are in contradiction with the relativity principle (this third statement is
a direct consequence of the second statement).
Initially,
these observations caused us to be suspicious about the Lorentz assumptions. Indeed,
in order to demonstrate the law  , we generally make use (as
Einstein did) of the law of conservation of the relativistic quantity of
motion in any inertial frame, which applies in relativity theory. If
Lorentz’s theory is reducible to Galilei’s, this law of
conservation cannot be used to demonstrate . Thus, at first sight, Lorentz
(Galileo) theory seemed incompatible with an important experimental fact,
i.e the variation of mass with speed.
Finally,
the fact that the Lorentz assumptions are not compatible with the
relativity principle seemed to constitute an important objection to this
approach. But, since that time, we have become aware that the relativity
principle does not strictly apply in the physical world 13,37,39. It only seems to apply provided that the measurements are subjected to the distortions mentioned above. So it can really be used, neither to demonstrate the law  , nor to
refute it. Indeed, if the laws of physics are not perfectly invariant,
then, there is no necessity for the total quantity of motion to be exactly
conserved in all "inertial" frames** .
So, an
important objection against Lorentz’s assumptions was removed.
In
fact, as demonstrated in 41, the law is rigorously exact
exclusively when the mass of a body in motion is compared with its rest
mass in the aether frame.
There
remained however some points to be clarified:
1. If
Lorentz’s law of composition of velocities is reducible to
Galilei’s, is it compatible with the existence of a limit velocity? At
first sight this does not seem to be the case!
2. Are
there today any experimental facts that make length contraction compulsory?
3. In the event these experimental
bases are proved, is length contraction necessary to explain other
experimental results: for example the ‘apparent’ (measured)
isotropy of the average two way speed of light? We will endeavour to answer
these three questions.
Suppose two inertial bodies, one having the speed v0 = 2 105
km/sec with respect to the Earth and the other v = 2 105 km/sec
with respect to the first, the three bodies being aligned.
According
to Einstein’s special relativity, the speed of the second with
respect to the Earth (v’) is
In
Galilean theory, the total speed would be:  , which is not agreement with
experiment.
But
this obstacle can be overcome if we assume that the speed of a body with
respect to the aether frame (V) is limited in such a way that V < C.
This
means that, if a body A moves at speed  with respect to the origin O of
a system of coordinates at rest with respect to the aether, the speed
relative to A of another body B moving along the direction OA will be
limited in such a way that:  .
Another
important point to be examined was the reality of length contraction and
its real compatibility with Lorentz’s theory (if we assume that it is
reducible to Galilei’s).
A
priori, a number of objections seemed to oppose these two concepts.
1. In Lorentz’s theory, length
contraction can be justified only if the law  applies 14.
At first sight this looked like an important objection since, as
demonstrated previously 9-11, the other Lorentz assumptions
seemed incompatible with one of them,  . But, as far as the relativity
principle does not strictly apply in the physical world, the objection no
longer holds.
2.
Lorentz contraction was never observed experimentally. The classical experiments
15 (Rayleigh, Brace, Trouton and Noble, Trouton and Rankine,
Chase, Tomashek, Wood Tomlison and Essex,
etc.) could not demonstrate it but, in ref.5 ,
Lorentz could explain these negative results by the increase of mass with
speed. (See also ref.14).
The more recent experiment by Sherwyn 16 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.
For these reasons Sherwyn's experiment proves unconvincing.
3. The
compressibility of matter is limited, and length contraction seems
difficult to justify at very high speeds. For example at 0.9999 C the ratio  would reduce
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 the present text realizes that, although not observed, there
exists some strong arguments lending support to Lorentz-Fitzgerald
contraction (see later)
Although the attempts made to measure the one-way speed of light must be analysed with much circonspection, they deserve to be examined without prejudice because they have been carried out carefully, and although numerous, they lead to results which are in agreement with each other. For example, using a Fizeau’s type toothed
wheels apparatus, Marinov (1984)17, brought to the fore the anisotropy of the one way speed
of light. (This kind of experiment does not fall within the category of
optical experiments forbidden by the Potier-Veltmann principle 18.)
The
principle of the toothed wheels experiment was simple. The apparatus
consisted of two identical circular steel plates with 40 round holes of
diameter = 6 mm.
They were mounted on a shaft of length L = 120 cm rotating with
angular velocity  . An argon laser illuminated the holes of
the first plate, and a silicon photocell detected the light passing through
the holes of the second plate. At the initial instant, the two wheels were
aligned with their holes exactly opposite. The transit time  for light
to travel the distance L, was determined by measuring the amount of light
received by the photocell as a function of the rotation rate "
(Wesley 36.)
According to the author, the
experiment demonstrated that the absolute velocity (v) of the solar system,
is of the order of 360 ± 40 km/sec, and that the speed of light is  in the
direction of motion of the solar system, and  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, for this reason, during a short time,
in most cases, the motion of the Earth with respect to the aether frame can
be identified with the motion of the solar system).
Note
that, contrary to the belief of many physicists, the experiment has
effectively been performed and the photograph of the apparatus is
available.
Silvertooth
for his part measured the standing waves formed by light beamed in opposite
directions, using two lasers. He claimed having observed changes in the
standing light wave patterns due to the absolute velocity of the laboratory
which was found to be 378 Km/sec. Silvertooth and Whitney21
confirmed the result by another experiment
performed in 1991.
More recently (April 2003) Cahill and Kitto21
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 measure the absolute velocity of the solar system (coupled mirrors experiment 36, anisotropy of the red shift of many distant galaxies 19, anisotropy of the 2.7°K microwave background 20, De Witte experiment (cited by Cahill), experiment of Wesley (muon flux anisotropy)21).
— On account of these results, it is easy to verify that a rod of length  aligned
along the direction of motion of the solar system, contracts in such a way
that  .
Consider
to this end a Michelson interferometer operating in vacuo whose
longitudinal arm slides at speed v along the x0 axis of a system
of coordinates S0(O, x0, y0, z0)
at rest in the Cosmic Substratum. A priori, we do not know if  or not.
But we know that Michelson’s experiment has brought to the fore a
fringe shift really too small to explain the existence of an aether wind of
about 300 km/sec, and then which can be ignored*** (see the review article by
Hayden 22.) Therefore, for our purpose, the two way transit time of
light along the two arms of the interferometer (t1, and t2)
can be regarded as identical. Along the longitudinal arm we have
(1)
Consider
now the arm perpendicular to the direction of motion. From the above
mentioned experiments we know that the speed of light is equal to C
exclusively in the aether frame. The signal starts from a point P of this
frame towards a point Q at the extremity of the arm and then comes back to
point P’. During that time, the interferometer has covered the path
vt2 (see figure 1.)
Figure 1
we have
 
so that
(2)
identifying t1 and t2 we obtain 
Hence
We
finally verify that length contraction can be deduced from
Michelson’s experiment and from the anisotropy of the one way speed
of light.
Now,
with the same premises, we will demonstrate that the
‘apparent’ (measured) average two way speed of light along the
longitudinal x0 axis, is equal to C independently of the speed
v.
Indeed, taking account of clock retardation, the ‘apparent’ two
way transit time of light (clock display) will be (from formula (2)):  .
Since
the arm is measured with a contracted meter stick, its length is found
equal to  and not to  , so that the
‘apparent’ average two way speed of light in the Earth frame will be found equal
to C. (This experimental value is in fact different from its real value
which, according to formula (1), is  ). Note also that in the absence
of length contraction the ‘apparent’ average two way speed of
light, would have been found different from C in contradiction with the
experiment.
We remark that, paradoxically, due to the measurement distortions, the apparent value of the speed of light in the Earth frame is equal to the real speed of light in the substratum.
Thus, two
meaningful results can be deduced from the experiments demonstrating the
anisotropy of the one way speed of light, in conjunction with
Michelson’s experiment.
But
this is not all. These experiments also explain the invariance of the two
way transit time of light along a rod, not only in two opposite directions, but
independently of the angle  assumed by the rod and of the speed of the
reference frame in which it is measured. (See ref. 11 section
II.7 or ref. 39.)
Also
the apparent constancy of the two way speed of light in any direction of
space and in any inertial frame can be deduced from these experiments 39 ****.
These results are highly meaningful.
Notes
We could have obtained the same results without length contraction by assuming that the speed of light is  in the direction of motion, and  in the
opposite direction. Let us examine such a hypothesis.
It is
obvious that the factor  would have resulted from the
proximity of the celestial bodies. In effect, at a distance from them, in a
reference frame not influenced by the presence of matter, there is no
possible explanation for the existence of such a term.
The
said factor, being multiplicative, indicates an increase of the speed of
light in the neighbourhood of the celestial bodies, which, in turn ,
implies a partial dragging of the aether.
Such a
hypothesis cannot be justified for the following reasons:
1. The
experiment of O. Lodge 24 demonstrated that the speed of light
is not changed at the neighbourhood of a rotating wheel.
2. A partial dragging of the aether
would have increased the speed of light in the direction of motion, and
reduced it in the opposite direction. And the multiplicative factor would
have been different in the two opposite directions, which is not the case
here.
3. On
the contrary there are no logical obstacles for Lorentz contraction. Moreover
there are theoretical bases for the process.
a -
Since several arguments demonstrate the anisotropy of the one way speed of
light, we are limited to two hypotheses (as we have seen in the body of
the text.) The elimination of the just mentioned hypothesis renders length
contraction necessary.
b -
According to Wilhelm : "Since matter consists of positive (nuclei) and
negative (electrons) charges, the contraction of their equipotential
surfaces causes Lorentz-Fitzgerald contraction".
c - The
anisotropy of the one way speed of light gives a physical basis to the
theoretical explanations of Larmor 4, Lorentz 5, and
several prominent modern physicists who assume length contraction (see ref.
23 and 25 to 33.) It invalidates the other
theories : Ritz theory 34 (ballistic), Einstein theory 7,
Stokes theory (completely dragged aether ) 35.
In the
light of this new data, the Lorentz assumptions appear today far better
founded than in the past. (Note nevertheless that this statement concerns
the Lorentz assumptions but not the usual interpretation of the Lorentz-Poincaré transformations as
will be seen later.)
We propose here to derive a set of
space-time transformations effective in all inertial frames. This purpose
is parallel but different from Selleri’s work 27, since,
here, we deal with the experimental transformations obtained from the usual
Einstein-Poincaré’s method of synchronization, or with the slow clock
transport method which is equivalent 40 (and not with the
absolute synchronization procedure of Mansouri and Sexl 29,
which is theoretical and would be difficult to apply experimentally.) ( Unlike the Lorentz-Poincaré transformations these transformations are not symmetrical, a fact that can solve the paradoxes inherent in these transformations caused by the reciprocity of observations. As we demonstrate in Ref 42, these transformations can be converted into a set of equations whose mathematical form is similar to the Lorentz Poincaré transformations but whose meaning is quite different because they don't suffer of this problem of symmetry.).
In
addition we demonstrate that the coordinates of these transformations,
although experimental, are fictitious and must be corrected in order to
obtain the real lengths, speeds, and times. Conversely, in order to derive
them we must make use of the Galilean transformations.
We have
to specify that these extended space-time transformations, apply to all
moving bodies and not exclusively in the special case of a light signal
(that was studied in detail in ref. 9.)
II - Derivation of the extended space-time transformations
Let us
use to this end the following thought experiment :
Consider
three "inertial" systems S0(O, x0, y0, z0),
S1 (O’, x1, y1, z1), S2(O’’,
x2, y2, z2), S0 is at rest in
the Cosmic Substratum (aether frame), S1 and S2 move
along the common x axis with a rectilinear uniform motion (see figure 2). We
propose to derive the space-time transformations between the frames S1
and S2.
Figure 2
When the body arrives at A it meets a
mirror firmly attached to frame S1.
A long
rigid rod O’’A at rest with respect to frame S2 is
aligned along the x2 axis.
At the
initial instant, the origins of the three frames O, O’ and
O’’ are coincident. At this very instant, a body M (coming from
the -x2 region) passes by O’’, and then continues on
its way along the rod with a rectilinear uniform motion towards point A.
Let us
designate as v01 the speed of reference frame S1 with
respect to S0 , v02the speed of S2 with
respect to S0 and v12 the speed of S2 with
respect to S1.
The
speed of the body with respect to S0 will be called V.
The
length of the rod O’’A would be  if frame S2 were at
rest with respect to the Cosmic Substratum. But, as a result of its motion,
O’’A is contracted and assumes the length  such that:
When
the body reaches point A, it meets a clock  equipped with a mirror, firmly
linked to frame S1 and standing at a point A’ in this
frame (so that when the body arrives at point A, A and A’ are
coincident).
The
real distance  covered by the body in frame S1
can easily be obtained: indeed, the ratio of the distances covered in
frames S1 and S2 is equal to the ratio of the speeds
with respect to these two frames. That is:
Thus:
(3)
Notice that real speeds are simply additive. On the contrary, as we will see, apparent speeds obey a law of composition of velocities different from the Galilean law.
Since
the distance O’A’ is measured with a contracted meter stick in
frame S1, the apparent distance  found by observer S1
will be:
(4)
Now, in
order to determine the time needed by the body to reach point A’, we
must beforehand synchronize two clocks placed at O’ and A’.
The
method of Poincaré-Einstein treats the clock display 
as the one way transit time of light. In reality, it is equal to half the ‘apparent' transit time of light from O' to A' and to O' again.
The
real transit time of light from O’ to A’ is in fact: 
(5)
and from A’ to O’: 
(6)
Taking
account of clock retardation in frame S1, the synchronism
discrepancy  between the clocks placed at O’ and
A’ is given by
From (3), (5), and (6) we obtain:
Now,
the real time needed by the body to cover the distance O’A’ in
frame S1 is
(from
(3))
This
time is the universal time which could be measured with clocks attached to frame S0
(in which there is no speed of light anisotropy and no clock retardation).
But in
frame S1 we must take account of the synchronism discrepancy
effect and of clock retardation, so that the experimental apparent time
obtained when we use the method of Poincaré-Einstein is:
(7)
From
expressions (4) and (7) we obtain
(8)
(Contrary to relativity, this expression of the composition of velocities law applies to apparent speeds and not to real speeds.)
Expressions
(4) and (7) can be expressed as functions of  and  . We note
that, since  is measured in frame S2 with a
contracted meter stick, we have
we also note that 
replacing  by this expression in (7) we obtain
(9)
and replacing  by in (4)
(10)
and conversely
We
remark that, contrary to special relativity,  and  , which are
the velocities of the frames S1 and S2 with respect
to the aether frame, are systematically omnipresent in the equations.
Expressions
(9) and (10) are the extended space time transformations, applicable to any
pair of "inertial" bodies receding uniformly with respect to one another
along the direction of motion of the solar system. Note that, during a
short time, the motion of the Earth with respect to the Cosmic Substratum
can be considered rectilinear and uniform. If this were not the case,
the bodies placed on its surface would be subjected to perceptible
accelerations.
Compared to the absolute motion of the sun, the orbital and the rotational
motions of the Earth are very slow. Thus, as a first approximation and during a
short time, the motion of the Earth can be identified with that of the solar
system. Practically S1 can be identified with the Earth frame,
and S2 with a vehicle moving on its surface in the direction of
motion of the Earth (for example a ship). Therefore, is the real speed of the Earth
with respect to the Cosmic Substratum (S0),and  the real
speed of the ship with respect to S0.
V is
the real speed, with respect to S0, of a body present on the
surface of the ship and moving in the same direction, and the apparent
speed of the body with respect to the Earth frame.
(Formula
(10) is identical to the expression of the space transformation which was given in a previous publication38. Formula
(9) represents a completely satisfactory expression of the time
transformation; indeed, it applies to all values of V. It replaces the
expression given in 38 which was limited to high values of V).
Note
It is
interesting to also express  as a function of 
From
 we
obtain
replacing this expression of V in (8) we obtain
Conversely,
from
we easly verify that
We can
notice that, in conformity with the ‘apparent’ (experimental)
speed of light invariance,  and conversely.
Important remark
It is
currently asserted that the total quantity of motion is exactly conserved
even when collisions occur at very high speeds. This seems eminently
questionable. Indeed, by means of the usual methods of clock synchronization, one makes
systematic errors in measuring the speeds. One finds the apparent speeds  in place
of the real ones.
III - Consistency of the transformations
The
consistency of the transformations can be checked. To this end, we need to
demonstrate their agreement with experimental data already known in
particular cases.
The
equations must take the mathematical form of the Lorentz-Poincaré transformations when  and they
must explain why the experimental measurement of the speed of
light by means of the usual methods, always gives C.
We note
that when , the frames S1 and S2
which were coincident at the initial instant, always remain coincident. This brings us back to the case of two "inertial" frames where one of them is at
rest in the Cosmic Substratum (aether frame). In this case, as expected,  = which is
the synchronism discrepancy effect defined by Prokhovnik 23. (More
exactly Prokhovnik takes  as a definition of the concept. But,
as demonstrated in 39, contrary to the opinion of this author,
this implies that the measurement is carried out with non retarded clocks).
One can
easily verify that, in this case, the space-time transformations connecting
the frames S0 and S1 assume the form of the
Lorentz-Poincaré transformations .
For  , the
apparent time and space co-ordinates reduce to:
and
In conformity with the experiment.
Now,
when  , S1 is at rest in the Cosmic
Substratum and then
and
After multiplication of the two fractions, we
obtain
Taking
account of the fact that
 and 
we obtain
and
(11)
Now,
from the expression 
we easily find from (11) that: 
Thus, as
expected, these expressions assume the form of the the Lorentz-Poincaré transformations when one of the
frames is at rest in the Cosmic Substratum. Yet their meaning is quite different since they connect the real co-ordinates of the preferred frame to the apparent co-ordinates of any other "inertial" frame breaking the symmetry of the Lorentz-Poincaré transfomations which is responsible of the paradoxical results inherent in these transformations.
Important remarks
 is not the real co-ordinate of point A
relative to S2 along the x2-axis, the real co-ordinate is  . The fact
that authors are not aware of this, is a source of much confusion.
It should also
be pointed out that, contrary to what is often believed, ,  and  are all
apparent (fictitious) co-ordinates.
IV - Conclusion
Starting
from the Galilean relationships and assuming the Lorentz postulates, we
obtained a set of transformations applicable to any pair of "inertial"
bodies aligned along the direction of motion of the solar system, even if
no one is at rest in the Cosmic Substratum (aether frame.) They take a
different form from the Lorentz-Poincaré transformations.
In
order to derive them we were compelled to modify the Galilean relationships
by making allowance for the systematic measurement errors.
Conversely,
they must be corrected in order to obtain the Galilean transformations,
which are the true transformations when no measurement errors are present.
The
consistency of the derivation is verified, since the relationships take the form of the Lorentz-Poincaré transformations when one of the frames they connect is the fundamental inertial frame. They also explain why the apparent (measured) velocity of light is found to be constant independently of the absolute speed of the solar system,
(however, after correction of the systematic measurement errors, they
show that the real velocity of light is constant
only in the fundamental frame.)
These
extended space-time transformations do contradict the application of the relativity
principle, but this does not disprove
them, since, as we have seen, real frames are not pefectly inertial, and the principle does not strictly apply in the physical world (see also ref. 13 and 42 )
(Yet, as we have noted in ref. 13, in different usual cases the conclusions of Galilei remain practically true.)
Now, as we remarked in the introduction, we must point out that the
questioning of the relativity principle, compels us to also relativize the
law of conservation of the total quantity of motion in any "inertial" frame. The
law exactly applies exclusively in the fundamental inertial frame. It is
only approximately true when the "inertial" frame in which a collision
occurs, moves at low speed with respect to this aether frame.  . (This
point of view is also shared by Wesley ref. 36 section 1.11.)
This is
not the case in conventional relativity, and this law was used by Einstein
to demonstrate the relationship  . (But, as demonstrated in ref 41,
the point of view of special relativity cannot be maintained).
Note
that the transformations just derived have not only theoretical interest,
they certainly reflect our actual situation in the cosmos since, according to the arguments listed in the previous chapters, the Earth moves with respect to the
Cosmic Substratum at about 300km/sec, and the Earth frame can be identified
with S1 17.
-
Another point of importance should be clarified. We know that in aether
theories, contrary to relativity, the kinetic energy possesses an absolute
character. It is defined with respect to the fundamental frame S0.
This means that the increase of kinetic energy of a body which moves from one "inertial"
frame S1 (different from S0) to another S2,
is different from the conventional value. Let us calculate it:
When
the body moves from S0 to S1, the kinetic energy
acquired is  (12)
If the
speed of the body is low  , expression (12) reduces to: 
when the body moves from S0 to S2  the kinetic
energy acquired is 
The
increase of kinetic energy from S1 to S2 is thus:  (13)
Since  ,we can
write  . Expression (13) then becomes
(14)
This
expression appears different from the conventional formula  . Indeed, unlike this, expression (14) depends on  which is
the speed of the Earth with respect to the fundamental frame. In ref.11
we viewed this result as an argument against aether theories, but in
the light of the new data which calls into question the application of the relativity
principle 13,37,39, we think that our previous position deserves further consideration. (And this, even if, due to measurement distortions which seem to corroborate the relativity principle, we find the conventional fomula)
When we analyse
expression (14) we notice that when  , the term depending on  can be
ignored and we are brought back to the conventional expression again. But,
very likely, in most cases  should not be ignored since it
is estimated at about 300 km/sec.
N.B-It should be noted that in conventional physics, when the strict application of the relativity
principle is assumed, the expression for kinetic energy is affected by an important
internal contradiction. Indeed, when a body moves from one inertial
system S0 to another S1, (assuming that  ) it
acquires the kinetic energy:
 (15)
Now
when the same body moves from S0 to S2, (with  ) the
kinetic energy acquired is
(16)
The
difference between (16) and (15) is 
which is different from 
(Note
that contrary to the fundamental aether theory,  is not the speed of the body
with the respect to a privileged frame). Suppose that  , assuming
that  , we easily check that the increase of
kinetic energy from S1 to S2 is  . But
according to the relativity principle it should be  since nothing
differentiates the three inertial frames.
This
important internal contradiction does not affect the fundamental aether theory.
The
paradox appears even more obvious if we note that the energy acquired when a body moves
from one "inertial" frame S0 to another S1 is not clearly defined,
and depends on the frame from which it is measured. Let us suppose that  and that S0
and S1 coincide, if  the kinetic energy acquired from
S1 to S2 will be considered equal to  . If we
measure the same kinetic energy from a frame S0 moving with
respect to S1 at 10 km/sec, we will find
This is
in contradiction with the idea that the energy needed to carry out a
certain work is well defined and cannot depend on the point from which it
is measured.
This
paradox is completely foreign to the fundamental aether theory where the energy
is perfectly defined, and depends on the speed of the body with respect to
the aether frame.
Appendix
Let us
consider the case of a light signal which starts from the common origin and
runs towards point A. When the signal reaches point A, it is reflected in a
mirror firmly attached to frame S1 at a point A’ in this
frame, which at this instant coincides with A. After reflection, the signal
comes back to O’.
1. Real co-ordinates
The
ratio of the distances covered by the signal from O’ to
A’ and from O’’ to A is equal to the ratio of the speeds
of the signal with respect to S1 and to S2:
Thus:
Therefore
the real time needed by the signal to cover the distance x1 is:
(17)
where  is the real velocity of light in frame S1.
Let us
point out that these lengths, times, and speeds, are not the parameters which are
measured experimentally.
2. Apparent co-ordinates
The
experimental (apparent) time can easily be obtained from the real time by
allowing for the systematic measurement distortions.
This is
also true for the apparent path. In effect, the distance x1
being measured with a contracted meter stick, appears longer than it really
is, the apparent distance is then:
As we
have seen, the real time needed by the light signal to move from O’ to
A’ is erroneously identified with the ‘apparent’ average
transit time which is equal to:
(18)
t1 is given by formula (17). We can see that in the reverse
direction (A’ O’) the light signal covers with
respect to S1 the same distance as from O’ to A’,
but with the speed C+ v01.
Thus:
and:
When this result is inserted in (18), the apparent average transit time of the signal in frame S1 reduces to:
- apparent speed of light
The
apparent (experimental) speed of light, as expected, emerges as  .
N.B:
the concurrent method of slow clock transport also appears affected by
similar systematic measurement distortions (consult ref 40).
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