Here is a simplification of the paradox.
In the left hand diagram above, a ray of light enters the chamber from the left, reflects off of a flat mirror, which is set at a 45o angle so the beam is reflected at a right angle, where it hits the ceiling directly above it's point of reflection.
The diagram on the right represents the same chamber accelerated to 0.87 C. Now everything is Lorentz contracted to half of it's original size in the direction of motion (green arrows). The mirror is no longer at a 45 degree angle. It's a 26.56o angle. so the light no longer hits the ceiling directly above the mirror. Or does it?