, should be
Copyright © 1999 David E. Rutherford
All Rights Reserved
The transformation matrix,
, should be
where the fourth row and column have been interchanged. The coordinate
transformations from an unprimed reference frame to a primed frame with
four-velocity 
relative to the unprimed frame are
and the inverse transformation equations are
where
The components of the electric field four-vector are
The components of the potential four-vector,
(not to be
confused with the transformation matrix
), is defined
here by
where the
are the components of the velocity four-vector of a charge distribution relative
to an unprimed frame of reference, and
is the scalar
potential of the distribution.
We will now find the components of the potential four-vector,
, for a simple
case. A charge distribution is at rest in a primed frame of reference in
uniform motion with four-velocity
as measured
by an observer in the unprimed frame. The potential four-vector,
, in the
primed frame is defined as
where, 
,
is the velocity of the charge distribution as measured by an observer in
the primed frame. Since the distribution is at rest in this frame, its
four-velocity is
, therefore
the potential four-vector in the primed frame is
.
The components of the potential four-vector in the unprimed frame,
, are related
to the components in the primed frame by
First, we will find the time component of the unprimed potential which is
but the only component of the primed potential that is nonzero is
so we have
simply
The remaining components in this case are found to be
By using Eqs. (6) and (7) we can see from Eqs. (10) that
but, since 
is the only component of the four-velocity in the primed frame it must
equal c, so we are left with
which shows that the scalar potential is invariant under transformation.
For a point charge,
, at rest
in a primed frame, the scalar potential,
, measured
by an observer at rest in the primed frame is
where 
is
the primed spacetime interval
If the primed frame is in uniform motion relative to an unprimed frame with
four-velocity measured by an observer at rest in the unprimed frame, we can find the
unprimed scalar potential,
, from the
primed scalar potential since
or
The primed spacetime interval is measured with
since we
make an instantaneous measurement of the distance from the charge to the
field point in the primed frame. To find the unprimed potential, we need
to put the primed variables of Eq. (16) in terms of the unprimed variables.
The unprimed spacetime interval,
, can be
found by substituting the values from Eqs. (2) into Eq. (15) and since
we can use Eqs. (28) and (29) of the "Original Article: May 7, 1999" on
this website to find that the unprimed charge is
, where
and
. Substituting
these quantities into Eq. (16) and remembering that the only nonzero components
of the velocity U are the x and t-components so that
we get
The primed charge,
, since it
is at rest in the primed frame, is the proper charge,
, so what
we really have in Eq. (18) is
where the unprimed spacetime interval,
, in this
case is
Since 
and
are scalar
tensor quantities, the quantities in the two center terms of Eq. (19) should
also be tensor quantities. This means the charges should be the rest or proper
charges and the distances should be the spacetime intervals, not simply the
space intervals in both frames of reference.
Since the electric field,
, transforms
as
we can write the primed components of the electric field four-vector,
, in terms
of the potential four-vector,
, by using
Eqs. (5) and (6) as
The components of the electric field tensor,
, can be
written using the components of the potential four-vector,
, in a way
that is completely equivalent to
. Using Eqs.
(5) and (6), we can write
and since 
is antisymmetric,
therefore
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net