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I am trying to get my head around the concept of redshifts and their relation to powers. Consider the observe $O$ at rest (who is measuring the power) and emitter $E$ in one of the three following situations:

  1. Moving in Minkowski space at a velocity $\vec{v}$.

  2. At rest (relative to $O$) in a non-Minkowski space (i.e. a non-expanding universe with a gravitational field.)

  3. At a fixed co-moving coordinate (relative to $0$) in a expanding flat universe.

And let us call the redshift associated with each $z_D$, $z_G$ and $z_C$. I know that in the case of cosmological redshift the flux goes like: $$\frac{1}{(1+z_C)^2}$$ since both the energy of the photons has change and the number which have arrive per unit time. My question is how does the power change in the first two cases? and how does this relate to the type of redshift?

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In all three cases the received power is decreased by exactly the same amount. There is no observational distinction between a cosmological redshift and a doppler redshift.

The proof of (1) can be easily found by transforming the E- and B-fields of a transverse wave and hence calculating the transformed Poynting vector. This shows that for an observer moving directly away at speed $v$ in frame $S^{\prime}$ from a source at rest in frame $S$ (or vice-versa) that the ratio of Poynting vectors in the two frames is $$\frac{N^{\prime}}{N} = \gamma^2 (1 - \frac{v}{c})^2 = \frac{1}{\gamma^2 (1 + v/c)^2} = \frac{1}{(1 + z_D)^2}$$

As for (2) well it doesn't matter that space is not universally flat; it is in the locality of the observer and so the power is reduced according to their locally measured doppler shift in exactly the same way. You cite the example of a gravitational field, say in the Schwarzschild metric. The same argument that you used for the cosmological redshift works there too. Time dilation means that an observer that is effectively at infinity receives photons at a rate slower by a factor $(1+z_G)$ from a source deeper into the gravitational potential, and each photon is redshifted in frequency by a factor $(1+z_G)$, leading to the same factor decrease in received power.

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