by T L Hurst
Thought Experiment continued...
So there is no discrepancy in how much time has actually passed for Alice and Betty (as indicated by their clocks), but the time signals received on adjacent devices from a distant transmitter differ by 1 second.
Let's now re-run the experiment, starting again at 10:00:00. This time, moments before they are adjacent, both girls move sideways such that at 10:00:05 each is at the exact position occupied by the other in the first experiment.
The small transverse movements should have no effect on when the girls receive the time signals, so Betty's device will still show "10:00:01" whilst Alice's will show "10:00:00". The girls are at the exact positions occupied by each other in the first run, yet each still receives the time signals according to their own relationship to the transmitter.
Similarly, we may add a third observer, Carol, who is travelling at 1/3 c away from the transmitter. Furthermore, if we assume that the girls could see the transmitter; at 10:00:05 Alice would see it as being 5 light seconds distant, Betty would see it as being 4 light seconds distant, and Carol would see it as being 3 light seconds distant.
Is there an actual difference in the distance? No. The girls see the transmitter as being at different distances because they are seeing it where it was (with respect to themselves) when the light that each receives was emitted. So even though the actual physical relationships are unaffected, the apparent relationships to a distant object can, and will, differ.
It is apparent that, in the time taken for the time signal "10:00:00.00" to reach Alice, she "moves" 1 light second towards the transmitter, and that time signal is 1 second earlier than the one Betty receives (when they are adjacent).