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?
Best Answer
We can imagine that two observers move relatively to each other and play some kind of ping pong with a photon. Each of them possesses a tube. Through this tube observer spits a photon out and another observer catches it. They have to tilt their tubes so that the photon passed through them.
So as a photon would pass through the both tubes, these observers must always tilt their tubes at corresponding angles.
For example, if both observers keep their tubes at right angle, they in no way will be able to see each other.
If observer A keeps his tube straight up and sends a photon, observer B has to keep his tube at oblique angle into front so as to catch it. He keeps his tube into front exactly as astronomers do looking at distant stars.
Astronomers keep their telescopes at small angle because of aberration of light. It takes some time for the light to pass through a tube, and if a tube is not tilted the photon will hit a wall.
That’s why a moving observer sees a source as if in front of him, though actually the source is exactly “below”.
These angles of directions of the tubes are always tied with each other by relativistic aberration formula
https://en.wikipedia.org/wiki/Relativistic_aberration
An observer can keep his tube at ANY arbitrary chosen angle. But, if they want to see each other, another observer must to adjust his tube accordingly. For example, they can ascribe themselves roughly half of relative velocity each. In this case they must tilt their tubes at equal angles. One observer tilts backward, another – forward.
Episode 3 – Rectilinear Motion. The both source and observer are “in motion”
https://www.youtube.com/watch?v=hnphFr2Iai4
But, we can consider this case either from reference frame of observer at rest or from the reference frame from moving observer.
If observer A is at rest in origin and keeps his tube straight up, a photon will travel along the y axis, and will come to a moving observer at oblique angle. Thus observer B has to keep his tube at oblique angle and to look into front. If observer B has a mirror he must have another tube which is tilted at the same angle back so as a photon would be able to jump out. Then photon travels back to A and comes back at right angle. This is Transverse Doppler Effect, and observer B is a moving clock. Episode 2 of this video.
https://www.youtube.com/watch?v=FQKp3FU8vR8
In observer B frame observer A keeps his tube straight up, but photon always moves together with observer A and passes through a tube. Good to note, that in this case photon hits observer B at the points of closest approach, exactly as in the previous episode. Please look at the Episode 1 of this video
https://www.youtube.com/watch?v=FQKp3FU8vR8
This video graphically demonstrates that angles of direction of tubes are equivalent in both episodes.
Good to note that a photon "brings back the same amount" of energy when it was released. If a photon was released straight up and had "green" frequency, the photon will be blue shifted at mirror. Mirror gains energy when swallows photon and immediately loses the same amount when reflects it. Thus, photon was "blue" at mirror but will red shift and will turn "green" again when will come back to the source.