A geometrically motivated coordinate system for exploring spacetime dynamics in numerical-relativity simulations using a quasi-Kinnersley tetrad

TL;DR

This paper introduces a quasi-Kinnersley tetrad and a geometrically motivated coordinate system for numerical relativity, improving wave extraction and visualization of spacetime dynamics in simulations, especially near black holes.

Contribution

It develops invariant coordinates and a preferred tetrad that reduce to known forms in Kerr spacetime, enhancing analysis of strong-field and wave-zone effects in NR simulations.

Findings

Coordinates and tetrad reduce to Boyer-Lindquist and Kinnersley in Kerr limit.

Improved wave-extraction accuracy near infinity and interior regions.

Useful for visualizing and analyzing junk radiation and merger dynamics.

Abstract

We investigate the suitability and properties of a quasi-Kinnersley tetrad and a geometrically motivated coordinate system as tools for quantifying both strong-field and wave-zone effects in numerical relativity (NR) simulations. We fix the radial and latitudinal coordinate degrees of freedom of the metric, using the Coulomb potential associated with the quasi-Kinnersley transverse frame. These coordinates are invariants of the spacetime and can be used to unambiguously fix the outstanding spin-boost freedom associated with the quasi-Kinnersley frame (resulting in a preferred quasi-Kinnersley tetrad (QKT)). In the limit of small perturbations about a Kerr spacetime, these coordinates and QKT reduce to Boyer-Lindquist coordinates and the Kinnersley tetrad, irrespective of the simulation gauge choice. We explore the properties of this construction both analytically and numerically, and we gain insights regarding the propagation of radiation described by a super-Poynting vector. We also quantify in detail the peeling properties of the chosen tetrad and gauge. We argue that these choices are particularly well suited for a rapidly converging wave-extraction algorithm as the extraction location approaches infinity, and we explore numerically the extent to which this property remains applicable on the interior of a computational domain. Using a number of additional tests, we verify that the prescription behaves as required in the appropriate limits regardless of simulation gauge. We explore the behavior of the geometrically motivated coordinate system in dynamical binary-black-hole NR mergers, and find them useful for visualizing features in NR simulations such as the spurious "junk" radiation. Finally, we carefully scrutinize the head-on collision of two black holes and, for example, the way in which the extracted waveform changes as it moves through the computational domain.