Time-asymmetric initial data for black holes and black-hole collisions

Abstract

As a first step in constructing initial data for dynamic black holes and general black-hole collisions, we study nonsingular vacuum Cauchy hypersurfaces with two isometric asymptotically flat ends connected by an Einstein-Rosen-type bridge. These hypersurfaces are assumed to be conformally flat and maximally embedded in spacetime but are neither spherically symmetric nor time symmetric. Three of the four constraints are solved explicitly for suitable extrinsic curvature tensors that possess linear momentum and/or intrinsic angular momentum. The complete initial data are shown to transform invariantly, modulo the sign of the extrinsic curvature tensor, under inversion through a minimal two-surface that represents the "throat" of the geometry. These and other properties show that the data represent a particular epoch in the history of a dynamic black hole. We describe the relation of our data to that of the Schwarzschild and Kerr black holes. Finally, we discuss the generalization to encounters of two or more black holes.

Asymptotically Flat Initial Data¶with Prescribed Regularity at Infinity

“…Since Ψ ab P is trace-free and divergence-free with respect to the flat metric, we could, of course, choose h ab to be the flat metric and Ψ ab R = 0. This would provide one of the conformally flat initial data sets discussed in [11]. We are interested in a more general situation.…”

“…Theorem 4.14 Assume that the metric h is smooth and Ψ ab is the solution of the momentum constraint determined in theorem 4.6. If (ii) Φ ab reg = r −3 Φ ab 1reg + Φ ab 2reg with Φ ab 1reg , Φ ab 2reg ∈ C ∞ (B a ) such that Φ ab 1reg = O(r 2 ), and x a x a Φ ab 1reg = r 2 Φ with some Φ ∈ C ∞ (B a ), then Ψ ab satisfies condition (11).…”

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

“…We will first recall the solution that represents BHs and next reintroduce the one suitable to model NSs. For BHs (S i = 0), Bowen and York [25] found that point-source solutions to (∆ L W) i = 0 are given by…”

“…In the BBH puncture approach [24], one only solves the Hamiltonian constraint for the conformal factor. The solution to the momentum constraint is given by the Bowen-York extrinsic curvature [25]. Each initial data set is then fully specified by the masses, spins and momenta of the BHs, and their separation.…”

Long gamma ray bursts from binary black holes

“…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. These are the controlled parameters in the first section of Table 1.…”