ICRC2023 Proceedings: Proposal of a gauge-invariant treatment of $l=0,1$-mode perturbations on the Schwarzschild background spacetime

TL;DR

This paper proposes a gauge-invariant method for treating the challenging l=0,1 modes in perturbations of Schwarzschild spacetime, deriving solutions and confirming a gauge-invariant version of the Kerr uniqueness theorem.

Contribution

It introduces a novel gauge-invariant approach to handle zero modes in Schwarzschild perturbations, addressing a longstanding problem in the field.

Findings

Derived solutions for l=0,1 modes with matter fields

Confirmed gauge-invariant linearized Kerr uniqueness theorem in vacuum

Provided a consistent gauge-invariant framework for these perturbations

Abstract

A gauge-invariant perturbation theory on a generic background spacetime is developing from 2003 and ``zero-mode problem'' for linear metric perturbations was proposed as the essential problem of this theory. In the perturbation theory on the Schwarzschild background spacetime, $l=0,1$ modes correspond to the above ``zero-mode'' and the gauge-invariant treatments of these modes is a famous non-trivial problem in perturbation theories on the Schwarzschild background spacetime. Due to this situation, a gauge-invariant treatment for these $l=0,1$-mode perturbations is proposed. Through this gauge-invariant treatment, the solutions to the linearized Einstein equation for these modes with a generic matter field are derived. In the vacuum case, the linearized version of uniqueness theorem of Kerr spacetime is confirmed in a gauge-invariant manner. In this sense, our proposal is reasonable.