, where
is the
Kronecker delta,
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
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 of a charged particle is no longer invariant, however, charge density is invariant. The electric field is described by a four-vector and the electromagnetic field tensor is replaced by the electric field tensor.
In special relativity, Einstein introduced two postulates: The first postulate 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 incorrect. 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 coordinate transformation equations and some of its consequences. The second postulate of special relativity 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 and the propagation of light in four dimensions rather than three.
Unlike the Maxwell tensor, its replacement, the electric field tensor, includes nonzero terms along the principle diagonal. These terms are responsible for the appearance of additional terms in many of the existing laws of physics. 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 the 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 magnitude of the spacetime separation between the
two events is called the spacetime interval,
, which
is defined as
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 square of the element of spacetime
interval, 
, we have
We can write the right-hand side of (3.3) in a more general way by using the Einstein summation convention as
where
and similarly for the left-hand side of (3.3). Subscripts are used, exclusively,
to conform with the convention that all indices appear as subscripts in Euclidean
space. The greek subscripts,
and
will
always range from 1 to 4 unless otherwise noted. 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,
for example,
) is in
uniform motion relative to the unprimed frame (indicated by the absence of
a prime over the quantity, for example,
).
Every vectorial quantity in spacetime will be represented by a four-dimensional
vector, or four-vector. We represent an arbitrary four-vector,
, in the
form 
.
The magnitude,
, of
is determined
in the same manner as the magnitude of the spacetime interval
If the spacetime interval,
, in either
frame is between events that are separated, solely, by a time interval, that
is 
, 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, that is
, 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 measurements. 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 (5.3) by
, we get
or, from (5.1),
We note that the right-hand side of (5.5) 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 (5.3) by
, then
rearranging terms to get
Using (5.1) and (5.7), 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 coordinate 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, initially,
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 (6.1) 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 (6.1), 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 "
"
and "
"
(this sign is "minus/minus", not "equals", in this context) signs preceeding
some terms in (6.1) can be taken as single "+" and "-" signs, although as
with the ""
and "" signs,
they are a combination or superposition of both signs. Multiplication of
combination signs is carried out by multiplying the corresponding parts of
each sign. That is, the top parts are multiplied together, separately, and
the bottom parts are multiplied together, separately. For example, if we
wish to multiply
"" and
"", we get
"". From
this point, for simplicity, all operations involving (6.1) will use only
one of the combination signs at a time. However, the combination sign associated
with each term of (6.1) will still be implied in any mathematical operations
carried out.
The Lorentz transformation matrix is REPLACED by (6.1).
The coordinate transformation equations from the unprimed frame to the primed
frame associated with the matrix,
, are
where
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 in 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 (6.1), 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
The transformation equations in (6.2) and (6.4) REPLACE the Lorentz transformation equations.
As in special relativity, we find a contraction of lengths in the direction
of motion. To show this, we compare lengths measured by observers 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 (6.4). Since the motion is in the x-direction, we can
take our lengths in the primed frame to be, simply,
and 
.
Measurements of length are made instantaneously, so we have
. Therefore,
from (6.4), 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 from (5.5) that
and since 
, in this case, we have
therefore, (7.2) 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 we find, by using (6.2), that an observer in the primed
frame measures the same contraction..
We can use similar methods to compare 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 that
. Using
(6.4), we get
and
But since we are comparing only time measurements, we have
or, from (7.4),
The coordinate
in this
case, is the proper
time,
. The coordinate
is the
unprimed observer's improper measurement of the elapsed time. 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 (6.4) to find the components of
. We replace
the coordinates
and
, in (6.4),
with the components of the velocities,
and
, respectively.
We then 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 (6.4), 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 , 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, in the unprimed frame is
where
and 
is
the spacetime interval between events
and
in the
unprimed frame from (3.2). Notice that the spatial part of
, can
be zero, while the time part remains non-zero. In this case, the potential
(10.1) does not become infinite at r = 0.
If we transform the components of the primed spacetime interval,
, using
(6.2) with
, since
the primed observer's measurement of the interval between the events is made
instantaneously in his frame, the spacetime interval
becomes
where
or
and
are the components of the spacetime interval between 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 P(2) are
where the 
are the components of the four-velocity of the element of charge
.
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 (10.9) into (10.8), we
get
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. We write the expressions for
the components of the generalized electric field four-vector as
or, written
out,
where the 
are the components of the potential field four-vector,
is the
scalar electric potential, and
means
We must remember that the signs associated with the terms on the right-hand side of (11.1) are, actually, combination signs reflecting the combination signs in (6.1). Therefore, before performing any operations with (11.1), we must reintroduce these combination signs. From here on, whenever we refer to the electric field four-vector, we will mean the generalized electric field four-vector with components as defined in (11.1), unless otherwise stated.
For a distribution of stationary charges, the components of the electric field four-vector (11.1) reduce to
We now define the electric field tensor,
, using
the components of the electric field four-vector
, from
(11.1), as
The electric field tensor,
, can
be used to construct the electric field equations by writing
where the 
are the components of the current density four-vector from (10.9). The
equations (12.2) are analogous to Maxwell's electromagnetic field equations.
Both the homogeneous and source equations are included in the single equation
(12.2). However, (12.2) contains terms related to the time component of the
electric field,
, unlike
Maxwell's equations.
The equations (12.2) reduce to the equations
In a similar manner, we can write the equations for the force density
four-vector,
, which
are analogous to the Lorentz force density equations. However, we will need
to use a variation of the electric field tensor (12.1). This variation,
, is defined
as
Using (12.4) we can write the force density four-vector as
Equations (12.5) include terms related to the time component of the electric
field four-vector, unlike the Lorentz equations. The left-hand side of (12.5)
can be put, entirely, in terms of the electric field by subsituting
from
(12.2) into (12.5). After changing dummy indices, we have
The energy-momentum tensor,
, can
be written in terms of the electric field tensor,
, and
its variation,
, as
We can also write (12.7) as
where the 
are the components of the electric field four-vector, and
is the
four-dimensional permutation symbol. If we now define the tensor,
, as
we can express (12.7) as
The energy-momentum tensor,
, is
symmetric, but instead of containing the components of the Poynting vector
in the fourth row and column (actually, the Poynting vector terms are included,
but they sum to zero), as in the conventional electromagnetic energy-momentum
tensor, we have terms including the time component of the electric field
four-vector,
.
Because of the invariance of the 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, inserting the equalities,
and
, into
(13.3), we have
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 (13.4), (14.1), (14.2) and (14.4), we have
or
This shows that the magnitude of the charge transforms in the same way as
lengths. 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
. We
can now write the relationship between the proper charge,
, and
the improper charge,
, as
We would like, now, to find the four-divergence of the current density of (12.2). This takes the form
After eliminating terms that cancel from the left and right-hand sides of (15.1), we are left with
where 
is an extra current density associated with the terms
by way
of
where 
is the time component,
, of the
electric field four-vector from (11.1).
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 (16.2), (16.3), and (16.4), we can say
Now, in the case that
in (16.4),
we have
In this case,
,
where
is the invariant proper time and, since
and
, we
can say that
Therefore, in this case, (16.5) becomes
We see that (16.2) and (16.9) 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 (16.2) and (16.9) and, therefore, (17.1) and (17.2) are similar in form to some of our previous equations. For a single stationary charge, (12.3) reduces to
or
Setting 
in (17.1), we get the four-dimensional wave equation
Interestingly, we see that (17.4) and (17.5) are equivalent, since from (10.7),
If we, now, multiply
in (17.5)
by 
we
get the four-dimensional wave equations associated with the potential four-vector
field
It is possible to multiply four-vectors algebraically by using the appropriate
conventions for the products of the orthonormal basis vectors,
. These
basis vectors satisfy the relations
for 
.
If 
, there
is no implied sum of the products, that is
The orientations of the products in (18.1) for
and
are
along the line described by the unused component. For example, the orientation
of 
is
along the line described by
. The
products must also satisfy the scalar and vector product rules of vector
multiplication. In a product where one of the basis vectors is
, for
and
, the
orientation is along the line described by the other basis vector. For example,
the orientation of
is along
the line described by
. The
orientation of the product where
and
, is
along the worldline or timeline described by
. For
example, the orientation of
, is
along the worldline described by
.
Although we have defined the orientations of the products, we have yet to define the directions of the products. Due to (18.1), there are eight sets of possible combinations of products - one for each possible direction (positive or negative) along the line of orientation of the product. One possible set of rules is
where the direction is described by the basis vector on the right. Another possible set is
and so on. Only one set of rules is used for a given four-vector product (there is an exception to this which will be dealt with later). These rules have been taken, partially, from Hamilton's quaternion rules for the products of basis vectors, but have been altered and expanded.
Using these rules, we can write the product of two four-vectors as a normal
algebraic product. For example, the product of two arbitrary four-vectors,
and
is
We can choose any one of the sets of the rules for the product. Let us choose, for this example, the rules in (18.3). Multiplying (18.5) algebraically, using (18.3) for the products of the basis vectors, we get
Consequently, the product of two four-vectors results in another four-vector.
The product
, in
(18.6), can be written more compactly as
where
Each of the products,
,
, and
on the
right-hand side of (18.7) can be either positive or negative independently
of the other products, resulting in eight possible results for
. If
we allow the
part
of to
be a product independent of
,
, and
, having
its own positive and negative values independent of the other products, then
there are sixteen possible results for
.
In actual use, the choice of rules will depend on the circumstances. For example, we can express the transformation equations (6.2) in four-vector notation as
where
and, in this case,
In (18.9), we have used a combination of two sets of rules, since in the transformation matrix (6.1), we have terms with indeterminate signs. We can write the inverse transformation (6.4) as
where the velocity four-vector
in (18.12)
is
Similarly, the electric field four-vector,
, from
(11.1) can be written equivalently, in four-vector notation , again using
a combination of two sets of rules, as
where
Of course, the left-hand side of (18.14) can also be written in condensed form, from (18.7), as
where
The product
is the
four-divergence of
,
is the
curl of 
, and 
is a new product we will call the evolution of
.
We can write the electric field equations (12.2) as
where 
. Notice that
, is
the electric field four-vector, here, not the electric field tensor,
as in (12.2). However, both (12.2) and (18.18) are completely equivalent.
The force density equations (12.5) can be written as
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 make several adjustments. We begin with the relation for the element of spacetime interval, which is the analog of (3.4)
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, (6.2) and (6.4),
become
and
respectively. Where previously it was unnecessary to distinguish between covariant and contravariant components of tensors, since there is no distinction 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, simply, 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, for
example, 
becomes 
where ";" indicates the covariant derivative of
with
respect to 
.
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
electric field tensor,
, or its
variation, 
, appear. 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 may be attributed to a reduction
in the magnitude of their charges. 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 (5.3), 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