[Paper Review] Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : -- Formulation and odd-mode perturbations

K. Nakamura|arXiv (Cornell University)|Oct 26, 2021

Astrophysical Phenomena and Observations8 references4 citations

TL;DR

This paper proposes a gauge-invariant perturbation framework for Schwarzschild spacetime that resolves longstanding challenges in treating $l=0$ and $l=1$ mode perturbations by decomposing metric perturbations using singular harmonic functions and regularizing singularities via boundary conditions. It derives linearized Einstein equations in a gauge-invariant manner for all $l \geq 0$, and demonstrates that odd-mode perturbations naturally include Kerr parameter perturbations, confirming physical consistency for these modes.

ABSTRACT

This is the Part I paper of our series of full papers on a gauge-invariant {\it linear} perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers [K.~Nakamura, Class. Quantum Grav. {\bf 38} (2021), 145010; K.~Nakamura, Letters in High Energy Physics {\bf 2021} (2021), 215.]. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the {\it higher-order} perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, a gauge-invariant treatments of $l=0,1$ mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for $l=0,1$ modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Based on this situation, we propose a strategy of a gauge-invariant treatments of $l=0,1$ mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of this singularity through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of $l\geq 0$ in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbation, which is physically reasonable.

Motivation & Objective

• To develop a gauge-invariant linear perturbation theory on the Schwarzschild background that can be extended to higher-order perturbations.
• To address the longstanding difficulty in constructing gauge-invariant variables for $l=0$ and $l=1$ mode perturbations when reconstructing the original metric perturbations.
• To propose a regularization strategy using singular harmonic functions and boundary conditions to handle the singularities inherent in $l=0,1$ modes.
• To derive linearized Einstein equations in a gauge-invariant form for all $l \geq 0$ modes.
• To demonstrate the physical reasonableness of odd-mode solutions, including the inclusion of Kerr parameter perturbations.

Proposed method

• Utilizes a general gauge-invariant perturbation framework previously developed for higher-order perturbations on generic spacetimes.
• Applies the framework to the Schwarzschild background by decomposing metric perturbations using spherical harmonics $Y_{lm}$ and identifying odd- and even-mode components.
• Introduces singular harmonic functions to represent $l=0$ and $l=1$ mode perturbations, which inherently carry singularities.
• Imposes boundary conditions on the Einstein equations to regularize the singularities in the $l=0,1$ modes, enabling consistent gauge-invariant treatment.
• Derives the linearized Einstein equations in a gauge-invariant form using the 2+2 formalism, with explicit expressions for the tensor components $H_{abc}[\mathcal{F}]$, $H_{ab}^{\;\.c}[\mathcal{F}]$, and $H_a^{\.bc}[\mathcal{F}]$.
• Employs the 2+2 decomposition of spacetime into radial and angular parts, with $A,B,C$ labeling angular components and $p,q,r$ radial components, to systematically derive the perturbation equations.

Experimental results

Research questions

• RQ1 How can gauge-invariant perturbations for $l=0$ and $l=1$ modes be consistently formulated on the Schwarzschild spacetime when standard reconstruction fails?
• RQ2 Can a gauge-invariant treatment of $l=0,1$ modes be achieved without abandoning the goal of reconstructing the original metric perturbations?
• RQ3 Do the odd-mode solutions derived through this method include physically meaningful quantities such as the Kerr parameter perturbation?
• RQ4 Is the proposed regularization of singularities via boundary conditions on the Einstein equations mathematically and physically consistent?
• RQ5 Does the resulting gauge-invariant formalism for $l \geq 0$ modes remain consistent with known results in the literature for $l \geq 2$?

Key findings

• The proposed method successfully derives linearized Einstein equations in a gauge-invariant manner for all $l \geq 0$ modes, including $l=0$ and $l=1$, by using singular harmonic functions and boundary condition regularization.
• The odd-mode perturbation solutions naturally include the Kerr parameter perturbation, which is physically reasonable and confirms the consistency of the approach.
• The framework is compatible with higher-order perturbation theory, as it is built upon a general gauge-invariant formalism previously developed for higher-order applications.
• The 2+2 formalism provides explicit component expressions for the tensors $H_{abc}[\mathcal{F}]$, $H_{ab}^{\.c}[\mathcal{F}]$, and $H_a^{\.bc}[\mathcal{F}]$, enabling systematic computation of perturbation equations.
• The regularization via boundary conditions effectively handles the singularities in $l=0,1$ mode perturbations, allowing for a consistent gauge-invariant treatment.
• The results for odd modes lay a foundation for the even-mode analysis in Part II and Part III, where physical reasonableness will be further confirmed.