under the link "New Transformation Equations and the Electric Field
Four-vector, Fifth Edition (PDF File)".
I've included the abstract, introduction, and conclusions below.
ABSTRACT:
In special relativity, spacetime can be described as Minkowskian.
We intend to show that spacetime, as well as the laws of
electromagnetism, can be described using a four-dimensional
Euclidean metric as a foundation. In order to formulate these laws
successfully, however, it is necessary to extend the laws of
electromagnetism by replacing the Maxwell tensor with an electric
field four-vector. In addition, to assure the covariance of the
new laws, we introduce equations that, completely, replace the
Lorentz transformation equations and Lorentz group. The above
replacements, we believe, lead naturally to a unification of the
electromagnetic field with the gravitational and nuclear fields.
We introduce, also, a new mathematical formalism which facilitates
the presentation of our laws.
INTRODUCTION:
Lorentz first derived his famous set of transformation equations
from the electromagnetic field equations of Maxwell. They assure
that Maxwell's equations will have the same form in any inertial
frame of reference. Unfortunately, if Maxwell's equations are
shown to be incomplete, then it is likely that the Lorentz
equations are incorrect. We intend to show that this is the case.
Maxwell's equations are, essentially, a set of
three-dimensional partial differential equations. That is,
each equation contains the partial derivatives with respect to
only three of the coordinates. In four-dimensional
spacetime, a three-dimensional description of anything is
inherently incomplete. We will extend Maxwell's equations so that
they form a set of four-dimensional equations. In so doing, it is
possible to encompass all of Maxwell's equations in a
single vector equation by introducing an electric field
four-vector. In addition to the electromagnetic field, we believe
the new equation incorporates the gravitational and nuclear
fields. This equation, however, is not Lorentz invariant
and requires a new set of transformation equations in order that
it has the same form in all inertial frames.
The Lorentz transformation equations forbid any contraction or
expansion of coordinates transverse to the direction of motion. We
present a new Euclidean set of transformation equations which
require a rotation of the coordinates transverse to the
direction of motion. There is also an analogous rotation in the
plane described by the direction of motion and the time
coordinate.
Due to the dependence of each of the Lorentz force equations on
only three of the components of the four-velocity, they also form
an incomplete set of equations. Therefore, we extend these
equations to four-dimensions, as well. The equation of motion then
follows naturally from our new force equation.
In expressing the force equations in terms of the fields, we
arrive at an energy-momentum tensor with components which include
the time component of our electric field four-vector. These
components offer, among other things, a new description of the
mechanism behind the flow of field energy.
A new mathematical formalism is introduced which substantially
simplifies the expression of our laws and helps give a deeper
understanding of the geometry behind them. This new formalism
borrows its structure, in part, from Hamilton's quaternions and
the Clifford algebras, but differs fundamentally from both.
CONCLUSIONS:
The special Lorentz transformation describes an imaginary rotation
of coordinate systems. We have described a real rotation of
coordinate systems including an automatic rotation
transverse to the direction of motion, which the Lorentz
transformations do not describe. These rotations are not
necessarily observable as rotations, however, but as precession,
time dilation, length contraction, etc.
In conventional theory, the magnitude of a charged particle is
invariant. In this paper, the magnitude of a charge is reduced as
its velocity is increased, however, the charge density in
the region of the charge is invariant. Thus, the quantity of
charge in a given volume does not change as long as no charge
enters or leaves the region. It is the invariance of charge
density, not the conservation of charge, which accounts for the
neutrality of atoms. This indicates that charge is most likely a
property of the field, rather than the particle itself.
It can be seen from the T_44 component of (103) that the
energy of the field can be negative. Rather than interpreting this
as a liability, we suggest the possibility that the field of a
particle is its antiparticle. We suspect that the creation
of a particle travelling forward in time is accompanied by the
creation of its antiparticle travelling backward in
time. We refer to particles (or antiparticles) travelling
forward in time as particles, since they have the characteristics
of particles, and particles (or antiparticles) travelling backward
in time as antiparticles having the characteristics of fields.
Since an antiparticle travelling backward in time may be said to
have negative energy and since the energy of the field from
T_44 is negative for E_t^2 < E_x^2 + E_y^2 + E_z^2, we
suggest that the antiparticle travelling backward in time appears
to us as the field of the particle. We are not referring,
for example, to the particle/antiparticle pair created in the
disintegration of an energetic photon. In that case, there is an
electron and an anti-electron (positron) created. However, both of
these ``particles'' are travelling forward in time.
Associated with each of these particles, is a field which we claim
is its antiparticle travelling backward in time. These
particle/field (or particle/antiparticle) pairs are the pairs to
which we refer. In this case, there are actually two
particle/antiparticle pairs created. The electron and its field
(antiparticle) comprise one particle/antiparticle pair, and the
positron and its field (antiparticle) comprise the other
particle/antiparticle pair. Since the antiparticles appear to us
as (and are) the fields of the particles, one might even entertain
the notion, that space and matter travelling backward in time are
one and the same thing.
Due to this apparent particle/antiparticle link, we also conclude
that there are no electric monopoles (since every charged
particle is accompanied by its field), just as there are
apparently no magnetic monopoles. These conclusions might offer a
logical explanation for the puzzling absence of antimatter in the
universe. If our suspicions are correct, this "missing" antimatter
exists all around us as the fields of matter and, possibly, as
space itself.
The equation for an expanding four-dimensional ``spherical'' light
wave front is (8), in this paper. The wave front
propogates in four-dimensions (3 space and 1 time), rather
than the usual three-dimensions (3 space), as in conventional
theory. This assures that the spatial speed of light in all
inertial reference frames is the invariant c.
The form and terminology of many of the equations in this paper
are, deceptively, similar to those of conventional physics,
however, they differ in several ways. It is important that one not
assume the equivalence of the definitions presented here with the
analogous definitions in conventional theory. In most cases, they
are not exactly the same. For example, the definition of the
scalar electric potential phi includes the spacetime interval
s in the denominator, not the spatial interval r. This
prevents an infinite potential at r = 0, a problem that plagues
conventional electromagnetic theory.
It would also be incorrect to assume that the electric field is
defined in the same manner as in conventional electromagnetic
theory. For example, it is easy to interpret the components of the
energy-momentum tensor T incorrectly as containing only the
components of the conventional electric field when, in fact, they
contain the components of the electric, magnetic, and possibly the
gravitational and nuclear fields, as well.
What we have provided, here, is to be considered only as a
foundation upon which a deeper and more unified understanding of
laws of nature might be built.
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I would apprectiate any comments or questions. I'm also looking for
someone to sponsor my submission of this paper to the LANL archives (I
have the Latex version that can be sent to LANL). Any help would be
greatly appreciated.