The Collapse of the Lorentz Transformation
Paul Marmet
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://www.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, 2401 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
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Original paper, May 2004
Revised paper, August 2004