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.
-
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:
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:
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:
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:
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:
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 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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