http://www.damtp.cam.ac.uk/user/tong/concepts/gr.pdf
David Tong: "The rocket has height h. It starts from rest, and moves with constant acceleration g. Light emitted from the top of the rocket is received below. By this time, the rocket is travelling at speed v=gt=gh/c. This gives rise to the Doppler effect. (Neglecting relativistic effects). f'=f(1+v/c)=f(1+gh/c^2), where f' is received frequency and f is emitted frequency."

Since f=c/λ (λ is the wavelength), we have

f' = f(1+v/c) = (c+v)/λ

where c'=c+v is the speed of the light relative to the receiver (the bottom of the rocket).

Clearly the speed of light (relative to the receiver) varies with the speed of the receiver, in violation of Einstein's relativity. This means that David Tong's subsequent derivation of gravitational time dilation is invalid. There is no gravitational time dilation, and accordingly there are no gravitational waves.

Pentcho Valev