Copyright © 1999 David E. Rutherford
All Rights Reserved
Transformation of Velocity
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
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net