An alternative approach to solving the Hamiltonian constraint

TL;DR

This paper introduces a novel method for solving Einstein's Hamiltonian constraint by focusing on an inverse power of the conformal factor, eliminating the need for excision or splitting into background and correction.

Contribution

The authors propose an alternative approach that solves for an inverse power of the conformal factor, which remains finite even with singular corrections, simplifying black hole initial data construction.

Findings

Method works without modification for rotating black holes in trumpet topology.

Avoids excision and background-correction split in solving the constraint.

Demonstrates robustness with singular corrections to the conformal factor.

Abstract

Solving Einstein's constraint equations for the construction of black hole initial data requires handling the black hole singularity. Typically, this is done either with the excision method, in which the black hole interior is excised from the numerical grid, or with the puncture method, in which the singular part of the conformal factor is expressed in terms of an analytical background solution, and the Hamiltonian constraint is then solved for a correction to the background solution that, usually, is assumed to be regular everywhere. We discuss an alternative approach in which the Hamiltonian constraint is solved for an inverse power of the conformal factor. This new function remains finite everywhere, so that this approach requires neither excision nor a split into background and correction. In particular, this method can be used without modification even when the correction to the conformal factor is singular itself. We demonstrate this feature for rotating black holes in the trumpet topology.