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

Kouji Nakamura

TL;DR

This paper extends gauge-invariant perturbation theory to even-mode perturbations on the Schwarzschild background, explicitly addressing the challenging $l=0$ and $l=1$ modes. It derives a Moncrief-type master variable $\Phi_{(e)}$ and a Zerilli-type master equation, and provides closed-form solutions for $l=0$ and $l=1$ in both vacuum and non-vacuum settings. In vacuum, the $l=0$ mode yields the linearized Schwarzschild mass perturbation $M_1$, realizing a linearized Birkhoff theorem in a gauge-invariant form, while $l=1$ even modes reduce to pure gauge (Lie-derivative) perturbations; non-vacuum cases introduce physically meaningful matter contributions that couple to the master variables. The results support the physical soundness of the gauge-invariant treatment and lay groundwork for higher-order perturbation theory and global solution construction (Part III).

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.

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

TL;DR

This paper extends gauge-invariant perturbation theory to even-mode perturbations on the Schwarzschild background, explicitly addressing the challenging and modes. It derives a Moncrief-type master variable and a Zerilli-type master equation, and provides closed-form solutions for and in both vacuum and non-vacuum settings. In vacuum, the mode yields the linearized Schwarzschild mass perturbation , realizing a linearized Birkhoff theorem in a gauge-invariant form, while even modes reduce to pure gauge (Lie-derivative) perturbations; non-vacuum cases introduce physically meaningful matter contributions that couple to the master variables. The results support the physical soundness of the gauge-invariant treatment and lay groundwork for higher-order perturbation theory and global solution construction (Part III).

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 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 modes. Furthermore, we explicitly derive the mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the -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.

Paper Structure

This paper contains 13 sections, 1 theorem, 174 equations.

Table of Contents

  1. Introduction
  2. Brief review of the general-relativistic gauge-invariant perturbation theory
  3. General framework of gauge-invariant perturbation theory
  4. Linear perturbations on spherically symmetric background
  5. Even-mode linearized Einstein equations
  6. Component treatment of the even-mode linearized Einstein equations
  7. $l=0$ mode perturbations on the Schwarzschild Background
  8. $l=0$ mode vacuum case
  9. $l=0$ mode non-vacuum case
  10. $l=1$ mode non-vacuum perturbations on the Schwarzschild Background
  11. $l=1$ mode vacuum case
  12. $l=1$ mode non-vacuum case
  13. Summary and Discussion

Key Result

Theorem 2.1

If the gauge-transformation rule for a perturbative pulled-back tensor field $h_{ab}$ to the background spacetime ${{\mathscr M}}$ is given by ${}_{{{\mathscr Y}}}\!h_{ab}$$-$${}_{{{\mathscr X}}}\!h_{ab}$$=$${\pounds}_{\xi_{(1)}}g_{ab}$ with the background metric $g_{ab}$ with spherical symmetry, th

Theorems & Definitions (2)

  • Conjecture 2.1
  • Theorem 2.1