It's a question that's been bugging me since I first read Einstein's paper on gravitational frequency shift. He derives it using the equivalence principle, and considers an accelerating source first.
Now, for an accelerating source the shift is approximately proportional to acceleration and the distance to observer.
$$ \Delta \nu \propto a L $$
But I never heard that cosmologists consider this kind of shift. I only hear about Doppler shift all the time, proportional to velocity.
$$ \Delta \nu \propto v $$
Even Hubble's law is explained by Doppler shift. But wouldn't it make more sense to explain it by shift due to acceleration of all the stars? It is proportional to the distance after all. And every star or galaxy is always accelerating.
Or am I completely wrong, and $L$ is not distance to observer in this case?