If we form the infinitesimal version of the Lorentz transformation of
coordinates:

(17.39) | |||

(17.40) | |||

(17.41) | |||

(17.42) |

Point is moving at velocity in frame , which is in turn moving at velocity with respect to the ``rest'' frame . We need to determine (the velocity of in ). We will express the problem, as usual, in coordinates and to the direction of motion, exploiting the obvious azimuthal symmetry of the transformation about the direction.

Note that
u_i = c dx_idx_0
for
. Then
u_&par#parallel;& = & c &gamma#gamma;(dx_1' + &beta#beta;dx_0')&gamma#gamma;(dx_0' +
&beta#beta;dx_1')

& = & c { dx_1'dx_0' + &beta#beta;
}{ 1 + &beta#beta;dx_1'dx_0' }

& = & u_&par#parallel;+ v1 + **u**' ·**v**c^2 .
Similarly,
(e.g. --
) is given by
u_2 & = & c dx_2'&gamma#gamma;(dx_0' + &beta#beta;dx_1')

& = & u_2'&gamma#gamma;(1 + **u**' ·**v**c^2
or
**u**_&perp#perp;= **u**_&perp#perp;&gamma#gamma;{ 1 + **u**'
·**v**c^2 } .
We see, then, that the velocity changes in *both* the
*and* the
directions.

Note also that if
and
, then
**u**' ·**v**c^2 « 1
and
&gamma#gamma;&ap#approx;1
so that we recover the Gallilean result,
**u**_&par#parallel;& = & **u**_&par#parallel;' + **v**

**u**_&perp#perp;& = & **u**_&perp#perp;'.

What about the other limit? If
, then
**u** = c
as you should verify on your own. *This is Einstein's second postulate!*
We have thus proven explicitly that the speed of light (and the speed of
anything else travelling at the speed of light) is *invariant* under
Lorentz coordinate transformations. This is their entire motivation.

We observe that the three spatial components of ``velocity'' do *not* seem
to transform like a four vector. Both the
and the
components are mixed by a boost. We can, however, make the velocity into a
four vector that does. We *define*
U_0 & = & dx_0d&tau#tau; = dx_0dt dtd &tau#tau;

& = & c &gamma#gamma;(u)

**U** & = & d**x**d&tau#tau; = d**x**dt
dtd&tau#tau;

& = & **u** &gamma#gamma;(u)
where
is evaluated using the magnitude of **u**. It is an
exercise to show that this transforms like the coordinate 4-vector
.

Now we can ``guess'' that the 4-momentum of a particle will be
.
To prepare us for this, observe that
U = (U_0,**U**) = (&gamma#gamma;_u c, &gamma#gamma;_u **u**)
are just the
-scaled ``velocities'' of the particle: