Second edition
A four-dimensional orthogonal transformation matrix is used as the starting point for the replacement of the Lorentz transformation equations, allowing the electric field to be described by a four-vector and forcing Maxwell's equations to include extra terms.
Copyright © 1999 by David E. Rutherford
All Rights Reserved
Second edition published: May 7, 1999
Lorentz introduced a set of coordinate transformation equations that revolutionized our perception of space and time. These equations assured that Maxwell's equations of the electromagnetic field have an invariant form under transformation between inertial reference frames. It has become apparent to me that a set of transformation equations exist which replace the Lorentz equations while still satisfying the requirement that the space-time interval remains invariant under transformation. However, in order that Maxwell's equations remain invariant under this transformation, new terms must be added to the existing equations. In addition, charge is no longer invariant and the electromagnetic field is no longer described by a second-rank tensor, but by an electric field four-vector.
In two dimensions, there are familiar orthogonal transformation matrices
involving components which are functions
of
and
. However, in three dimensions, there are no
orthogonal matrices in which all components are functions of
eitheror. To find orthogonal transformation matrices
that satisfy this condition, we must go to four dimensions. Although there
are several combinations of components that fulfill this requirement, we
choose, for reasons to be explained later,
where 
. Any
combination of components in which the principle diagonal consists entirely
of terms and
all other terms are either
or
- the last term
appearing exactly once in each row and column, and the negative signs placed
so that an orthogonal matrix is formed - will work.
Now, let a primed reference frame with coordinates,
xµ', move with uniform velocity, v, along the
x1 axis of an unprimed reference frame with coordinates,
. If we make
the angle, 
,
a function of the velocity, v, so that
we find, by simple calculation, that
and
are
where 
, and
c is the speed of light. By substituting Eqs. (3) into Matrix (1)
we get for the transformation matrix,
, as a function
of 
The equations for the transformation of the coordinates, xµ, using Matrix (4), are now
Written out in full, these become
where the xµ can be identified as either
or
The inverse relations are obtained by reversing the primed and unprimed coordinates and replacing v with -v. Eqs. (6) describe rotations of the primed coordinate system relative to the unprimed system in four dimensions. Contrary to the Lorentz transformation, there is a rotation of the y'-z' plane relative to the y-z plane.
It can be verified by simple calculations that the space-time interval,
remains invariant, as it must, under transformation by Eqs. (6). If we impose the additional condition that the space-time interval for light must equal zero, then we must choose Eqs. (8), rather than Eqs. (7), as our coordinates. Our first choice for the placement of the "i" terms in Matrix (1) would have been along the diagonal from a41 to a14 for symmetry reasons, but we chose instead the combination of components as they appear in Matrix (1) so that the components a11, a14, a41, and a44 in Matrix (4) would match the analogous components in the Lorentz transformation matrix.
In order to express the velocity in four-vector form, we need to use the
invariant proper
time,
, in
the denominator rather than the improper time, t, so that
where, U, is the velocity four-vector which transforms according to
Any four-vector, including the light velocity four-vector, has the same magnitude in all reference frames, so that the velocity of light is c in all frames.
Let there be a scalar electric potential,
, due to the
presence of a charge, q', at rest in the primed frame. We use the
four-gradient, 
,
of this potential to define the components of an electric field four-vector,
Eµ', in the primed frame as
where
The unprimed components can be obtained easily by eliminating the primes. Making use of the transformation properties of the four-gradient under Eqs. (6),
and the invariance of the scalar electric potential,
, we find
that the transformed electric field is
where
We now construct a matter density tensor, M, from the four-gradient,
, of the electric
field four-vector, E,
where 
is a constant
of proportionality. Next, we form a flow density tensor, S, by multiplying
Eq. (17) by the velocity four-vector, U,
By setting 
,
we get
where 
, and
is the current
density four-vector. These are Maxwell's source equations containing the
additional terms
Multiplying again by the velocity four-vector, U, we get the energy density tensor, T,
It can be deduced from the transformation of the left-hand side of Eq. (17),
that the charge density,
, is an invariant.
We start by setting
then, due to the orthogonality of
, we have
where 
is the
Kronecker delta. Now, remembering that
and
we reach the
conclusion that,
Since the charge density in the primed frame,
, is
where, V', is the volume containing the charge, q', in the
primed frame, we can find the relationship between the magnitudes of the
charges as measured by observers in the two frames. Using the Jacobian,
J, of with
(since we are
making an instantaneous measurement of the volume in the
unprimed frame),
we see that the volume in the primed frame is related to the volume in the unprimed frame by
Inserting this into Eq. (26) and remembering the equality in Eq. (25) we have
or
The charge in the primed frame, q', can also be referred to as the invariant proper charge, q0, since it is the same at rest in any reference frame. It is interesting to note that the magnitude of the charge, q, at v = c is zero.
We can create a relation for the force four-vector, F, by combining the invariant proper charge, q0 (or q'), and the electric field four-vector, E,
which transforms as
An expression for the current four-vector, I, can be formed from the invariant charge, q0, and the velocity four-vector, U,
which, under transformation, becomes
The analog of the Maxwell tensor,
(not the same
as the force four-vector, F, above), in this theory is the tensor
However, unlike the Maxwell tensor, the terms F11,
F22, F33, and F44 are
not zero and the generator of the tensor is not
. It is now possible
to derive an expression for the analog of the Lorentz force four-vector,
f, by contracting the
of Eqs. (35)
with the current four-vector, I
These analogs contain components which do not appear in the Maxwell or Lorentz
equations due to the additional terms along the principle diagonal of
.
The primed coordinates described by Matrix (1) are orthogonal in four-dimensional
space-time, however, if we take only the portion described by
(the spatial
part) they are not orthogonal. This is analogous to taking what we,
in the unprimed frame, see as an instantaneous snapshot of space-time.
The problem is that what we see as instantaneous is not
instantaneous from the point of view of an observer at rest in the primed
frame. From our viewpoint, the positions of the primed coordinates are neither
linear nor orthogonal - they form cones centered on the unprimed coordinates.
The primed coordinates actually are these cones to us, but
to the observer in the primed frame, these coordinates are completely linear
and orthogonal. The best way to visualize the primed coordinates from the
unprimed observer's viewpoint, using Matrix (1) as our reference, is to imagine
the y'-axis rotated by the
angle,, about
the origin relative to the y-axis in any direction. Then, keeping
the orientation of the y'-axis constant, we spin the
x'-z' plane about the y'-axis and, while the
x'-z' plane spins, we rotate the y'-axis about the
y-axis 
,
keeping constant.
The figure traced out by the combination of all of these actions is what
the unprimed observer sees as the primed system of coordinates which,
interestingly, resembles quantum-mechanical electron orbitals.
According to this theory, the decreased reaction of a moving charged particle to an electric or magnetic field with increased velocity is due, at least in part, to a decrease in the magnitude of its charge rather than to an increase in its mass. The concept of a magnetic field is no longer necessary. It can be explained, instead, by a rotation of the primed electric field relative to the unprimed electric field. Furthermore, there is no need for induced magnetic or electric fields. All forces are due, simply, to the rotated electric field, E'. A change in the velocity of a particle causes a change in the rotation of its field, and therefore, a change in the components of that field relative to the unprimed frame. In the case of a system of charged particles that have no net spatial electric field when at rest, a field may arise when the charges are put in motion due to a rotation of the time component of the moving charges. This component now has a projection along the unprimed spatial axes where, initially, there was none.
Copyright © 1999 by David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net