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

TL;DR

This paper develops a gauge-invariant, linear perturbation framework on the Schwarzschild background and addresses the long-standing zero-mode problem by introducing singular harmonic functions that render the $l=0,1$ modes treatable in parity-decomposed perturbations. The authors construct gauge-invariant variables within a 2+2 (spherical) decomposition, derive the linearized Einstein equations in a gauge-invariant form for all modes, and formulate an explicit strategy for odd-mode perturbations, showing the Kerr parameter perturbation naturally arises in the $l=1$ sector. A key innovation is the use of kernel-augmented harmonics $k_{(\hat{\Delta})}$ and $k_{(\hat{\Delta}+2)}$ to include zero modes, with a subsequent regularization by boundary conditions to recover physically reasonable solutions. The Part I results establish the framework and provide full odd-mode solutions, laying the groundwork for Part II (even modes) and Part III (exact solutions), and enabling extensions to higher-order perturbations relevant for EMRI modeling and precision tests of gravity. Overall, the work advances a covariant, gauge-invariant perturbation theory on spherical backgrounds and supplies physically meaningful interpretations of perturbations, including spin perturbations and their relation to the Kerr geometry.

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.

Figures (4)

• Figure 1: The first kind gauge is a coordinate system of a single manifold. The points $r$ and $s$ and its coordinates $\{x^{\mu}(s),x^{\mu}(r)\}$ and $\{y^{\mu}(s),y^{\mu}(r)\}$ are used in the explanations at the paragraph of Eq. (\ref{['eq:tensor-pull-back-first-kind-gauge-def']}).
• Figure 2: The second kind gauge is a point-identification between the physical spacetime ${{\mathscr M}}_{{\rm ph}}={{\mathscr M}}_{\epsilon}$ and the background spacetime ${{\mathscr M}}$ on the extended manifold ${{\mathscr N}}$. Through Eq. (\ref{['eq:variable-symbolic-perturbation']}), we implicitly assume the existence of a point-identification map between ${{\mathscr M}}_{\epsilon}$ and ${{\mathscr M}}$. However, this point-identification is not unique by virtue of the general covariance in the theory. We may choose the gauge of the second kind so that $p\in{{\mathscr M}}$ and "$p$"$\in{{\mathscr M}}_{\epsilon}$ is same (${{\mathscr X}}_{\epsilon}$). We may also choose the gauge so that $q\in{{\mathscr M}}_{0}$ and "$p$"$\in{{\mathscr M}}_{\epsilon}$ is same (${{\mathscr Y}}_{\epsilon}$). These are different gauge choices. The gauge transformation ${{\mathscr X}}_{\epsilon}\rightarrow{{\mathscr Y}}_{\epsilon}$ is given by the diffeomorphism $\Phi_{\epsilon}={{\mathscr X}}_{\epsilon}^{-1}\circ{{\mathscr Y}}_{\epsilon}$.
• Figure 3: A second kind gauge transformation induces a coordinate transformation. The diffeomorphism $\psi_{\alpha}\circ{{\mathscr X}}_{\epsilon}^{-1}$ maps the open set ${{\mathscr X}}_{\epsilon}(O_{\alpha})\subset{{\mathscr M}}_{{\rm ph}}$ to a open set on ${{\mathbb R}}^{4}$. If we change the gauge choice from ${{\mathscr X}}_{\epsilon}$ to ${{\mathscr Y}}_{\epsilon}$, this change induces the coordinate transformation $\psi_{\alpha}\circ{{\mathscr X}}_{\epsilon}^{-1}$ to $\psi_{\alpha}\circ{{\mathscr Y}}_{\epsilon}^{-1}$.
• Figure 4: Consider the $n$-dimensional physical manifolds ${{\mathscr M}}_{\epsilon}$ and the background ${{\mathscr M}}$. We may introduce the coordinate transformation on the physical spacetime ${{\mathscr M}}_{\epsilon}$, even if we completely fix the second-kind gauge as ${{\mathscr X}}_{\epsilon}$. Actually, we may introduce the diffeomorphism $\psi_{\alpha}$ from the open set $O_{\alpha}$ to an open set on ${{\mathbb R}}^{n}$ and the diffeomorphism $\psi_{\beta}$ from the open set $O_{\beta}$ to an open set on the other ${{\mathbb R}}^{n}$. If $O_{\alpha}\cap O_{\beta}\neq\emptyset$, we can consider the coordinate transformation $\psi_{\beta}\circ\psi_{\alpha}^{-1}$ which transforms the coordinate system $(O_{\alpha},\psi_{\alpha})$ to $(O_{\beta},\psi_{\beta})$. If we choose the gauge-choice of the second-kind by ${{\mathscr X}}_{\epsilon}$, this gauge-choice induce the coordinate systems $\{{{\mathscr X}}_{\epsilon}^{-1}O_{\alpha},\psi_{\alpha}\circ{{\mathscr X}}_{\epsilon}\}$ and $\{{{\mathscr X}}_{\epsilon}^{-1}O_{\beta},\psi_{\beta}\circ{{\mathscr X}}_{\epsilon}\}$ on ${{\mathscr M}}$. Furthermore, the coordinate transformation is given by $(\psi_{\beta}\circ{{\mathscr X}}_{\epsilon})\circ(\psi_{\alpha}\circ{{\mathscr X}}_{\epsilon})^{-1}$$=$$\psi_{\beta}\circ\psi_{\alpha}^{-1}$.

Theorems & Definitions (2)

• Conjecture 2.1
• Theorem 4.1