Constructing a boosted, spinning black hole in the damped harmonic gauge

TL;DR

This paper develops a method to construct a single boosted, spinning black hole in damped harmonic coordinates, enabling more accurate initial data for binary black hole simulations in numerical relativity.

Contribution

The authors derive and solve nonlinear elliptic equations to produce a boosted, spinning black hole in damped harmonic gauge, facilitating improved initial data construction.

Findings

Successfully constructed a single black hole in damped harmonic coordinates.

Provided a set of elliptic equations for the transformation.

Enables more accurate initial data for binary black hole simulations.

Abstract

The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein's equations, and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium BH in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of 4 coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.