by T L Hurst
Velocity Effect continued...
At time t0: A is a distance s0 from the transmitter, and is travelling away from it at velocity v. B is stationary with respect to the transmitter at a distance s2. In B's rest frame, the time signal "t0" is approaching him at the velocity c, such that he will receive the time signal "t0" at time t2.
At time t1: A is a distance s1 from the transmitter when the time signal "t1" is emitted. In A's rest frame, A is stationary and the time signal is approaching him at the velocity c, such that he will receive the "t1" time signal at time t2.
At time t2: A and B are adjacent, but A receives the time signal "t1", whilst B receives time signal "t0".
So, we can say that the time taken for the t1 signal to reach A is t2 - t1. In that time, A moves s2 - s1 at velocity v (with respect to the transmitter). Therefore:
(t2 - t1) = (s2 - s1)/v
(s2 - s1) = v(t2 - t1)
- The magnitude of the difference in the time signals received (in seconds) is equal to the distance "moved" by the observer (in light seconds) between the time that the time signal is emitted and when it is received.
- A velocity towards the transmitter causes the observer to receive an earlier time signal than that received by a "stationary" observer (who is co-located when the time signals are received).