, where
is the Kronecker
delta,
Third edition
The Lorentz transformation equations are replaced by a new set of transformation equations, the electric field is described by a four-vector, and the analog of Maxwell's electromagnetic field tensor contains nonzero terms along the main diagonal, causing any relation that contains its components to include extra terms.
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
Third edition published: September 22, 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 covariant form under transformation between inertial reference frames. It has become apparent that a set of transformation equations exist which replace the Lorentz equations while still satisfying the requirement that the spacetime interval remains invariant under transformation. However, in order that Maxwell's equations remain covariant under this transformation, new terms must be added to the existing equations. In addition, the magnitude of the charge is no longer invariant, but charge density is. The electric field is described by a four-vector and the antisymmetric electromagnetic field tensor is replaced by the electric field tensor.
In special relativity, Einstein introduced two postulates: The first postulate is that the speed of light is invariant for all inertial observers. In this theory, unlike in relativity, we use a Euclidean metric to describe spacetime, leading to a redefinition of the invariance of the speed of light in terms of a four-vector, rather than a three-vector. The second postulate of relativity is that the laws of physics are the same for all inertial observers. This requires that the laws have a covariant form under a Lorentz transformation between inertial reference frames. Our intention is to show that the Lorentz equations are incomplete. Since these equations are used to determine the covariance of the laws of physics, any change in their form requires a change in the form and scope of these laws. We will introduce the new version of the Lorentz transformation equations and some of its consequences.
Unlike the Maxwell tensor, its analog, the electric field tensor, includes nonzero terms along the main diagonal. These terms are responsible for the appearance of additional terms in many of the existing laws of physics, for example, Maxwell's electromagnetic field equations, the Lorentz four-force equations, and the electromagnetic energy-momentum tensor.
The Euclidean nature of spacetime, in this theory, leads directly to the invariance of charge density as well as the dependence of the magnitude of a charge on its velocity.
In special relativity, spacetime is described by the Minkowski metric. Here,
it is described by a four-dimensional Euclidean metric,
, where
is the Kronecker
delta,
We introduce now the concept of events in spacetime. These are the analogs
in four-dimensional spacetime of points in three-dimensional space. An event
is something that occurs at a specific place and at a specific time in a
particular reference frame. We represent an event in spacetime by
, where
and
are the
coordinates of the event. If we have a second event
in the
same reference frame, the interval between the two events is called the spacetime
interval, 
. The square of the spacetime interval between these two events in spacetime
is then
where 
is
the speed of light. This is, simply, the extension of the Pythagorean theorem
to four dimensions. We have included
to make
the units consistent throughout. It is required that the spacetime interval
be invariant for any inertial observer. That is, any observer moving with
uniform four-velocity will measure the same spacetime interval between the
events. An observer in a primed reference frame might measure the same two
events at 
and 
in his frame. Using the expression for the element of interval,
, we have
To indicate a sum, we will use the Einstein summation convention which states,
and,
for a double sum,
. This
means that whenever an index is repeated in a given term, we are to sum over
that index from 1 to m or 1 to n. The greek subscripts (indices), will range
from 1 to 4 unless otherwise noted. This leads to the general formula for
the element of spacetime
interval
where
In comparing measurements, we will frequently refer to primed and unprimed
frames of reference. It will always be the case that the primed frame (indicated
by a prime over the quantity, i.e.
) is in
uniform motion relative to the unprimed frame (indicated by the absence of
a prime over the quantity, i.e.
). We can
simplify things, considerably, by assuming that the first event takes place
at the origin of both reference frames, thus referring to it as
and
in the
two frames, then referring to the second event, simply, as
and
in the
two frames. In this case, the expression for the invariant spacetime interval
becomes
We represent an arbitrary four-vector in the form
, where
is the
four-vector, and
and
are its
components. The magnitude,
, of an arbitrary
four-vector, 
, can be found from
In three-dimensional coordinates, the position vector is used to locate a point in space. We will use the same concept, here, to locate an event in spacetime. The spacetime vector in four-dimensional spacetime is analogous to the position vector in three-dimensional space. We now define the spacetime vector.
Two events in the unprimed frame with coordinates
and
are separated
by the spacetime vector
from
to
, where
The same two events in the primed frame have coordinates
and
, respectively.
So the spacetime vector from
to
in the
primed frame is
, where
If the spacetime interval,
, in either
frame is between events that are separated, solely, by a time interval, i.e.
, then
, where
is the
proper time. Since the proper time
is the only
component in the interval, we have
. Similarly,
if the events are separated solely by a space interval, i.e.
, then
, where
is the
proper length, hence,
. The proper
time, 
,
and the proper length,
, are always
the maximum measurements of time and length made between events in
any frame. We put no primes on the proper time or length, since they are
the same in all reference frames. If there are space and time components
of the spacetime interval between events in a reference frame, then
neither component is proper and both will be less than the proper
value. However, an observer can always determine the proper values from his
own components. In general, the proper value of any quantity, in this
theory, will be its maximum value.
The components of the velocity four-vector,
, of a body
in motion are
where 
is
the proper time and
We will represent the velocity four-vector by
. The magnitude
of the four-velocity of any body, as we will now show, is invariant.
Let two events occur at the same place, but at different times in an inertial
frame. Since an observer at rest in the frame measures no space interval
between the events, his time measurement is the proper time,
, and his
four-velocity is the four-velocity of the frame. The element of interval
in this case is then
Dividing both sides of Eq. (12) by
, we get
or
We note that the right-hand side of Eq. (14) is the square of the magnitude
of the four-velocity,
, so we
have
Since 
represents an arbitrary four-velocity, we conclude that the magnitude of
the four-velocity of any body is always equal to
. Therefore,
every body is perpetually travelling at the speed of light through
spacetime. The magnitude of
can never
be changed; only its direction can be altered. Therefore, any acceleration
is at right angles to the four-velocity. A body which appears to be stationary
has four-velocity,
, and a
body moving so that
has
four-velocity,
, where
.
The velocity,
, can be
derived from the four-velocity,
, by first
dividing Eq. (12) by
, then
rearranging terms to get
Using Eqs. (10) and (16), we can write
or
We see that 
is always greater than
, and that
as 
approaches
, the magnitude
of 
approaches
infinity.
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 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 is in
the clockwise direction. The effect is reciprocal since an observer in the
primed frame sees a clockwise rotation of the y-z plane when looking
in the direction of relative motion of the unprimed frame. The inverse
transformation equations from the primed frame to the unprimed frame are
As in special relativity, we find a contraction of lengths in the direction
of motion. To show this, we compare lengths in two reference frames in uniform
relative motion. The primed frame is in uniform motion with four-velocity
relative
to the unprimed frame. To compare length measurements in the direction of
motion, we use Eqs. (22). Since the motion is in the x-direction,
we can take our length in the primed frame to be, simply,
. Measurements
of length are made instantaneously, so we have
. Therefore,
the unprimed coordinates are
and
But, we are interested only in the length, or x-coordinate in the unprimed frame, so our comparison of lengths in the primed and unprimed frames gives us
Remembering now that
and, since 
, we have
therefore, Eq. (24) can be written as
The coordinate
, in this
case, is the proper length,
, since
all other coordinates in the primed frame are zero. The coordinate
is the
improper length measured by the observer in the unprimed frame and is less
than the primed observer's proper length,
. This
represents a contraction of length in the direction of motion. The effect
is reciprocal, because an observer in the primed frame finds the same contraction
when performing the transformation using Eqs. (20).
We can use similar methods to find the comparison of elapsed times in two
reference frames. The frames are in uniform motion with four-velocity
relative
to each other. An observer at rest in the primed frame makes his time
measurement, 
, at the same place in his frame, so
. Using
Eqs. (22), we have
and
But since we are comparing only time measurements, we have
or
The coordinate 
in this case, is the proper
time,
.
The coordinate 
is the unprimed observer's improper measurement of the elapsed time in
the primed frame. This constitutes a dilation of time. In other words, an
observer in the unprimed frame says that the rate at which clocks run in
the primed frame is slower than in his own frame. Again, the effect is
reciprocal.
A body is moving with uniform four-velocity,
, as measured
by an observer at rest in the primed frame. The primed frame, in turn, is
moving with uniform four-velocity,
, relative
to the unprimed frame. We wish to find the four-velocity,
, of the
body as measured by an observer at rest in the unprimed frame. Since the
components of all four-vectors must transform like the coordinates
of an event in spacetime, we can use Eqs. (22) to find the components of
. With the
components of the velocities,
and
, replacing
the coordinates,
and
, respectively,
we find the components of the four-velocity of the body as measured by the
unprimed observer to be
If the four-velocity of the primed frame is
, and the
four-velocity of the body relative to the primed frame is,
, the
components of the four-velocity
of the
body relative to the unprimed frame, using Eqs. (22), are
and
At spatial velocities much less than c,we have
, and we
get the Galilean result
. The signs
in the previous sentence are "approximately equal to", not "equal to". We
see that, for low velocities, most of the four-velocity is still in the time
direction in both primed and unprimed frames of reference. Incidentally,
any time the magnitude of the spatial part of a four-velocity is less than
c, we will also have a time component, and vice versa.
For velocities in transverse directions, for example
and
, we have
For combined velocities along the x-axis at the speed of light,
and
, we get
, but
In the unprimed frame, there are no spatial components in the transformed
four-velocity,
, but there
is a component directed along the negative t-axis. The magnitude of
V, however, is still
Its direction is rotated 180 degrees relative to the direction of the
four-velocity of the unprimed frame, which we will call
, where
, since
it is at "rest" in space, but not in time.
For spatial velocities at the speed of light in transverse directions, for
example 
and 
,
we have 
and
The scalar electric potential
, or
for short,
at an arbitrary field event
, or
for short,
due to a stationary point charge
at an event
, or
for short,
is
where 
is
the spacetime interval between the events
and
.
To find the scalar potential
due to
a distribution of stationary charges, we need to sum the contributions
from each of the individual elements of charge. For the contribution of an
element of charge
at an event
, we make
use of the fact that
, where
is the
charge density at the event
, and
is the
element of volume containing
. The general
equation for the scalar potential
at an arbitrary
field event 
due to a distribution of stationary charges is
The components of the potential field four-vector
, not to
be confused with the transformation matrix
, at the
field event 
due to a distribution of moving charges at the field event
are,
where the 
are the components of the four-velocity of the element of charge
, and the
spacetime interval
, in this
case, is
where
or
and
are the components of the spacetime interval between events
and
. The components
of Matrix (41) were obtained by transforming the coordinates of the primed
frame with 
, since the primed observer's measurement of the interval between the events
is made instantaneously in his frame.
Noting that the components of the current density four-vector,
, at the
event 
are
we can write the general relation for the components of the potential field
four-vector 
at an arbitrary field event
due to
a distribution of moving charges. By inserting Eqs. (44) into Eqs. (39),
we get
It may help, when determining the spacetime interval,
of Eq.
(40), to remember that
The potential field tensor,
, in this
theory, is
where the 
are the components of the potential field four-vector and the
are the
components of the four-gradient,
Written out in full, Eq. (47) is
We define the static electric field four-vector in the same way as in three-dimensional space, but with an additional time component. This applies to a distribution of charge which is static in space, but may vary in time. The components of the electric field four-vector are
Written out, in full, Eqs. (51) are
Because of the invariance of the Euclidean inner product, we can show that
the charge density,
, is the
same in all inertial reference frames. First, we transform the gradient of
the electric field
Then, by setting
, we have
Now, due to the orthogonality of
, we get
And, finally, since
and
, we conclude
that
The charge density,
, in the
primed frame, is defined as
where 
is
the magnitude of the charge and
is the
volume containing the charge as measured by an observer at rest in the primed
frame. Using the Jacobian,
, of
, we can
find the magnitude of the charge as measured by an observer at rest in the
unprimed frame. Since we are making an instantaneous measurement of the volume,
we take 
, so that
where
Therefore, the volume in the unprimed frame is
Using Eq. (58) and remembering the equalities in Eqs. (57) and (61), we have
or
This shows that the magnitude of the charge transforms in the same way as
lengths and times. The magnitude of the charge at rest in the primed frame,
, in this
case, is the rest or proper charge. The proper charge will be referred
to, from this point on, as
. Any proper
or rest quantity, from now on, will be denoted by a zero subscript. We can
now write the relationship between the proper charge,
, and the
improper charge,
, as
A current four-vector can be formed by combining the proper charge,
, and its
four-velocity,
. The current
four-vector will, normally, be used in reference to a test charge moving
in the presence of electric and potential fields, rather than in reference
to the distribution of charge that creates the fields. The components of
the current four-vector,
, are
or in terms of the improper charge,
, we have
We will now introduce the analogs of Maxwell's field equations and the Lorentz four-force equations. First, we will expand the analogs of Maxwell's field equations
We expand the first set of equations,
, to get
or
The second set of equations,
, obtained
by setting 
in Eqs. (69), is
The right-hand side of the last equality in Eqs. (71) can be reduced to one term by noting that
so that two of the terms cancel. The remaining term we interpret as
where the 
are the components of the current density four-vector. The relations for
the analogs of Maxwell's inhomogeneous equations now become
Inserting the components of
into Eqs.
(74), we have
where there is an additional implied sum over the repeated index,
. In addition
to the usual field terms of Maxwell's equations, we have, in Eqs. (75), the
terms, 
,
not found in Maxwell's equations. The
part of
, is the
trace of 
, as well as the four-divergence of the potential field. These terms imply
the existence of an extra induced current density
The analogs of the Lorentz four-force equations are
Remembering that
, where
, is the
proper magnitude of the test charge,
is its
four-velocity, and
, where
is the
potential field four-vector at the position of
due to
all charges other than
, we expand
the first set of equations,
,
or
The second set of equations,
, obtained
by setting 
in Eqs. (78), is
Since
and similarly
we see that two of the terms on the right-hand side of the last equality in Eqs. (80) cancel. The remaining term we identify as
where the 
are the components of the momentum four-vector. We can now write the complete
analogs of the Lorentz four-force equations as
where the 
are the components of the force four-vector. Writing out the components
of 
, we
get
and, as before, there is an additional implied sum over the repeated index,
. Here,
again, there are the additional terms,
, which
do not appear in the Lorentz equations. These terms imply the existence of
an extra force
We would like to look further into the relationship between the terms in
Eq. (83), in hopes of finding a connection between mass, charge, and the
scalar electric potential. In the case where the potential
is created
by a single point charge, we can write the components of the potential
four-vector,
, as
where the 
are the components of the four-velocity of the point charge. Noting that
and starting
with Eq. (83), we get
or
We conclude from Eq. (89) that the proper mass of a particle is proportional
to its proper charge and the potential of the field at its location. The
improper mass,
, of a
particle varies with the magnitude of the charge, therefore, the improper
mass is
This states that mass is velocity dependent, however, its magnitude due to the particle's velocity is decreased, rather than increased, as it is in special relativity, therefore,
Our intention, now, is to present the electric field in its most general
form and to express the field and force equations and energy-momentum tensor
in terms of the generalized electric field. First, we find the general form
of the components of the electric field four-vector,
. Since the
electric field four-vector is a four-gradient it must transform as a covariant
vector, therefore
where the 
are the components of the stationary electric field of Eq. (51). Remembering
now that, for a single moving point charge, the components of the potential
four-vector, 
, in the unprimed frame are given by Eq. (87), we can write the general
expressions for the components of the electric field four-vector in terms
of the potential field four-vector as
At this point, we would like to introduce a new operator; the four-dimensional analog of the three-dimensional curl of a vector field. This operator we call the "turn" of a four-vector field, which we define as
where 
,
here, can be thought of as an arbitrary four-vector field and i,
j, k, and l are unit vectors in the x, y, z, and t
directions, respectively. Another way of writing
is
, where
the operator,
, is not
to be mistaken as the D'Alembertian operator. Using this shorthand, we can
express Eqs. (102) as a single four-vector equation by writing either
or,
equivalently,
.
If we now define the electric field tensor,
, using the
components of 
from Eqs. (102), as
we can include both of the field equations, Eqs. (68), in a single equation by writing
where the terms not containing repeated indices are associated with the homogeneous field equations and the remaining terms are related to the inhomogeneous field equations.
In a similar manner, both of the force equations, Eqs. (77), can be combined by writing
where the 
are the components of the current four-vector, the
are the
components of the momentum four-vector, and the
are the
components of the force four-vector. As before, the terms separate into the
homogeneous and inhomogeneous force equations.
The energy-momentum tensor,
, can be
written in terms of Eq. (104) as
which can also be separated into homogeneous and inhomogeneous equations.
If we now define the tensor
as
we can express Eq. (107) as
We would like, now, to find the four-divergence of the current density of Eq. (105). This takes the form
After eliminating terms that cancel from the left and right-hand sides of Eq. (110), we are left with
where 
is
the time component of the generalized electric field four-vector,
, from Eq.
(102). Simplifying Eq. (111) further, we get
, which
is equivalent to Eq. (76). Since
is the negative
four-divergence of the potential four-vector, it is a scalar quantity. We
can say then that
, where
is a scalar
quantity with the units of force per unit charge. The term,
, on the
right-hand side of Eq. (111) is the four-divergence of the extra current
density. It is, therefore, a scalar quantity with the units of charge per
unit four-dimensional volume or four-dimensional charge density. To simplify
this term, we say that
, where
is the
four-dimensional charge density. In order to simplify Eq. (111) still further,
we introduce the operator
,
which is the four-dimensional Laplacian operator. Putting all of this together, Eq. (111) becomes
where 
is
a constant of proportionality. Since
represents
a matter density, we suspect that
represents
a scalar potential.
Now, remembering that the proper mass is
we can replace
in Eq.
(114) with the four-dimensional mass density,
, and
with the
four-dimensional charge density,
, to get
Multiplying both sides of Eq. (113) by
, we have
Inserting Eq. (115) into Eq. (116) and setting
, Eq. (116)
becomes
where 
is
the gravitational potential.
We can derive an equation for the energy of a particle from the equation for the transformation of charge. The equation for transformation of charge, Eq. (64) is
Squaring Eq. (118), we get
or
Now, multiplying both sides of Eq. (120) by the square of the static electric
potential, 
, we have
Remembering, from Eq. (87), that the potential four-vector for a single moving
point charge is
, we can
write Eq. (121) as
Taking the square root of both sides of Eq. (122), we get
or, in terms of the total energy,
, and proper
mass, 
,
Noting that the spatial part of the four-momentum is
, where
, we can
write Eq. (124) as
We notice from Eq. (125) that the total energy,
, of a particle
can be either positive or negative. In addition, from Eq. (121) we see that,
when 
, the
total energy due to translatory motion is zero. However, a particle may possess
energy due to its rotation in addition to the energy of its translational
motion, even though the magnitude of its spatial four-velocity,
, is the
speed of light,
. This
additional energy,
, due to
the frequency of rotation,
, is described
by the Planck relation,
, where
is Planck's
constant.
We would like to introduce an analogy to the element of spacetime interval,
involving the second partial derivative of an arbitrary scalar function of
the coordinates,
,
which we call the spacetime derivative. Since the spacetime interval,
, and the
scalar function,
, are invariant
under transformation between arbitrary inertial reference frames, we can
say that 
and 
, therefore,
In the primed frame
so that, from Eqs. (127), (128), and (129), we can say
Now, in the case that
in Eq. (129), we have
In this case,
, where
is the
invariant proper time and, since
and
, we can
say that
Therefore, in this case, Eq. (127) becomes
We see that Eqs. (127) and (134) can easily be put in the form of four-dimensional wave equations by subtracting the terms on the right-hand side of each equation from both sides, obtaining
and
respectively, where c is the speed of propogation of the waves. These waves propogate in four-dimensional spacetime rather than three-dimensional space.
We notice that Eqs. (127) and (134) and, therefore, Eqs. (135) and (136)
are similar in form to some of our previous equations, for example, the
differential equation for the scalar electric potential,
,
or
Setting 
in Eq. (135), we get
Interestingly, we see that Eqs. (138) and (139) are equivalent, providing
so that Eq. (137), in the form of Eq. (139) after inserting Eq. (140), is actually a four-dimensional wave equation. Similarly, we see that Eqs. (113) and (117) can be written in the form of the wave equations
and
respectively. In general, any equation which has the same form as Eqs. (113) and (117) can be written as wave equations. That is, any equation, in this theory, whose left-hand side is the four-dimensional Laplacian, Eq. (112), of a scalar potential function of the coordinates and whose right-hand side is a source term, can be written as a four-dimensional wave equation.
Until now, we have limited our study to four-dimensional rectangular coordinates in Euclidean spacetime. However, in order to describe our laws in general four-dimensional coordinate systems in Euclidean spacetime we must introduce the general covariant forms of the laws. We begin with the relation for the element of spacetime interval, which is the analog of Eq. (4)
where the 
are the components of the metric tensor. In order to handle more general
coordinate systems, every previous occurrence of
must be
replaced by . Our formulations of the coordinate transformation equations, Eqs. (20)
and (22), become
and
respectively. Where previously it was unnecessary to distinguish between covariant and contravariant components of tensors, since they are equivalent in rectangular coordinates, we must now specify which we are using. To comply with convention, superscripts will indicate contravariant components and subscripts will indicate covariant components.
In rectangular coordinates, the derivative of a tensor is identical to the ordinary derivative. But in general coordinates, we have terms which include nonzero Christoffel symbols. These terms vanish in rectangular coordinates since the components of the metric tensor are constants. In general coordinates, however, the components of the metric tensor are not always constant, thus, the Christoffel symbols do not always vanish. Because of this, all ordinary derivatives of tensors must be replaced by their covariant derivatives. The potential field tensor, Eq. (47), in general coordinates, therefore, is
where the ";" indicates the covariant derivative of
with respect
to 
. Any
relation involving
in general
coordinates must use Eq. (146) rather than Eq. (47) in order to assure
covariance.
All of the equations in Eqs. (67), except the first two, are already covariant
in general coordinates providing we use Eq. (146) rather than Eq. (47) for
. In order
to make the two exceptions covariant, we need to replace the ordinary derivatives
by the covariant derivatives. The first equation in Eqs. (67) then becomes
and the second becomes
Similarly, we must write Eqs. (105) as
and Eq. (110) as
Since we have altered the Maxwell tensor so that it includes nonzero terms along the main diagonal, additional terms appear in any equation where the potential field tensor appears. The effects of these extra terms should be readily testable and verifiable. We have shown, also, that the magnitude of a charge is reduced as its velocity is increased. The resistance of high velocity charged particles to acceleration in particle accelerators might better be attributed to a reduction in the magnitude of their charges, than to an increase in the magnitude of their masses. The variability of the magnitude of the charge is not forbidden, since the charge density is invariant.
The equation for an expanding four-dimensional spherical light wave front is Eq. (12), in this theory. The wave front propogates in four-dimensions (3 space and 1 time), rather than the usual three-dimensions (3 space), as in relativity. This assures that the spatial velocity of light in any inertial reference frame is invariant.
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net