Black hole data via a Kerr-Schild approach

Abstract

We present a new approach for setting initial Cauchy data for multiple black hole space-times. The method is based upon adopting an initially Kerr-Schild form of the metric. In the case of non-spinning holes, the constraint equations take a simple hierarchical form which is amenable to direct numerical integration. The feasibility of this approach is demonstrated by solving analytically the problem of initial data in a perturbed Schwarzschild geometry.

Solving the Initial Value Problem of Two Black Holes

“…There are no space slices for which the spatial 3-metric of a single Kerr (non-zero spin) black hole can be written in a conformally flat way [7], and recent work on sequences of initial data sets for circular orbits casts some doubt on the physical realism of conformally flat approaches to black hole binaries [4]. One way to overcome this problem is to use a Kerr-Schild slicing of spacetime [8]. Matzner, Huq, and Shoemaker [9] proposed a method that bases the initial data on a background metric and extrinsic curvature that is a superposition of Kerr black hole metrics written in ingoing Eddington-Finkelstein coordinates.…”

Kerr Schild-type initial data for black holes with angular momenta

“…Let us briefly review Bishop et al's proposal [6] to solve the constraint equations of general relativity: The assumption is that at the initial slice, the metric and its time derivative are of Kerr-Schild type,…”

“…II, uses a Kerr-Schild type of metric in order to construct the initial data. While in [6] the superposition of two Schwarzschild black holes is discussed, we generalize their proposal to spinning black holes in Sec. III.…”

“…Therefore, this data is likely to contain "junk" radiation which, at least when the two black holes are very far from each other or have merged and settled down to a stationary black hole, might not reflect a realistic astrophysical scenario [18]. More recent proposals relax the requirement for the initial data to be conformally flat [6,7,8] and result in new constructions that are able to naturally incorporate spinning black holes.…”

“…In the present work, we construct initial data following an approach introduced by Bishop et al [6] and further elaborated in [9,10] which is not directly based on the York-Lichnerowicz decomposition. The advantage of this approach is that one can easily obtain closed analytic expressions for the solution to the constraint equations in some limiting cases, including the close limit approximation.…”