Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations

TL;DR

This paper derives a gauge-invariant method for measuring black hole spin in 3D numerical relativity simulations using surface integrals of Weinberg's pseudotensor, independent of lapse and shift functions.

Contribution

It establishes a coordinate-invariant approach to compute black hole spin from surface integrals, connecting the flat-space rotational Killing vector method with Weinberg's pseudotensor in Gaussian normal coordinates.

Findings

The method yields the Komar angular momentum in axisymmetric spacetimes.

It does not depend explicitly on the lapse and shift functions.

The approach is gauge-invariant and applicable in 3D simulations.

Abstract

We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse $\alpha$ and shift $\beta^i$ on the respective timeslice, as they are fixed to Gaussian normal coordinates, while leaving the coordinate labels of the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ unchanged. Such gauge fixing endows the method with coordinate invariance, which is not present in integral expressions using Weinberg's pseudotensor, as they normally rely on the explicit use of Cartesian coordinates.