Lorentz
Contraction – Version 2
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A rod is
at rest in the inertial frame of reference S’. A light emitter is located on
one end of a rod and a light detector is located at the other end of a rod. The
emitter emits a continuous laser beam to a mirror which is at rest in S’ and
located as shown in Fig. 1 below.
After
the light is reflected from the mirror it strikes the detector at the other end
of the rod. S’ moves in the +x
direction with speed v with respect to the inertial frame S’. S and
S’ are in standard configuration. The laser beam can be thought of as composed
of a stream of photons. As viewed from S the system appears as in Fig. 2 below.
The mirror is omitted for clarity.
The
velocity transformation relations between S and S’ are given by
The
emitter-mirror beam will be referred to as beam A and the mirror-detector beam
will be referred to as beam B. The components of the velocity of the photons in
frame S’ along beam A are
In frame
S the components of the velocity of the photons of beam A in frame S are
Along
beam B the components of the velocity of the photons in frame S’ are
In frame
S the components of the velocity of the photons of beam A in frame S are
The time
it takes for a photon to go from emitter to mirror is tA
while the time for a photon to go from mirror to detector tB.
These times are related to the velocities above as
The time
between emission and detection is therefore mirror is t = tA
+ tB.
The distance from the location the light is emitted to the location that the
light is detected is therefore
This
distance is the length, L, of the moving rod plus vt, the distance
traveled by the end of the rod during the flight time of the photons. That is
Solving
for L gives
Substituting
the values from above gives
Therefore
since g
> 1 it follows that a moving rod is shorter than the same rod at rest.