The Collapse of the Lorentz
Transformation
Paul Marmet
(
Last checked 2018/01/16 - The estate of Paul Marmet )
Eine deutsche Übersetzung dieses Artikels finden Sie hier.
The Breakdown of the Lorentz Transformation
Abstract.
Following the observation that the velocity of light with
respect to a moving observer appears constant in all frames,
independently of the velocity of the moving frame, Lorentz
proposed a transformation of coordinates of space and time to
allow for the velocity of the moving frame. However, we
show that the solution found by Lorentz, does not lead to a
constant velocity of light. On the contrary, we show that
the Lorentz solution is an average velocity between light
traveling in two directions, and that the velocity of light in
each direction is never equal to the velocity c just as with the
Galilean coordinates. The difference between the Galilean
transformation and the Lorentz transformation is that, in the
latter, the average velocity is constant after two light paths,
traveling in opposites directions. This result is
certainly not compatible with the general definition of a
velocity in physics. We also present a numerical example
to the Lorentz transformation, which confirms that the velocity
of light is not constant for the observer in the moving
frame. After calculating that the constant velocity of
light is not compatible with the Lorentz transformations, we see
that no other acceptable mathematical functions can solve that
problem of a constant one-way velocity of light in all
directions, unless the time and length dilation factors change
with the direction light is traveling. Such a solution is
not acceptable in physics. A realistic solution is found
in compatibility with a new interpretation of the
Michelson-Morley experiment, in which secondary phenomena are
taken into account. We can see how the constant velocity
of light in a moving frame is only apparent. It is found
that an isotropic length dilation or contraction (g times) coupled with the usual
slowing down of clocks (g
times) leads to a complete realistic solution of the problem
compatible with all observational data.
1 - Lorentz Transformation.
The Lorentz transformations (1) published in 1904,
were first discovered by Woldemar Voigt in 1887 when studying
the elastic theory of light, even if they are generally
attributed to Lorentz. Joseph Larmor (2) arrived
at the same conclusion in 1900 and also Henri Poincaré in
1905. The aim of the Lorentz
transformations (1) is to calculate the relationships
between the lengths and time units between a frame supposedly at
rest and another frame in motion, assuming that the same
velocity of light is measured in both frames.
Let us
assume that a pulse of light is emitted at time to=0 from the origin of coordinates of a frame Fo at rest, at the same instant (to=0), the origin of a moving frame FV passes at that same location. This is illustrated on
figure 1.
The
observers in each frames use the proper units existing in their
own frame. The indexes o or v, characterize the parameters
measured in the rest and moving frames. Figure 1 gives an
illustration of the initial conditions when both frames are
superimposed, at the moment light is emitted from the origins of
coordinates.
At the
instant t=0, the clocks, presumably running at the same rate in
both frames, are synchronized to show the same clock
display. We have:
 |
1 |
On figure
1, at the instant a pulse of light is emitted from the origin 0o of the rest frame Fo,
the origin 0v of the moving
frame Fv moving at velocity v,
is superimposed on origin 0o of
the rest frame Fo. Later,
after a time interval, figure 2 shows the relative position of the
two frames and the curved wavefront of the transmitted wave.
Light emitted at an angle d with
respect to the Xo-Xv axis, reaches the coordinates (xo,yo) after some time
interval, as shown on figure 2.
In order
to simplify the illustration, a rotation has been made around the
Xo-Xv axis, so that any Zo
component (perpendicular to the paper sheet) becomes zero.
Figure
2
shows the Xo-0-Yo and the Xv-0-Yv planes, and also the
spherical wavefront, as it exists “at a given instant”. The
aim of the Lorentz problem is to test the invariance of the
velocity of light, comparing observations in the moving frame with
respect to the observations in the frame at rest. It is
assumed that both observers find, that the measured velocity of
light is always given by the same number (~300 000 Km/s) in both
frames.
There is a difficulty in the usual Lorentz formulation due to the
fact that in that conventional transformation, it is clearly
established that light moves to location (xo, yo), but it is not
shown how light carries on from that point (xo, yo) to each
observers in order to be detected. If light does not go back
to the observers, no experiment is possible. We will see
below that changing the velocity of light from +c to –c after
reflection should have been considered. Consequently, since
this condition is unspecified in the Lorentz calculation, the
reader can speculate, about the behavior of light which reaches
the observers, during the last phase of the experiment.
Logically, we must then consider that “something” reflects or
diffuses light at location (xo,
yo) towards the observer at the
moving origin.
2 – Lorentz Problem along the X-axis.
On the
rest frame, we know that the distance ro traveled by light is:
 |
2 |
However,
it is assumed that the velocity of light is also c in the moving
frame. Consequently, in the moving frame, the distance rv traveled by light is, using the moving coordinates:
 |
3 |
In
Galilean geometry, the distance ro traveled in the rest frame (see figure 2) is given by
the mathematical relationship:
 |
4 |
Equations
2 and 4 give:
 |
5 |
Similarly
to equation 5, but in the moving system of coordinates Yv-O-Xv, (that uses the
observations made in the moving frame), the corresponding distance
is written:
 |
6 |
We
consider here that the velocity of a moving frame is along the
X-axis. Therefore the Y and Z axis are unaltered by the
motion along the X-axis. This gives:
 |
7 |
After
substitution of equation 7 in equations 5 and 6, the difference
between these two equations gives:
 |
8 |
We also
know that, in the rest frame, the distance traveled is xo given by:
 |
9 |
In the
moving frame, the corresponding coordinate is:
 |
10 |
The
solution to equations 8, 9 and 10 is:
 |
11 |
and
 |
12 |
The
definition of g is:
 |
13 |
It can be
verified that equations 11, 12 and 13 are the solutions to these
equation by substituting equations 11 and 12 into the right hand
side of equation 8. Since there is no velocity component
along the Y and Z axis, there is no change of coordinates along
these axes. We have:
 |
14 |
and
 |
15 |
3 - Origin of the Error.
We have
mentioned above that, in order for light to be observable by a
moving observer, light must necessarily reach a remote location
and come back to the local observer. That motion of light
in the forward and backward directions, where the speeds are
respectively (c-v) and (c+v) is not taken into account in the
Lorentz transformation. It is not taken into account that
the velocity of light passes from +c to –c after
reflection. Only the “square of the function” is
considered. Let us examine equation 8 above, which
is:
|
16 |
In
equation 16, we find that the velocity of light in the moving
frame must be compatible with 
. Let us
examine a mathematical identity to that term . We have:
 |
17 |
Simple
mathematical transformations show that the mathematical identity
of equation 17 is always valid. Therefore, the term does not necessarily imply a constant velocity
of light in the moving frame. On the right hand side of
equation 17, we see that it implies a variable velocity equal to
(c-v) in one direction (with the expected factor 1/2) and (c+v) in
the other direction (with the other factor 1/2). Therefore
the interpretation of a constant velocity of light currently given
to the left hand side of equation 17, is in fact a variable
velocity equal to (c-v) in the forward direction and (c+v) in the
backward direction, just as expected in Galilean
coordinates. The physical reality corresponding to this
mathematical identity will also be illustrated with a numerical
example below. Consequently we must understand that the
length contraction suggested by Lorentz does not have to be
compatible with the constant velocity of light, as seen in
equation 17, because it depends on whether light is going forward
or backward.
This
error can be best seen when using a numerical example. In
the following section, we calculate the velocity of light as it is
implied using the Lorentz transformation. We will see that
in the moving frame, using the moving units, the Lorentz equations
do not give a constant one-way velocity of light in the moving
frame. In fact, the Lorentz transformations predicts only
the transformation that gives an “average” velocity of light equal
to c, which means that the velocity of light is slower in the
forward direction and faster in the backward direction, in the
moving frame, just as illustrated in equation 17. This is
certainly not compatible with the hypothesis of a constant
velocity of light.
4 - Calculation.
In
order to simplify the demonstration, we consider only one
dimension, along the X-axis. Therefore, in our
numerical test, we consider the Lorentz transformation along the
X-axis. In the numerical calculation, we assume that the
moving frame Lo was initially
100 meters long, when previously at rest and measured in the
rest frame.
We also
assume in our demonstration that the velocity of the moving frame
is equal to one tenth of the velocity of light. We have:
 |
19 |
Therefore
the value of g defined in equation 13
is:
In our
example, the moving frame always travels at velocity v, along the
positive values of the X-coordinates.
Let us calculate the distance traveled by light when
the distance traveled in the frame moving at velocity v, from the
origin O, is equal to 100 meters.
Symbol 
means that, light travels toward the
right hand side direction.
On figure 3, the moving frame is illustrated three
times for more clarity. Let us consider the distance
traveled by light between frame locations A and B. The
symbol [rest] means that the distance traveled by light is
calculated in the rest frame. Before (the assumed)length
contraction, the distance traveled by light before reaching the
other end of the moving frame is:
 |
21 |
Correspondingly, when light travels toward the left hand
side 
, and light travels 100 meters in the
moving frame, the distance traveled by light, between frame
locations B and C (figure 3) as measured in the rest frame, is:
 |
22 |
However,
according to Lorentz, since the frame is moving along the X-axis,
its length is contracted g times. When
the Lorentz “length contraction” is taken into account, this is
represented by the symbol (L.C.) and illustrated on figure
4. Therefore, the observer at rest, measures a length g times shorter.
From equation 20 and 21, when light moves in the
direction , after taking into account the
length contraction, the distance traveled by light, illustrated on
figure 4, is:
 |
23 |
Similarly, from equation 22, after taking into account length
contraction, when light moves in the opposite direction , we get:
 |
24 |
We know
that the distances traveled by light in both equations 23 and 24,
divided by c, gives the time for light to travel across the width
of the moving frame. However, according to Lorentz, in the
moving frame the time (local clocks) runs g
times slower. Using equation 23, taking into account
the "Lorentz time dilation" (T.D.), we find that the time between
frame locations A and B (figure 4) for the moving observer is:
 |
25 |
Using
equation 22, also taking into account time dilation, we find the
time between frame locations B and C (figure 4) on the moving
frame is:
 |
26 |
In
physics, when a body moves at a variable velocity, the "average"
velocity 
is defined as the sum (S ) of distances, divided by the sum (S ) of time intervals taken to travel that
distance. We have:
 |
27 |
Let us
calculate the average velocity of light (complete trip) (between
frame locations A and C on figure 4), as measured on the moving
frame. Using equations 25 and 26, the average velocity [mov]
of light traveling 100 meters in both directions in the moving
frame is:
 |
28 |
Equation
28 shows that the “average velocity of light”,
during the sum of the two light paths in the forward and backward
directions, is equal to c. However, during these two trips,
at no moment, light travels to velocity c for the moving
observer.
Let us
calculate the exact velocities in each direction. We know that due
to the velocity of the frame, lengths along the X-axis are
contracted g times. Therefore,
not only the frame, but also the standard reference unit of
length, moving with the frame is contracted g
times. Using the local reference meter, the length measured
by the moving observer is therefore 100 local meters. We
have:
 |
29 |
Using
equations 25 and 29, we find, the velocity of light “Light ” measured on the moving frame at velocity
v[mov], is:
 |
30 |
Similarly, using equations 26 and 29, we find, the velocity of
light “Light ” measured on the moving
frame at velocity v[mov] is:
 |
31 |
Equations
30 and 31 show that, using Lorentz length contraction and time
dilation, the velocity of light is not constant in the moving
frame. For the moving observer, the velocity of light is
slower in the forward direction, and faster in the backward
direction. Following the Lorentz transformations, the only
thing that has been changed, is that the “average”
velocity of light is equal to c, when a two way measurement of
velocity of light is done.
This
numerical example is identical to the mathematical identity given
in equation 17. It is quite clear that the velocity of light
"measured" in the moving frame is not c, but it is (c-v) and (c+v)
depending on the direction light is traveling. This is
mathematically compatible with equation 17. The Lorentz
equation does not solve the problem of a constant velocity of
light in a moving frame. On the contrary, the Lorentz
equations correspond to finding a solution so that the quadratic
quantity
is constant. That is quite a
different matter. That solution is not compatible with the
concept of a “constant velocity of light” in a
moving frame.
5 - Consequences Related to the Defective Lorentz
Transformation.
5-A
Non-constant one-way velocity of light in the Lorentz
transformation.
After a
century, it is astonishing to discover that the Lorentz
transformation, that requires a distortion between the X and Y
axis, does not lead to a constant (one-way) velocity of light
when "measured" in the moving frame. Even the Lorentz
“time distortion” added to a “length distortion” between the X
and Y axis does not lead to a constant velocity of light in the
moving frame. However, we must accept this fact, which is
clearly demonstrated here. The solution to the quadratic terms
used by Lorentz leads to an “average”, rather than a “real”
velocity of light, between two light paths traveling in opposite
directions (as shown mathematically in equation 17, and
illustrated on figure 4). This result is certainly not
compatible with an authentic constant velocity of light.
If we try finding a mathematical solution to a real one-way
constant velocity of light, we can find some other esoteric
solutions, but they are not acceptable in physics. For
example, a mathematical solution requiring the hypothesis that
“time” and “lengths” in the moving frame are a function of the
direction of light is not acceptable. Instead of
considering such a non-sense, it is preferable to look for
another more rational solution, which in fact already
exists.
5-B
Comparison with the Michelson-Morley Experiment.
Looking for Another Solution.
The
incompatibility between the Lorentz solution and the constant
velocity of light appears surprising to many people, because the
Lorentz transformation seems to agree with the well accepted
Michelson-Morley experiment, which also involves the constancy
of the velocity of light. In the past, the asymmetric
length distortion between the X and Y axes predicted by Lorentz
“appeared” as a natural explanation to the fact, that there is
no shift of interference fringes in the Michelson-Morley
experiment. However, since we see now that the Lorentz
transformation does not lead to a genuine constant velocity of
light, it is imperative to consider the Lorentz transformation
and the Michelson-Morley simultaneously, in order to guarantee
their compatibilities. Since the asymmetric
distortion between the X and Y axis predicted by Lorentz does
not solve the problem of a constant velocity of light in the
moving frame, it is important to inquire whether there exists
any other solution capable of solving that problem. Of
course, one solution is that the velocity of light is not
constant in moving frames. The “apparent” velocity of
light would be just an “illusion” due to an erroneous
interpretation of the measurement. Since the assumed
constant velocity of light has been accepted under the influence
of the Michelson-Morley experiment, we must reexamine that
experiment.
5-C
Investigating the Validity of the Interpretation of the M-M
Experiment.
It has
usually been claimed that the Michelson-Morley experiment proves
that space-time distortion needs to exist in order to explain
the absence of shift of interference fringes in the
Michelson-Morley experiment. This is an error.
It has been shown (3) that, in the
calculation of the Michelson-Morley experiment, some fundamental
phenomena have been ignored. For example, it has been
totally overlooked that the angle of reflection on a moving
mirror at 45 degrees is not 90 degrees. Due to the
velocity of the mirror, there is an anomalous angle of
reflection. This has been clearly demonstrated (3) using the Huygens principle. It seems
that this anomalous angle of reflection on a moving mirror was
previously unknown. Furthermore, the Bradley aberration,
which changes the angle at which light is moving across the
moving frame, has also been ignored. It is shown (3) that, when we take into account the Huygens
principle and the Bradley effect, we find that there should
exist no shift of interference fringes, when the
Michelson-Morley interferometer is rotated, (considering that
the asymmetric Lorentz length contraction does not exist).
Consequently, the absence of any asymmetric Lorentz length
contraction is perfectly compatible with the zero shifts of
interference fringes in the Michelson-Morley experiment in a
Galilean space. As a result, when taking into account the
phenomena overlooked in the Michelson-Morley calculation, we
find that the Michelson-Morley data are in perfect agreement
with a symmetric distortion in the Lorentz transformation.
The esoteric (space-time distortion) explanation suggested by
the Lorentz transformation is useless. Therefore, there is
no need to look for a new solution to the Lorentz equations,
because the data obtained in the Michelson-Morley experiment is
already compatible with a non-asymmetric distortion in the
Lorentz transformation. There exists no asymmetric
distortion between axes.
5-D
Experiments Proving that the Velocity of Light is Equal to
c±v in the Moving Frame?
We must
definitively look for experiments showing that the velocity of
light in a moving frame is equal to c±v. Until now, it is
generally believed that the velocity of light is c in a moving
frame, but we must examine how this is an illusion. It has
been demonstrated previously that using the GPS, (4),
the velocity of light is c±v in perfect compatibility with the
velocity of rotation of the Earth. The fact that one must
make a correction (4) involving the velocity
v of Earth rotation in the GPS, which is the same as in the
Sagnac effect, is one of the proofs that the velocity of light
is c±v at the surface of the rotating Earth.
Unfortunately, most people failed to see that the need to use
the velocity v of the Earth surface in the calculation implies a
velocity equal to c±v.
5-E
How the Moving Clock Shows a Different Clock Rate!
The
fact that the one-way velocity of light equal to c is only
apparent, has been explained (5, 6, 7) previously. This
illusion is due to a phenomenon involving the increase of
kinetic energy needed to carry (the atoms of) the clock from the
rest frame to the moving frame. Using quantum mechanics,
it has been shown (5, 6, 7) that using the principle of
mass-energy conservation, the increase of velocity (kinetic
energy) produces a change of energy (quantum levels) to the
electrons in atoms, which is responsible for a shift of quantum
levels of all atoms in the moving frame. That shift of
quantum levels (5, 6, 7) makes the moving clocks run at a different
rate.
Since
the time in the moving frame is determined using a clock that
has been moved from the rest frame to the moving frame, the
change of clock rate is unnoticeable to the moving observer,
because everything (all matter) in the moving frame is submitted
to that change of Bohr radius. However, that natural
change of clock rate due to the acquisition of kinetic energy in
atoms, is responsible (5, 6,
7) in the measurements, for the
difference between the apparent value c, and the real velocity
c±v. As demonstrated previously (5, 6, 7) in the moving
frame, the velocity appears to be c, while in fact it is
c±v. Therefore, using quantum mechanics, the illusion of
the constant velocity of light is well explained, due to
mass-energy conservation, which changes the Bohr radius, that
changes the size of the atoms and also the energy of the quantum
states, which finally, changes the clock rate.
Therefore, it is totally useless to look for an asymmetric
distortion between the X and Y-axes that might lead to a
constant velocity of light in the moving frame, since anyhow,
the velocity of light is not constant in the moving frame.
The constant velocity of light in the moving frame is nothing
but an illusion. That error has been erroneously supported
(3) by the erroneous belief that the
null result in the Michelson-Morley experiment proves that the
velocity of light is constant in the moving frame.
5-F
Since Asymmetric Distortions between Axes are Useless, what
Kind of symmetric Solutions Are Acceptable?
There
is another question that must be resolved. Since the
asymmetric Lorentz length contraction is useless to explain the
apparent constant velocity of light, do we need any symmetric
dilation or contraction to explain all the physical
observations? We can see that any symmetric dilation of
contraction along the three axes can become compatible with the
observed apparent constant velocity of light in a moving frame
(and therefore a null result in the Michelson-Morley
experiment). For example, if we consider moving cubes
(Axes along, X, Y and Z) having various sizes, we can see that,
using the Michelson-Morley experiment, any size of moving cubes
is compatible with the apparent constant velocity of light in
that moving frame. The Michelson-Morley data will be
independent of the size of the moving cubes. Therefore any
symmetric dilation or contraction along all three axes gives a
null result in the Michelson-Morley experiment.
5-G
What Physical Principle Can we Use to Find the Correct
Parameter for Length Dilation or Contraction?
Since
any symmetric length dilation or contraction (or none) are
compatible with the Michelson-Morley experiment, does any change
of size exist, when a body is accelerated to high
velocities? The answer has been already solved using the
principle of mass-energy conservation and quantum mechanics. Due
to the increase of kinetic energy as a function of velocity, the
mass of particles, like atoms, electrons and protons retains
that energy (5, 6, 8), which is added as mass to the carrying
particle. That increase of mass of electrons inside atoms
changes the quantum states of these atoms. We can see that
in the case of the hydrogen atom, mass energy conservation
requires a slight change of electron mass, which leads to a
change of Bohr radius when we apply quantum mechanics, which
then changes the Bohr radius. Calculation shows (5, 6, 8),
that this change of size of the Bohr radius increases exactly as
the parameter g. Furthermore,
at the same time, we find that the energy of transition of these
new quantum states is reduced g
times (5, 6, 8). Therefore, similarly, atomic clocks
are slowed down g times. This
is in perfect agreement with the slowing down of time (change of
clock rate) usually considered in relativity.
Consequently, we see now that the principle of mass-energy
conservation and quantum mechanics provides us with a realistic
coefficient of dilation of bodies as a function of
velocity. As a function of velocity, we see that the
physical length of bodies increases g
times due to the increase of the Bohr radius following the
kinetic energy transferred to the electrons. Of course,
just as the shape of the orbitals in quantum mechanics, the size
of the wave functions increases symmetrically, so that all the
X, Y, and the Z axis increases equally by the same quantity g. This is demonstrated in the
papers (5, 6, 8).
We must
conclude that there is no need of any weird interpretation
requiring non-realistic physics and the denial of conventional
logic. There is no need of space contraction, or time
dilation. The size of matter changes, due to the change of
Bohr radius, that also makes clocks run at a different
rate. Everything can be explained naturally using
conventional logic, mass-energy conservation, and the equations
of quantum mechanics. Finally, we can see that these
explanations are complete and coherent without having to
hypothesize the existence of ether.
The
author wishes to thank Dennis O’Keefe and G. Y. Dufour for
reading this paper and giving useful comments and suggestions.
6 - References.
1 - H. A. Lorentz, Proc. Acad.
Sci. (Amsterdam), 6, 809 (1904).
Also: http://galileo.phys.virginia.edu/classes/252/lorentztrans.html
2 - J. Larmor, “Aether and
Matter”, (Cambridge University Press, 1900)
3 – P. Marmet, “The
Overlooked Phenomena in the Michelson-Morley Experiment”
To be published.
http://www.newtonphysics.on.ca/michelson/index.html
4 – “The GPS and the
Constant Velocity of Light” Paul Marmet.
On the Web at: http://www.newtonphysics.on.ca/illusion/index.html
also:
P. Marmet, Explaining the Illusion of
the Constant Velocity of Light, Meeting "Physical
Interpretations of Relativity Theory VII" University of
Sunderland, London U.K., 15-18, September 2000. Conference
Proceedings "Physical Interpretations of Relativity Theory VII"
p. 250-260 Ed. M. C. Duffy, University of Sunderland.
Also: "The GPS and the Constant Velocity of
Light. Acta Scientiarum", Universidade Estadual De
Maringá, Maringá-Paraná-Brazil, Vol. 22 No: 5, page
1269-1279, December 2000. Also: "The GPS and the
Constant Velocity of Light". NPA Meeting University
of Conn. Storrs, Connecticut in June 2000.
Also, "The GPS and the Constant Velocity of
Light", Galilean Electrodynamics Vol. 14, No: 2, p.
23-30, March/April 2003.
5 – P. Marmet, “Einstein's
Theory of Relativity Versus Classical Mechanics” pp.
200 pages, Ed. Newton Physics Books, Ogilvie Road, Ottawa,
Ontario, Canada, K1J 7N4
On the Web at: http://www.newtonphysics.on.ca/einstein/index.html
6 – “Natural Length
Contraction Mechanism Due to Kinetic Energy”.
P. Marmet
On the Web at: http://www.newtonphysics.on.ca/kinetic/index.html
Also: Invited paper, Journal of New Energy, ISSN
1086-8259, Vol. 6, No: 3, pp. 103-115, Winter 2002.
7 – “A Detailed Classical
Description of the Advance of the Perihelion of Mercury”.
P. Marmet
On the Web at: http://www.newtonphysics.on.ca/mercury/index.html
A similar paper has been published under the
title: "Classical Description of the Advance of the
Perihelion of Mercury" in Physics Essays, Vol. 12, No:
3, 1999, P. 468-487.
Paper presented at the International Meeting:
"Galileo Back in Italy II" Bologna Italy, 26-30 May 1999,
Title: "Einstein's Mercury Problem Solved
in Galileo's Coordinates". This paper is printed in
the proceedings: "Galileo Back in Italy"
Istituto di Chimica, "G. Ciamician", Via
Selmi 2 - Bologna, Italy. P. 352 to 359.
Also, invited speaker, meeting of the “Society for
Scientific Exploration” Albuquerque, June 3-5, 1999.
Title: "A Logical and Understandable
Explanation to the Advance of the Perihelion of Mercury"
8 – “Natural Physical
Length Contraction Due to Gravity” P. Marmet
On the Web at: http://www.newtonphysics.on.ca/gravity/index.html
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Papers on the Web
Original paper, May 2004
Revised paper, August 2004