Red shift / Blue Shift - Gravity Well Roundtrip

[D.S.Beyer]

TL;DR Summary

Does wavelength change from trips through gravity wells?

If I shine a white light out into space from Earth, and it bends around a black hole and comes straight back to me so I can see it in my telescope, is the light red or blue shifted, or neither?

For sake of the thought experiment, let’s leave out cosmic expansion that would stretch it on it’s journey, and let’s say my frame of reference doesn’t move so we can leave out any doppler shift. I am purely interested if there is any effect on the light from going in and out of gravity wells. Does it return to its original wavelength (if viewed from the reference frame of its origination)?

 

[Bandersnatch]

Does it return to its original wavelength

Yes. The photon gains just as much energy falling into a gravity well as it later loses climbing out of it. The picture changes with large structures under certain kinds of expansion, where gravity wells evolve during a photon's journey (the Sachs-Wolfe effect). But in the simplest scenario it's just energy conservation.

 

[Halc]

I am purely interested if there is any effect on the light from going in and out of gravity wells. Does it return to its original wavelength (if viewed from the reference frame of its origination)?

The only thing that would have an effect is if the black hole (or mirror for that matter) is changing its distance from you, in which case there will be red/blue shift. The gravity doesn't have any effect at all from your perspective since you're measuring the light at the same potential as where it left you.

That said, if I shine a light at a (small, distant) black hole and it comes back to me, it will take longer to make the round trip (as measured by me) than light from a mirror placed at approximately the equivalent distance. But assuming no relative motion, both will come back at the original wavelength.