# Speed of gravitational wave in strong gravity

I have a question about gravitational waves which I can't really find an answer to.

My current understanding is that one assumes there is some frame in which the metric has the form $$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$$ where $$h$$ is a small perturbation. One then expands everything to first order in the perturbation and ends up with the wave equation $$\Box \bar h_{\mu\nu} = 0$$ where $$\bar h$$ is the trace inverse of $$h$$ (assuming zero stress-energy tensor everywhere). This clearly describes a wave that travels at the speed of light.

What I can't seem to understand is what happens when the gravitational field is strong and the first order approximation fails. Is there any other way to argue that the gravitational wave travels at the speed of light which does not involve perturbation theory? Actually, how would one even define a gravitational wave without this linear approximation?

I also heard that the LIGO collaboration detection of last year showed that GW should travel at the speed of light because we received a gamma burst exactly at the same time as the GW from the same source. But here I'm interested more in a theoretical explanation rather than a "it was measured so it's true" explanation.

Thank you!

• it was measured so it's true I feel I must point out that in physics that is the gold standard for knowledge and the useful application of theory. – StephenG Feb 15 '19 at 13:56
• @StephenG Of course, but I was interested in knowing if it comes out of the theoretical apparatus of GR or not! That's why that answer would not have helped. – Angelo Brillante Romeo Feb 15 '19 at 14:37
• @AndrewSteane: I don't think that's quite right. You can't prove a general proposition based on one example. In fact it's false or undefined in general that gravitational waves travel at c, if you consider large amplitudes. – Ben Crowell Feb 15 '19 at 16:02
• @BenCrowell Thanks; I'll modify my comment. I guess this is to do with the non-lineariity. – Andrew Steane Feb 15 '19 at 18:47
• This modifies an earlier comment. There are cases of propagation at $c$ for an exact solution, and Rindler discusses one. But non-static metrics are hard to interpret, and GR is non-linear, so the general answer is unclear as Ben Crowell says. – Andrew Steane Feb 15 '19 at 18:49

In general it is either false or undefined that gravitational waves propagate at $$c$$.

As a counterexample, GR predicts that a gravitational-wave pulse propagating on a background of curved spacetime develops a trailing edge that propagates at less than $$c$$. See Misner, Thorne, and Wheeler, p. 957. This effect is weak when the amplitude is small or the wavelength is short compared to the scale of the background curvature.

For a general spacetime that includes high-amplitude gravitational waves, there need not be any meaningful definition of the speed of propagation of such waves. It's only in the low-amplitude limit that we can meaningfully talk about splitting a metric into a background term and a wave term. In general such a split is not uniquely defined, because there is in general no notion of adding metrics the way we add fields in SR. To add metrics you have to add them point by point, but there is in general no way to decide which point to add to which point. So for a high-amplitude gravitational wave, we don't necessarily have a background metric, and without a background metric we don't have any way to define what we mean by the speed at which a wave propagates.

Andrew Steane wrote in a comment:

Rindler considers an extract treatment of a plane wave, and I expect it's in MTW and other well known books. Owing to the subtlety of non-static metrics it's fairly zany stuff but basically, yes, it travels at c.

I think the example you're talking about is in Rindler, Relativity: Special, General, and Cosmological, 2nd ed., ch. 13. Rindler shows that the wave propagates with coordinate velocity $$dx/dt=c$$, where the $$x$$ and $$t$$ coordinates are defined such that, in the flat region of spacetime that has not yet been visited by the wave, they are the standard Minkowski coordinates. He doesn't seem to explicitly justify the assumption that this coordinate velocity is correctly interpreted as the velocity of propagation, but given the setup, it's pretty plausible: observers in the region that hasn't yet been visited can synchronize clocks, etc. But this is just one example, and you can't prove a general rule from one example. MTW's example is a counterexample.

It would be interesting to know whether there are counterexamples where the velocity is equally unambiguous and is greater than $$c$$, rather than less than $$c$$ as in the MTW example. I would be surprised (and disturbed) if there were. Maybe there are general theorems that rule this out, but it's not obvious to me what is even the best formulation of such a conjecture.

The propagation character of $$h_{\mu \nu}$$ field is determined solely by the kinetic term in the corresponding action.

In principle, one must expand the equations of motion (EOM) to all orders in $$h$$. But the only term that matters for propagation, is the term linear in $$h$$. The term that is linear in $$h$$ in EOM comes from the term quadratic in $$h$$ in the action. Terms that are quadratic in $$h$$ in EOM come from terms that are cubic in $$h$$ in the action. These describe (graviton self) interactions, not propagation.

Near the wave-emitting source, gravity is strong and curvature of spacetime is high. This curvature causes backscattering of the propagating gravitational waves.

The interaction of wave with itself and the curved background will delay the propagation over a finite distance. The self-interactions are suppressed more and more (by Planck mass in the denominator) at each higher order. Also, locally, since we can always go to a locally inertial coordinate system and subsequently use linearized theory, gravity travels at the speed of light.

Also, the measurements of 17 Aug 2017 binary neutron star merger did not conclude that the gravitational waves and gamma rays arrived at exactly the same time. The gamma ray burst was detected 1.74 seconds later. One can then place a bound on the difference between speed of gravity $$v_{GW}$$ and speed of light $$v_{EM}$$:

$$-3 \times 10^{-15} \leq \frac{\Delta v}{v_{EM}} \leq 7 \times 10^{-16}$$

where $$\Delta v = v_{GW} - v_{EM}$$.

Reference: pg 272-273, Michele Maggiore, Gravitational Waves: Volume 1: Theory and Experiments

• The LIGO data are not relevant to the question, both because the OP is asking about theory and because the OP is asking about high-amplitude waves. Re the theoretical argument you present using expansions in powers of $h$, you don't really say explicitly what you're claiming. It can't be a proof to all orders that waves propagate at $c$, because MTW p. 957 would then be a counterexample. Is it intended to be a proof to all orders in the case where the background is flat? – Ben Crowell Feb 15 '19 at 16:43
• @BenCrowell Thanks for pointing this out. The data is relevant because it corrects the wrong comment made in the question. As stated in the answer, the cause of a field's propagation is its kinetic term. Scattering from self-interactions and background does occur, but linearized theory remains accurate, as mentioned in MTW. Locally, one can always go to a frame of flat space. Also see Maggiore's book; pages added in the answer. – Avantgarde Feb 15 '19 at 20:16