Proposal of a gauge-invariant treatment of l=0,1-mode perturbations on Schwarzschild Background Spacetime

TL;DR

This paper introduces a gauge-invariant method for analyzing monopole and dipole perturbations in Schwarzschild spacetime, deriving solutions that include Kerr and mass perturbations, and confirming a linearized Birkhoff's theorem.

Contribution

It presents a novel gauge-invariant approach for l=0,1 perturbations in Schwarzschild spacetime, including solutions with matter fields and vacuum cases.

Findings

Derived gauge-invariant solutions for monopole and dipole modes.

Included Kerr parameter and mass perturbations in solutions.

Confirmed linearized Birkhoff's theorem gauge-invariantly.

Abstract

A gauge-invariant treatment of the monopole- ($l=0$) and dipole ($l=1$) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the $l=1$ odd modes and the additional mass parameter perturbations of the Schwarzschild mass in the $l=0$ even modes. The linearized version of Birkhoff's theorem is also confirmed in a gauge-invariant manner. In this sense, our proposal is reasonable.