Relativistic
Optics
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Consider
a mirror that is at rest the xz-plane in the inertial frame of reference
S. Let a ray of light impinge on the mirror as shown in Fig. 1 below
The
velocity of the light of the incident beam in frame S is
The
velocity of the light of the reflected bean in frame S is, according to the law
of incidence and reflection
Now
consider a frame of reference S’ which is moving in the +x-direction
with respect to S with speed v as show below in Fig. 2
The velocity
transforms from frame S to frame S’ as
For the
incident beam we have
For the
reflected beam we have
The
tangent of the incident angle in S’ is given by
The
tangent of the reflected beam in S’ is given by
Therefore
Thus the
law of the equality of the angle of incidence and the angle of reflection holds
for this orientation of the mirror.
Now consider a plane
mirror at rest in S but lying in the yz-plane. Let a ray of light impinge
on the mirror as shown in Fig. 3 below
The
velocity of the light in the incident beam in frame S is
The
velocity of the light in the reflected bean in frame S is, according to the law
of incidence and reflection
Now
consider a frame of reference S’ which is moving in the +x-direction
with respect to S with speed v as show below in Fig. 4
For the
incident beam we find
For the
reflected beam we find
The
tangent of the incident beam in S’ is
The
tangent of the reflected beam in S’ is
Therefore
