Black hole initial data from a non-conformal decomposition

TL;DR

This paper introduces a novel method for generating initial data in general relativity that avoids conformal decomposition, using a unit vector field and scalar, with solutions demonstrated for axisymmetric perturbations of Schwarzschild.

Contribution

It proposes an alternative to conformal decomposition for initial data in general relativity, with numerical and analytic solutions for axisymmetric Schwarzschild perturbations.

Findings

Successfully solved nonlinear elliptic equations numerically.

Derived solutions for perturbations of Schwarzschild black holes.

Validated the approach with both numerical and linearized analytic methods.

Abstract

We present an alternative approach to setting initial data in general relativity. We do not use a conformal decomposition, but instead express the 3-metric in terms of a given unit vector field and one unknown scalar field. In the case of axisymmetry, we have written a program to solve the resulting nonlinear elliptic equation. We have obtained solutions, both numerically and from a linearized analytic method, for a general perturbation of Schwarzschild.