Gravitational Wave Extraction and Outer Boundary Conditions by Perturbative Matching

Abstract

We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three-dimensional numerical relativity code. [S0031-9007(98)05380-0] PACS numbers: 04.25. Dm, 04.30.Db, 04.70.Bw Numerical relativity represents the only currently viable method for obtaining solutions to Einstein equations for highly dynamical and strong field sources of gravitational radiation. Using these techniques to study coalescing black hole binaries is the purpose of the multi-institutional Binary Black Hole "Grand Challenge" Alliance effort [1] which is presently underway in the United States. This effort is also motivated by the prospect of observations with the next generation of gravitational wave detectors.In addition to tremendous demands on computational resources, implementing the standard 3 1 1 [2,3] formulation of Einstein theory as a Cauchy problem [4] is complicated considerably by the necessity of imposing boundary conditions which maintain numerical accuracy and the physical correctness of the solution. Both inner and outer boundary conditions have received considerable attention. Recent efforts on interior boundaries have focused on the excision of the interior of the black hole from the computational domain (see, for example, [5]). This paper will concentrate on the problem of outer boundary conditions applied at a finite radius around a source of gravitational waves.Proper boundary conditions on spacelike slices of asymptotically flat spacetimes are essential for the accurate computation of the gravitational wave forms produced in the strong field region that represent the observationally relevant aspect of the computation. Since it is not feasible to simulate on spacelike slices out to arbitrarily large distances from the source, it is necessary to extract gravitational waves comparatively near the strong field region and to have boundary conditions that allow radiation to pass cleanly off the mesh. If poor outgoing boundary conditions are imposed, spurious radiation is produced which can contaminate the computed gravitational wave form. Additionally, the outer boundary is usually close enough to the isolated source that backscatter of radiation from curvature is significant. This source of incoming radiation needs to be built into the outer boundary conditions. An approach to the extraction of gravitational wave information and the computation of outer boundary conditions that exploits the matching of the interior numerical solution with an exterior perturbative solution on spacelike slices has been developed during the past decade and applied to a number of different physical scenarios [6][7][8]. Extension of these techniques to three-dimensional (3D) simulations has been one of the ef...

Testing outer boundary treatments for the Einstein equations

“…In [46,47], boundary conditions for the full nonlinear Einstein equations on a finite domain are obtained by matching to exact solutions of the linearized field equations at the boundary. Alternatively, the interior code could be matched to an 'outer module' that solves the linearized field equations numerically [48][49][50][51]. Other approaches involve matching the interior nonlinear Cauchy code to an outer characteristic code (see [52] for a review) or using hyperboloidal spacetime slices that can be compactified towards null infinity (see [53] for a review).…”

Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations

“…These solutions exhibit non-trivial dynamics which exercise the boundaries, but for which the source terms of the Einstein equations are negligible. The particular initial data which we use are the quadrupole Teukolsky waves [28], which have been used as a testbed in a number of numerical studies [29][30][31][32]. The particular solution which we use follows Eppley [33] in combining incoming and outgoing wave packets so as to produce a solution which is regular everywhere in the spacetime.…”

Characteristic Evolution and Matching

“…Most perturbative-numerical matching schemes that have been implemented in general relativity have been based upon perturbations of a Schwarzschild background using the standard Schwarzschild time slicing [1, 4, 2, 3, 181, 180, 156]. It would be interesting to compare results with an analytic-numeric matching scheme based upon the true null cones.…”

“…Most of the effort in numerical relativity has centered about the Cauchy {3+1} formalism [226], with the gravitational radiation extracted by perturbative methods based upon introducing an artificial Schwarzschild background in the exterior region [1, 4, 2, 3, 181, 180, 156]. These wave extraction methods have not been thoroughly tested in a nonlinear 3D setting.…”