Does the phase between electric and magnetic waves of light change if the path is bent by 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).
Has any experiment been done to measure the phase of electric and magnetic waves as light passes by a gravitational mass? I'm thinking that a path bent by gravity represents stored energy and a phase shift is how energy is stored in a dielectric (which also bends light).
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).
I’ve just been reading that an observer that is far from the gravitational mass does see the speed of light slow down as the light passes close by the gravitational mass. Wouldn’t that also mean that he sees a phase shift between electric and magnetic waves? An observer close to the gravitational mass doesn’t see the light slow so he wouldn’t see the phase shift.
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.
That's not how it works. A dielectric bends the path of a beam of light at its boundary because the wavelength at a given frequency is different inside the dielectric than outside; this is a general property of waves of any sort at the boundary between different mediums. The gravitational deflection of a beam of light is a completely different phenomenon, one that probably shouldn't be thought of as "bending" at all. At every point along its trajectory, the light is travelling in a locally straight line; the equations that govern its local propagation, including the relative phases of the electrical and magnetic fields, are those that apply when there is no gravity. There's no dielectric effect at work.
He doesn't actually "see the speed of light slow down"; he calculates a coordinate velocity that is less than ##c##, but this value has no physical significance. No matter how he calculates this quantity, however, the distance travelled and the time elapsed is the same for the electrical and magnetic components so there is no possibility of a phase shift between them.
I guess I’m making the assumption that if I see something moving slower than c then it must be carrying some rest mass. The only way I know how is to shift the phase. But you are saying that it is just my ruler and clock that make the light appear like it is moving slower – it is still light so the phase must be the same. But doesn’t that contradict the basic principle that light travels at a velocity of c irrespective of reference frame?
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.
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