constant speed elevator light beam paradox I can't get my head around this problem.  Suppose there is an elevator moving up at a constant speed relative to an observer outside of the elevator, far away from any gravitational effects. Suppose a laser gun with a 1 meter long narrow barrel with a diameter of .10 cm emits a laser beam from one side of the elevator's wall, parallel to the floor of the elevator. A spot of light should appear on the opposite wall just below the line of sight of the laser gun as viewed from the outside observer.   But the observer inside the elevator would see the spot appear exactly opposite of the barrel. How is this paradox resolved? 
 A: The spot of light isn't below the line of sight of the laser gun, and the outside observer shouldn't expect that to be the case.
The laser gun is attached to the elevator wall, so according to the outside observer, the momentum of photons as they come out of the laser gun must have a non-zero upward component, or else conservation of momentum would be violated.  From the outside observer's perspective, the upward component of the photon's momentum is just enough to make the photon land on the line of sight of the laser gun as of when the photon lands.
A: If the elevator is going up rather fast, the light from the back of the elevator takes longer to reach the observer than the light from the front.
This means that I see the back of the elevator at an "earlier time" - put differently, it looks to be a bit lower. The angle by which it appears to be lower is given by the velocity of the elevator divided by the velocity of light.
Exactly the same angle by which you expect the beam to be low. So yes - the beam will appear angled, but it will hit a "low" target. All is well.
