Updated on November 15th 2005 and August 30th 2008
Abstract
As we
have seen in previous papers 1,2, the space-time transformations
derived from the Lorentz postulates conceal hidden variables. So, a priori,
they seemed at variance with the law of variation of mass with speed  At first
sight this result appeared as an important objection against the Lorentz
assumptions, but we demonstrate here that the objection can be overcome.
Now, we
will realize that the law  is completely exact exclusively
when the measurement compares the mass of a moving body with its mass when it is at rest in the aether frame.
It is only approximately exact when the measurement is made from the Earth
frame, whose speed with respect to the aether frame is low in comparison
with the speed of light.
Some
results of this approach differentiate completely the fundamental aether
theory from the conventional theory of relativity.
I - Introduction
As
demonstrated in ref 1, the space-time transformations
derived from the Lorentz assumptions conceal hidden variables. This is also the case for the fundamental
extended space-time transformations studied in ref 2. Indeed,
after correction of the systematic measurement distortions entailed by
length contraction, clock retardation and arbitrary clock synchronization,
they reduce to the Galilean relationships,  , .
This
poses a problem: if the real transformations are the Galilean
relationships, the total relativistic momentum, in the course of a
collision, is not conserved in any inertial frame. But this law of
conservation is considered as a necessary condition to demonstrate . This
seems, a priori, an important objection against the fundamental approach
based on the Lorentz postulates,2,3 which, at first sight,
rendered them suspect to us.
Nevertheless,
in ref 4 and 5, we have become aware that the
relativity principle is not strictly applicable in the physical world, and,
for this reason, provided that the errors inherent in the usual measurement procedures are corrected, the laws of physics must not be perfectly invariant. Then,
there is no necessity for the total relativistic quantity of motion to be
exactly conserved in any inertial frame*.
This, I
realize, will surprise the readers and needs an explanation: in fundamental
theories, an aether wind exists which blows the opposite way from the
absolute motion of the reference frame in which a collision occurs. In the
course of the collision (between marbles or particles), these are slowed
down by the aether wind, and therefore their total quantity of motion
cannot be exactly the same before and after the collision. A part of the momentum being transferred to the aether.
This is
particularly true when the particles move at high speed with respect to the
aether frame (V>105 km/sec), but, at low speeds (V<<C),
the action of the aether wind should, most often, be ignored. (It is probably
for this reason that the aether wind does not perceptibly disturb the
orbital motion of the Earth).
Note
that, if the total quantity of motion is not conserved, the so called
“law of conservation of the total relativistic momentum” can be
used, neither to demonstrate, nor to disprove it. In fact,
using other arguments, we will demonstrate that the law  applies,
but, contrary to relativity, is the rest mass in the
fundamental frame: it is not the rest mass in all inertial frames.
For the
following it is important to note that, in the particular cases where the
total quantity of motion is exactly conserved, the conservation must be
effective for all observers, no matter if they are at rest or in motion
with respect to the system in which the collision occurs (see the example
given below).This fact will be used to demonstrate the law  without
relying on relativistic arguments. We will then deduce .
*Note that as demonstrated in ref 9 (second article) the relativity principle may seem to apply because of the systematic distortions that affect the usual measurement procedures, and as a consequence the law seems to apply identically in any "inertial"frame. But this result is wrong and does not maintain when the measurement errors are corrected.
II - Demonstration of  without
relativistic arguments
Consider
a body at rest in the fundamental frame S0, which emits N
identical photons simultaneously in two opposite directions (+ x and - x),
see figure 1. (For this demonstration, we will follow one given in ref 6,
but with different assumptions).
Figure 1
Consider
now another "inertial"* frame S moving along the x axis at speed v. In frame S0,
the total momentum is conserved. This must also be true for any observer
moving with respect to frame S0. With respect to frame S, we
have:
 (1)
where is the initial momentum, and  the final
momentum of the body. The other terms are the momenta of the photons
altered by the Doppler shift. (Note that  is a formula of classical
electrodynamics independent of relativity).
The
loss in source momentum  viewed from frame S will be :
(2)
Since,
obviously, the source is at rest in frame S0 both before and
after emission, with respect to frame S it must have the speed v both
before and after emission, so:
(3)
Now,
according to the energy conservation law:
(4)
(5)
 is the variation of energy resulting from
the emission of the photons. From (3) and (5) we obtain
(6)
Note
that this mass-energy equivalence formula has been obtained without the
help of the Lorentz-Poincaré transformations.
III - Variation of
mass with speed.
Consider
a body at rest in the fundamental frame, which is subjected to a
force F. The elementary expression of the kinetic energy is.
(7)
where  is the work carried out by the force F in
the displacement  . (We suppose that F and  are
aligned).
This
expression can also be written as follows :
(8)
From
F= we obtain
(9)
and
(10)
and finally 
(11)
This
expression, which connects the kinetic energy and the momentum was derived
by Lewis 7 in
1908.
Now, as
seen previously, the equivalence of mass and energy takes the form
(12)
From
the expression of the momentum  (13)
and that of the energy, we can write:
(14)
From
(11) and (12) we have:
(15)
replacing E and  by their expressions given in (12) and
(13), we obtain
(16)
so
(17)
designating  as u so that  we find successively
(18)
 (19)
so
 (20)
for
 so
(21)
 is the rest mass. As we will now see,
expression (21) is completely exact exclusively when  is the
mass at rest in the fundamental frame.
IV - Variation of mass
with speed in relativity and in the fundamental aether theory.
In
relativity, since no absolute frame exists, the mass of a body at rest in a
given "inertial" frame, viewed by an observer of this frame, is always
identical, whatever this "inertial" frame may be. This mass is defined as the
proper mass or the rest mass of the body.
If the
body moves with respect to a reference frame S with velocity v, its mass
with respect to S is supposed to be:
(22)
whatever the reference frame S may be.
The
point of view of the fundamental theory is completely different. Indeed,
consider a body having the mass in the fundamental frame S0.
Since it is necessary to provide this body with the kinetic energy  in order to
move it from S0 to any other "inertial" frame S1,
the rest mass of the body in this frame will be .
So, a
hierarchy of rest masses, as a function of the absolute speed of the body,
exists.
(Note
that it is necessary to distinguish the real mass from the measured mass,
which can be falsely estimated. In effect, if we measure the mass  of a body
in the fundamental frame by comparison with a standard  , if  and are transported
in another inertial frame, they are changed in the same ratio. Therefore
the mass appears not to have changed, which is
inexact).
In
other words, the real mass of the body in frame S cannot be measured by an
observer at rest in this frame. In all cases the measurement gives the value
which is the mass of the body in th aether frame.
Let us
now examine the consequences of these results in the following example. Consider
three "inertial" frames S0, S1 and S2,
and let three bodies of masses  ,  and  be
respectively at rest in these three frames. The said masses were initially
identical in reference frame S0 and equal to  , before
being transported in their respective reference frame. We propose to
determine the effect of motion on these masses (see figure 2).
Figure 2
1/Point
of view of the conventional theory of relativity.
Measured
by an observer at rest with respect to one of the bodies, its mass remains,
in all cases, equal to  . Therefore, for observer S1,
we have
(23)
where  designates the relativistic mass of body b2 as measured
by observer S1 and v12 designates the relative speed
of reference frames S1 and S2.
If we
suppose that  , expression (23) can be written to first
order as follows
(24)
So
that, viewed by observer S1, the energy of body b2 is
(25)
(This
corresponds to the sum of the rest energy and the kinetic energy needed by
b2 to move from S1 to S2).
For an
observer at rest in reference frame S0, the energy of b2 is
different. Designating as the mass of body b2
as measured by an observer in S0, we have, (for ):
(26)
and the energy of body b1 is assumed to be
(27)
so that, for observer S0 the kinetic energy needed by the body b2
to move from S1 to S2 is
(28)
This
result is different from the measurement made by observer S1,  , although, obviously,
it should be the same.
2/Point
of view of the fundamental aether theory
In our
book “Relativité et Substratum Cosmique 8”,
the results below were seen as a stumbling block for the
fundamental aether theory, because they lead to an expression for
kinetic energy different from the usual expression. Nevertheless, objections to the application of the relativity principle and present-day arguments in favour of the
aether and of the anisotropy of the speed of light, compel us to reassess
our past point of view.
Let us
reconsider the figure with the three bodies, and suppose that S0
is the fundamental inertial frame, and S1 and S2 two
inertial frames aligned with S0.
According
to the fundamental aether theory,  and  have no
meaning. A body at rest in a given "inertial" frame has only one real mass. The
mass of the body b2 is:
(29)
and the mass of b1:
(30)
Conversely,
as we will see, the rest mass of a body will not be  in the different
"inertial" frames, from (29) and (30) we obtain
(31)
If one
supposes that  ,  reduces to
(32)
(33)
This
expression is different from (24). It contains a term depending on  which
vanishes when S1 is at rest with respect to S0. We also realize that
expression (31) which connects any couple of inertial frames, assumes a
mathematical form different from (29) and (30). These results also confirm that the relativity principle does not strictly apply in the physical world. (The principle seems to apply as a result of systemetic measurement distortions, which demonstrates its contingent character(see Ref 9))
We also
note that, when  or in other words when  , the
terms depending on  and in expression (32) cancel. Thus,
 represents the rest mass assumed by the
aforementioned bodies when they stand in reference frame S1. This
is different from special relativity for which the rest mass is  in any
"inertial" frame.
Nevertheless,
we must distinguish the absolute rest mass from the other rest masses
standing in "inertial" frames which are in motion with respect to the aether
frame.
Note
however that when  , and  expression (31) reduces to
(34)
and since  , we obtain
(35)
It is
the case of particles moving at high speed with respect to the Earth frame,
(while the Earth moves with respect to the fundamental frame at low speed ( 300
km/sec)). In such cases the Earth can be considered as almost at rest with
respect to the fundamental frame. So the relativistic approach and the
fundamental approach lead to practically equivalent results.
3/The
question of reciprocity.
This
question makes a great difference between relativity and fundamental
theories. Indeed, according to relativity, when a mass is transported
from one"inertial" system S0 to another S1, viewed
from S0, this mass is supposed to be
(36)
but conversely, if the mass comes back to S0, viewed from S1
it will also appear equal to 
(37)
According
to the fundamental theory, suppose that S0 is the fundamental
frame. If the mass is at rest in frame S1, we also have
(38)
 is  . Indeed we have been
compelled to supply energy in order to move from S0 to S1,but
if the mass comes back to S0, the energy is restored. All
observers (including the observer at rest in frame S1) will conclude that the
real mass is equal to  .
(39)
This
result is completely in contradiction with relativity, but it is the only
result which is in accordance with mass-energy conservation.
Important remarks.
In
fundamental theories, we must distinguish the total available energy of a
body (which is equal to the sum of the rest energy  and the
kinetic energy with respect to the fundamental frame), from the available
energy of the body with respect to any other inertial frame, which is
weaker than the previous one, and takes another mathematical form.
In the
example previously quoted, the total available energy of body b2
is
+ small terms of higher
order
(40)
(This
notion has no equivalence in conventional relativity for which the energy
of a body is completely relative and depends on its speed with respect to
another body).
And the
available energy with respect to frame S1 is
(41)
4/Possible
measurement of the absolute speed of an "inertial" system
Assuming
that  , the kinetic energy needed to move from S1
to S2 reduces to:
 (42)
knowing  and  , it is theoretically possible
to measure the absolute speed,  , of the "inertial" system S1,
that is:
(43)
This
result is also in contradiction with the relativity principle.
5/Conservation
of energy
In our
opinion, the mass-energy conservation law should not be questioned and
should apply exactly in any "inertial" frame. Note nevertheless that, at high
speeds, the role of the aether wind would not be completely negligible and
should be taken into account in any event where an exchange of energy
occurs.
References
1 - J.
Levy, Hidden variables in Lorentz transformations, Physical Interpretations
of Relativity Theory (PIRT) (1998), supplementary papers p 86.
A symposium sponsored by the British Society for the Philosophy of Science,
M.C Duffy Chairman. Updated in the web site
www.levynewphysics.com
2 -
ibid, Extended space-time transformations for a fundamental aether theory. PIRT
(2002), M.C Duffy Editor. Updated in the web site www.levynewphysics.com
3 -
ibid, Relativity and Cosmic Substratum, PIRT (1996), M.C Duffy Etidor,
precirculated proceeding p 231. Updated in the web site
www.levynewphysics.com
4 -
ibid, Is simultaneity relative or absolute, in Open questions in
relativistic physics, F. Selleri Editor ; Apeiron, 4405 rue St Dominique, Montreal, Quebec
H2W 2B2 Canada. E-mail:apeiron@vif.com,
updated in the web site
www.levynewphysics.com
5 -
ibid, Is the relativity principle an unquestionable concept of physics,
PIRT (1998), M.C Duffy Editor, late papers p 156. Updated in the web site www.levynewphysics.com
6 - F.
Rohrlich, Am J Phys, 58(4), p 348, (1990).
7 - G. N Lewis , phil mag, 16, 705, (1908).
8 - J. Levy, Relativité et Substratum Cosmique, a book
of 230 p, Lavoisier, Cachan, France (1996), E-mail: edition@Lavoisier.fr
9 - J. Levy," Basic concepts for a fundamental aether theory", and "Aether theory and the principle of relativity" in Ether space-time & cosmology Volume 1 Michael C. Duffy and J. Levy Editors, PD Publications, Liverpool, UK, March 2008.
*We call "inertial" the frames in which a body at rest is not submitted to any perceptible external force, a term sanctionned by use. But we must be aware that insofar as an aether drift exists, real frames are never strictly inertial.
* The main ideas of this
manuscript were registered at the French Society of authors on march 20th
2001.
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