Similarly, the components of are

Now we can write Eqs. (1346)-(1349) in the form , , , and , where

(1360) |

(1361) | |||

(1362) | |||

(1363) |

As in the transformation of coordinates, we can obtain the inverse transform by interchanging primed and unprimed symbols, and replacing with . Thus,

Equations (1364)-(1366) can be regarded as giving the resultant,
, of two velocities,
and
, and are therefore usually referred
to as the relativistic *velocity addition formulae*. The following
relation between the magnitudes
and
of the velocities
is easily demonstrated:

According to Eq. (1367), if then , no matter what
value takes: *i.e.,* the velocity of light is invariant
between different inertial frames. Note that the Lorentz transform
only allows *one* such invariant velocity [*i.e.*, the
velocity which appears in Eqs. (1346)-(1349)]. Einstein's relativity
principle tells us that any disturbance which propagates through a
vacuum must appear to propagate at the same velocity in all inertial
frames. It is now evident that *all* such disturbances must propagate
at the velocity . It follows immediately that
all electromagnetic waves must propagate through the vacuum with
this velocity, irrespective of their wavelength.
In other words, it is impossible for
there to be any dispersion of electromagnetic waves propagating through
a vacuum. Furthermore, gravity waves must also propagate with the
velocity .

The Lorentz transformation implies that the velocities of propagation of all physical
effects are limited by in deterministic physics. Consider a general
process by which an event *causes* an event at a
velocity in some frame . In other words, *information*
about the event appears to propagate to the event with a
superluminal velocity. Let us choose coordinates such that these
two events occur on the -axis with (finite) time and distance separations
and , respectively. The time separation in
some other inertial frame is given by [see Eq. (1349)]

(1368) |