(Article submitted to Physical Review Letters on March 1, 1999)
Copyright © 1999 by David E. Rutherford
All Rights Reserved
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 Lorentz's equations while still satisfying the requirement that the space-time interval remain 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.
Let a primed reference frame with coordinates, x', y', z', and t', move with uniform velocity, v, along the x-axis of an unprimed reference frame with coordinates, x, y, z, t. It is a necessary condition that the space-time interval,
remain invariant in both frames of reference and that all effects be reciprocal. The Lorentz equations may be altered without conflicting with these requirements as follows
where 
. Eqs.
(2) may also be written
where
and
The inverse relations are obtained by reversing the primed and unprimed components and replacing v with -v.
Holding the signs of a11, a14, a22, a23, a32, a33, a41, and a44 fixed in aµ - for reasons to be explained below - we are free to vary the signs of the remaining terms while keeping the space-time interval, Eq. (1), invariant. Eqs. (2) describe rotations of the primed frame relative to the unprimed frame - the direction of rotation depending on the signs of the terms. Contrary to the Lorentz equations, there is a rotation of the y'-z' plane relative to the y-z plane. Furthermore, if we assume that these variable terms may take on both signs concurrently, there appears to be a potential for simultaneous rotations of the primed axes in different directions. In this case, each of the "±" and "" signs may be thought of as a kind of single sign with the characteristics of both "+" and "-", but having opposite effects. A positively charged particle might experience the effects of the "+" part of the sign, while a negatively charged particle experiences the effects of the "-" part, or vice versa.
A body in uniform motion with velocity components ux', uy ,' and uz' relative to the primed frame of reference has the velocity components ux, uy, and uz relative to the unprimed frame defined by
By using Eq. (6) and the inverses of Eqs. (2) we have,
It is required that the velocity of light along any axis in both the primed and unprimed frames equals c. Restricting the motion of a photon to the primed axis in question, we get the desired result,
As mentioned above, certain signs in aµ must be held fixed. This is necessary to assure that both Eqs. (1) and (8) are satisfied. Variations exist for the placement of the variable signs which also satisfy the above conditions, but those in Eq. (2) seem to be the most symmetrical. Interestingly, Eq. (1) is also satisfied by x4 = ct, but Eq. (8) is not.
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, 
, 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. (2),
and the invariance of the scalar potential,
,we find
that the transformed electric field is
where
We now construct a matter density tensor, M, from the derivatives of the electric field four-vector, E,
where 
is a
constant of proportionality. Next, we form a flow density tensor, S,
from the derivatives of the electric field four-vector, E, and the
velocity four-vector, v,
By setting 
,
we get
where 
, and
S is the current density four-vector. These are Maxwell's source equations
containing the additional terms
Multiplying again by the velocity four-vector, v, we get the energy density tensor, T,
To simplify matters, the terms on the left-hand side of Eq. (18) can be referred
to as a tensor D, so that we have
The concept of a magnetic field is no longer necessary. 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.
The scalar electric potential,
, in the primed
frame is represented by,
where r' is the primed radial distance from the charge,
Noting that an observer in the unprimed frame measures the orthogonal projections of the primed components of r' onto his unprimed axes, we find that
which is
Remembering that the scalar potential,
, is an invariant,
we can write
and, therefore,
The proper magnitude of the charge, q', at rest in the primed frame is greater than its magnitude, q, in motion in the unprimed frame, but the radial distance, r', in the primed frame is also greater by the same factor, so observers in both frames measure the same potential. The decreased reaction of a moving charged particle to an electric or "magnetic" field is due, at least in part, to a decrease in the magnitude of its charge rather than to an increase in its mass.
It can be deduced from the transformation of Eq. (14), that the charge density,
, is an invariant.
We start with
then, due to the orthogonality of aµ, we have
where 
is the
Kronecker delta. Now, remembering that
and
we reach the
conclusion that,
It would appear that the electric current, (icq, I ), is a four-vector and, therefore, must transform as
The second-rank force tensor, F, can now be obtained by combining the electric field four-vector, E, and the electric current four-vector, I,
which transforms according to
