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...

We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3 + 1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a 3-dimensional Cartesian (x, y, z) coordinate grid which covers (say) the y = 0 plane, but is only one finite-difference-molecule-width thick in the y direction. The field variables in the central y = 0 grid plane can be updated using normal (x, y, z)-coordinate finite differencing, while those in the y = 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3 + 1 numerical general relativity, involving both black holes and collapsing gravitational waves.

The dynamics of apparent and event horizons of various black hole spacetimes, including those containing distorted, rotating and colliding black holes, are studied. We have developed a powerful and efficient new method for locating the event horizon, making possible the study of both types of horizons in numerical relativity. We show that both the event and apparent horizons, in all dynamical black hole spacetimes studied, oscillate with the quasinormal frequency.Comment: 4 pages, 94-

In this paper we study a new family of black hole initial data sets corresponding to distorted ''Kerr'' black holes with moderate rotation parameters, and distorted Schwarzschild black holes with even-and odd-parity radiation. These data sets build on the earlier rotating black holes of Bowen and York and the distorted Brill wave plus black hole data sets. We describe the construction of this large family of rotating black holes. We present a systematic study of important properties of these data sets, such as the size and shape of their apparent horizons, and the maximum amount of radiation that can leave the system during evolution. These data sets should be a very useful starting point for studying the evolution of highly dynamical black holes and can easily be extended to 3D. ͓S0556-2821͑96͒04114-8͔PACS number͑s͒: 04.25. Dm, 95.30.Sf, 97.60.Lf

We have developed a numerical code to study the evolution of distorted, rotating black holes. This code is used to evolve a new family of black hole initial data sets corresponding to distorted "Kerr" holes with a wide range of rotation parameters, and distorted Schwarzschild black holes with oddparity radiation. Rotating black holes with rotation parameters as high as a/m = 0.87 are evolved and analyzed in this paper. The evolutions are generally carried out to about t = 100M , where M is the ADM mass. We have extracted both the even-and odd-parity gravitational waveforms, and find the quasinormal modes of the holes to be excited in all cases. We also track the apparent horizons of the black holes, and find them to be a useful tool for interpreting the numerical results. We are able to compute the masses of the black holes from the measurements of their apparent horizons, as well as the total energy radiated and find their sum to be in excellent agreement with the ADM mass.PACS numbers: 04.30.+x, 95.30.Sf, 04.25.Dm Typeset using REVT E X

Dynamic black hole spacetimes are studied by examining the evolution of apparent horizons surrounding the holes. We performed numerical evolutions of three different initial data sets: nonrotating black holes distorted by time symmetric (Brill) gravitational waves, distorted rotating black holes, and the time symmetric two black hole Misner data. Although the initial data sets represent different physical problems, the results for these systems are strikingly similar. At early times in the evolution, the apparent horizons may be very distorted and nonspherical (or disjoint in the case of two black holes), but the systems quickly settle down to a nearly spherical or oblate (in the case of rotating holes) configuration and the horizons are then seen to oscillate at the quasinormal frequency of the final black hole. In the case of two black holes with disjoint horizons, we see the appearance of a larger horizon surrounding both holes as they collide. From this point the horizon dynamics is very similar to the single distorted black hole systems. The wavelength and damping time of the quasinormal modes and the rotation parameter in the rotating cases can be read off directly from oscillations in the geometry of the black hole horizons. The apparent horizon is thus shown to be a powerful tool in the study of black hole spacetimes.PACS number(s): 04.30. Db, 04.20.Ex, 04.25.Dm, 04.70.B~

We have developed a new numerical code to study the evolution of distorted, rotating black holes. We discuss the numerical methods and gauge conditions we developed to evolve such spacetimes. The code has been put through a series of tests, and we report on (a) results of comparisons with codes designed to evolve non-rotating holes, (b) evolution of Kerr spacetimes for which analytic properties are known, and (c) the evolution of distorted rotating holes. The code accurately reproduces results of the previous NCSA non-rotating code and passes convergence tests. New features of the evolution of rotating black holes not seen in non-rotating holes are identified. With this code we can evolve rotating black holes up to about t = 100M , depending on the resolution and angular momentum. We also describe a new family of black hole initial data sets which represent rotating holes with a wide range of distortion parameters, and distorted non-rotating black holes with odd-parity radiation. Finally, we study the limiting slices for a maximally sliced rotating black hole and find good agreement with theoretical predictions.PACS numbers: 04.30.+x, 95.30.Sf, 04.25.Dm

We present a series of test beds for numerical codes designed to find apparent horizons. We consider three apparent horizon finders that use different numerical methods: one of them in axisymmetry, and two fully three-dimensional. We concentrate first on a toy model that has a simple horizon structure, and then go on to study single and multiple black hole data sets. We use our finders to look for apparent horizons in Brill wave initial data where we discover that some results published previously are not correct. For pure wave and multiple black hole spacetimes, we apply our finders to survey parameter space, mapping out properties of interesting data sets for future evolutions. 04.25.Dm, 04.30.Db, 95.30.Sf, 97.60.Lf