I have thought up a situation that I cannot understand with my understanding of special relativity. I don't know general relativity, but as the situation doesn't involve gravity or acceleration, I'm not sure if it is needed.
Imagine there are 2 digital count up timers, A and B, separated by 100 light days in the same frame of reference. There is a person at each timer. Both timers are turned on at the same time (this could be done via a signal at a point C which is the same distance away from both A and B).
At this point, a person at A will see the timer at B being 100 days less than the timer at A (due to the time light takes to travel from A to B). Vice versa for a person at B.
Now imagine a ship is traveling at 0.99c through A to B. The ship also has a timer on it, which is initially off.
When it passes past A the pilot notes timer A is at 1000 days. The pilot then starts the ship's timer.
Eventually, the ship will go past B. My questions are:
1) What time will people A and B see on the their timer and the ship's timer when the ship passes past B?
2) What time will the ship see on A and B when it passes past B?
Here are the issues I have with this:
From B's perspective, the ship passes past A when B's timer is 1100 days. If you ignore relativity, you would expect a ship traveling at 0.99c to take
100 / 0.99 = 101.01 days to get to B. Wouldn't that mean from B's perspective, it only takes 1.01 days for the ship to get from A to B? I thought length contraction would explain that, but A and B aren't moving relative to each other. S would appear to be moving faster than the speed of light, which doesn't make sense.
From A's perspective, S will experience time 7.088 slower than A. So when it passes B, the ships timer will only be
101.01 / 7.088 = 14.25 days. Taking 101.01 days to get to B, it will see B's timer being 1101.01 days. It will see this time when it's timer is 1201.01 days. That means it will take 201.01 days for the ship's timer to advance by 14.24 days, a time dilation of
201.01 / 14.24 = 14.12, which is different to 7.088? I must be double counting somewhere, I don't know.