Chapter Seven
The Lorentz Transformations in Three
Dimensions.
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 |
|
