Can we use curved spacetime to measure the one-way speed of light?

Question

This is related to Measuring one-way speed of light with gravitational lensing and Measuring the one-way speed of light with a black hole?

The idea is to shine a beam of light from a clock towards a heavy mass, like a black hole, in some manner such that the curvature of spacetime around the mass is sufficient to "bend" the light directly back to the clock. That is, to send the light along a self-intersecting geodesic using a sufficiently curved spacetime. Note that in this case the source and the detector are the same object, so no clock synchronization is required. Hypothetically, this should measure the one-way speed of light.

None of the given refutations from the similar posts on this site are convincing. Here are some of the main refutations, as I understand them, and my counter to them:

(1) In being curved by spacetime, the light is actually changing direction, hence this is not the one-way speed.

To my understanding of General Relativity, the light travels along a geodesic; that is, a curve such that, ignoring technicalities of Differential Geometry, its tangent direction is constant. There is no reflection or acceleration. So in this sense, how can the light with a constant tangent direction be said to change direction? One might say that the path deviates from a straight line in Euclidean space, but given that spacetime is not Euclidean, has no extrinsic structure, and has no global notion of direction, this is the only definition of straight or "one direction" that makes sense. Thus, either the light does not change direction and is "one-way," or we need to be clearer about what exactly we mean by "one-way" in the context of General Relativity.

(2) Such an experiment would be impossible or impractical.

I think some posters were saying this because the original questions implied that the clock would have to be at the event horizon of a black hole, which is of course impossible or impractical; however, I don't think such an extreme closeness to the black hole is required. Given that the photon sphere of a black hole, where light orbits in a circle, is outside of the event horizon, shining the light at some distance outside of the photon sphere should be sufficient to curve the light back to the clock, regardless of the distance of the clock from the black hole (see my rough illustration). At the extreme, we could place the clock directly on the photon sphere, which should be theoretically possible. Regardless, the question is not about practicality, but whether the one-way speed is theoretically measurable.

Update:

Essentially what this argument boils down to, I think, is that the average speed of light along a single self-intersecting geodesic may differ from the average speed of light along a non-geodesic self-intersecting path, which is always c. Put that way, however, it does seem a little ridiculous. It is hard to see how the anisotropy of the speed could be defined in such a way as to be consistent with this pathological case.

Presumably, we could approximate a single self-intersecting geodesic with a non-geodesic path arbitrarily closely, so, logically, the average speeds over each path should be approximately the same, i.e., c. Furthermore, how would the "preferred direction" of the speed of light behave around a black hole, for instance? Regardless, I still think this experiment does actually solve the problem for the given notion of "direction". Though the problem is usually posed in the flat spacetime of Special Relativity, with a global notion of direction, so the apparent unanswerability of this question should be seen as a limitation of Special Relativity only, which is then solved by General Relativity.

Answer 1

(1) In being curved by spacetime, the light is actually changing direction, hence this is not the one-way speed.

(2) Such an experiment would be impossible or impractical.

I agree with you that neither of these two objections are satisfactory. The light travels on a null geodesic so it is straight and it is just a thought experiment so impracticality is irrelevant.

I am not sure if a strict categorization as one-way or two-way is important in curved spacetime. This experiment shares characteristics of both types of experiments. As with a one-way experiment the light travels on a geodesic, but as with a two-way experiment it uses a single clock. However, I think that you make a mistake here:

in this case the source and the detector are the same object, so no clock synchronization is required

Unfortunately, this is not correct. In order to get a speed you need both a time and also a distance. The arrangement of your experiment makes the time independent of synchronization, but the distance is not.

Suppose that we have a rope around the black hole and that the rope is laid out so that the light pulse follows its length the whole way around. Now, general relativity is a four-dimensional geometric theory, so in 4D spacetime that rope forms a cylinder. We can put marks at regular points along the rope, and those marks form lines going along the length of the cylinder, and we can put one mark next to the clock as the reference mark.

Then, the pulse of light forms a helix which winds around the cylinder, starting and ending at the reference mark. The intersection of the helix with the reference mark forms a pair of events, but the only frame-invariant measurements are the spacetime intervals along the reference line and the helix between those two events. The reference line interval gives us a time, but the helix interval is null so it does not give us a distance.

To get a distance we have to draw a set of spacelike lines circumferentially around the cylinder, and then measure the interval around the cylinder along those lines. The issue is that many different sets of lines are valid. We can have some that cut straight across the cylinder and others that slice it at an angle, and even other sets that are more exotic. The interval around the cylinder depends on which set of lines we choose. Each set of lines represents a different valid simultaneity convention. That convention will determine both the overall global length as well as the local one-way speed of light at each point.

Now, you may claim that there is only one natural set of lines to use, specifically the one cutting straight across the cylinder which would be the proper length of the rope. That is true, but the whole discussion about the one-way speed of light is not about naturalness. Indeed, it is clear that it is unnatural to assume an anisotropic one-way speed of light given the isotropic two-way speed of light. So naturalness is not at issue, the question is whether other unnatural conventions are nevertheless consistent with the observable data as predicted by the natural convention. In this case, they are.

So, although this approach does have some similarities with a flat-spacetime one-way measurement, it does not avoid the key problem inherent to all one-way measurements: the result depends on your chosen synchronization convention.