Quasinormal modes of a Schwarzschild black hole within the Bondi-Sachs framework

TL;DR

This paper investigates Schwarzschild black hole quasinormal modes using the Bondi-Sachs framework, applying a characteristic formulation and Leaver's method to recover standard modes and identify the algebraically special mode.

Contribution

It introduces a novel approach using the Bondi-Sachs characteristic formulation to compute black hole QNMs, including the algebraically special mode.

Findings

Successfully recovers standard Schwarzschild QNMs.

Identifies the algebraically special mode under outgoing boundary conditions.

Demonstrates the effectiveness of the characteristic formulation for QNM analysis.

Abstract

Studies of quasinormal modes (QNMs) of black holes have a long and well established history. Predominantly, much research in this area has customarily focused on the equations given by Regge, Wheeler and Zerilli. In this work we study linearized perturbations of a Schwarzschild black hole using the Characteristic formulation of numerical relativity, with an emphasis on the computation of QNMs. Within this formalism, the master equation describing gravitational perturbations is known to satisfy a fourth order differential equation. We analyse the singular points of this master equation, and obtain series solutions whose coefficients are given by three term recurrence relations, from which Leaver's continued fraction method can be applied. Using this technique, we recover the standard Schwarzschild quasinormal modes. In addition, we find that imposing purely outgoing boundary conditions, a natural feature of the Bondi-Sachs framework, leads to the recovery of the algebraically special mode.