Take two clocks, one in a circular orbit around a Schwarzschild black hole and another with a distant observer.
The time dilation factor between the two clocks is said (at https://en.wikipedia.org/wiki/Gravitational_time_dilation ) to be given by $\sqrt{1 - 3GM/rc^2}$ (versions of the proof can be found at Time Dilation in Orbits in the Schwarzschild Metric and Is gravitational time dilation different from other forms of time dilation? ).
What is troubling me is whether this represents some sort of average? Surely a clock in orbit around the black hole is periodically travelling towards/away from the distant observer (unless the orbital plane is at right angles), which will result in a time-dependent time-dilation factor (and indeed a time-dependent doppler shift).
Can someone enlighten me?
Answer
Time-dilation is defined independent of retardation. That is, you are assumed to use some kind of range sensitive detector (perhaps a really big phased array) to collect the light of distant events and the reconstruct their "real" time and position in your frame of reference before computing the dilation.
A result of this is that red- or blue-shift of a light source alone is not sufficient to give you the time-dilation factor in all cases. (Note that if it was then time-dilation would depend on whether the source was approaching or receding.)
So for distant observers at rest with respect to the black hole (or more precisely to the barycenter of the hole and clock), I believe the orientation of the orbit doesn't change the fact that the relative speed of the observer and the orbiting clock is constant, and that therefore the orientation of the orbit doesn't matter.
The situation is complicated if the observer moves with respect to the hole, because then the relative velocity between clock and observe does vary in time (in some orbital orientations).
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