Nonexistence of conformally flat slices of the Kerr spacetime

Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restrictions, we show that no conformally flat slices can exist.Comment: 5 LaTeX pages; no figures; to be submitted to Phys. Rev.

Evolving the Bowen-York initial data for spinning black holes

The Bowen-York initial value data typically used in numerical relativity to represent a spinning black hole are not those of a constant-time slice of the Kerr spacetime. If Bowen-York initial data are used for each black hole in a collision, the emitted radiation will be partially due to the “relaxation” of the individual holes to Kerr form. We compute this radiation by treating the geometry for a single hole as a perturbation of a Schwarzschild black hole, and by using second order perturbation theory. We discuss the extent to which Bowen-York data can be expected accurately to represent Kerr holes. © 1998 The American Physical Society

Collision of boosted black holes: Second order close limit calculations

We study the head-on collision of black holes starting from unsymmetrized, Brill-Lindquist type data for black holes with non-vanishing initial linear momentum. Evolution of the initial data is carried out with the “close limit approximation,” in which small initial separation and momentum are assumed, and second-order perturbation theory is used. We find agreement that is remarkably good, and that in some ways improves with increasing momentum. This work extends a previous study in which second order perturbation calculations were used for momentarily stationary initial data, and another study in which linearized perturbation theory was used for initially moving holes. In addition to supplying answers about the collisions, the present work has revealed several subtle points about the use of higher order perturbation theory, points that did not arise in the previous studies. These points include issues of normalization, and of comparison with numerical simulations, and will be important to subsequent applications of approximation methods for collisions. © 1999 The American Physical Society

Colliding Black Holes: How far can the close approximation go?

We study the head-on collision of two equal-mass momentarily stationary black holes, using black hole perturbation theory up to second order. Compared to first-order results, this significantly improves agreement with numerically computed wave forms and energy. Much more important, second-order results correctly indicate the range of validity of perturbation theory. This use of second-order corrections to provide “error bars” to the first-order results makes perturbation theory a viable tool for providing benchmarks for numerical relativity in more generic collisions and, in some range of collision parameters, for supplying wave form templates for gravitational wave detection. © 1996 The American Physical Society

Gravitational radiation from Schwarzschild black holes: The second-order perturbation formalism

The perturbation theory of black holes has been useful recently for providing estimates of gravitational radiation from the black-hole collisions. Second-order perturbation theory, relatively undeveloped until recently, has proved to be important both for providing refined estimates and for indicating the range of validity of perturbation theory. Here we review the second-order formalism for perturbations of Schwarzschild spacetimes. The emphasis is on practical methods for carrying out the second-order computations of the outgoing radiation. General issues are illustrated throughout with examples from close-limit results, perturbation calculations in which black holes start from the small separation. © 2000 Elsevier Science B.V

Second-order perturbations of a Schwarzschild black hole

We study the even-parity ℓ = 2 perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first-order perturbations. This provides a formalism to address the validity of many first-order calculations of interest in astrophysics. © 1996 IOP Publishing Ltd

Second order perturbations of a Schwarzschild black hole: inclusion of odd parity perturbations

We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the $\ell=2$ axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge-Wheeler-Zerilli approach. This formulation is useful, for instance, for the treatment in the ``close limit approximation'' of the collision of counterrotating black holes.Comment: 12 pages RevTe

Colliding black holes: how far can the close approximation go?

We study the head-on collision of two equal-mass momentarily stationary black holes, using black hole perturbation theory up to second order. Compared to first-order results, this significantly improves agreement with numerically computed waveforms and energy. Much more important, second-order results correctly indicate the range of validity of perturbation theory. This use of second-order, to provide ``error bars,'' makes perturbation theory a viable tool for providing benchmarks for numerical relativity in more generic collisions and, in some range of collision parameters, for supplying waveform templates for gravitational wave detection.Comment: 6 pages, RevTeX, 2 figures included with eps

Making use of geometrical invariants in black hole collisions

We consider curvature invariants in the context of black hole collision simulations. In particular, we propose a simple and elegant combination of the Weyl invariants I and J, the {\sl speciality index} ${\cal S}$. In the context of black hole perturbations $\cal S$ provides a measure of the size of the distortions from an ideal Kerr black hole spacetime. Explicit calculations in well-known examples of axisymmetric black hole collisions demonstrate that this quantity may serve as a useful tool for predicting in which cases perturbative dynamics provide an accurate estimate of the radiation waveform and energy. This makes ${\cal S}$ particularly suited to studying the transition from nonlinear to linear dynamics and for invariant interpretation of numerical results.Comment: 4 pages, 3 eps figures, Revte