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?


In the case of a constant gravitational field and a non-relativistically moving observer, the redshift $z$, i.e. the relative change in frequency $z=\frac{\nu_e}{\nu_o}-1$, is (there's a nice short explanation here under 'Gravitational Redshift'): $$z=\frac{gL}{c^2}$$

The analog in cosmology would be Hubble's law: $$v = H_0D$$ with $H_0$ being Hubble's constant (it has dimensions of frequency, by the way). Since $v=cz$ (implicitly assuming reasonably small $D$ here): $$z = \frac{H_0D}{c}$$

$D$ and $L$ are both the distance to the source, so equating the two expressions for redshift above and re-arranging:

$$g = H_0c$$

So the expansion rate of the Universe (described by the FLRW metric), which here we've approximated as being $\sim$constant, causes the same redshift as a constant gravitational field of magnitude $H_0c$. Hopefully you can see the analogy with your first expression (the proportionality to distance is obvious, anyway).

This is in contrast to the other component of redshift often mentioned in cosmology, that caused by the peculiar velocity of an object, which is simply explained as a Doppler shift, analogous to your second expression.

  • $\begingroup$ Thanks, but I do not see the answer to my question here. Is there a shift due to accelerated movement of stars and galaxies? Do cosmologists consider it? I do not see how Doppler shift is separated from gravitational one when explaining the observed Hubble's law. You answer doesn't tell me anything new. Of course I've seen the analogy, hence my question. I will look up the links you provided though. $\endgroup$ – Yuriy S Dec 17 '15 at 20:15
  • $\begingroup$ What I'm trying to show is a close analogy between the frequency shifts due to (i) acceleration in a constant gravitational field and (ii) the metric expansion of the Universe. I'm arguing that this is the same shift you're talking about, just cast differently. For objects at cosmological distances, you can't separate a shift due to (peculiar) acceleration from the one due to peculiar velocity, because the peculiar velocity is not independently measurable without a redshift independent distance measurement (we don't have sufficiently accurate distance estimators for the job). $\endgroup$ – Kyle Oman Dec 17 '15 at 21:27
  • $\begingroup$ There may be examples outside of cosmology (gravitational redshift near a black hole? or simply the Harvard tower experiment). The whole point of the equivalence principle, after all, is that an observer in a gravitational field is the same as an accelerated observer. $\endgroup$ – Kyle Oman Dec 17 '15 at 21:29

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