Is there well know formulas (or some articles) from General Relativity how photon (light) is changing (how frequency is changing) while reflecting from the accelerated mirror?
I suspect it should be different from the reflecting from just moving mirror with constant speed because accelerated mirror is Non-inertial reference frame and general relativity is applicable in that case.
After some comments, I want to clarify the question. What will happen in the picture below? Let's say I have a harmonically oscillating mirror x(t) = Sin(t) on my desk. The blue line is the mirror, red/orange circle is a photon. I put values of coordinate x, speed v and acceleration a on top of the image.
At the point x=1, v=0 and a=1. It looks strange, how is it possible v=0, but a is not. But it is the same if we throw a stone up. Acceleration is always g, but at the top point, speed is zero. At this point where x=1, v=0, a=1 photon is falling on the mirror.
The question is what will happen with photon frequency from my side (if this oscillating mirror is on my desk)? What will I see?
I change colour from red to orange (blueshift), because I assumed, that acceleration direction is from right to left, that means fictitious force will be in opposite direction, which is equivalent to gravitational force from left to right and photon is reflecting to the source of gravitational force (to the source of the gravity). Or it is not correct? Maybe I will not see any changes and all changes will be visible only for the observer on the mirror?
Also what if the photon hits the mirror when speed is not zero and acceleration is not zero? Will I see only Doppler shift or something different?
Of course, I understand, that in real life with the real oscillating mirror on the desk with any achievable speed any difference in frequency will be undetectable, but I just want to know theoretically with the assumption, that I can detect very very small changes in frequency.