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The Collapse of the Lorentz Transformation
Paul Marmet
( Last checked 2018/01/16 - The estate of Paul Marmet )

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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.

 Lo=100 m 18
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:
 g=1.005037815 20
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 (56, 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 (56, 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, 68), 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 (568).  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 (56, 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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Original paper, May 2004
Revised paper, August  2004