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

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

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Thank you for your help!