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

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.

(v1) 10 pages, no figure; (v2) 16 pages, Section 2 is added for the explanation of our previous works, Additional explanations are added, references are added and re-organized; (v3) Some explanations are added, Typos are corrected

References in corpus (5)

• Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --
• High-order gauge-invariant perturbations of a spherical spacetime
• Consistency of Equations in the Second-order Gauge-invariant Cosmological Perturbation Theory
• Gauge-invariant variables in general-relativistic perturbations: globalization and zero-mode problem
• Recursive structure in the definitions of gauge-invariant variables for any order perturbations

Cited by in corpus (6)

• Backreaction of scalar waves on black holes at low frequencies
• Backreaction of Mass and Angular Momentum Accretion on Black Holes: General Formulation of the Metric Perturbations and Application to the Blandford-Znajek Process
• Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime
• Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : -- Formulation and odd-mode perturbations
• Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -- Even-mode perturbations
• Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: -- Realization of exact solutions