This question pertains to EM signals exchanged between an inertial central clock and an orbiting clock i.e. non-inertial frame. (I edited out the variables connected with gravity and separation between bodies)
This is how I found the frequency and wavelength received by clock (orbiting) as sent from clock (central). Hereafter let $$c=1$$ and angular speed as the proportion of c $$rω/c=β$$ so that the length contraction factor $$\gamma$$ for each instant in a rotating frame is given as $$γ=\sqrt{1-β^2}$$
Clock 2 is situated at the centre of clock 1’s uniform rotation. We can use the simplified formula $$f_1=f_2/\sqrt{1-β^2}$$ to determine the frequency of a signal that rotating clock 1 receives from central clock 2's emitted frequency.
Let central clock 2’s frequency=1, so $$f_2=1$$ Therefore, a familiar relation appears $$f_1 = 1/\sqrt{1 - β^2}$$ or just $$f_1=1/γ$$
The frequency of a signal and its wavelength in relation to the speed of light is given by $$c=fλ$$ Let's view this relation from rotating clock 1's frame. So, $$c_1=(1/γ*f_1)(γλ_1)$$ but from central clock 2's frame, we get the reciprocal $$c_2=(γf_2)(1/γ*λ_2)$$
Is this correct?
