A new method to compute quasi-local spin and other invariants on marginally trapped surfaces

TL;DR

This paper introduces a new invariant surface integral method for accurately computing quasi-local spin and invariants on marginally trapped surfaces in numerical relativity, avoiding the need for solving Killing equations.

Contribution

The authors develop a novel, efficient technique using invariant surface integrals to determine quasi-local quantities without solving Killing equations or relying on approximate Killing vectors.

Findings

Method accurately computes quasi-local spin and invariants in simulations.

Results show good agreement with traditional methods during black hole ringdown.

The technique requires only a few hundred surface points for reliable calculations.

Abstract

We accurately compute the scalar 2-curvature, the Weyl scalars, associated quasi-local spin, mass and higher multipole moments on marginally trapped surfaces in numerical 3+1 simulations. To determine the quasi-local quantities we introduce a new method which requires a set of invariant surface integrals, allowing for surface grids of a few hundred points only. The new technique circumvents solving the Killing equation and is also an alternative to approximate Killing vector fields. We apply the method to a perturbed non-axisymmetric black hole ringing down to Kerr and compare the quasi-local spin with other methods that use Killing vector fields, coordinate vector fields, quasinormal ringing and properties of the Kerr metric on the surface. Interesting is the agreement with the spin of approximate Killing vector fields during the phase of perturbed axisymmetry. Additionally, we introduce a new coordinate transformation, adapting spherical coordinates to any two points on the sphere like the two minima of the scalar 2-curvature on axisymmetric trapped surfaces.