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I. Overview
Another SA asked me to elaborate on a remark I made to the effect that frequency shift phenomena always (even in Minkowski vacuum) involve at least the following ingredients:
Given these ingredients, one can compute a energy-momentum four-vector all along C'', representing a "photon" whose world line is C''. Then by taking the inner product of this four-vector at A with the unit tangent vector to C and comparing with the inner product at A' with the unit tangent vector to C', one can compute "photon energy" (or equivalently, frequency or wavelength) at the emission and reception events.
In simple cases, one can often make a reasonable decomposition of the net frequency shift into a kinematic Doppler shift and a gravitational frequency shift, but one should not expect this to be feasible or even insightful in every case.
Another SA asked me to elaborate on a remark I made to the effect that frequency shift phenomena always (even in Minkowski vacuum) involve at least the following ingredients:
- two (proper time parameterized) timelike curves C, C'
- an event A on C ("emission event")
- an (affinely parameterized) null geodesic C'' from A to an event A' on C' ("reception event").
Given these ingredients, one can compute a energy-momentum four-vector all along C'', representing a "photon" whose world line is C''. Then by taking the inner product of this four-vector at A with the unit tangent vector to C and comparing with the inner product at A' with the unit tangent vector to C', one can compute "photon energy" (or equivalently, frequency or wavelength) at the emission and reception events.
In simple cases, one can often make a reasonable decomposition of the net frequency shift into a kinematic Doppler shift and a gravitational frequency shift, but one should not expect this to be feasible or even insightful in every case.