# Some Mathematical Questions on Gravitational Waves and Numerical Relativity

Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived.

The wave-forms of the detected gravitational waves are said to have been compared with predicted wave-forms produced by a combination of post-Newtonian approximations and revamped methods in numerical relativity, such as the puncture method and the excision technique.

Here then are my questions.

Question 1. In a discussion on the evolution of binary black holes, I suppose that one assumes the ADM formalism, in which space-time is divided into time slices. This reminds me of a famous result by Robert Geroch that a space-time is globally hyperbolic if and only if it admits a foliation by Cauchy hyper-surfaces. Hence, when performing computer simulations of binary black holes, does one assume that space-time is globally hyperbolic?

Question 2. I’ve read that the existence of gravitational waves involves analyzing what happens at ‘null infinity’, the definition of which presumably depends on some kind of asymptotic structure on space-time. Hence, when performing computer simulations of gravitational waves emanating from a black-hole merger, what asymptotic structure does one assume on space-time? Besides, I’ve heard that a definition of black holes has been satisfactorily given only for asymptotically flat space-times.

Question 3 (Not very mathematically precise). Can the movement of space-time singularities be shown to be consistent with the notion/axiom of general relativity that the world-lines of point particles are geodesics (after perhaps having performed some de-singularization process)?

Question 4. Does anyone here know of papers written to establish rigorous error bounds for the approximation methods used in numerical relativity?

Question 5. One of the reasons mentioned for justifying the use of the excision technique is that signals originating in a black hole cannot propagate out of it. Now, I understand that light signals cannot propagate out of a black hole because the world-lines of photons are null geodesics, and a black hole is defined as a portion of space-time not contained in the causal past of null infinity. However, in the context of full non-linear general relativity, a gravitational wave is not something that propagates with respect to a background metric but is intimately tied to the metric itself. The Cauchy problem in general relativity says that, due to the nature of hyperbolic PDE’s, perturbing metrical data on a portion of a Cauchy hyper-surface (subject to particular constraint conditions) affects the metric only within the future light-cone of that portion. However, if global hyperbolicity is not assumed (supposing that the answer to Question 1 is negative), then one may not have a Cauchy hyper-surface. Is there, then, an analog of finite propagation speed in any black hole?

Thank you for your help!

Question 3:

You should not think of the singularity (corresponding to a black hole) as moving in space-time. It is not. So you are asking the wrong question if your motivation is gravitational waves.

The answer to the question you did ask however is "yes", see the work of Einstein-Infeld-Hoffman.

Question 5:

First, finite propagation speed always holds, by virtue of Einstein's equations being essentially hyperbolic. (This is a purely local property, whereas global hyperbolicity, as its name suggests, is a global property.)

Second, you are correct that the definition of a black hole is teleological: you only know what a black hole is if you know what the null infinity looks like. However, for numerical computations a much more acceptable, local substitute is used. Instead of the event horizon (which is defined as the boundary of the past of future null infinity), it is much more common to use the apparent horizon as a proxy for the boundary of the black hole. The apparent horizon will always sit within the black hole, and captures a local notion of "no escape". And in particular global geometry does not come into play in the excision process. (For more about apparent horizons, Wikipedia has a fairly readable lay discussion.)

• Hello Willie. Thanks for your response. I understand that singularities aren’t part of space-time, so there’s no such thing as singularities moving in space-time. However, in computer simulations that show the evolution of binary black holes, it appears that singularities are represented as points in $\mathbb{R}^{3}$ with time-dependent coordinates. I was asking my question in that context. – Transcendental Oct 24 '17 at 7:33
• @Transcendental: can you point me to the source where you found that "singularities are represented as points in $\mathbb{R}^3$"? The only method I have even a passing familiarity with is the excision method, and I had thought that the excised region is most of the black hole, and we don't go near the singularity at all. If you believe weak cosmic censorship, there's not much point of keeping track of the singularity at all for gravitational waves. – Willie Wong Oct 24 '17 at 12:57
• @Transcendental: also, I think you misunderstood my answer: the key idea I try to mention is not that the singularity is not part of the space-time, but that singularities are expected to be (by some) to be generically space-like or null, and it doesn't make too much sense to speak of them as moving. (In the Schwarzschild picture, the singularity is the future terminus of all observers who enter the horizon. It is not a point you move toward because you travel in space, but a line you move toward because you travel in time.) – Willie Wong Oct 24 '17 at 13:01
• Thanks for the clarification. As for the source that you’ve requested, it’s this video clip created by the SXS Project. After reading your comments, I watched it more carefully and realized (after all this time) that its use of a color-enhanced rubber-sheet analogy indeed doesn’t account for regions very close to the singularities. This wasn’t apparent at first because the (black) excised regions were blocked from view. Beneath the clip is an explanation of the color scheme, but it doesn’t mention anything about the excised regions. – Transcendental Oct 24 '17 at 18:05
• @Transcendental: regarding the final question, "not quite". The method (as far as I understand) used by numerical relativists work because they already started with black hole solutions (from which light/signals/information cannot classically escape). My comment about weak cosmic censorship concerns the hypothetical question you may have had "what if I want to know about how singularities interact with gravitational waves?" To which I would respond "if you believe in WCC then this is a moot point." Does that make sense? – Willie Wong Oct 25 '17 at 2:30

Only partial answers / attempts at answers, hopefully to get a more extensive discussion started.

1. Yes, by construction, numerical relativity solves Einstein's equations as an initial value problem, restricting one selves to globally hyperbolic spacetimes. The Strong Cosmic Censorship Hypothesis asserts that all of spacetime outside black holes is globally hyperbolic.
1. Coordinate conditions are fixed on an ingoing null hypersurface, and then propagated outwards. This is a complication for the identification of gravitational waves, which can only be unambiguously definied at future null-infinity, but that asymptotic condition is not imposed on the numerical calculation.