Synchronization of clocks and observations of time in special relativity

Question

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.

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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.

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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?

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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.

Answer 1

I think the thing you have to realize is that in relativity, the time that an observer sees something happen - that is, the time at which the light signal of the event reaches the observer - is not the time at which it actually happened. If you're going to talk about these relativistic effects like time dilation, you have to correct for the time it takes light signals to travel, or only compare events (e.g. timer readings) which happen at the same location in space and at the same time.

Regarding your issues:

• As observed by B, the ship does take 101.01 days to get from A to B. Now, suppose a light signal is emitted from A when the ship passes it. That light signal takes 100 days to get to B, and the ship takes 1.01 days more than that. So B sees a delay of 1.01 days between the image of the ship arriving at A and the image of the ship arriving at B. But that doesn't mean the ship actually made it from A to B in 1.01 days.
• The ship passes A when A's timer reads 1000 days, and B when B's timer reads 1101.01 days. I don't know where you're getting 1201.01 days from - that value doesn't show up anywhere in the analysis.