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.
Best Answer
There are two events of particular interest. One is where a pulse of light is emitted when the source and absorber are closest together; the other is when a pulse of light is absorbed when the source & absorber are closest together.
Transverse redshift: Light emitted at the instant of closest approach will be redshifted. The light pulse will appear to the absorber have originated from the closest point along the emitter's trajectory. At the time of absorption the distance between the source and absorber will be increasing.
Transverse blueshift: Light absorbed at the instant of closest approach will be blueshifted. The light pulse will appear to the absorber to have originated from a point earlier along the emitter's trajectory than the closest point. At the time of emission the distance between the source and absorber was decreasing.
Let: γ = $ \frac{1}{\sqrt{1-v^2}} $ fe = emitted frequency measured in the emitter's frame, fa = absorbed frequency measured in the absorber's frame.
In case #1, in the absorber's frame there is no classical Doppler shift, only the relativistic time dilation factor γ-1. (fa = fe·1·γ-1)
In case #2, in the absorber's frame there is a classical Doppler shift of γ2 which cancels the relativistic time dilation of γ-1, leaving a net shift of γ. (fa = fe·γ2·γ-1)
In case #1, in the emitter's frame there is a classical Doppler shift of γ-2 which cancels the relativistic time dilation factor γ, leaving a net shift of γ-1. (fa = fe·γ-2·γ)
In case #2, in the emitter's frame there is no classical Doppler shift, only the relativistic time dilation of γ. (fa = fe·1·γ)