Einstein's Theory of Relativity

               versus Classical Mechanics

                                        by   Paul Marmet 
Chapter Seven
The Lorentz Transformations in Three Dimensions. 
          Important Note:
          We must recall, that there are two aspects in the Lorentz Transformations. There is the mathematical aspect, which consists in the mathematical solution of the equations established by Lorentz.  This is discussed in the paper: http://www.newtonphysics.on.ca/lorentz/index.html
          In that paper we show that the transformations calculated by Lorentz are compatible "only" with an "average" velocity equal to c, when light makes a complete travel in the moving frame.  It is shown that the transformations calculated by Lorentz do not lead to a constant velocity of light, when light makes a one-way travel in the moving frame. 
          The second aspect of the Lorentz transformations, is related to the physics involved, so that the velocity of light "measured" in a moving frame, appears to be equal to c in any direction.  In that case, we demonstrate two previously ignored secondary phenomena taking place in the Michelson-Morley experiment.  The existence of secondary phenomena in the Michelson-Morley experiment is demonstrated in the paper:  http://www.newtonphysics.on.ca/michelson/index.html
          Following these above considerations, and in order to be compatible with the experimental observations about the "observed" one-way constant velocity of light in a moving frame, we need a transformation of matter which is described here.  This chapter 7  is required due to the principle of mass-energy conservation and quantum mechanics.  The relevance of chapter 7 can be understood after the study of the two above mentioned papers. 
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7.1 - Basic Principles of a Transformation.
        The Lorentz transformations are usually considered as nothing more than a transformation of coordinates between a rest frame and a moving frame. They appear as geometrical transformations of coordinates. Let us consider the fundamental meaning of such transformations. Let us first have a look at the geometrical transformation of Cartesian coordinates into spherical coordinates. We find that the equation of a sphere in spherical coordinates is:
r = constant  7.1
        In Cartesian coordinates, the same sphere is represented by:
x2+y2+z2 = r2 7.2
        Equations 7.1 and 7.2 represent the same physical or geometrical object. Such a transformation does not change anything to the physical system described. Absolutely no physics is involved in such a change of coordinates because these transformations are purely mathematical. However, one system of coordinates (the spherical coordinates) can be more suitable mathematically to study rotational motion or a particular orientation in space.
        Geometrical transformations used to transform coordinates between a moving frame (at velocity ux) and an initial frame supposedly at rest are called Galilean. When the velocity of an object is given by Vx, Vy and Vz with respect to a frame at rest, the velocity components Vx' , Vy' and Vz' of the same object with respect to the moving frame are:
Vx' = Vx-ux 7.3
Vy' = Vy 7.4
Vz' = Vz 7.5
        The description given by the parameters Vx', Vy' and Vz' is quite identical to the description given by Vx, Vy and Vz knowing that the moving frame has velocity ux. Therefore these transformations of coordinates involve no physics at all. They represent the same physical object using a different system of coordinates. They are just mathematical transformations.
        However, in some other cases, physical phenomena necessarily accompany a change of coordinates meaning that some physical changes are related to a change of frame of reference. Let us consider an example of transformation of coordinates in which there is a physical phenomenon taking place at the same time as a change of coordinates. This is the case of a boat sinking at sea. Inside the boat, there are five spherical balloons inflated with air, glued to each other along a vertical line (Y axis). At the surface of the sea, the diameter "yo" of each balloon is one meter. Therefore the row of balloons is five meters long. As the boat sinks to great depths, due to the increase of pressure the gas inside the balloons is compressed and the diameters get smaller as a function of depth. Consequently, the length of the row gets more and more contracted with depth. We know that the relationship between the volume of a gas and its pressure at a constant temperature is given by:
PV = constant  7.6
        We also know that the volume of a constant amount of air as a function of pressure (and therefore depth D) is given by:
7.7
        where D is the depth in meters from the surface, Vo is the volume of the balloon at atmospheric pressure when located at the surface of the sea and V is the volume of the gas at different depths. At normal atmospheric pressure, the value of A equals 9.8. The relationship between the diameter y and the volume V is:
7.8
        From equations 7.7 and 7.8, we get:
7.9
        Equation 7.9 gives the relationship between the diameter y of each balloon as a function of the depth D.
        Let us consider a moving frame of reference y' going down with the sinking ship and having its origin at one end of the row of balloons. Since the initial length (at Do=0) of the row of balloons is Yo = 5 meters, the length Y' of the axis at depth D is given by:
7.10
        The important point to notice is that when the balloons sink into the sea, there is not only a change of coordinates of the balloons with respect to the original frame, there is also a change in the length of the row of five balloons due to the compression of the gas which is a function of the distance of the balloons from the surface. This is an example where the relationship giving a transformation of coordinates is necessarily related to a physical phenomenon.
        Let us now complete these considerations for the other axes. We need again to consider the physical phenomenon involved to show that the X and Z diameters of the balloons decrease simultaneously when the pressure contracts the gas. This gives:
7.11
7.12
        where Xo and Zo are equal to one meter. Equations 7.11 and 7.12 can be written only because we know the exact physical phenomenon taking place (a compressed balloon contracts equally on all three axes). A mathematical transformation of coordinates alone cannot describe whether the other axes X and Z will also be contracted. Physics is needed to give information about what happens in the X and Z directions. Equations 7.11 and 7.12 are quite conclusive because we know the physical phenomenon that accompanies the mathematical transformation.

7.2 - The Lorentz Transformations.
        Let us now consider the case of the Lorentz transformations. We have seen that they are not pure geometrical transformations since there are physical conditions involved with the transformations. There is a change of mass of the electron due to the kinetic energy of the particle. Of course, the experiment with the balloons is quite different from the change of size of atoms when they acquire kinetic energy. However, both experiments have in common that the size of the objects depends on a well identified physical phenomenon and not on a simple change of coordinates. For the balloons, the pressure changes their size by compressing the gas in them. For atoms, the change of kinetic energy changes their size and the inter atomic distance in molecules.
        Quantum mechanics predicts that the distribution of the wave function of an electron around the nucleus does not get flattened when the electron mass increases. The increase of the electron mass changes the size of the wave function equally in all directions.
        The hypothesis of Lorentz and Einstein that the other axes do not change and that the transformations are purely geometrical is not compatible with the physics implied in the calculations of quantum mechanics. It is quite clear that the change of the electron mass changes the distribution along all three directions. Nobody in quantum mechanics has ever suggested flatter wave functions (and flatter atoms and molecules) when the electron mass is larger. Consequently, when an atom is accelerated in one direction, the size of the atom or the length of the intermolecular distance changes in all three directions. Therefore the assumption in relativity that there is no change of size of the coordinates Y and Z while the coordinate X is changing is an error that must be corrected.

7.3 - The Equations.
        We have seen that in the direction of the velocity (the X direction) there is a physical mechanism leading to the Lorentz equation for the X axis given in equation 3.55:

x' = g(x-uxt)  7.13
        Since this result comes from quantum mechanics which predicts a symmetry in all three directions when the electron mass (which is a scalar) changes, we must conclude that the phenomenon of length dilation is just as valid in the transverse directions than in the longitudinal direction. Using Lorentz and Einstein's choice of coordinates x, y and z, let us write the transformation of coordinates for the transverse directions y and z due to the change of the Bohr radius as given by quantum mechanics. From equation 7.13 with uy = 0 and uz = 0, we find:
y' = g 7.14
        and
z' = g 7.15
        We conclude that the previous description given by Lorentz and Einstein which assumes a transformation in only one dimension (which has never been observed in any experiment) is erroneous because it is not compatible with quantum mechanics and with the principle of mass-energy conservation.

7.4 - Symbols and Variables.

 
D depth of the balloon
V volume of the balloon
Vo volume of the balloon at sea level
y diameter of the balloon
yo diameter of the balloon at sea level
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