B EM Phase in Gravity

An electromagnetic wave in a vacuum travels in what is locally a straight line, which in curved space-time is a null geodesic. As far as I know, this has no significant effect on the relative phase. Note that if the gravitational field is static, the frequency of the wave (as seen by a fixed observer) is also the same at all points, although the wavelength may vary slightly depending on the coordinate speed of light (as when curved space-time is mapped to flat space coordinates, it is not possible for the same scale factor to apply everywhere).

 

Maxwell's equations say that for a free space wave the phases are synchronized.

The energy of an electromagnetic wave is not affected by a static gravitational field. (The coordinate momentum typically changes because the coordinate speed of light changes).

 

It's not actually a matter of how far the observer is, but rather what coordinate system the observer chooses. If the observer chooses a coordinate map which matches the observer's own local space locally, which is the usual convention, and the observer is far from the mass, then the speed of light relative to that coordinate system near a gravitational mass will indeed be slower than the standard local value.

But that doesn't affect the phase, and according to a local observer the speed of the wave is still the usual speed of light. It's just that if you try to construct a coordinate map which includes different potentials, the scale has to be different at different places.

 

Gravity curves space-time. Locally, light still travels at c everywhere. However, you can't make a flat map of curved space-time which matches the scale of local space-time everywhere, in the same way that you can't make a flat map of a large area of the earth without some amount of distortion. If you want for example to describe orbits around a mass, the amount of local space and time which corresponds to the space and time coordinates on the map varies with location.

The usual convention is to use a map where local rulers appear to be a fraction smaller and local clocks appear to run a fraction slower than their coordinate values. In "weak field" cases (which go up to much stronger fields than are found in the solar system) that fraction is approximately GM/rc^2 where M is the source mass and r is the distance from it.