In Horizon Penetrating Coordinates: Kerr Black Hole Metric Perturbation Construction and Completion

TL;DR

This paper develops a method to construct and complete Kerr black hole metric perturbations using horizon-penetrating coordinates, enabling better analysis of perturbation behavior across the horizon.

Contribution

It introduces a novel approach to metric perturbation construction in Kerr spacetime using horizon-penetrating coordinates, including handling an extra singularity and providing explicit formulas.

Findings

Successfully constructed metric perturbations in horizon-penetrating coordinates.

Identified the role of an additional singularity in the radial derivative for convergence.

Maintained the physical boundary conditions without regularity constraints.

Abstract

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington-Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.