0

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.

aleph
  • 103
  • 3
  • The Wikipedia page on the relativistic Doppler effect covers many scenarios including uniform circular motion: https://en.wikipedia.org/wiki/Relativistic_Doppler_effect – Er Jio Feb 27 '24 at 13:56
  • The twin paradox is unrelated to the Doppler effect so it is not clear what you are asking. – John Rennie Feb 27 '24 at 14:05
  • You asked a previous question that you deleted after I closed it as a duplicate. Have you read the question I linked? I and several other site members went to considerable trouble to explain exactly how the twin paradox works in detail. – John Rennie Feb 27 '24 at 14:07
  • @JohnRennie Yes, I read it and found the answer to that question. Thank you! The question I am asking here is in fact not relevant only to the twin paradox (I was only explaining my motivation for asking it). The question is this: does the gravitational Doppler shift arise due to accelerated motion? Is the Doppler shift that Stella observes equal to the standard (kinematic) Doppler shift, or is there a correction to this shift due to acceleration? – aleph Feb 27 '24 at 15:26
  • @JohnRennie As for the twin paradox, there are ways of analyzing it that rely entirely on the Doppler effect, that is what I meant. For example, this paper: https://www.nature.com/articles/180976a0. Or this link (among many others): https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html. When applying this analysis of the twin problem with an arbitrary trajectory using only kinematic Doppler, I get the right difference in aging between Terrence and Stella. So it seems like there should not be any other Doppler shift. But, I am not sure if my reasoning is correct. – aleph Feb 27 '24 at 15:28
  • 2
    Does this help? Blueshift in a rotating frame. You can attribute the frequency shift to kinematic time dilation, or to gravitational time dilation, or to some mixture, depending on the reference frame that you use. But if you are working in an inertial frame, then there is no gravitational time dilation. – Sten Feb 27 '24 at 15:44
  • @Sten Yes and no :) It does address (a special case of) the problem I am interested in. But would you perhaps know some reference(s) where I could find a more detailed explanation/derivation of why only the standard Doppler suffices in the analysis and no corrections are needed due to acceleration? (I.e., why there is no need to equate acceleration to gravity and account for the gravitational redshift in addition to the kinematic one.) Thank you for the answer and the link in any case. – aleph Feb 27 '24 at 15:55
  • 1
    From https://physics.stackexchange.com/q/704658/123208 "the clock postulate of special relativity says that time dilation (among other relativistic effects) is caused only by relative velocity and that acceleration itself has no direct effect on it". – PM 2Ring Feb 27 '24 at 17:03

1 Answers1

3

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.

Dale
  • 99,825
  • Thank you! Would you mind clarifying the "Time dilation is the transverse Doppler" statement? In the scenario when Stella travels along a straight line away from and back to Terrence, there is no transverse Doppler, but there is time dilation, am I right? – aleph Feb 27 '24 at 18:01
  • @aleph in classical physics the transverse Doppler shift does not exist. The relativistic Doppler shift is the classical Doppler shift times the transverse Doppler/time dilation. So in a straight line away or towards there is both transverse and classical Doppler. At 90 degrees the classical Doppler is 0 and only the transverse Doppler remains (hence the name), but the effect is present (time dilation) in all directions. – Dale Feb 27 '24 at 23:06