Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications

TL;DR

This paper introduces a hyperboloidal slicing method for wave equations on Kerr-Schild backgrounds, enabling direct access to null infinity and improved numerical stability, with applications to studying late-time Kerr tails.

Contribution

It develops a hyperboloidal transformation approach for wave equations in Kerr-Schild spacetimes, enhancing numerical simulations and analysis of decay rates at null infinity.

Findings

Effective numerical access to null infinity

Improved understanding of Kerr tail decay rates

Numerical stability near null infinity

Abstract

We present new results from two open source codes, using finite differencing and pseudo-spectral methods for the wave equations in (3+1) dimensions. We use a hyperboloidal transformation which allows direct access to null infinity and simplifies the control over characteristic speeds on Kerr-Schild backgrounds. We show that this method is ideal for attaching hyperboloidal slices or for adapting the numerical resolution in certain spacetime regions. As an example application, we study late-time Kerr tails of sub-dominant modes and obtain new insight into the splitting of decay rates. The involved conformal wave equation is freed of formally singular terms whose numerical evaluation might be problematically close to future null infinity.