Copyright © 1999 David E. Rutherford
All Rights Reserved
The Transformation Equations
We wish to find the most general set of transformation equations that assure
the covariance of the laws of physics. Since the spacetime interval is described
by a Euclidean metric, in this theory, the transformation matrix we will
use must be orthogonal. We are making a transformation from a stationary
unprimed frame of reference to a uniformly moving primed frame, so we assume
that the components of the transformation matrix contain the components of
the four-velocity,
, of the
moving frame. But, the components must be dimensionless so that the transformed
quantity has the same units as the original quantity. Therefore, we divide
each component of the transformation matrix by the invariant speed of light,
. We also
suspect that each component of the transformed quantity should depend on
all of the components of the original quantity. With this in mind,
we choose the most simple combination of components that satisfy these
requirements for the transformation matrix,
,
where the 
are the components of the four-velocity of the primed frame relative to
the unprimed frame. The
"
" and
"
" signs
preceding some of the terms in (19) imply that these terms can be
either plus or minus without affecting the orthogonality. That is,
the signs of these terms may be taken as the top sign in one transformation
and the bottom sign in another transformation, while the terms preceded by
"+" or "-" signs remain fixed. However, the choice of sign must be consistent,
in other words, we must take either the top sign, throughout, or the bottom
sign, throughout. For example, if we take the "+" part of the
component
in (19), we must take the "-" part of
, the "-"
part of 
,
the "+" part of
, etc..
Due to the seemingly indeterminate nature of these terms, we suspect that
they exist as a kind of mixture, or superposition, of both signs,
simultaneously. It is only when the sign of a term needs to be specified,
that the term becomes associated with one sign or the other. Therefore, these
signs might, more properly, be called plus and minus, rather than
plus or minus signs, until one or the other sign is specified. However,
even though both of these signs (plus and minus, and minus and plus) are
combinations of both plus and minus, they retain their opposite nature. In
addition, the specification of the sign of one of these indeterminate terms
immediately establishes the signs of all the other indeterminate terms
completely.
The Matrix (19) replaces the Lorentz transformation matrix. The coordinate
transformation equations from the unprimed frame to the primed frame associated
with the matrix,
, are
where
The transformation equations in Eqs. (20) replace the Lorentz transformation
equations. For a velocity in the x-direction, we have a rotation of
the x'-t' plane relative to the x-t plane, but we also have,
contrary to relativity, a rotation of the y'-z' plane relative to
the y-z plane. The axes are rotated by the angle
. Looking
from the unprimed frame in the direction of the positive x-axis, the
rotation of the y'-z' plane relative to the y-z plane, due
to the indeterminate terms in (19), is in both the clockwise and
counter-clockwise directions, simultaneously, until one direction or the
other is determined by additional conditions. These rotations are not necessarily
observable as rotations. They may be manifested as seemingly unrelated properties
of bodies in motion, such as precession, time dilation, length contraction,
etc.
The inverse transformation equations from the primed frame to the unprimed
frame are
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net