In most books to explain transverse Doppler effect the following example is given:
Consider a source that emits flashes at frequency f0 (in its own frame), while moving across your field of vision at speed v. There are two reasonable questions we may ask about the frequency you observe:
• Case 1: At the instant the source is at its closest approach to you, with what frequency do the flashes hit your eye?
• Case 2: When you see the source at its closest approach to you, with what frequency do the flashes hit your eye?
In the first case we observe from the trains frame, while in the second we do not. The explanation for doing this is given as follows. If we observe from the ground frame the following error is supposed occur:
The error can be stated as follows. The time dilation result, ∆t = γ·∆t0, rests on the assumption that the ∆x0 between the two events is zero. This applies fine to two emissions of light from the source. However, the two events in question are the absorption of two light pulses by your eye (which is moving in the source frame), so ∆t = γ·∆t0 is not applicable. Instead, ∆t0 = γ·∆t is the relevant result, valid when ∆x = 0.
Here x0, t0 is the observation in the moving frame, and γ is the dilation factor.
My question is, for what events and why is ∆x0 not equal to 0. And why when we observe from the moving frame ∆x is supposedly 0.