TITLE: Do light experience Doppler shift along and against frame dragging?
QUESTION [2 upvotes]: Imagine 2 photons traveling along the equator in opposite directions, of course I'm not suggesting light goes into orbit I'm just wandering if these photons will be blue and red shifted ever so slightly as massive body such as Earth spins in general relativity?

REPLY [0 votes]: Arguably, no. Here's my reasoning:
First, note that when talking about frequency shifts in a generic general-relativistic setting, you need to specify your observers (emitter and absorber) as well as the photon trajectory. The generalized Doppler shift is then nothing but the difference in the time-component of photon momentum as measured in the two reference frames. This will yield special-relativistic Doppler shift as well as gravitational and cosmological redshift (which can be phenomenologically attributed to time dilation and spatial expansion, respectively).
Now, there are two different circular photon orbits in the equatorial plane of a Kerr black hole (cf Wikipedia), one prograde and one retrograde. An observer sitting at fixed Boyer–Lindquist coordinates along either of the orbits will always measure the same photon energy, no matter how often the photon has circled the black hole.
However, when you consider more complicated trajectories (eg when starting at an arbitrary radius and shooting photons in opposite directions around the hole), this story will generally change - but you will no longer compare apples to apples due to the inequivalent positions of the absorbers.
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