Copyright © 1999 David E. Rutherford
All Rights Reserved
Two four-vectors may be multiplied together by employing the usual algebraic methods if the appropriate conventions for the multiplication of unit vectors are followed. We begin by defining the unit vectors which lie in the directions of the orthogonal coordinate axes. The unit vector i is directed along the positive x-axis, j along the y-axis, k along the z-axis, and l (this is the lower case letter "el", not the number "one") along the t-axis. The rules for the multiplication of these vectors have been taken, partially, from Hamilton's quaternions. However, in addition, we've included the unit vector along the time axis and its interactions with the other unit vectors. These vectors multiply as follows
and
The relations in Eqs. (2) might be used for the gradient, divergence, D'Alembertian, etc..
We now define an arbitrary four-vector A as
where Ax, Ay, Az, and At are the components of A along the coordinate axes. Defining a second four-vector B as
we will form a third four-vector C by multiplying the two four-vectors A and B together using the familiar rules of algebraic multiplication and the rules for multiplication of unit vectors listed above.
The components of C can also be obtained using matrix multiplication
by arranging the components of A in a 4x4 matrix,
, and multiplying
by the components of B
The matrix, 
,
can be viewed as a transformation matrix between B and C. We
might also imagine that its components are the components of a velocity,
v. Any velocity measurement in a particular frame of reference
is a combination of the coordinates in that frame, so that
where
But, these components are not the components of a four-vector. To
put the components in the correct form, we must have the invariant proper
time, 
, in
the denominator. We will need to find the improper time, dt, in terms
of the proper time,
, which we
can get from
Now, we divide by
and rearrange
terms to get
where
and
We can now put the
in terms
of the
by
using Eqs. (10) and (12)
where
But, in order that B and C have the same units, we must make
dimensionless.
This is accomplished by dividing Eqs. (13) by an invariant velocity, the
velocity of light c. Finally, we have the components of the transformation
matrix, ,
The matrix, ,
might be used as a substitute for the transformation matrix,
, and the
Lorentz transformation. Its advantage is that, unlike
, it contains
no complex terms, but is still orthogonal.
We will now use
in the transformation of coordinates
which is in the same form as Eq. (6) with C being replaced by x' and B by x. For a primed frame of reference moving with uniform velocity, U, in the x direction of an unprimed frame there is, contrary to the Lorentz transformation, a rotation of the y' and z' axes relative to the y and z axes, in addition to the rotation of the x' and t' axes relative to the x and t axes. The inverses of Eqs. (16) can be obtained by reversing the primed and unprimed coordinates and the signs of Ux, Uy, and Uz, but Ut remains positive. The relations between the unit vectors in Eqs. (1) and (2) also change accordingly.
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net