Some Mathematical Questions on Gravitational Waves and Numerical Relativity

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.
• 2. 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.
• 4. For example, Template Banks for Binary black hole searches with Numerical Relativity waveforms.

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.)