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(1655) |
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(1656) |
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(1657) |
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(1658) |
Equations (1661)-(1663) 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 Equation (1664), if
then
, no matter what
value
takes: that is, the speed of light is invariant
between different inertial frames. Note that the Lorentz transform
only allows one such invariant speed [i.e., the
speed
that appears in Equations (1643)-(1646)]. Einstein's relativity
principle tells us that any disturbance that propagates through a
vacuum must appear to propagate at the same speed in all inertial
frames. It is now evident that all such disturbances must propagate
at the speed
. It follows immediately that
all electromagnetic waves must propagate through the vacuum with
this speed, 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 at the
speed
.
The Lorentz transformation implies that the propagation speeds 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 Equation (1646)]
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(1663) |