Mathematical Rel.
The Mathematical Relationship
The scenario we wish to model is where the velocity of light is constant with respect to the target. The relationships between macro objects conform to Newtonian relativity. I.e. All motion is relative, and there is no time dilation or distance contraction.
The following diagram describes the scenario:
Note: For convenience, the graphic displays the scenario as viewed from the rest frame of the transmitter, and velocities are generally given with respect to the transmitter. However, it must be borne in mind that, in A's rest frame, he is stationary and the transmitter is moving away from him at velocity v. At time t2 A and B are shown graphically next to each other, but this is intended to represent them being the same distance s2 from the transmitter.
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".
The idea of adjacent observers receiving different time signals from a distant transmitter/clock may seem illogical, even contradictory. Certainly, if we envisage space as a physical entity and light as particles or waves travelling through that space, it would be impossible. We present a conceptual resolution of this apparent contradiction below, but here we seek to quantify the phenomenon, not deal with it's validity per se.
So, we can say:
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
or
(s2 - s1) = v(t2 - t1)
We noted here that:
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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.
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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).
- A velocity away from the transmitter causes the observer to receive a later time signal than that received by a "stationary" observer (who is co-located when the time signals are received).
This was referred to as the "velocity of light effect", which we shall denote with the symbol o. So:
o = (s2 - s1) = v(t2 - t1) (where s1 and s2 are measured in light seconds, and v is measured in light seconds per second). Or:
o = (s2 - s1)/c = v(t2 - t1)/c (where s1, s2 and v are is SI units).
