A Simple Construction of Initial Data for Multiple Black Holes

Abstract

We consider the initial data problem for several black holes in vacuum with arbitrary momenta and spins on a three space with punctures. We compactify the internal asymptotically flat regions to obtain a computational domain without inner boundaries. When treated numerically, this leads to a significant simplification over the conventional approach which is based on throats and isometry conditions. In this new setting it is possible to obtain existence and uniqueness of solutions to the Hamiltonian constraint. [S0031-9007(97)03144-X] PACS numbers: 04.20. Ex, 04.25.Dm, 04.70.Bw, 95.30.Sf Binary black hole spacetimes are one of the great challenges for numerical general relativity, even if no matter sources are present. Here we consider the problem of finding initial data for several black holes in vacuum with arbitrary momenta and spins. In general relativity, initial data on a hypersurface cannot be specified freely, because the Einstein equations give rise to four equations, three momentum constraints, and the Hamiltonian constraint that the initial data has to satisfy. The purpose of this Letter is to introduce a novel approach which is significantly simpler than the conventional method based on throats and conformal imaging.In all that follows we will assume vacuum, that the metric is conformally flat, and that the extrinsic curvature is tracefree. A convenient form of the constraints of general relativity can be obtained by rescaling the physical three-metric g The momentum constraint becomesand the Hamiltonian constraint becomes an elliptic equation for the scalar field c,where the covariant derivatives are defined by the flat metric g ab , which is also used to raise and lower indices (see [1,2]). In order to obtain black hole vacuum data, one has to introduce a nontrivial topology. The first calculations were performed by Einstein and Rosen [3] in their work on point particles in general relativity. Various constructions for black holes based on Einstein-Rosen bridges and "wormholes" were given in, e.g., [4][5][6][7]. The spatial slice typically consists of two or more copies of R 3 with several spheres removed and identifications of the various spherical inner boundaries. In this way several asymptotically flat regions are obtained that are connected by bridges or "throats."The simplest example derives from the Schwarzschild spacetime in quasi-isotropic coordinates. Considered as a problem on R 3 minus the point r 0, the constraint equations (2) and (3) are solved bywhere m is the mass and r the isotropic radius. To make contact with the throat picture, recall that there exists an isometry given by r ! m 2 ͞4r which leaves the coordinate sphere r m͞2 invariant and which maps the entire exterior asymptotically flat space into that sphere. Consequently, there exists a second asymptotically flat region near r 0. Equivalently, one can represent this solution to the constraints on a space consisting of two copies of R 3 with a sphere excised and appropriate identification at the spheres.For N black holes a...

Approximate initial data for binary black holes

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“…We would like to solve the Hamiltonian constraint (5) for a boosted black hole with bare mass M and momentum P located at C. We follow the puncture approach of [22] and write the conformal factor as…”

“…The past year, however, has seen dramatic progress, and at this point independent codes using very different approaches and techniques can reliably model the coalescence, merger and ring-down of binary black holes [17][18][19][20][21]. Some of these calculations [18,19,21] treat the black hole singularity with the so-called puncture method [22] (see also [23,24]). Given that puncture initial data for binary black holes are most easily constructed in the transversetraceless formalism (see [4] and compare [25,26]), this development has renewed some interest in transversetraceless initial data.…”

Long gamma ray bursts from binary black holes

“…The initial data are provided by the TwoPunctrures thorn (Ansorg et al 2004). This module numerically solves the binary puncture equations for a pair of BHs (Brandt & Brügmann 1997). The initial state of space time is described by the extrinsic curvature in the Bowen-York form (Bowen & York 1980), for a given mass, momentum, and spin of each puncture.…”

Bowen-York-type initial data for binaries with neutron stars

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