Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part II: -- Even-mode perturbations

Abstract

This is the Part II paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory and the proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations on the Schwarzschild background spacetime in the Part I paper [K.~Nakamura, arXiv:2110.13508 [gr-qc]], we examine the linearized Einstein equations for even-mode perturbations. We discuss the strategy to solve the linearized Einstein equations for these even-mode perturbations including $l=0,1$ modes. Furthermore, we explicitly derive the $l=0,1$ mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for $l=0$ mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the $l=0,1$-mode problem is physically reasonable due to the realization of the additional Schwarzschild mass parameter perturbation and the Kerr parameter perturbation in the Part I paper.

arXiv comment34 pages, no figure, The Part II paper of the full paper version of the previous short papers arXiv:2102.00830v3[gr-qc] (v1); references are corrected. (v2); 33 pages, no figure, some explanations are added (v3); minor revision in explanations (v4); minor revision (v5)

References in corpus (18)

• Self-force and radiation reaction in general relativity
• Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond
• Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism
• Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions
• Gauge-invariant Non-spherical Metric Perturbations of Schwarzschild Black-Hole Spacetimes
• Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --
• Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion
• Analytic Solutions of the Teukolsky Equation and their Properties
• Gauge-invariant and coordinate-independent perturbations of stellar collapse I: the interior
• Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations
• Gauge-invariant and coordinate-independent perturbations of stellar collapse II: matching to the exterior
• High-order gauge-invariant perturbations of a spherical spacetime
• Second Order Gauge Invariant Perturbation Theory -- Perturbative curvatures in the two-parameter case --
• Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status
• Perturbations of Matter Fields in the Second-order Gauge-invariant Cosmological Perturbation Theory
• Construction of gauge-invariant variables of linear metric perturbations on an arbitrary background spacetime
• Proposal of a gauge-invariant treatment of l=0,1-mode perturbations on Schwarzschild Background Spacetime
• Formal solutions of any-order mass, angular-momentum, dipole perturbations on the Schwarzschild background spacetime