How can a beam of light have components of momentum in more than one dimension? Regarding Einstein's famous thought experiment: How can a beam of light emitted from a rocket traveling at .5c maintain the inertial velocity of the rocket when the beam is perpendicular to the rocket's direction?
I understand how that would work with a bouncing ball on a moving train but how can light have an additional component of momentum when it must always move with the maximum possible velocity in one dimension?
Is this what is claimed by Special Relativity?
 A: Einstein’s second postulate is that the speed of light is c in an inertial frame. There is no constraint on the direction. The direction of the light can be determined by the relativistic aberration formula: $$\theta’ =\cos^{-1} \left( \frac{\cos \theta -v/c}{1-(v/c)\cos \theta} \right)$$
With this formula the light you describe will have the correct component of velocity parallel to the ship as well as the correct speed.
A: Imagine you are standing on a platform and I am standing on a train moving slowly past you.
If you walk perpendicularly across the platform towards its edge you are moving diagonally in my frame of reference.
From the perspective of a third observer in a train moving in the opposite direction to mine, you are moving diagonally in the opposite sense.
You believe you are moving perpendicular to the length of the platform, but each observer sees you moving at an angle to it.
It is the same with the light and the rocket. The observer on the rocket believes them-self to be stationary and to be directing the beam of light in a given direction. Any observers moving relative to the rocket will see the beam at a different angle in their frame of reference. It is simply a matter of perspective.
