I’m curious if the magnitude of the displacement of light by a gravity well is variable to any property of the photons (frequency, polarization, etc).
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As you can see from the angular deviation formula (it is $\theta=\frac{4GM}{rc^2}$, you can find it on Wikipedia for instance: https://en.wikipedia.org/wiki/Gravitational_lens), the deflection does not depend on light's energy, polarization or whatever. The calculation has been done for a central mass (times $c^2$) much bigger than the energy of the light ray involved, since it would be very hard to find the motion of light in a metric different from Schwarzschild/Kerr/RN etc.; This is of course a reasonable approximation, since the curvature of spacetime due to a planet/star/BH is much bigger than the curvature induced by the light ray, even if it had several TeV energy.
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$\begingroup$ the deflection depends on light's energy, polarization or whatever. Don't you mean the deflection does not depend on these factors? $\endgroup$– user4552Commented Aug 4, 2018 at 0:16
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$\begingroup$ Yes, of course, it was a typo :D $\endgroup$ Commented Aug 4, 2018 at 1:13
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All massless objects follow a trajectory in spacetime given by an equation called the null geodesic equation. Annoyingly there is no Wikipedia article specifically on the null geodesic equation, but the article on geodesics in general is pretty good. Light, being massless, follows trajectories described by the null geodesic equation, and other factors like polarisation and frequency make no difference. The bottom line is that the deflection of light in a gravitational field is unaffected by any property of the light.
answered Aug 3, 2018 at 19:15
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$\begingroup$ that is correct from the GR perspective, while from the QM perspective the larger wavelength should be subject to larger tidal forces (also see Aneikei's answer), while in GR photons are point particles who experience no tidal forces $\endgroup$– YukterezCommented Jul 17, 2022 at 13:24
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One way of stating Einstein's equivalence principle is that if two different test particles are released in the same gravitational field with the same initial position and velocity vector, their motion will be the same, regardless of factors such as their internal composition. This also applies to rays of light, which can be considered as massless test particles. Therefore properties such as frequency and polarization can't make any difference here.
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Nope, frequency doesn't make a big difference. We know that photons, as massless particles, follow null geodesics, and for a given source location and direction there's only one distinct null geodesic, independent of the frequency.
You may have read several stuff about how gravitational lensing is independent of wavelength (though there are small variations for extremely large wavelengths, I'm not sure how those work). It's the same mechanism.
Also relevant: Is light of different colors affected differently by gravity?
The shape of the geodesic does not depend on the photon's wavelength (i.e. on the light's color). It only depends on the mass distribution of the object that is causing space to be curved. When other objects (photons, particles, etc.) move through this curved space, the particular path that they follow will depend on their velocity, but since all photons have the same velocity they will all follow the same paths and there should be no dispersive effect.
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Does light’s deflection by a gravitational well vary depending on frequency or other properties of the wave?
Yes it does, as the length scale around the sun exists at different magnitudes depending on the radius from the Sun (the curvature of space) and because different wavelengths of light have - well different lengths. Then 600nm of light will have a greater length than 400nm of light.
As such, the 600nm light will be exposed to a greater difference of space than the 400nm of light and thus will traverse the space above the Sun in slightly different orientations. This is precisely why prisms are able to separate the colors of white light.
However, this effect is virtually unnoticeable around the Sun but it does exist as all lenses suffer from Chromatic aberration which is caused by dispersion: the refractive index of the lens elements varies with the wavelength of light.
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$\begingroup$ I appreciate the answer, and it jives with my intuition (though the intuitive seems often to not be the friend of GR, QM, et al). Hmmm, 4v1... My confirmation bias is tingling. Can you point me at some sources that could help in understanding your PoV better? $\endgroup$ Commented Aug 10, 2022 at 2:09
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Einstein started his general theorem of relativity with stating that one couldn’t discern a constantly accelerating frame with a frame with constant gravity. Therefore light should move the same in both frames. How much light seems to bend in a constantly accelerating frame depends only on the acceleration of that frame, so logically that would mean that how much light bends as a result of gravity only depends on how strong gravity’s attraction is.
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$\begingroup$ I've removed a comment thread that discussed a previous version of this answer. $\endgroup$– rob ♦Commented Aug 5, 2018 at 13:40