by T L Hurst
A Thought Experiment
In a region of space far away from other material objects there is a radio transmitter and two girls, Alice and Betty. The transmitter emits a hundred time signals per second. At 10:00:00 the transmitter sends the time "10:00:00.00". Alice and Betty each have a device capable of displaying the time signals as they are received. Each girl also has a synchronized clock co-located with them. Note: There is no time dilation in this scenario, so the clocks always indicate the same time as the transmitter.
Betty is at rest with respect to the transmitter and 4 light seconds distant from it. So the time displayed on Betty's device is always 4 seconds behind the actual time (as displayed on her clock). I.e. At 10:00:00 Betty's device displays "09:59:56.00". Whilst at 10:00:00 Alice is 5 light seconds distant from the transmitter and approaching it at 0.2c (i.e. one fifth of the speed of light). At this moment, the time signal displayed on Alice's device is not known.
In 5 seconds time (at 10:00:05) Betty's device will display the time signal "10:00:01.00". Meanwhile, during the 5 seconds. Alice will have travelled 1 light second and be adjacent to Betty. You might therefore expect that at 10:00:05 she too would receive the time signal "10:00:01.00". However, in Alice's rest frame, she is stationary and the transmitter is approaching her at 0.2c. At 10:00:00, when it is 5 light seconds distant from her, the transmitter emits the time signal "10:00:00.00" and continues on its way. The time signal approaches her at the speed of light, so Alice will receive the time signal "10:00:00.00" at 10:00:05.
So at 10:00:05, even though they are adjacent, Betty receives the time signal "10:00:01.00" and Alice receives the time signal "10:00:00.00". Furthermore, Alice, will see Betty's device displaying "10:00:01.00", and Betty will see Alice's device displaying "10:00:00.00". Also, as five seconds have passed since the time signal "10:00:00.00" was transmitted, both their clocks will display "10:00:05.00".