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Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: -- Realization of exact solutions

Kouji Nakamura

TL;DR

This paper tests the gauge-invariant perturbation framework for the Schwarzschild background by connecting the previously derived even-mode $l=0,1$ solutions to two exact spacetimes: the Lemaître-Tolman-Bondi (LTB) dust solutions and the non-rotating C-metric. By constructing perturbations around Schwarzschild and employing a 2+2 harmonic decomposition, the authors show that the linearized LTB solution is realized as a perturbation in the $l=0$ sector up to a gauge Lie derivative, while the linearized C-metric is realized within the $l=0,1$ and $l\ge 2$ sectors through explicit harmonic decompositions and relations among the gauge-invariant and gauge-variant parts. The analysis yields explicit expressions for the perturbative variables, stresses, and Moncrief master variables, demonstrating the physical reasonableness and internal consistency of the gauge-invariant approach for both the $l=0,1$ even modes and higher-order, providing a concrete bridge between perturbative and exact solutions. Overall, the results validate the gauge-invariant treatment of $l=0,1$ even-mode perturbations on a Schwarzschild background and illustrate how exact solutions are realized within this framework, with implications for higher-order perturbation theory in black-hole spacetimes.

Abstract

This is the Part III 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 [K.~Nakamura, arXiv:2110.13508 [gr-qc]], we examine the problem whether the $l=0,1$ even-mode solutions derived in the Part II paper [K.~Nakamura, arXiv:2110.13512 [gr-qc]] are physically reasonable, or not. We consider the linearized versions of the Lemaître-Tolman-Bondi solution and the non-rotating C-metric. As the result, we show that our derived even-mode solutions to the linearized Einstein equations actually realize above two linearized solutions. This fact supports that our derived solutions are physically reasonable, which implies that our proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations are also physically reasonable. We also briefly summarize our conclusions of our series of papers.

Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: -- Realization of exact solutions

TL;DR

This paper tests the gauge-invariant perturbation framework for the Schwarzschild background by connecting the previously derived even-mode solutions to two exact spacetimes: the Lemaître-Tolman-Bondi (LTB) dust solutions and the non-rotating C-metric. By constructing perturbations around Schwarzschild and employing a 2+2 harmonic decomposition, the authors show that the linearized LTB solution is realized as a perturbation in the sector up to a gauge Lie derivative, while the linearized C-metric is realized within the and sectors through explicit harmonic decompositions and relations among the gauge-invariant and gauge-variant parts. The analysis yields explicit expressions for the perturbative variables, stresses, and Moncrief master variables, demonstrating the physical reasonableness and internal consistency of the gauge-invariant approach for both the even modes and higher-order, providing a concrete bridge between perturbative and exact solutions. Overall, the results validate the gauge-invariant treatment of even-mode perturbations on a Schwarzschild background and illustrate how exact solutions are realized within this framework, with implications for higher-order perturbation theory in black-hole spacetimes.

Abstract

This is the Part III 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 [K.~Nakamura, arXiv:2110.13508 [gr-qc]], we examine the problem whether the even-mode solutions derived in the Part II paper [K.~Nakamura, arXiv:2110.13512 [gr-qc]] are physically reasonable, or not. We consider the linearized versions of the Lemaître-Tolman-Bondi solution and the non-rotating C-metric. As the result, we show that our derived even-mode solutions to the linearized Einstein equations actually realize above two linearized solutions. This fact supports that our derived solutions are physically reasonable, which implies that our proposal on the gauge-invariant treatments for mode perturbations are also physically reasonable. We also briefly summarize our conclusions of our series of papers.

Paper Structure

This paper contains 17 sections, 1 theorem, 239 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. Realization of LTB solution as a perturbation of Schwarzschild spacetime
  7. Perturbative expression of the LTB solution on Schwarzschild background spacetime
  8. Expression of the perturbative LTB solution in static chart
  9. Realization of the linearized non-rotating C-metric
  10. The linearized non-rotating C-metric
  11. Components of metric perturbation of the linearized non-rotating C-metric
  12. Harmonic decomposition of the perturbative non-rotating C-metric
  13. Realization of $l\geq 2$ mode perturbations
  14. Realization of $l=0$ mode perturbations
  15. Realization of $l=1$ mode perturbations
  16. ...and 2 more sections

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 spherically symmetry,

Theorems & Definitions (2)

  • Conjecture 2.1
  • Theorem 2.1