Lorentz transform of the components of the velocity
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A photon moves at an angle θ with respect to the x' axis in frame S'.
Frame S' moves at speed v with respect to frame S (along the x' axis).
Calculate the components of the photon's velocity in S, and verify that
the speed is c.
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Solution Summary
The expert explains the solution of this problem starting from the Lorentz transform. The components of the photon's velocity in S is verified.
Solution Preview
In the following I'll work in c = 1 units. An event at coordinates (t',x',y',z') in S' will have coordinates (t,x,y,z) in S where:
x = gamma (x' + v t')
y = y'
z = z'
t = gamma (t' + v x')
where
gamma = 1/sqrt(1-v^2)
To find the rules for transforming velocities, we can consider an object moving at some velocity in frame S and take two events to be the object being at some position at some time and then at an infinitesimally later time at some infinitesimally different position. The components of the velocity are then given by the ratio of the components of the change in the position vector w.r.t. the change in time in the limit where the differences tend to zero. From the above ...
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