# Light dispersion in gravitational theories

GR predicts no Ricci curvature in vacuum (or at least when we can ignore the cosmological constant). Would theories that violate this lead to observable light dispersion in solar system tests of gravity, or in light lensed around a galaxy?

The field equation for light travelling in curved space-time, $$\nabla^a\nabla_a A_b = {R^a}_b A_a$$ makes it look like there would be dispersion in light propagation when there is non-zero Ricci curvature. The reasoning being that as the frequency increases, the Ricci term become negligible, so maybe it could kind of act like a dispersion term.

So I'm curious:

1. What are the experimental limits of light dispersion in light travelling long distances through curved space-time?

2. Can this somehow be transferred to experimental limits on Ricci curvature of empty space?

The Parameterized post-Newtonian formalism is used to test theories of gravity, and provides a formalism for testing how well experiment can constrain to agreement with GR. But it was not clear to me which of the PPN parameters connect to this. Or maybe that is not the correct approach, and deviation from GR here would mean deviation from energy conservation somehow. Regardless, I'd like to know how well we can currently experimentally test the GR prediction that $R_{ab} = 0$ in empty space using measurements on electromagnetic waves.

• In GR, $\nabla_a T_{ab} = 0$ is true even if the cosmological constant is non-zero, and the Einstein field equations are in some sense the unique second order equations of the metric to allow this. So maybe deviation from $R_{ab}=Λg_{ab}$ in vacuum might show up in the PPN parameters dealing with energy conservation. Wikipedia says the parameters $\zeta_1,\zeta_2,\zeta_3,\zeta_4,\alpha_3$ measure the extent and nature of breakdowns in global conservation laws. That may be a good path forward, but I don't know enough about PPN to know for sure. – JJMalone Sep 18 '17 at 17:24

These Ricci-curvature terms appear also in other equations; they are unavoidable in spinor wave equations and can be added to the scalar wave equation. However, they are mostly not measurable as I will argue.

Let us pass to the eikonal approximation. We put $A_{\mu}(x^\nu) = a_\mu(x^\nu) \exp(i S (x^\nu))$ and the wave-equation then reads $$-a_\mu S_{,\alpha}S_,^{\;\alpha}+ i a_\mu S_{;\alpha}^{\;\;\alpha} + 2 i a_{\mu;\alpha} S_,^{\;\alpha} + a_{\mu;\alpha}^{\;\;\;\;\alpha} - R^\nu_{\;\mu}a_\nu = 0$$ The standard assumption of the eikonal approximation is that the wavelength of the light is much smaller than the curvature scale of the space-time and we end up obtaining the two leading-order equations $$S_{,\alpha} S_,^{\;\alpha} = 0$$ $$a_\mu S_{;\alpha}^{\;\;\alpha} + 2 a_{\mu;\alpha} S_,^{\;\alpha} = 0$$ These can then be solved as a self-contained system of equations. The first one corresponds to the Hamilton-Jacobi problem for the trajectory of a massless particle and is to be solved first, and the second one corresponds to the evolution of the amplitude along the wave.

Of course, we have lost the Ricci-proportional term along the way. The reason for that is it is actually a term which would usually be assumed to be of smallest order! This is because $R \sim 1/\ell_c^2$ where $\ell_c$ is the curvature scale of the space-time. For this term to become relevant in light propagation, the curvature scale would have to be at around the wavelength of the light we are interested in.

Remember that even when Ricci curvature does not show up directly in the equations, it does always affect motion nearby because it sets a "boundary condition" for the surrounding vacuum in very much the same way that gravitating sources do. It is then reasonable to assume that the hypothetical gravitational theory produces the Ricci curvature at most at the order of magnitude as is produced by the physical gravitating objects nearby because otherwise the phenomenology would break down.

The Sun is the densest thing around and thus will have the shortest Ricci curvature length, it is easy to compute the length as $\sim 10^{11} m$. I.e., you will be out of luck in the Solar system because we certainly do not observe light at these wavelengths. Similarly in any other thinkable system it is essentially impossible to measure these effects.

As for vacuum light dispersion caused in other ways, there are proposed quantum-gravity effects which have recently been constrained and could be understood also as a marginal bound on the effect you are proposing, see "A limit on the variation of the speed of light arising from quantum gravity effects" as published in 2009 in Nature.

• In the high frequency limit, a mass term is also negligible, so this feels like the wrong approximation if we want to discuss dispersion as it is throwing away the terms we'd like to experimentally limit. And yes I understand that the effect would be small, but that doesn't mean we can't get an experimental limit on it at all. Experiments with electromagnetism are some of our most precisely measured null experiments. For instance experiment has constrained the photon mass to less than $10^{-14} \ \ \text{eV}/c^2$. – JJMalone Sep 18 '17 at 17:40
• Said another way, if the Ricci curvature of empty space was huge, I'd expect we'd be able to notice easily. We don't, so we should be able to put some kind of experimental limit on the prediction that the Ricci curvature of empty space is near zero. One might even interpret this as a very weak upper limit of direct detection of the cosmological constant. – JJMalone Sep 18 '17 at 17:46
• Well, in very much the same way I discuss it in the post, we can compute that the mass term of e.g. the electron is most relevant at wavelengths $\sim 10^{-13} m$ and longer. I.e., by this argument we would be able to see mass at the order of the electron mass in the behavior of the photon all the way from the longest radio waves to gamma rays. That is, you have to run the numbers to weigh the terms to see if the eikonal approximation holds and I have done that. From this analysis (as given above), the Ricci term seems to be irrelevant in most thinkable cases. – Void Sep 18 '17 at 21:07
• But yes, you can take the various measurements of photon mass also as an experimental bound on the "polarization average" of the Ricci tensor in vacuum in the given experiment (assuming that the experiments were polarization insensitive). A bound $10^{-14} eV/c^2$ on the mass term is equivalent to a bound $10^{-28} eV/c^4$ on the polarization average of the Ricci tensor in the region where the photons were flying. – Void Sep 18 '17 at 21:08

General Relativity, with or without the cosmological constant, predicts no dispersion at all due to gravitation, curved spacetime or any other way you might call it. All light and any zero mass particle wave propagation (such as gravitational waves) travel at the same velocity regardless of frequency. It always travels in local light cones, at all frequencies. It always travels in null geodesics.

One way in which it has been confirmed is in the detection of gravitational waves from merging Black Holes (BH) where no dispersion was observed, even for waves that traveled 3 billion light years. The observed frequencies arrived simultaneously. See a simple summary in Wired from the third merging BH observations in January of 2017 in the June issue at https://www.wired.com/2017/06/physicists-find-another-gravitational-wave-prove-einstein-right/

There are many other observations, including those for light, that have placed pretty stringent limits. For gravitational waves the limits were on the order of 1 part in $10^{19}$. See Wikipedia at https://en.m.wikipedia.org/wiki/Speed_of_gravity. See the LIGO results and analysis in figure 3 at http://ligo.org/science/Publication-GW170104/index.php

Some of the alternative theories of gravity indeed can show dispersion (for both light and gravitational waves) indicating a non constant speed of light. Many have been ruled out because of it

You mentioned the PPN formalism. The wiki article at https://en.m.wikipedia.org/wiki/Alternatives_to_general_relativity shows all the models till the last 10 years when cosmological observations are bringing forth some alternative theories of gravity to explain dark matter or dark energy. Up till then almost all alternatives except Cartan's had been ruled out.

It is somewhat more complex now, and you can google alternative theories of gravity. A relatively recent review is at Living Reviews at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255900/. It's not just the constancy of c that is tested, it's all the other GR predictions. With BH gravitational waves things are getting more constrained. So far LIGO's results have not found any discrepancy with GR.

Still because of the possibility and the possible effects of higher dimensions, string theory and nonlocal gravity (the AdS/CFT conjecture about holographic gravity), it continues being researched.

• Much of this is off topic. Your first paragraph is relevant, but it would be nice if you proved your statements from the equations. If you work out the math, I can see what the OP is saying. In GR with the cosmological constant, in vaccum $R_{ab} = \Lambda g_{ab}$. So the electrodynamics equation becomes: $\nabla^a \nabla_a A_b = g^{ca} R_{cb} A_a = g^{ca} \Lambda g_{cb} A_a = \Lambda A_b$. In other words, $(\Box - \Lambda) A_a = 0$. This looks like dispersion to me. – PPenguin Sep 15 '17 at 2:08
• This doesn't seem to actually answer either part of the question. Given the basic rearrangement shown by PPengin, I worry that your starting point is wrong. I also see no reason to assume that in alternative or test theories of gravity that gravitational waves must have the same dispersion as light. How do you hope to transfer this information to limits on Ricci curvature of vacuum? I feel trying to discuss gravitational waves in alternative theories over complicates this. My question isn't about gravitational waves, and can be handled considering just static backgrounds. – JJMalone Sep 15 '17 at 22:30