Acceleration in flat space-time and gravitational redshift

Question

Consider two observers in flat space-time, of which one, called Terrence, is stationary, while the other, called Stella, moves in an accelerated way. I am particularly interested in the case where Stella moves at a constant speed $v$ along a closed trajectory (hence the acceleration), but the question below is more general. Suppose also that Terrence and Stella are sending photons to each other. The expressions for the (kinematic) Doppler factors for communication in both directions are well-known and can be found in standard texts on relativity.

Question: Does the gravitational Doppler effect also somehow come into play due to the acceleration (by invoking the equivalence principle)? Would the Doppler shift that Stella observes be a combination of the kinematic and gravitational Doppler shifts in this scenario?

Motivation: I am currently studying the standard analysis of the twin paradox via the Doppler effect. When this analysis is applied to an arbitrary closed trajectory using only kinematic Doppler factors, I get the right aging difference between Terrence and Stella (a factor of $\sqrt{1 - (v/c)^2}$), so it seems like there should not be any additional changes in the Doppler factors due to the acceleration. But I am still wondering if that argument is correct, i.e., if some weird things can happen that are caused by acceleration.

If possible, I would also appreciate a concrete reference where relevant problems were analyzed.

P.S. Please forgive my ignorance, I am not a physicist by training, but I did look in many textbooks on special relativity and found no explanation of this.

Answer 1

Question: Does the gravitational Doppler effect also somehow come into play due to the acceleration (by invoking the equivalence principle)? Would the Doppler shift that Stella observes be a combination of the kinematic and gravitational Doppler shifts in this scenario?

The idea that (with respect to an inertial frame) there is no additional time dilation due to acceleration is called the clock hypothesis. In the 1970's Bailey did some experiments where they took muons that were going in a circular loop at relativistic speeds and measured their decay rates to determine their proper time. This experiment confirmed the clock hypothesis up to about $10^{18}\ g$.

Bailey et al., "Measurements of relativistic time dilation for positive and negative muons in a circular orbit," Nature 268 (July 28, 1977) pg 301.

Bailey et al., Nuclear Physics B 150 pg 1–79 (1979).

So in Terrence's inertial frame, Stella's time dilation is due only to her velocity and there is no additional dilation that must be accounted for due to her acceleration. Now, of course, Stella's frame is more complicated and cannot use the standard time dilation formula. However, due to the manifest covariance of the laws of physics, we are guaranteed that with the calculation of the correct time dilation formula, Stella's frame will obtain the same result as Terrence's frame for all measurable outcomes.

Note, the above analysis is focused on the time dilation while the question asked about the Doppler shift. Time dilation is the transverse Doppler, so they are closely related. As you go around a closed path, the non-transverse parts of Doppler cancel out and all you are left with is the transverse Doppler, or time dilation.