What is the difference between the way light travels in a gravitational field and the way it travels through an expanding universe?
In the former situation spacetime is curved across space, in the latter situation spacetime is curved over time.
A photon is travelling through space. As it travels, the space through which it travels is expanding. Does this describe:
A - a photon travelling from a distant galaxy to earth
B - a photon travelling from a point of higher to a point of lower gravity
C - either
It describes A. It can also be likened to a photon travelling from a point of low gravitational potential to a point of high gravitational potential. Hence we talk of gravitational redshift, and cosmological redshift.
If like me you think the answer is C, can you explain why light should travel differently through a gravitational field to how it travels through an expanding universe? In other words, why in one case does the speed of the photon vary, while in the other case it doesn't?
Who says it doesn't? We can find Einstein saying the speed of light varies with gravitational potential year after year:
1912: “On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential”.
1913: “I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis”.
1914: “In the case where we drop the postulate of the constancy of the velocity of light, there exists, a priori, no privileged coordinate systems”.
1915: “the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned”.
1916: “In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity”.
1920: “Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.
We can also find Irwin Shapiro saying the same thing in 1964. See line 7 of his paper in the fair-use excerpt here. It's even on Wikipedia: “the speed of a light wave depends on the strength of the gravitational potential along its path”. For something more recent see Is The Speed of Light Everywhere the Same? by PhysicsFAQ editor Don Koks. He says “Einstein talked about the speed of light changing in his new theory” and "a "global" observer can say that ceiling light does travel faster than floor light". Also see Ned Wright’s Deflection and Delay of Light, where you can read this: “In a very real sense, the delay experienced by light passing a massive object is responsible for the deflection of the light".
Of course, some people will claim that c=1 and the speed of light is truly constant, but they're contradicting Einstein and not understanding the tautology. John Moffat and João Magueijo explained that in their Comments on “Note on varying speed of light theories”. Contemporary physicists use the local motion of light to define the second and the metre, then use them to measure the local speed of light. Hence they'll say the local speed of light is 299,792,458 m/s, at both the floor and the ceiling. Moffat and Magueijo (and others) were the authors of variable speed of light theories as an alternative to inflation. See Wikipedia. They said the speed of light was much faster in the early universe. Sadly I think they got that back to front, but never mind.
Alternatively can you explain why you don't think it's C?
It isn't C because space isn't expanding in a gravitational field. Instead it's "neither homogeneous nor isotropic". That's what Einstein said in 1920. Note though that when space is expanding, it's inhomogeneous over time. Note this paper: Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. Einstein referred to inhomogeneous vacuum whilst most modern authors refer to curved spacetime, but they're talking about the same thing.