Bernd Brügmann

A Simple Construction of Initial Data for Multiple Black Holes

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 ph ab and its extrinsic curvature K ph ab by a conformal factor c,

Gauge conditions for long term numerical black hole evolutions without excision

We extend previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that are able to respond naturally to the spacetime dynamics. We show that the combination of excision and gauge conditions we use is able to drive highly distorted, rotating black holes to an almost static state at late times, with well behaved metric functions, without the need for any special initial conditions or analytically prescribed gauge functions. Further, we show for the first time that one can extract accurate waveforms from these simulations, with the full machinery of excision or no excision and dynamic gauge conditions. The evolutions can be carried out for long times, far exceeding the longevity and accuracy of even better resolved 2D codes. While traditional 2D codes show errors in quantities such as apparent horizon mass of over 100% by t ≈ 100M, and crash by t ≈ 150M, with our new techniques the same systems can be evolved for more than hundreds of M’s in full 3D with errors of only a few percent.

Supermassive recoil velocities for binary black-hole mergers with antialigned spins

Recent calculations of the recoil velocity in binary black-hole mergers have found the kick velocity to be of the order of a few hundred km/s in the case of nonspinning binaries and about 500 km/s in the case of spinning configurations, and have lead to predictions of a maximum kick of up to 1300 km/s. We test these predictions and demonstrate that kick velocities of at least 2500 km/s are possible for equal-mass binaries with antialigned spins in the orbital plane. Kicks of that magnitude are likely to have significant repercussions for models of black-hole formation, the population of intergalactic black holes, and the structure of host galaxies.

Towards a stable numerical evolution of strongly gravitating systems in general relativity: The conformal treatments

We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless ~CT! approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.

Geometry and Regularity of Moving Punctures

Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.

Simple excision of a black hole in 3¿1 numerical relativity

We describe a simple implementation of black hole excision in 3+1 numerical relativity. We apply this technique to a Schwarzschild black hole with octant symmetry in Eddington-Finkelstein coordinates and show how one can obtain accurate, long-term stable numerical evolutions.

Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries : The orbital hang-up case

We compare results from numerical simulations of spinning binaries in the ‘‘orbital hang-up’’ case, where the binary completes at least nine orbits before merger, with post-Newtonian results using the approximants Taylor T1, T4, and Et. We find that, over the ten cycles before the gravitational-wave frequency reaches M! ¼ 0:1, the accumulated phase disagreement between numerical relativity (NR) and 2.5 post-Newtonian (PN) results is less than three radians, and is less than 2.5 radians when using 3.5PN results. The amplitude disagreement between NR and restricted PN results increases with the black holes’ spin, from about 6% in the equal-mass case to 12% when the black holes’ spins are Si=M 2 ¼ 0:85. Finally, our results suggest that the merger waveform will play an important role in estimating the spin from such inspiral waveforms.

SYMMETRY WITHOUT SYMMETRY: NUMERICAL SIMULATION OF AXISYMMETRIC SYSTEMS USING CARTESIAN GRIDS

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 three-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.

3D Grazing Collision of Two Black Holes

We present results for two colliding black holes (BHs), with angular momentum, spin, and unequal mass. For the first time, gravitational waveforms are computed for a grazing collision from a full 3D numerical evolution. The collision can be followed through the merger to form a single BH, and through part of the ringdown period of the final BH. The apparent horizon is tracked and studied, and physical parameters, such as the mass of the final BH, are computed. The total energy radiated in gravitational waves is shown to be consistent with the total initial mass of the spacetime and the apparent horizon mass of the final BH.

Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity

We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture that they are responsible for instabilities encountered in numerical evolutions of the ADM formulation. These zero speed modes are of two kinds: pure gauge modes and constraint violating modes. We show how the decoupling of the gauge by a conformal rescaling can eliminate the problem with the gauge modes. The zero speed constraint violating modes can be dealt with by using the momentum constraints to give them a finite speed of propagation. This analysis sheds some light on the question of why some recent reformulations of the 3+1 evolution equations have better stability properties than the standard ADM formulation.