Can you measure the one-way speed of light using gravity?

Question

I recently learned that you can't measure the speed of light one way, only back and forth. I thought you might be able to use gravity to measure the speed one way.

If you have to stations at the same altitude and shine a laser from one station to another, then see how far the beam of light has dropped. You can then use the acceleretation of gravity and calculate how long it took for the light to get to the station. When we know the time we can use that to calculate the speed of light.

For example if the light dropped 20m we know that the light took about 2 seconds to reach the station.

I know it would very hard to measure accurately but I just want to know if it would theoretically possible.

Answer 1

A non-Lorentz coordinate transformation on Minkowski space can always change one-way speeds in various directions (exercise for OP: write down a transformation law that changes the speed of light in positive $x^1$ and negative $x^1$ directions).

That of course has absolutely no physical significance because all coordinate systems are equally good for describing nature. The invariant quantities also depend on the metric tensor which has a different form in the original and transformed coordinate systems.

Every physically measurable quantity must be invariant under coordinate transformations according to the principle of relativity. As we saw above, one-way speed of light is not invariant. Therefore it can't possibly be physically measurable in GR.