I was thinking the other day about the simple example used to demonstrate time dilation effects and to derive the Lorrentz factor - where the time it takes for a light pulse to be emitted, bounce of a mirror, and then detected, is seemingly different to someone in the reference frame of the light pulse and somebody for whom that reference frame is moving.
Thinking about this example, though, I noticed something that I couldn't quite figure out. It's been several years since I took a class on relativity, and I'm sure there's an answer, but I couldn't find it.
My issue is this: for the person in the reference frame, the motion of the light makes sense - it is shot out of the emitter, travels in a straight line, bounces off the mirror, and is received by the sensor. But for the outside observer, for whom the reference frame of the light pulse is moving, the motion of the light doesn't make sense. When the light leaves the emitter, it will be at one point. However, by the time the light reaches the mirror, that mirror will be at a different location than where it was when the light was emitted from the emitter. As far as this observer is concerned, the emitter was emitting the light in some direction, say perpendicular to the motion of the reference frame, and yet that light still hit the mirror, a path which would not have been perpendicular to the motion. This would suggest that either the light was emitted at an angle, which it shouldn't be because the direction of the emitter should be unaltered, or that the light somehow carried with it some momentum from its reference frame that allowed it to travel diagonally, which it obviously shouldn't.
So how do we rectify this dilemma? How do we explain the motionless observer seeing light travel in a diagonal direction when it should be going straight?