. These
basis vectors satisfy the relations
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
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
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net