Well-Posed Initial-Boundary Evolution in General Relativity

The waveform emitted in the inspiral and merger of a relativistic binary is theoretical input crucial to the success of the fledgling gravitational wave observatories. A computational approach is necessary to treat the highly nonlinear regime of a black hole or neutron star collision. Developing this computational ability has been the objective of the Binary Black Hole (BBH) Grand Challenge [1] and other world wide efforts. The Grand Challenge built a code based on the Arnowitt-Deser-Misner [2] formulation to solve Einstein’s equations by Cauchy evolution. The numerical instabilities encountered with that code have been traced, at least in the linear regime, to the improper application of boundary conditions [3]. Other groups have encountered similar difficulties in treating boundaries (see [4] for a recent discussion) and the working practice is to forestall problems by placing the outer boundary at a large distance from the region of physical interest (see e.g. [5]). This deficiency extends beyond numerical relativity to a lack of analytic understanding of the initial-boundary value problem for general relativity. The local-version of the initial-boundary value problem is schematically represented in Fig. 1. Given Cauchy data on a spacelike hypersurface S and boundary data on a timelike hypersurface B, the problem is to determine a solution in the appropriate domain of dependence. Whereas there is considerable mathematical understanding of the gravitational initial value problem (for recent reviews see [6–8]), until recently the initial-boundary value problem has received little attention. Indeed, only relatively recently have methods been available in the mathematical literature, in particular the technique of maximally dissipative boundary conditions, which can be applied to the nonlinear initial-boundary problem of the type arising in general relativity [9–12]. Friedrich and Nagy [13] have applied these mathematical tools to give the first demonstration of a well-posed initial-boundary value formulation for Einstein’s equations. The Friedrich-Nagy work is of seminal importance for introducing the maximally dissipative technique into general relativity. However, their formulation, which uses an orthonormal tetrad, the connection and the curvature tensor as evolution variables, is quite different from the metric formulations implemented in current numerical codes designed to tackle the BBH problem. Although it is not apparent how to apply the details of the Friedrich-Nagy work to other formalisms, the general principles can be carried over provided Einstein’s equations are formulated in the symmetric hyperbolic form ∑