I'm trying to understand the concepts of time dilation and spatial compression. I've been using the classic example of firing a photon of light inside a ship (spaceship, boxcar, etc.) moving at a constant velocity to the second observer. But when I do the Lorentz, I get different values depending upon the direction of travel of the photon relative to the ship.
My understanding is that time dilation for the two observers should be a constant, as should spatial compression, since their relative velocities are constant (at least for the duration of the experiment).
When I fire the photon in the direction of travel of the ship, the measurements all work out as expected. Time on the ship is passing slower. Then when I fire the photon backward in the direction of travel of the ship, all the values that should be constant have different values. So I assumed I'd done the transforms wrong. I redid them and still got different values. hen I simplified the equations by picking values that make the transforms very basic. Ship length = 1 light second for the ship observer. Ship speed = 0.1 light year for the second observer. Even then I get variances in the values that I expect to be constants.
My problem might be in measuring the distance the photon traveled for each observer. In firing forward, the second observer sees the photon travel the compressed length of the ship plus the distance the ship travels. The ship observer see it travel the length of the uncompressed ship. When firing backwards, the second observer sees the photon travel the compressed length of the ship minus the distance the ship travels. Since the ship observer sees the photon travel the same distance and amount of time regardless of direction, the second observer must also see it travel the same time in both directions. But the distance is shortened for the secondary observer.
So either the time dilation or the spatial compression would have to change for the equations to work. But my original premise was those are constants because the relative velocity is assigned to be constant for the experiment! Am I still doing the transforms wrong, or do I have a bad assumption in which values become constants by forcing relative velocity to be constant?
I've reviewed dozens of the time paradox explanations. But those arise from the two observers returning to one frame of reference. I've let my ship sail on beyond the end of the experiment indefinitely, purposely to avoid those problems from interfering in learning how to do the transforms correctly. Now I'm not sure I didn't step into a different paradox. Maybe this should be solved in GR instead?