Stuffed Black Holes

Initial data corresponding to spacetimes containing black holes are considered in the time symmetric case. The solutions are obtained by matching across the apparent horizon different, conformally flat, spatial metrics. The exterior metric is the vacuum solution obtained by the well known conformal imaging method. The interior metric for every black hole is regular everywhere and corresponds to a positive energy density. The resulting matched solutions cover then the whole initial (Cauchy) hypersurface, without any singularity, and can be useful for numerical applications. The simpler cases of one black hole (Schwarzschild data) or two identical black holes (Misner data) are explicitly solved. A procedure for extending this construction to the multiple black hole case is also given, and it is shown to work for all time symmetric vacuum solutions obtained by the conformal imaging method. The numerical evolution of one such 'stuffed' black hole is compared with that of a pure vacuum or 'plain' black hole in the spherically symmetric case.Comment: 12 pages, Latex, 4 postscript figures, corrected some typos, new section about physical interpretatio

Towards a Singularity-Proof Scheme in Numerical Relativity

Progress in numerical relativity has been hindered for 30 years because of the difficulties of avoiding spacetime singularities in numerical evolution. We propose a scheme which excises a region inside an apparent horizon containing the singularity. Two major ingredients of the scheme are the use of a horizon-locking coordinate and a finite differencing which respects the causal structure of the spacetime. Encouraging results of the scheme in the spherical collapse case are given.Comment: 9 page

Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision

We present long-term-stable and convergent evolutions of head-on black hole collisions and extraction of gravitational waves generated during the merger and subsequent ring-down. The new ingredients in this work are the use of fixed mesh-refinement and dynamical singularity excision techniques. We are able to carry out head-on collisions with large initial separations and demonstrate that our excision infrastructure is capable of accommodating the motion of the individual black holes across the computational domain as well as their their merger. We extract gravitational waves from these simulations using the Zerilli-Moncrief formalism and find the ring-down radiation to be, as expected, dominated by the l=2, m=0 quasi-normal mode. The total radiated energy is about 0.1 % of the total ADM mass of the system.Comment: Revised version, 1 figure added, accepted for publication in Phys.Rev.D, 15 pages, 10 figures, revtex 4.

Complex Wave Numbers in the Vicinity of the Schwarzschild Event Horizon

This paper is devoted to investigate the cold plasma wave properties outside the event horizon of the Schwarzschild planar analogue. The dispersion relations are obtained from the corresponding Fourier analyzed equations for non-rotating and rotating, non-magnetized and magnetized backgrounds. These dispersion relations provide complex wave numbers. The wave numbers are shown in graphs to discuss the nature and behavior of waves and the properties of plasma lying in the vicinity of the Schwarzschild event horizon.Comment: 21 pages, 9 figures, accepted for publication Int. J. Mod. Phys.

The Lazarus Project. II. Spacelike extraction with the quasi-Kinnersley tetrad

The Lazarus project was designed to make the most of limited 3D binary black-hole simulations, through the identification of perturbations at late times, and subsequent evolution of the Weyl scalar $\Psi_4$ via the Teukolsky formulation. Here we report on new developments, employing the concept of the ``quasi-Kinnersley'' (transverse) frame, valid in the full nonlinear regime, to analyze late-time numerical spacetimes that should differ only slightly from Kerr. This allows us to extract the essential information about the background Kerr solution, and through this, to identify the radiation present. We explicitly test this procedure with full numerical evolutions of Bowen-York data for single spinning black holes, head-on and orbiting black holes near the ISCO regime. These techniques can be compared with previous Lazarus results, providing a measure of the numerical-tetrad errors intrinsic to the method, and give as a by-product a more robust wave extraction method for numerical relativity.Comment: 17 pages, 10 figures. Journal version with text changes, revised figures. [Note updated version of original Lazarus paper (gr-qc/0104063)

Initial Data and Coordinates for Multiple Black Hole Systems

We present here an alternative approach to data setting for spacetimes with multiple moving black holes generalizing the Kerr-Schild form for rotating or non-rotating single black holes to multiple moving holes. Because this scheme preserves the Kerr-Schild form near the holes, it selects out the behaviour of null rays near the holes, may simplify horizon tracking, and may prove useful in computational applications. For computational evolution, a discussion of coordinates (lapse function and shift vector) is given which preserves some of the properties of the single-hole Kerr-Schild form

Numerical Evolution of axisymmetric vacuum spacetimes: a code based on the Galerkin method

We present the first numerical code based on the Galerkin and Collocation methods to integrate the field equations of the Bondi problem. The Galerkin method like all spectral methods provide high accuracy with moderate computational effort. Several numerical tests were performed to verify the issues of convergence, stability and accuracy with promising results. This code opens up several possibilities of applications in more general scenarios for studying the evolution of spacetimes with gravitational waves.Comment: 11 pages, 6 figures. To appear in Classical and Quantum Gravit

Collisions of boosted black holes: perturbation theory prediction of gravitational radiation

We consider general relativistic Cauchy data representing two nonspinning, equal-mass black holes boosted toward each other. When the black holes are close enough to each other and their momentum is sufficiently high, an encompassing apparent horizon is present so the system can be viewed as a single, perturbed black hole. We employ gauge-invariant perturbation theory, and integrate the Zerilli equation to analyze these time-asymmetric data sets and compute gravitational wave forms and emitted energies. When coupled with a simple Newtonian analysis of the infall trajectory, we find striking agreement between the perturbation calculation of emitted energies and the results of fully general relativistic numerical simulations of time-symmetric initial data.Comment: 5 pages (RevTex 3.0 with 3 uuencoded figures), CRSR-107

Coordinate Singularities in Harmonically-sliced Cosmologies

Harmonic slicing has in recent years become a standard way of prescribing the lapse function in numerical simulations of general relativity. However, as was first noticed by Alcubierre (1997), numerical solutions generated using this slicing condition can show pathological behaviour. In this paper, analytic and numerical methods are used to examine harmonic slicings of Kasner and Gowdy cosmological spacetimes. It is shown that in general the slicings are prevented from covering the whole of the spacetimes by the appearance of coordinate singularities. As well as limiting the maximum running times of numerical simulations, the coordinate singularities can lead to features being produced in numerically evolved solutions which must be distinguished from genuine physical effects.Comment: 21 pages, REVTeX, 5 figure

The Evolution of Distorted Rotating Black Holes II: Dynamics and Analysis

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 odd-parity 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.Comment: 26 pages, LaTeX with RevTeX 3.0 macros. 27 uuencoded gz-compressed postscript figures. Also available at http://jean-luc.ncsa.uiuc.edu/Papers/ Submitted to Physical Review