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 oneway 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 MichelsonMorley experiment. The existence of
secondary phenomena in the MichelsonMorley 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" oneway 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 massenergy conservation and quantum
mechanics. The relevance of chapter 7 can be understood
after the study of the two above mentioned papers.

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:
In
Cartesian coordinates, the same sphere is represented by:
x^{2}+y^{2}+z^{2} = r^{2} 
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 u_{x}) and an initial frame
supposedly at rest are called Galilean. When the velocity of an
object is given by V_{x}, V_{y} and V_{z}
with respect to a frame at rest, the velocity components V_{x}'
, V_{y}' and V_{z}' of the same object with
respect to the moving frame are:
The
description given by the parameters V_{x}', V_{y}'
and V_{z}' is quite identical to the description given by
V_{x}, V_{y} and V_{z} knowing that the
moving frame has velocity u_{x}. 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 "y_{o}" 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:
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, V_{o} 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 D_{o}=0) of the row
of balloons is Y_{o} = 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 X_{o}
and Z_{o} 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:
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 u_{y} = 0 and u_{z} = 0, we
find:
and
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 massenergy conservation.
7.4  Symbols and
Variables.
D 
depth of the balloon 
V 
volume of the balloon 
V_{o} 
volume of the balloon at sea level 
y 
diameter of the balloon 
y_{o} 
diameter of the balloon at sea level 
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