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Figure 15.2:
has
in frame. is
moving in the 1 direction at . changes frames. We
want
.
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If we form the infinitesimal version of the Lorentz transformation of
coordinates:
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|
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(15.66) |
|
|
|
(15.67) |
|
|
|
(15.68) |
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|
|
(15.69) |
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
|
(15.70) |
for
. Then
Similarly, (e.g. -- ) is given by
or
|
(15.73) |
We see, then, that the velocity changes in both the and the directions.
Note also that if
and
, then
|
(15.74) |
and
|
(15.75) |
so that we recover the Gallilean result,
What about the other limit? If
, then
|
(15.78) |
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
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
|
(15.81) |
are just the -scaled ``velocities'' of the particle:
Figure 15.3:
Note that
so that each component of the 4-velocity
is always ``larger'' than associated Cartesian components, even though (as
usual) the length of the four velocity is invariant. What is its invariant
length?
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Next: Relativistic Energy and Momentum
Up: Special Relativity
Previous: Proper Time and Time
Contents
Robert G. Brown
2007-12-28