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 of gauge-invariant treatments for $l=0,1$-mode perturbations on the Schwarzschild background spacetime in the Part I paper, 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.

1. Introduction

Gravitational wave observations are now being conducted through ground-based detectors [1,2,3,4]. Furthermore, the projects of future ground-based gravitational wave detectors [5,6] are also progressing to obtain more sensitive detectors. In addition to these ground-based detectors, some projects on space gravitational wave antenna are also progressing [7,8,9,10]. Among them, the Extreme-Mass-Ratio Inspiral (EMRI), which is a source of gravitational waves from the motion of a stellar-mass object around a supermassive black hole, is a promising target of the Laser Interferometer Space Antenna [7]. To describe gravitational waves from EMRIs, black hole perturbations are used [11]. Furthermore, the sophistication of higher-order black hole perturbation theories is required to support these gravitational-wave physics as a “precise science”. The motivation of this paper is in such theoretical sophistication of black hole perturbation theories on higher-order perturbations for wide physical situations.

Although realistic black holes have angular momentum and we have to consider the perturbation theory of a Kerr black hole for direct applications to the EMRI, we may say that further sophistication is possible even in perturbation theories on the Schwarzschild background spacetime. Since the pioneering works by Regge and Wheeler [12] and Zerilli [13,14], there have been many studies on perturbations in the Schwarzschild background spacetime [15,16,17,18,19,20,21,22,23,24,25,26,27]. In these works, perturbations on the Schwarzschild spacetime are decomposed through the spherical harmonics $Y_{lm}$ because of the spherical symmetry of the background spacetime, and the $l=0$ and $l=1$ modes should be separately treated. Furthermore, “gauge-invariant” treatments for $l=0$ and $l=1$ even modes are unknown.

Owing to this situation, in previous papers [28,29], we proposed the strategy of the gauge-invariant treatments of these $l=0,1$-mode perturbations, which is declared as Proposal 1 in Section 2 of this paper below. One of the important premises of our gauge-invariant perturbations is the distinction between the first-kind gauge and the second-kind gauge. The first-kind gauge is the choice of the coordinate system in a single manifold, and we often use this first-kind gauge when we predict or interpret the measurement results of experiments and observations. On the other hand, the second-kind gauge is the choice of point identifications between points in the physical spacetime ${\mathcal{M}}_{ϵ}$ and the background spacetime $\mathcal{M}$. This second-kind gauge has nothing to do with our physical spacetime, $\mathcal{M}$. Based on this distinction between the first- and the second-kind gauges, we are developing the general formulation of a higher-order gauge-invariant perturbation theory on a generic background spacetime for unambiguous, sophisticated, nonlinear, general relativistic perturbation theories [30,31,32]. The proposal in the Part I paper [29] is a part of this development. This general formulation of the higher-order gauge-invariant perturbation theory was already applied to the second-order cosmological perturbation theory [33,34,35]. Even in cosmological perturbation theories, the same problem exists as in the above $l=0,1$-mode problem. This lies in the gauge-invariant treatments of homogeneous modes of cosmological perturbations. We have to re-investigate the arguments of the previous cosmological perturbation theory [33,34,35] in the near future. In this sense, we can expect that the proposal in the previous paper [29] is not only for perturbations on the Schwarzschild background spacetime but will also be a clue to the same problem in the gauge-invariant perturbation theory on the generic background spacetime.

Furthermore, in the previous Part I paper, we derived the linearized Einstein equations in a gauge-invariant manner following Proposal 1 of the gauge-invariant treatments of $l=0,1$-mode perturbations on the Schwarzschild background spacetime. From the parity of the perturbations, we could classify the perturbations on the spherically symmetric background spacetime into even- and odd-mode perturbations. In the Part I paper [29], we also gave a strategy to solve the odd-mode perturbations, including $l=0,1$-mode ones. Moreover, we also derived the explicit solutions for the $l=0,1$ odd-mode perturbations to the linearized Einstein equations, following Proposal 1.

This paper is the Part II paper of our series of papers on the application of our gauge-invariant perturbation theory to the Schwarzschild background spacetime. This series of papers is the full-paper version of our short paper [28]. In this Part II paper, we discuss a strategy to solve the linearized Einstein equation for even-mode perturbations, including $l=0,1$-mode perturbations. We also derive the explicit solutions to $l=0,1$-mode perturbations with a generic, linear-order energy–momentum tensor. As a result, we show the additional Schwarzschild mass parameter perturbation in the vacuum case. This is the realization of the Birkhoff theorem [36] at the linear-perturbation level in a gauge-invariant manner. This result is physically reasonable, and it also implies that Proposal 1 is also physically reasonable. The other supports for Proposal 1 are also given by the realization of exact solutions with matter fields which will be discussed in the Part III paper [37]. Furthermore, brief discussions on the extension to higher-order perturbations are given in the short paper [38].

The organization of this Part II paper is as follows. In Section 2, after a brief review of the framework of the gauge-invariant perturbation theory, we summarize our proposal in refs. [28,29]. Then, we also summarize the linearized even-mode Einstein equation for the Schwarzschild background spacetime, which was derived in ref. [29] following Proposal 1. In Section 3, following Proposal 1, we discuss a strategy to solve these even-mode Einstein equations, including $l=0,1$-mode perturbations. In Section 4, we derive the explicit solutions to the linearized Einstein equation for $l=0$-mode perturbations in both the vacuum and the non-vacuum cases. In Section 5, we also derive the explicit solutions to the linearized Einstein equation for $l=1$-mode perturbations in both the vacuum and the non-vacuum cases. The final section (Section 6) is devoted to our summary and discussions.

We use the notation used in the previous papers [28,29,38] and the unit $G=c=1$, where G is Newton’s constant of gravitation and c is the velocity of light.

2. A Brief Review of the General Relativistic Gauge-Invariant Perturbation Theory

In this section, we review the premise of the series of our papers [28,29,37] and this paper. In Section 2.1, we briefly review the framework of the gauge-invariant perturbation theory [30,31]. This is an important premise of our series of papers. In Section 2.2, we review linear perturbation on spherically symmetric background spacetimes and include our proposal in ref. [28,29]. In Section 2.3, we review the linearized Einstein equations for even-mode perturbations on the Schwarzschild background spacetime. These equations are to be solved within this Part II paper.

2.1. General Framework of Gauge-Invariant Perturbation Theory

When considering any perturbation theory in general relativity, we always keep two different spacetime manifolds in mind. The first one is the physical spacetime $({\mathcal{M}}_{\mathrm{ph}},{\overline{g}}_{ab})$, which we want to describe by perturbations. In many cases, the physical spacetime $({\mathcal{M}}_{\mathrm{ph}},{\overline{g}}_{ab})$ is identified with our nature itself. The second one is the background spacetime $(\mathcal{M},g_{ab})$, prepared as our reference for the perturbative analyses. We note that this background spacetime does not exist in our nature. This background spacetime is just a virtual manifold. Note that these two spacetimes are distinct. Furthermore, in any general relativistic perturbation theory, we always write the equation

$$Q(“p”)=Q_0\left(p\right)+\delta Q\left(p\right)$$

(1)

as the definition of the perturbation of the variable Q. Equation (1) is a curious equation in the sense that this equation gives a relation between variables in different manifolds. Actually, $Q(“p”)$ on the left-hand side of Equation (1) and is a variable in the physical spacetime ${\mathcal{M}}_{ϵ}={\mathcal{M}}_{\mathrm{ph}}$, whereas $Q_0\left(p\right)$ and $\delta Q\left(p\right)$ on the right-hand side of Equation (1) are variables in the background spsacetime $\mathcal{M}$. Because we regard Equation (1) as a field equation, Equation (1) includes an implicit assumption of the existence of a point identification map, ${\mathcal{X}}_{ϵ}$:$\mathcal{M}\to {\mathcal{M}}_{ϵ}$: $p\in \mathcal{M}\mapsto “p”\in {\mathcal{M}}_{ϵ}$. This point identification map is called a “gauge choice” in general relativistic perturbation theories. This is the notion of the “second-kind gauge” from Sachs [39] and the important notion in our general relativistic gauge-invariant perturbation theory. Note that this second-kind gauge is different from the degree of freedom of the coordinate transformation in a single manifold. We call the degree of freedom of the coordinate transformation in a single manifold the “first-kind gauge” [29,35] to distinguish the second-kind gauge.

Perturbations are the comparison of the variable Q in the physical spacetime ${\mathcal{M}}_{ϵ}$ and its background value $Q_0$ in the background spacetime $\mathcal{M}$. To conduct this comparison, we use the pull-back ${\mathcal{X}}_{ϵ}^{*}$ of the above identification map ${\mathcal{X}}_{ϵ}$:$\mathcal{M}\to {\mathcal{M}}_{ϵ}$ and we evaluate the pulled-back variable ${\mathcal{X}}_{ϵ}^{*}Q$ in the background spacetime $\mathcal{M}$. Furthermore, in perturbation theories, the pulled-back variable ${\mathcal{X}}_{ϵ}^{*}Q$ in the background spacetime is expanded by the infinitesimal parameter $ϵ$ for the perturbation as follows:

$$\begin{array}{c}\hfill {\mathcal{X}}_{ϵ}^{*}Q=Q_0+ϵ{}_{\mathcal{X}}{}^{\left(1\right)}Q+O\left({ϵ}^2\right).\end{array}$$

(2)

We emphasize that Equation (2) is evaluated in the background spacetime $\mathcal{M}$. When we have two different gauge choices, ${\mathcal{X}}_{ϵ}$ and ${\mathcal{Y}}_{ϵ}$, “gauge-transformation” can be considered the change in the point identification map ${\mathcal{X}}_{ϵ}\to {\mathcal{Y}}_{ϵ}$. This gauge transformation is given by the diffeomorphism ${\Phi }_{ϵ}$$:=$ ${\left({\mathcal{X}}_{ϵ}\right)}^{-1}\circ {\mathcal{Y}}_{ϵ}$:$\mathcal{M}$ →$\mathcal{M}$ of the change in the point identification map. Actually, the diffeomorphism ${\Phi }_{ϵ}$ induces a pull-back from the representation ${\mathcal{X}}_{ϵ}^{*}{Q}_{ϵ}$ to the representation ${\mathcal{Y}}_{ϵ}^{*}{Q}_{ϵ}$ as ${\mathcal{Y}}_{ϵ}^{*}{Q}_{ϵ}={\Phi }_{ϵ}^{*}{\mathcal{X}}_{ϵ}^{*}{Q}_{ϵ}$. From general arguments of the Taylor expansion [40], the pull-back ${\Phi }_{ϵ}^{*}$ is expanded as

$$\begin{array}{ccc}\hfill {\mathcal{Y}}_{ϵ}^{*}{Q}_{ϵ}& =& {\mathcal{X}}_{ϵ}^{*}{Q}_{ϵ}+ϵ{\pounds }_{{\xi }_{\left(1\right)}}{\mathcal{X}}_{ϵ}^{*}{Q}_{ϵ}+O\left({ϵ}^2\right).\hfill \end{array}$$

(3)

Here, ${\xi }_{\left(1\right)}^a$ in Equation (3) is defined as the generator of ${\Phi }_{ϵ}$. From Equations (2) and (3), the gauge transformation for the first-order perturbation ${}^{\left(1\right)}Q$ is given by

$$\begin{array}{ccc}\hfill {}_{\mathcal{Y}}{}^{\left(1\right)}Q-{}_{\mathcal{Y}}{}^{\left(1\right)}Q& =& {\pounds }_{{\xi }_{\left(1\right)}}{Q}_0.\hfill \end{array}$$

(4)

In our gauge-invariant perturbation theory, we employ “order-by-order gauge invariance” as a concept of gauge invariance [34]. We call the kth-order perturbation ${}_{\mathcal{X}}{}^{\left(k\right)}Q$ gauge-invariant if and only if

$$\begin{array}{c}\hfill {}_{\mathcal{X}}{}^{\left(k\right)}Q={}_{\mathcal{Y}}{}^{\left(k\right)}Q\end{array}$$

(5)

for any gauge choice ${\mathcal{X}}_{ϵ}$ and ${\mathcal{Y}}_{ϵ}$. This is the basic setup of our gauge-invariant perturbation theory.

Based on the above setup, we proposed a procedure to construct gauge-invariant variables for general relativistic higher-order perturbations [30,31]. In our proposal, we first expand the metric in the physical spacetime ${\mathcal{M}}_{ϵ}$. This metric in the physical spacetime is pulled back to the background spacetime $\mathcal{M}$ through the gauge choice ${\mathcal{X}}_{ϵ}$ as ${\mathcal{X}}_{ϵ}^{*}{\overline{g}}_{ab}$ and its expansion is given by

$$\begin{array}{ccc}\hfill {\mathcal{X}}_{ϵ}^{*}{\overline{g}}_{ab}& =& g_{ab}+ϵ{}_{\mathcal{X}}{h}_{ab}+O\left({ϵ}^2\right).\hfill \end{array}$$

(6)

We have to emphasize that Expression (6) depends entirely on the gauge choice ${\mathcal{X}}_{ϵ}$. However, henceforth, we do not explicitly express the index of the gauge choice ${\mathcal{X}}_{ϵ}$ in the expression if there is no possibility of confusion. Our proposal starts from the following conjecture [30,31] for the linear metric perturbation $h_{ab}$:

Conjecture 1. 

If the gauge-transformation rule for the perturbative, pulled-back tensor field $h_{ab}$ to the background spacetime $\mathcal{M}$ is given by ${}_{\mathcal{Y}}{h}_{ab}$−${}_{\mathcal{X}}{h}_{ab}$= ${\pounds }_{{\xi }_{\left(1\right)}}{g}_{ab}$ with the background metric $g_{ab}$, there then exists a tensor field, ${\mathcal{F}}_{ab}$, and a vector field, $Y^a$, such that $h_{ab}$ is decomposed as $h_{ab}$ $=:$ ${\mathcal{F}}_{ab}$ + ${\pounds }_Y{g}_{ab}$, where ${\mathcal{F}}_{ab}$ and $Y^a$ are transformed as ${}_{\mathcal{Y}}{\mathcal{F}}_{ab}$− ${}_{\mathcal{X}}{\mathcal{F}}_{ab}$= 0 and ${}_{\mathcal{Y}}{Y}^a$ −${}_{\mathcal{X}}{Y}^a$=${\xi }_{\left(1\right)}^a$ under the gauge transformation, respectively.

This is the crucial premise of our general framework of the general relativistic, higher-order, gauge-invariant perturbation theory. We call ${\mathcal{F}}_{ab}$ and $Y^a$ the “gauge-invariant” and “gauge-dependent” parts of $h_{ab}$, respectively.

The proof of Conjecture 1 is highly nontrivial [32], and it is found that the gauge-invariant variables are essentially non-local. Despite this non-triviality, once we accept Conjecture 1, we can decompose the linear perturbation of an arbitrary tensor field, ${}_{\mathcal{X}}{}^{\left(1\right)}Q$, whose gauge transformation is given by Equation (4), through the gauge-dependent part, $Y_a$, of the metric perturbation in Conjecture 1 as

$$\begin{array}{c}\hfill {}_{\mathcal{X}}{}^{\left(1\right)}Q={}^{\left(1\right)}\mathcal{Q}+{\pounds }_{{}_{\mathcal{X}}Y}{Q}_0.\end{array}$$

(7)

As examples, the linearized Einstein tensor ${}_{\mathcal{X}}{}^{\left(1\right)}G_a^b$ and the linear perturbation of the energy–momentum tensor ${}_{\mathcal{X}}{}^{\left(1\right)}T_a^b$ are also decomposed as

$$\begin{array}{c}\hfill {}_{\mathcal{X}}{}^{\left(1\right)}{G}_a^b={}^{\left(1\right)}{\mathcal{G}}_a^b\left[\mathcal{F}\right]+{\pounds }_{{}_{\mathcal{X}}Y}{G}_a^b,{}_{\mathcal{X}}{}^{\left(1\right)}{T}_a^b={}^{\left(1\right)}{\mathcal{T}}_a^b+{\pounds }_{{}_{\mathcal{X}}Y}{T}_a^b,\end{array}$$

(8)

where $G_{ab}$ and $T_{ab}$ are the background values of the Einstein tensor and the energy–momentum tensor, respectively. Then, using the background Einstein equation $G_a^b=8\pi T_a^b$, the linearized Einstein equation ${}_{\mathcal{X}}{}^{\left(1\right)}{G}_{ab}=8\pi {}_{\mathcal{X}}{}^{\left(1\right)}{T}_{ab}$ is automatically given in the gauge-invariant form

$$\begin{array}{c}\hfill {}^{\left(1\right)}{\mathcal{G}}_a^b\left[\mathcal{F}\right]=8\pi {}^{\left(1\right)}{\mathcal{T}}_a^b\end{array}$$

(9)

even if the background Einstein equation is nontrivial. We also note that, in the case of a vacuum background case, i.e., $G_a^b=8\pi T_a^b=0$, Equation (8) shows that the linear perturbations of the Einstein tensor and the energy–momentum tensor are automatically gauge-invariant in the sense of the second kind.

We can also derive the perturbation of the divergence of ${\overline{\nabla }}_a{\overline{T}}_b^a$ of the second-rank tensor ${\overline{T}}_b^a$ in $({\mathcal{M}}_{\mathrm{ph}},{\overline{g}}_{ab})$. Through the gauge choice ${\mathcal{X}}_{ϵ}$, ${\overline{T}}_b^a$ is pulled back to ${\mathcal{X}}_{ϵ}^{*}{\overline{T}}_b^a$ in the background spacetime $(\mathcal{M},g_{ab})$, and the covariant derivative operator ${\overline{\nabla }}_a$ in $({\mathcal{M}}_{\mathrm{ph}},{\overline{g}}_{ab})$ is pulled back to the derivative operator ${\overline{\nabla }}_a(={\mathcal{X}}_{ϵ}^{*}{\overline{\nabla }}_a{\left({\mathcal{X}}_{ϵ}^{-1}\right)}^{*})$ in $(\mathcal{M},g_{ab})$. Note that the derivative ${\overline{\nabla }}_a$ is the covariant derivative associated with the metric ${\mathcal{X}}_{ϵ}{\overline{g}}_{ab}$, whereas the derivative ${\nabla }_a$ in the background spacetime $(\mathcal{M},g_{ab})$ is the covariant derivative associated with the background metric $g_{ab}$. Bearing in mind the difference in these derivatives, the first-order perturbation of ${\overline{\nabla }}_a{\overline{T}}_b^a$ is given by

$$\begin{array}{c}\hfill {}^{\left(1\right)}\left({\overline{\nabla }}_a{\overline{T}}_b^a\right)={\nabla }_a{}^{\left(1\right)}{\mathcal{T}}_b^a+H_{ca}^a\left[\mathcal{F}\right]T_b^c-H_{ba}^c\left[\mathcal{F}\right]T_c^a+{\pounds }_Y{\nabla }_a{T}_b^a.\end{array}$$

(10)

The derivation of Formula (10) is given in ref. [31]. If the tensor field ${\overline{T}}_b^a$ is the Einstein tensor ${\overline{G}}_a^b$, Equation (10) yields the linear-order perturbation of the Bianchi identity. Furthermore, if the background Einstein tensor vanishes with $G_a^b=0$, we obtain the identity

$$\begin{array}{c}\hfill {\nabla }_a{}^{\left(1\right)}{\mathcal{G}}_b^a\left[\mathcal{F}\right]=0.\end{array}$$

(11)

By contrast, if the tensor field ${\overline{T}}_b^a$ is the energy–momentum tensor, Equation (10) yields the continuity equation of the energy–momentum tensor. Furthermore, if the background energy–momentum tensor vanishes with $T_{ab}=0$, we obtain the linearized version of the continuity equation for the linear perturbation of the energy–momentum tensor

$$\begin{array}{c}\hfill {\nabla }_a{}^{\left(1\right)}{\mathcal{T}}_b^a=0.\end{array}$$

(12)

We should note that the decomposition of the metric perturbation $h_{ab}$ into its gauge-invariant part ${\mathcal{F}}_{ab}$ and into its gauge-dependent part $Y^a$ is not unique [34,35]. As explained in the Part I paper [29], for example, the gauge-invariant part ${\mathcal{F}}_{ab}$ has six components and we can create the gauge-invariant vector field $Z^a$ through these components of the gauge-invariant metric perturbation ${\mathcal{F}}_{ab}$ such that the gauge transformation of the vector field $Z^a$ is given by ${}_{\mathcal{Y}}{Z}^a$−${}_{\mathcal{X}}{Z}^a$= 0. Using the gauge-invariant vector field $Z^a$, the original metric perturbation can be expressed as $h_{ab}$ = ${\mathcal{F}}_{ab}$ − ${\pounds }_Z{g}_{ab}$ +${\pounds }_{Z+Y}{g}_{ab}$ $=:$ ${\mathcal{H}}_{ab}$+${\pounds }_X{g}_{ab}$. The tensor field ${\mathcal{H}}_{ab}:={\mathcal{F}}_{ab}-{\pounds }_Z{g}_{ab}$ is also regarded as the gauge-invariant part of the perturbation $h_{ab}$ because ${}_{\mathcal{Y}}{\mathcal{H}}_{ab}-{}_{\mathcal{X}}{\mathcal{H}}_{ab}=0$. Similarly, the vector field $X^a:=Z^a+Y^a$ is also regarded as the gauge-dependent part of the perturbation $h_{ab}$ because ${}_{\mathcal{Y}}{X}^a$−${}_{\mathcal{X}}{X}^a$= ${\xi }_{\left(1\right)}^a$. This non-uniqueness appears in the solutions derived in Section 4 and Section 5, as in the case of the $l=1$ odd-mode perturbative solutions in the Part I paper [29]. The non-uniqueness of the gauge-invariant variable can be regarded as the first-kind gauge, as explained in the Part I paper [29], i.e., the degree of freedom of the choice of the coordinate system in the physical spacetime ${\mathcal{M}}_{ϵ}$. Since we often use the first-kind gauge when we predict and interpret the measurement results of observations and experiments, we should regard that this non-uniqueness of the gauge-invariant variable of the second kind may have some physical meaning [29].

2.2. Linear Perturbations on Spherically Symmetric Background

Here, we consider the 2 + 2 formulation of the perturbation of a spherically symmetric background spacetime, which was originally proposed by Gerlach and Sengupta [19,20,21,22]. Spherically symmetric spacetimes are characterized by the direct product $\mathcal{M}={\mathcal{M}}_1\times S^2$, and their metric is

$$\begin{array}{ccc}\hfill g_{ab}& =& y_{ab}+r^2{\gamma }_{ab},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill y_{ab}& =& y_{AB}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b,{\gamma }_{ab}={\gamma }_{pq}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b,\hfill \end{array}$$

(14)

where $x^A=(t,r)$; $x^p=(\theta ,\varphi )$; and ${\gamma }_{pq}$ is the metric in the unit sphere. In the Schwarzschild spacetime, the metric (13) is given by

$$\begin{array}{ccc}\hfill y_{ab}& =& -f{\left(dt\right)}_a{\left(dt\right)}_b+f^{-1}{\left(dr\right)}_a{\left(dr\right)}_b,f=1-{\displaystyle \frac{2M}{r}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\gamma }_{ab}& =& {\left(d\theta \right)}_a{\left(d\theta \right)}_b+{sin}^2\theta {\left(d\varphi \right)}_a{\left(d\varphi \right)}_b={\theta }_a{\theta }_b+{\varphi }_a{\varphi }_b,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\theta }_a& =& {\left(d\theta \right)}_a,{\varphi }_a=sin\theta {\left(d\varphi \right)}_a.\hfill \end{array}$$

(17)

For this background spacetime, $(\mathcal{M},g_{ab})$, the components of the metric perturbation are given by

$$\begin{array}{c}\hfill h_{ab}=h_{AB}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b+2h_{Ap}{\left(d{x}^A\right)}_{(a}{\left(d{x}^p\right)}_{b)}+h_{pq}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\end{array}$$

(18)

Here, we note that the components $h_{AB}$, $h_{Ap}$, and $h_{pq}$ are regarded as components of the scalar, vector, and tensor in $S^2$, respectively. In many studies, these components are decomposed through the decomposition from [41,42,43] using the spherical harmonics $S=Y_{lm}$ as follows:

$$\begin{array}{ccc}\hfill h_{AB}& =& \sum _{l,m}{\tilde{h}}_{AB}S,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill h_{Ap}& =& r\sum _{l,m}\left[{\tilde{h}}_{\left(e1\right)A}{\widehat{D}}_{p}S+{\tilde{h}}_{\left(o1\right)A}{ϵ}_{pq}{\widehat{D}}^{q}S\right],\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill h_{pq}& =& r^2\sum _{l,m}\left[{\displaystyle \frac{1}{2}}{\gamma }_{pq}{\tilde{h}}_{\left(e0\right)}S+{\tilde{h}}_{\left(e2\right)}\left({\widehat{D}}_p{\widehat{D}}_q-{\displaystyle \frac{1}{2}}{\gamma }_{pq}{\widehat{D}}^r{\widehat{D}}_r\right)S+2{\tilde{h}}_{\left(o2\right)}{ϵ}_{r(p}{\widehat{D}}_{q)}{\widehat{D}}^{r}S\right],\hfill \end{array}$$

(21)

where ${\widehat{D}}_p$ is the covariant derivative associated with the metric ${\gamma }_{pq}$ in $S^2$; ${\widehat{D}}^p={\gamma }^{pq}{\widehat{D}}_q$; and ${ϵ}_{pq}={ϵ}_{\left[pq\right]}=2{\theta }_{[p}{\varphi }_{q]}$ is the totally antisymmetric tensor in $S^2$.

If we employ the decomposition (19)–(21) with $S=Y_{lm}$ with the metric perturbation $h_{ab}$, special treatments for $l=0,1$ modes are required [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. This is due to the fact that the set of harmonic functions

$$\begin{array}{c}\hfill \left\{S,{\widehat{D}}_{p}S,{ϵ}_{pq}{\widehat{D}}^{q}S,{\displaystyle \frac{1}{2}}{\gamma }_{pq}S,\left({\widehat{D}}_p{\widehat{D}}_q-{\displaystyle \frac{1}{2}}{\gamma }_{pq}\right)S,2{ϵ}_{r(p}{\widehat{D}}_{q)}{\widehat{D}}^{r}S\right\}\end{array}$$

(22)

loses its linear independence for $l=0,1$ modes. Actually, the inverse relation of the decomposition Formulae (19)–(21) requires the Green functions of the derivative operators $\widehat{\Delta }:={\widehat{D}}^r{\widehat{D}}_r$ and $\widehat{\Delta }+2:={\widehat{D}}^r{\widehat{D}}_r+2$. Since the eigenmodes of these operators are $l=0$ and $l=1$, respectively, this is the reason why the special treatments for these modes are required. However, these special treatments become an obstacle when we develop the higher-order perturbation theory [44].

To resolve this $l=0,1$-mode problem, in the Part I paper [28,29], we chose the scalar function S as

$$\begin{array}{c}\hfill S=S_{\delta }=\left\{\begin{array}{ccccc}{Y}_{lm}& & \mathrm{for}& & l\ge 2;\\ k_{(\widehat{\Delta }+2)m}& & \mathrm{for}& & l=1;\\ k_{\left(\widehat{\Delta }\right)}& & \mathrm{for}& & l=0.\end{array}\right.\end{array}$$

(23)

and used the decomposition Formulae (19)–(21), where the functions $k_{\left(\widehat{\Delta }\right)}$ and $k_{(\widehat{\Delta }+2)}$ satisfy the equation

$$\begin{array}{c}\hfill \widehat{\Delta }{k}_{\left(\widehat{\Delta }\right)}=0,\left(\widehat{\Delta }+2\right)k_{(\widehat{\Delta }+2)}=0,\end{array}$$

(24)

respectively. As shown in the Part I paper [29], the set of harmonic functions (22) becomes the linear-independent set including $l=0,1$ modes if we employ

$$\begin{array}{ccc}\hfill k_{\left(\widehat{\Delta }\right)}& =& 1+\delta ln{\left({\displaystyle \frac{1-cos\theta }{1+cos\theta }}\right)}^{1/2},\delta \in \mathbb{R},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill k_{(\widehat{\Delta }+2,m=0)}& =& cos\theta +\delta \left({\displaystyle \frac{1}{2}}cos\theta ln{\displaystyle \frac{1+cos\theta }{1-cos\theta }}-1\right),\delta \in \mathbb{R},\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill k_{(\widehat{\Delta }+2,m=\pm 1)}& =& \left[sin\theta +\delta \left(+{\displaystyle \frac{1}{2}}sin\theta ln{\displaystyle \frac{1+cos\theta }{1-cos\theta }}+cot\theta \right)\right]e^{\pm i\varphi }.\hfill \end{array}$$

(27)

These choices guarantee one-to-one correspondence between the components $\{h_{AB},$$h_{Ap},$$h_{pq}\}$ and the mode coefficients $\{{\tilde{h}}_{AB},$${\tilde{h}}_{\left(e1\right)A},$${\tilde{h}}_{\left(o1\right)A},$ ${\tilde{h}}_{\left(e0\right)},$${\tilde{h}}_{\left(e2\right)},$${\tilde{h}}_{\left(o2\right)}\}$ with the decomposition Formulae (19)–(21), owing to the linear independence of the set of the harmonic function (22) when $\delta \ne 0$. Then, the mode-by-mode analysis including $l=0,1$ is possible when $\delta \ne 0$. However, the mode functions (25)–(27) are singular if $\delta \ne 0$. When $\delta =0$, we have $k_{\left(\widehat{\Delta }\right)}\propto Y_{00}$ and $k_{(\widehat{\Delta }+2)m}\propto Y_{1m}$. Using the above harmonics function $S_{\delta }$ in Equation (23), we propose the following strategy:

Proposal 1. 

We decompose the metric perturbation $h_{ab}$ on the background spacetime with the metric (13)–(16) through Equations (19)–(21) with the harmonic function $S_{\delta }$ given by Equation (23). Then, Equations (19)–(21) become invertible including $l=0,1$ modes. After deriving the mode-by-mode field equations such as the linearized Einstein equations by using the harmonic function $S_{\delta }$, we choose $\delta =0$ as the regular boundary condition for solutions when we solve these field equations.

As shown in the Part I paper [29], once we accept Proposal 1, Conjecture 1 becomes the following statement:

Theorem 1. 

If the gauge-transformation rule for a perturbative tensor field, $h_{ab}$, pulled back to the background spacetime $\mathcal{M}$ is given by ${}_{\mathcal{Y}}{h}_{ab}$−${}_{\mathcal{X}}{h}_{ab}$= ${\pounds }_{{\xi }_{\left(1\right)}}{g}_{ab}$ with the background metric $g_{ab}$ with spherical symmetry, there then exists a tensor field, ${\mathcal{F}}_{ab}$, and a vector field, $Y^a$, such that $h_{ab}$ is decomposed as $h_{ab}$$=:$${\mathcal{F}}_{ab}$+ ${\pounds }_Y{g}_{ab}$, where ${\mathcal{F}}_{ab}$ and $Y^a$ are transformed as ${}_{\mathcal{Y}}{\mathcal{F}}_{ab}$− ${}_{\mathcal{X}}{\mathcal{F}}_{ab}$= 0 and ${}_{\mathcal{Y}}{Y}^a$ −${}_{\mathcal{X}}{Y}^a$=${\xi }_{\left(1\right)}^a$ under the gauge transformation, respectively.

Furthermore, including $l=0,1$ modes, the components of the gauge-invariant part ${\mathcal{F}}_{ab}$ of the metric perturbation $h_{ab}$ is given by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{AB}& =& \sum _{l,m}{\tilde{F}}_{AB}{S}_{\delta },\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{Ap}& =& r\sum _{l,m}{\tilde{F}}_A{ϵ}_{pq}{\widehat{D}}^q{S}_{\delta },{\widehat{D}}^p{\mathcal{F}}_{Ap}=0,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{pq}& =& {\displaystyle \frac{1}{2}}{\gamma }_{pq}{r}^2\sum _{l,m}\tilde{F}{S}_{\delta }.\hfill \end{array}$$

(30)

Thus, we have resolved the zero-mode problem in perturbations on the spherically symmetric background spacetime. Here, we note that the variables in Equations (28)–(30) are the gauge-invariant variables associated with the Regge–Wheeler gauge, although the gauge-invariant variables are not unique, as emphasized above. Actually, the Regge–Wheeler gauge is realized by the choice of the gauge-dependent variable $Y^a=0$ in Theorem 1. Through the gauge-invariant variables (28)–(30), we derived the linearized Einstein equations in the Part I paper [29].

2.3. Even-Mode Linearized Einstein Equations

Since odd-mode perturbations are discussed in the Part I paper [29], we consider the linearized even-mode Einstein equations for the Schwarzschild background spacetime in this paper. The Schwarzschild spacetime is vacuum solution to the Einstein equation $G_a^b=0=T_a^b$. Since we proved Theorem 1 in the spherically symmetric background spacetime, the linearized Einstein equation is given in the following gauge-invariant form as Equation (9). To evaluate the Einstein Equation (9) through mode-by-mode analysis including $l=0,1$, we also consider the mode decomposition of the gauge-invariant part ${}^{\left(1\right)}{\mathcal{T}}_{ab}$$:=$ $g_{bc}{}^{\left(1\right)}{\mathcal{T}}_a^c$ of the linear perturbation of the energy–momentum tensor through the set (22) of the harmonics as follows:

$$\begin{array}{ccc}\hfill {}^{\left(1\right)}{\mathcal{T}}_{ab}& =& \sum _{l,m}{\tilde{T}}_{AB}{S}_{\delta }{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b+r\sum _{l,m}\left\{{\tilde{T}}_{\left(e1\right)A}{\widehat{D}}_p{S}_{\delta }+{\tilde{T}}_{\left(o1\right)A}{ϵ}_{pr}{\widehat{D}}^r{S}_{\delta }\right\}2{\left(d{x}^A\right)}_{(a}{\left(d{x}^p\right)}_{b)}\hfill \end{array}$$

$$\begin{array}{ccc}& & +r^2\sum _{l,m}\left\{{\tilde{T}}_{\left(e0\right)}{\displaystyle \frac{1}{2}}{\gamma }_{pq}{S}_{\delta }+{\tilde{T}}_{\left(e2\right)}\left({\widehat{D}}_p{\widehat{D}}_q{S}_{\delta }-{\displaystyle \frac{1}{2}}{\gamma }_{pq}{\widehat{D}}_r{\widehat{D}}^r{S}_{\delta }\right)\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & \left.+{\tilde{T}}_{\left(o2\right)}2{ϵ}_{r(p}{\widehat{D}}_{q)}{\widehat{D}}^r{S}_{\delta }\right\}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\hfill \end{array}$$

(31)

Since the background spacetime is a vacuum, the pulled-back divergence of the energy–momentum tensor is given by Equation (12) and the even-mode components of Equation (12), in terms of the mode coefficients defined by Equation (31), are given by

$$\begin{array}{ccc}& & {\overline{D}}^C{\tilde{T}}_C^B+{\displaystyle \frac{2}{r}}\left({\overline{D}}^{D}r\right){\tilde{T}}_D^B-{\displaystyle \frac{1}{r}}l(l+1){\tilde{T}}_{\left(e1\right)}^B-{\displaystyle \frac{1}{r}}\left({\overline{D}}^{B}r\right){\tilde{T}}_{\left(e0\right)}=0,\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & {\overline{D}}^C{\tilde{T}}_{\left(e1\right)C}+{\displaystyle \frac{3}{r}}\left({\overline{D}}^{C}r\right){\tilde{T}}_{\left(e1\right)C}+{\displaystyle \frac{1}{2r}}{\tilde{T}}_{\left(e0\right)}-{\displaystyle \frac{1}{2r}}(l-1)(l+2){\tilde{T}}_{\left(e2\right)}=0.\hfill \end{array}$$

(33)

Owing to the linear independence of the set (22) of harmonics, we can evaluate the gauge-invariant linearized Einstein Equation (9) through mode-by-mode analyses including $l=0,1$ modes. As summarized in the Part I paper [29], the traceless even part of the $(p,q)$-component of the linearized Einstein Equation (9) is given by

$$\begin{array}{c}\hfill {\tilde{F}}_D^D=-16\pi r^2{\tilde{T}}_{\left(e2\right)}.\end{array}$$

(34)

Using this equation, the even part of the $(A,q)$-component, and equivalently the $(p,B)$-component, of the linearized Einstein Equation (9) yields

$$\begin{array}{c}\hfill {\overline{D}}^D{\tilde{\mathbb{F}}}_{AD}-{\displaystyle \frac{1}{2}}{\overline{D}}_A\tilde{F}=16\pi \left[r{\tilde{T}}_{\left(e1\right)A}-{\displaystyle \frac{1}{2}}{r}^2{\overline{D}}_A{\tilde{T}}_{\left(e2\right)}\right]=:16\pi S_{\left(ec\right)A}\end{array}$$

(35)

through the definition of the traceless part ${\tilde{\mathbb{F}}}_{AB}$ of the variable ${\tilde{F}}_{AB}$:

$$\begin{array}{c}\hfill {\tilde{\mathbb{F}}}_{AB}:={\tilde{F}}_{AB}-{\displaystyle \frac{1}{2}}{y}_{AB}{\tilde{F}}_C^C.\end{array}$$

(36)

Using Equations (34) and (35) and the background Einstein equations, the trace part of the $(p,q)$-component of the linearized Einstein Equation (9) yields Equation (33).

Finally, through Equations (34) and (35) and the background Einstein equations, the trace part of the $(A,B)$-component of the linearized Einstein Equation (9) is given by

$$\begin{array}{ccc}& & \left({\overline{D}}_D{\overline{D}}^D+{\displaystyle \frac{2}{r}}\left({\overline{D}}^{D}r\right){\overline{D}}_D-{\displaystyle \frac{(l-1)(l+2)}{r^2}}\right)\tilde{F}-{\displaystyle \frac{4}{r^2}}\left({\overline{D}}_{C}r\right)\left({\overline{D}}_{D}r\right){\tilde{\mathbb{F}}}^{CD}=16\pi S_{\left(F\right)},\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill S_{\left(F\right)}& :=& {\tilde{T}}_C^C+4\left({\overline{D}}_{D}r\right){\tilde{T}}_{\left(e1\right)}^D-2r\left({\overline{D}}_{D}r\right){\overline{D}}^D{\tilde{T}}_{\left(e2\right)}-(l(l+1)+2){\tilde{T}}_{\left(e2\right)}.\hfill \end{array}$$

(38)

On the other hand, the traceless part of the $(A,B)$-component of the linearized Einstein Equation (9) is given by

$$\begin{array}{ccc}& & \left[-{\overline{D}}_D{\overline{D}}^D-{\displaystyle \frac{2}{r}}\left({\overline{D}}_{D}r\right){\overline{D}}^D+{\displaystyle \frac{4}{r}}\left({\overline{D}}^D{\overline{D}}_{D}r\right)+{\displaystyle \frac{l(l+1)}{r^2}}\right]{\tilde{\mathbb{F}}}_{AB}\hfill \end{array}$$

$$\begin{array}{cc}& +{\displaystyle \frac{4}{r}}\left({\overline{D}}^{D}r\right){\overline{D}}_{(A}{\tilde{\mathbb{F}}}_{B)D}-{\displaystyle \frac{2}{r}}\left({\overline{D}}_{(A}r\right){\overline{D}}_{B)}\tilde{F}\hfill \end{array}$$

$$\begin{array}{cc}=& 16\pi S_{\left(\mathbb{F}\right)AB},\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill S_{\left(\mathbb{F}\right)AB}& :=& T_{AB}-{\displaystyle \frac{1}{2}}{y}_{AB}{T}_C^C-2\left({\overline{D}}_{(A}\left(r{\tilde{T}}_{\left(e1\right)B)}\right)-{\displaystyle \frac{1}{2}}{y}_{AB}{\overline{D}}^D\left(r{\tilde{T}}_{\left(e1\right)D}\right)\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +2\left(\left({\overline{D}}_{(A}r\right){\overline{D}}_{B)}-{\displaystyle \frac{1}{2}}{y}_{AB}\left({\overline{D}}^{D}r\right){\overline{D}}_D\right)\left(r{\tilde{T}}_{\left(e2\right)}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +r\left({\overline{D}}_A{\overline{D}}_B-{\displaystyle \frac{1}{2}}{y}_{AB}{\overline{D}}^D{\overline{D}}_D\right)\left(r{\tilde{T}}_{\left(e2\right)}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +2\left(\left({\overline{D}}_{A}r\right)\left({\overline{D}}_{B}r\right)-{\displaystyle \frac{1}{2}}{y}_{AB}\left({\overline{D}}^{C}r\right)\left({\overline{D}}_{C}r\right)\right){\tilde{T}}_{\left(e2\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & +2y_{AB}\left({\overline{D}}^{C}r\right){\tilde{T}}_{\left(e1\right)C}-r{y}_{AB}\left({\overline{D}}^{C}r\right){\overline{D}}_C{\tilde{T}}_{\left(e2\right)}.\hfill \end{array}$$

(40)

Equations (34), (35), (37), and (39) are all independent linearized Einstein equation for even-mode perturbations. These equations are coupled equations for the variables ${\tilde{F}}_C^C$, F, and ${\tilde{\mathbb{F}}}_{AB}$ and the energy–momentum tensor for the matter field. When we solve these equations, we have to take into account the continuity Equations (32) and (33) for the matter fields. We note that these equations are valid not only for $l\ge 2$ modes but also for $l=0,1$ modes in our formulation. For $l\ge 2$ modes, we can derive the Zerilli equation, while we can derive formal solutions for $l=0,1$ modes. The derivations of these formal solutions for $l=0,1$ modes are the main ingredients of this paper.

3. Component Treatment of Even-Mode Linearized Einstein Equations

To summarize the even-mode Einstein equations, we consider the static chart of $y_{AB}$ as Equation (15). In this chart, the components of the Christoffel symbol ${\overline{\Gamma }}_{AB}^C$ associated with the covariant derivative ${\overline{D}}_A$ are summarized as

$$\begin{array}{c}\hfill {\overline{\Gamma }}_{tt}^t=0,{\overline{\Gamma }}_{tr}^t={\displaystyle \frac{f^{\prime }}{2f}},{\overline{\Gamma }}_{rr}^t=0,{\overline{\Gamma }}_{tt}^r={\displaystyle \frac{f{f}^{\prime }}{2}},{\overline{\Gamma }}_{tr}^r=0,{\overline{\Gamma }}_{rr}^r=-{\displaystyle \frac{f^{\prime }}{2f}},\end{array}$$

(41)

where $f^{\prime }:={\partial }_{r}f$.

First, Equation (34) is a direct consequence of the even-mode Einstein equation. Here, we introduce the components $X_{\left(e\right)}$ and $Y_{\left(e\right)}$ of the traceless variable ${\tilde{\mathbb{F}}}_{AB}$ by

$$\begin{array}{c}\hfill {\tilde{\mathbb{F}}}_{AB}=:X_{\left(e\right)}\left\{-f{\left(dt\right)}_A{\left(dt\right)}_B-f^{-1}{\left(dr\right)}_A{\left(dr\right)}_B\right\}+2Y_{\left(e\right)}{\left(dt\right)}_{(A}{\left(dr\right)}_{B)}.\end{array}$$

(42)

Through the components $X_{\left(e\right)}$ and $Y_{\left(e\right)}$, the t- and r- components of Equation (35) are given by

$$\begin{array}{ccc}& & {\partial }_t{X}_{\left(e\right)}+f{\partial }_r{Y}_{\left(e\right)}+f^{\prime }{Y}_{\left(e\right)}-{\displaystyle \frac{1}{2}}{\partial }_t\tilde{F}=16\pi S_{\left(ec\right)t},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{f}}{\partial }_t{Y}_{\left(e\right)}+{\partial }_r{X}_{\left(e\right)}+{\displaystyle \frac{f^{\prime }}{f}}{X}_{\left(e\right)}+{\displaystyle \frac{1}{2}}{\partial }_r\tilde{F}=-16\pi S_{\left(ec\right)r}.\hfill \end{array}$$

(44)

The source term $S_{\left(ec\right)A}$ is defined by

$$\begin{array}{c}\hfill S_{\left(ec\right)A}:=r{\tilde{T}}_{\left(e1\right)A}-{\displaystyle \frac{1}{2}}{r}^2{\overline{D}}_A{\tilde{T}}_{\left(e2\right)}.\end{array}$$

(45)

Furthermore, the evolution Equation (37) for the variable $\tilde{F}$ is given by

$$\begin{array}{c}\hfill -{\partial }_t^2\tilde{F}+f{\partial }_r\left(f{\partial }_r\tilde{F}\right)+{\displaystyle \frac{2}{r}}{f}^2{\partial }_r\tilde{F}-{\displaystyle \frac{(l-1)(l+2)}{r^2}}f\tilde{F}+{\displaystyle \frac{4}{r^2}}{f}^2{X}_{\left(e\right)}=16\pi Gf{S}_{\left(F\right)}.\end{array}$$

(46)

The source term $S_{\left(F\right)}$ is defined by

$$\begin{array}{ccc}\hfill S_{\left(F\right)}& :=& {\tilde{T}}_E^E+4\left({\overline{D}}_{D}r\right){\tilde{T}}_{\left(e1\right)}^D-2r\left({\overline{D}}_{D}r\right){\overline{D}}^D{\tilde{T}}_{\left(e2\right)}-(l(l+1)+2){\tilde{T}}_{\left(e2\right)}\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& =& -{\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}+4f{\tilde{T}}_{\left(e1\right)r}-2rf{\partial }_r{\tilde{T}}_{\left(e2\right)}-(l(l+1)+2){\tilde{T}}_{\left(e2\right)}.\hfill \end{array}$$

(48)

The component expression of Equation (39) with the constraints (43) and (44) is given by

$$\begin{array}{ccc}& & {\partial }_t^2{X}_{\left(e\right)}-f{\partial }_r\left(f{\partial }_r{X}_{\left(e\right)}\right)-{\displaystyle \frac{2(1-2f)f}{r}}{\partial }_r{X}_{\left(e\right)}-{\displaystyle \frac{(1-f)(1-5f)-l(l+1)f}{r^2}}{X}_e\hfill \\ & & -{\displaystyle \frac{(1-3f)f}{r}}{\partial }_r\tilde{F}\hfill \\ & =& -16\pi \left(S_{\left(\mathbb{F}\right)tt}+{\displaystyle \frac{2f(3f-1)}{r}}{S}_{\left(ec\right)r}\right),\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & {\partial }_t^2{Y}_{\left(e\right)}-f{\partial }_r\left(f{\partial }_r{Y}_{\left(e\right)}\right)-{\displaystyle \frac{2(1-2f)f}{r}}{\partial }_r{Y}_{\left(e\right)}-{\displaystyle \frac{(1-f)(1-5f)-l(l+1)f}{r^2}}{Y}_{\left(e\right)}\hfill \\ & & +{\displaystyle \frac{1-3f}{r}}{\partial }_t\tilde{F}\hfill \\ & =& 16\pi \left(f{S}_{\left(\mathbb{F}\right)tr}-{\displaystyle \frac{2(1-2f)}{r}}{S}_{\left(ec\right)t}\right).\hfill \end{array}$$

(50)

Here, we note that the $\left(rr\right)$-component of Equation (39) with the constraint (44) is equivalent to Equation (49). The source terms $S_{\left(\mathbb{F}\right)tt}$ and $S_{\left(\mathbb{F}\right)tr}$ in Equations (49) and (50) are given by

$$\begin{array}{ccc}\hfill S_{\left(\mathbb{F}\right)tt}& =& {\displaystyle \frac{1}{2}}\left({\tilde{T}}_{tt}+f^2{\tilde{T}}_{rr}\right)-r{\partial }_t{\tilde{T}}_{\left(e1\right)t}-3f^2{\tilde{T}}_{\left(e1\right)r}-r{f}^2{\partial }_r{\tilde{T}}_{\left(e1\right)r}\hfill \\ & & +{\displaystyle \frac{r^2}{2}}{\partial }_t^2{\tilde{T}}_{\left(e2\right)}+{\displaystyle \frac{r^2}{2}}{f}^2{\partial }_r^2{\tilde{T}}_{\left(e2\right)}+3r{f}^2{\partial }_r{\tilde{T}}_{\left(e2\right)}+2f^2{\tilde{T}}_{\left(e2\right)},\hfill \end{array}$$

(51)

$$\begin{array}{ccc}\hfill S_{\left(\mathbb{F}\right)tr}& =& {\tilde{T}}_{tr}-r{\partial }_t{\tilde{T}}_{\left(e1\right)r}-r{\partial }_r{\tilde{T}}_{\left(e1\right)t}-{\tilde{T}}_{\left(e1\right)t}+{\displaystyle \frac{1-f}{f}}{\tilde{T}}_{\left(e1\right)t}\hfill \\ & & +r^2{\partial }_t{\partial }_r{\tilde{T}}_{\left(e2\right)}+2r{\partial }_t{\tilde{T}}_{\left(e2\right)}-{\displaystyle \frac{r(1-f)}{2f}}{\partial }_t{\tilde{T}}_{\left(e2\right)}.\hfill \end{array}$$

(52)

The components of Equation (32) are given by

$$\begin{array}{ccc}& & -{\partial }_t{\tilde{T}}_{tt}+f^2{\partial }_r{\tilde{T}}_{rt}+{\displaystyle \frac{(1+f)f}{r}}{\tilde{T}}_{rt}-{\displaystyle \frac{f}{r}}l(l+1){\tilde{T}}_{\left(e1\right)t}=0,\hfill \end{array}$$

(53)

$$\begin{array}{ccc}& & -{\partial }_t{\tilde{T}}_{tr}+{\displaystyle \frac{1-f}{2rf}}{\tilde{T}}_{tt}+f^2{\partial }_r{\tilde{T}}_{rr}+{\displaystyle \frac{(3+f)f}{2r}}{\tilde{T}}_{rr}-{\displaystyle \frac{f}{r}}l(l+1){\tilde{T}}_{\left(e1\right)r}-{\displaystyle \frac{f}{r}}{\tilde{T}}_{\left(e0\right)}=0,\hfill \end{array}$$

(54)

where Equation (53) is the t-component and Equation (54) is the r-component, respectively. Furthermore, Equation (33) is given by

$$\begin{array}{c}\hfill -{\partial }_t{\tilde{T}}_{\left(e1\right)t}+f^2{\partial }_r{\tilde{T}}_{\left(e1\right)r}+{\displaystyle \frac{(1+2f)f}{r}}{\tilde{T}}_{\left(e1\right)r}+{\displaystyle \frac{f}{2r}}{\tilde{T}}_{\left(e0\right)}-{\displaystyle \frac{f}{2r}}(l-1)(l+2){\tilde{T}}_{\left(e2\right)}=0.\end{array}$$

(55)

From the time derivative of Equations (43) and (44), we obtain

$$\begin{array}{ccc}& & {\partial }_t^2{X}_{\left(e\right)}-f{\partial }_r\left(f{\partial }_r{X}_{\left(e\right)}\right)-2{\displaystyle \frac{1-f}{r}}f{\partial }_r{X}_{\left(e\right)}-{\displaystyle \frac{(1-3f)(1-f)}{r^2}}{X}_{\left(e\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{1}{2}}{\partial }_t^2\tilde{F}-{\displaystyle \frac{1}{2}}f{\partial }_r\left(f{\partial }_r\tilde{F}\right)-{\displaystyle \frac{1-f}{2r}}f{\partial }_r\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & -16\pi {\partial }_t{S}_{\left(ec\right)t}-16\pi {\partial }_r\left(f^2{S}_{\left(ec\right)r}\right)=0.\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & -{\partial }_t^2{Y}_{\left(e\right)}+f{\partial }_r\left(f{\partial }_r{Y}_{\left(e\right)}\right)+2f{\displaystyle \frac{1-f}{r}}{\partial }_r{Y}_{\left(e\right)}+{\displaystyle \frac{(1-3f)(1-f)}{r^2}}{Y}_{\left(e\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{1-f}{2r}}{\partial }_t\tilde{F}-f{\partial }_r{\partial }_t\tilde{F}-16\pi {\partial }_r\left(f{S}_{\left(ec\right)t}\right)-16\pi f{\partial }_t{S}_{\left(ec\right)r}=0.\hfill \end{array}$$

(57)

From Equations (49) and (56), we obtain

$$\begin{array}{ccc}& & 4f{\partial }_r\left(f{X}_{\left(e\right)}\right)+{\displaystyle \frac{2}{r}}l(l+1)\left(f{X}_{\left(e\right)}\right)+r{\partial }_t^2\tilde{F}+rf{\partial }_r\left(f{\partial }_r\tilde{F}\right)+(5f-1)f{\partial }_r\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& =& -32\pi r\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}\right].\hfill \end{array}$$

(58)

Furthermore, from Equations (50) and (57), we obtain

$$\begin{array}{ccc}& & 4f{\partial }_r\left(f{Y}_{\left(e\right)}\right)+{\displaystyle \frac{2}{r}}l(l+1)\left(f{Y}_{\left(e\right)}\right)-2rf{\partial }_r{\partial }_t\tilde{F}-(5f-1){\partial }_t\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& =& 32\pi r\left[f{S}_{\left(\mathbb{F}\right)tr}+f{\partial }_t{S}_{\left(ec\right)r}-{\displaystyle \frac{1-3f}{r}}{S}_{\left(ec\right)t}+f{\partial }_r{S}_{\left(ec\right)t}\right].\hfill \end{array}$$

(59)

Equations (43) and (59) yield

$$\begin{array}{ccc}\hfill l(l+1)Y_{\left(e\right)}& =& r{\partial }_t\left(2X_{\left(e\right)}+r{\partial }_r\tilde{F}\right)+{\displaystyle \frac{3f-1}{2f}}r{\partial }_t\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & +16\pi r^2\left[S_{\left(\mathbb{F}\right)tr}+{\partial }_t{S}_{\left(ec\right)r}-{\displaystyle \frac{1-f}{rf}}{S}_{\left(ec\right)t}+{\partial }_r{S}_{\left(ec\right)t}\right].\hfill \end{array}$$

(60)

Similarly, Equations (46) and (58) yield

$$\begin{array}{ccc}& & 4f{\partial }_r\left(f{X}_{\left(e\right)}\right)+{\displaystyle \frac{2}{r}}[l(l+1)+2f]\left(f{X}_{\left(e\right)}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +2f{\partial }_r\left(rf{\partial }_r\tilde{F}\right)+(5f-1)f{\partial }_r\tilde{F}-{\displaystyle \frac{(l-1)(l+2)}{r}}f\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& =& -32\pi r\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right].\hfill \end{array}$$

(61)

Thus, we may regard that the independent components of the Einstein equations for the even-mode perturbations are summarized as Equations (46), (49), (60) and (61).

As shown in many studies [13,14,15,16,17], it is well known that Equations (46), (49), and (61) are reduced to the single master equation for a single variable. We trace this procedure.

Equation (61) is an initial value constraint for the variables $(X_{\left(e\right)},\tilde{F})$, while Equations (46) and (56) are evolution equations. Equation (60) directly yields that the variable $Y_{\left(e\right)}$ is determined by the solution $(X_{\left(e\right)},\tilde{F})$ to Equations (46), (56), and (61), if $l\ne 0$. If the initial value constraint (61) is reduced to the equation of the variables ${\Phi }_{\left(e\right)}$ and $\tilde{F}$, we may expect that ${\Phi }_{\left(e\right)}$ linearly depends on $f{X}_{\left(e\right)}$, $\tilde{F}$, and $rf{\partial }_r\tilde{F}$. To show this, we introduce the variable ${\Phi }_{\left(e\right)}$ as

$$\begin{array}{c}\hfill \alpha {\Phi }_{\left(e\right)}:=f{X}_{\left(e\right)}+\beta \tilde{F}+\gamma rf{\partial }_r\tilde{F},\end{array}$$

(62)

where $\alpha$, $\beta$, and $\gamma$ may depend on r. Substituting Equation (62) into Equation (61), we obtain

$$\begin{array}{ccc}\hfill 0& =& -4rf{\alpha }^{\prime }{\Phi }_{\left(e\right)}-2[l(l+1)+2f]\alpha {\Phi }_{\left(e\right)}-4rf\alpha {\partial }_r{\Phi }_{\left(e\right)}+4\left(\gamma -{\displaystyle \frac{1}{2}}\right)rf{\partial }_r\left[rf{\partial }_r\tilde{F}\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & +\left[4\beta +4rf{\gamma }^{\prime }+2\left\{l(l+1)+2f\right\}\gamma -(5f-1)\right]rf{\partial }_r\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & +\left[4rf{\beta }^{\prime }+2\left\{l(l+1)+2f\right\}\beta +(l-1)(l+2)f\right]\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & -32\pi r^2\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right].\hfill \end{array}$$

(63)

Here, we choose

$$\begin{array}{c}\hfill \gamma ={\displaystyle \frac{1}{2}}\end{array}$$

(64)

to eliminate the term of the second derivative of $\tilde{F}$. Owing to this choice, we obtain

$$\begin{array}{ccc}\hfill 0& =& -4rf{\alpha }^{\prime }{\Phi }_{\left(e\right)}-2[l(l+1)+2f]\alpha {\Phi }_{\left(e\right)}-4rf\alpha {\partial }_r{\Phi }_{\left(e\right)}+[4\beta +\Lambda ]rf{\partial }_r\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & +\left[4rf{\beta }^{\prime }+2\left\{l(l+1)+2f\right\}\beta +(l-1)(l+2)f\right]\tilde{F}\hfill \end{array}$$

$$\begin{array}{ccc}& & -32\pi r^2\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right].\hfill \end{array}$$

(65)

Here, we choose $\beta$ as

$$\begin{array}{c}\hfill \beta =-{\displaystyle \frac{1}{4}}\Lambda :=-{\displaystyle \frac{1}{4}}[(l-1)(l+2)+3(1-f)],\Lambda :=(l-1)(l+2)+3(1-f)\end{array}$$

(66)

to eliminate the term of the first derivative of $\tilde{F}$. Due to this choice, we obtain

$$\begin{array}{ccc}\hfill l(l+1)\Lambda \tilde{F}& =& -8rf{\partial }_r\left(\alpha {\Phi }_{\left(e\right)}\right)-4[l(l+1)+2f]\alpha {\Phi }_{\left(e\right)}\hfill \\ & & -64\pi r^2\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right].\hfill \end{array}$$

(67)

This equation yields that the variable $\tilde{F}$ is determined by the single variable ${\Phi }_{\left(e\right)}$ and the source terms if $l\ne 0$ and if the coefficient $\alpha$ is determined.

At this moment, the variable ${\Phi }_{\left(e\right)}$ is determined up to its normalization, $\alpha$, as

$$\begin{array}{c}\hfill \alpha {\Phi }_{\left(e\right)}:=f{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}\Lambda \tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}.\end{array}$$

(68)

Eliminating $X_{\left(e\right)}$ in Equation (46) through Equation (68), we obtain

$$\begin{array}{c}\hfill -{\partial }_t^2\tilde{F}+f{\partial }_r\left(f{\partial }_r\tilde{F}\right)+{\displaystyle \frac{1}{r^2}}3(1-f)f\tilde{F}+{\displaystyle \frac{4}{r^2}}f\alpha {\Phi }_{\left(e\right)}=16\pi f{S}_{\left(F\right)}.\end{array}$$

(69)

Similarly, eliminating $X_{\left(e\right)}$ in Equation (49) through Equations (67)–(69), we obtain

$$\begin{array}{ccc}\hfill 0& =& -\alpha {\partial }_t^2{\Phi }_{\left(e\right)}+\alpha f{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]+2\alpha \left[{\displaystyle \frac{{\alpha }^{\prime }}{\alpha }}+{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{r\Lambda }}3(1-f)\right]f^2{\partial }_r{\Phi }_{\left(e\right)}\hfill \\ & & +\left[{\alpha }^{″}f+{\alpha }^{\prime }{\displaystyle \frac{1}{r}}(1+f)+{\alpha }^{\prime }{\displaystyle \frac{1}{r\Lambda }}3(1-f)2f+\alpha {\displaystyle \frac{3(1-f)\left\{l(l+1)+2f\right\}}{r^2\Lambda }}\right.\hfill \\ & & \left.-\alpha {\displaystyle \frac{(l-1)(l+2)+1+f}{r^2}}\right]f{\Phi }_{\left(e\right)}\hfill \\ & & +16\pi f{\displaystyle \frac{\Lambda +3(1-f)}{\Lambda }}\left[S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4f^2}{r}}{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right]\hfill \\ & & -16\pi f{S}_{\left(\mathbb{F}\right)tt}-32\pi {\displaystyle \frac{3f-1}{r}}{f}^2{S}_{\left(ec\right)r}-16\pi (-{\displaystyle \frac{1}{4}}\Lambda )f{S}_{\left(F\right)}-16\pi r{\displaystyle \frac{1}{2}}f{\partial }_r\left[f{S}_{\left(F\right)}\right].\hfill \end{array}$$

(70)

We determine $\alpha$ so that the terms proportional to ${\partial }_r{\Phi }_{\left(e\right)}$ vanish. Then, we obtain the equation for $\alpha$ as

$$\begin{array}{c}\hfill {\displaystyle \frac{{\alpha }^{\prime }}{\alpha }}+{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{r\Lambda }}3(1-f)=0.\end{array}$$

(71)

From this equation, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\alpha }}={\displaystyle \frac{Cr}{\Lambda }},\end{array}$$

(72)

where C is a constant of integration. In this paper, we choose $C=1$. Then, we obtain

$$\begin{array}{c}\hfill {\Phi }_{\left(e\right)}:={\displaystyle \frac{1}{\alpha }}\left[f{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}\Lambda \tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\right]={\displaystyle \frac{r}{\Lambda }}\left[f{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}\Lambda \tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\right].\end{array}$$

(73)

This is the Moncrief variable.

From Equation (72), we obtain

$$\begin{array}{c}\hfill {\alpha }^{\prime }=-{\displaystyle \frac{1}{r}}\alpha -{\displaystyle \frac{1}{\Lambda }}3{\displaystyle \frac{1-f}{r}}\alpha ,{\alpha }^{″}=+{\displaystyle \frac{2}{r^2}}\alpha +{\displaystyle \frac{12(1-f)}{\Lambda r^2}}\alpha .\end{array}$$

(74)

Then, using

$$\begin{array}{c}\hfill \mu :=(l-1)(l+2),\Lambda =\mu +3(1-f),\end{array}$$

(75)

Equation (70) is given by

$$\begin{array}{ccc}& & -{\displaystyle \frac{1}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]-{\displaystyle \frac{1}{r^2{\Lambda }^2}}\left[{\mu }^2[(\mu +2)+3(1-f)]+9{(1-f)}^2\left(\mu +1-f\right)\right]{\Phi }_{\left(e\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& =& 16\pi {\displaystyle \frac{r}{\Lambda }}\left[-{\partial }_t{S}_{\left(ec\right)t}-f^2{\partial }_r{S}_{\left(ec\right)r}+2f{\displaystyle \frac{f-1}{r}}{S}_{\left(ec\right)r}+{\displaystyle \frac{r}{2}}f{\partial }_r{S}_{\left(F\right)}+{\displaystyle \frac{1}{2}}{S}_{\left(F\right)}-{\displaystyle \frac{1}{4}}\Lambda S_{\left(F\right)}\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & \left.+{\displaystyle \frac{3(1-f)}{\Lambda }}\left[-S_{\left(\mathbb{F}\right)tt}-{\partial }_t{S}_{\left(ec\right)t}-f^2{\partial }_r{S}_{\left(ec\right)r}-{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}+{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right]\right].\hfill \end{array}$$

(76)

This is the Zerilli equation for the Moncrief variable (73).

Here, we summarize the equations for even-mode perturbations. We derive the definition of the Moncrief variable as Equation (73), i.e.,

$$\begin{array}{c}\hfill {\Phi }_{\left(e\right)}:={\displaystyle \frac{r}{\Lambda }}\left[f{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}\Lambda \tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\right],\end{array}$$

(77)

where $\Lambda$ is defined by Equation (66) and we use Equation (75). This definition of the variable ${\Phi }_{\left(e\right)}$ implies that if the variables ${\Phi }_{\left(e\right)}$ and $\tilde{F}$ are determined, the component $X_{\left(e\right)}$ of the metric perturbation is determined through the equation

$$\begin{array}{c}\hfill f{X}_{\left(e\right)}={\displaystyle \frac{1}{r}}\Lambda {\Phi }_{\left(e\right)}+{\displaystyle \frac{1}{4}}\Lambda \tilde{F}-{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}.\end{array}$$

(78)

As the initial value constraint for the variables $\tilde{F}$ and $Y_{\left(e\right)}$, we have Equations (60) and (67) as

$$\begin{array}{ccc}\hfill l(l+1)Y_{\left(e\right)}& =& r{\partial }_t\left(2X_{\left(e\right)}+r{\partial }_r\tilde{F}\right)+{\displaystyle \frac{3f-1}{2f}}r{\partial }_t\tilde{F}+16\pi r^2{S}_{\left(Y_{\left(e\right)}\right)},\hfill \end{array}$$

(79)

$$\begin{array}{ccc}\hfill l(l+1)\Lambda \tilde{F}& =& -8f\Lambda {\partial }_r{\Phi }_{\left(e\right)}+{\displaystyle \frac{4}{r}}\left[6f(1-f)-l(l+1)\Lambda \right]{\Phi }_{\left(e\right)}-64\pi r^2{S}_{(\Lambda \tilde{F})},\hfill \end{array}$$

(80)

where the source terms $S_{(\Lambda \tilde{F})}$ and $S_{\left(Y_{\left(e\right)}\right)}$ are given by

$$\begin{array}{ccc}\hfill S_{(\Lambda \tilde{F})}& :=& S_{\left(\mathbb{F}\right)tt}+{\partial }_t{S}_{\left(ec\right)t}+f^2{\partial }_r{S}_{\left(ec\right)r}+{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}-{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& =& {\tilde{T}}_{tt}+r{f}^2{\partial }_r{\tilde{T}}_{\left(e2\right)}+2f(f+1){\tilde{T}}_{\left(e2\right)}+{\displaystyle \frac{1}{2}}f(l-1)(l+2){\tilde{T}}_{\left(e2\right)},\hfill \end{array}$$

(82)

and

$$\begin{array}{ccc}\hfill S_{\left(Y_{\left(e\right)}\right)}& :=& S_{\left(\mathbb{F}\right)tr}+{\partial }_t{S}_{\left(ec\right)r}-{\displaystyle \frac{1-f}{rf}}{S}_{\left(ec\right)t}+{\partial }_r{S}_{\left(ec\right)t}\hfill \end{array}$$

(83)

$$\begin{array}{ccc}& =& {\tilde{T}}_{tr}+r{\partial }_t{\tilde{T}}_{\left(e2\right)}.\hfill \end{array}$$

(84)

Equation (80) implies that the variable $\tilde{F}$ of the metric perturbation is determined if the variable ${\Phi }_{\left(e\right)}$ and source term $S_{(\Lambda \tilde{F})}$ are specified. Equation (79) implies that the component $Y_{\left(e\right)}$ of the metric perturbation is determined if the variables $X_{\left(e\right)}$ and $\tilde{F}$ and the source term $S_{\left(Y_{\left(e\right)}\right)}$ are specified.

Thus, apart from the source terms, the component $\tilde{F}$ of the metric perturbation is determined through Equation (80) if the Moncrief variable ${\Phi }_{\left(e\right)}$ is specified. The component $X_{\left(e\right)}$ of the metric perturbation is determined through Equation (78) if the variables ${\Phi }_{\left(e\right)}$ and $\tilde{F}$ are specified. Finally, the component $Y_{\left(e\right)}$ of the metric perturbation is determined through Equation (79) if the variables $\tilde{F}$ and $X_{\left(e\right)}$ are specified. Namely, the components $X_{\left(e\right)}$, $Y_{\left(e\right)}$, and $\tilde{F}$ of the metric perturbation are determined by the Moncrief variable ${\Phi }_{\left(e\right)}$. The Moncrief variable ${\Phi }_{\left(e\right)}$ is determined by the master equation

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]-V_{even}{\Phi }_{\left(e\right)}=16\pi {\displaystyle \frac{r}{\Lambda }}{S}_{\left({\Phi }_{\left(e\right)}\right)},\end{array}$$

(85)

where the potential function $V_{even}$ is defined by

$$\begin{array}{ccc}\hfill V_{even}& :=& {\displaystyle \frac{1}{r^2{\Lambda }^2}}\left[{\mu }^2[(\mu +2)+3(1-f)]+{\left(3(1-f)\right)}^2\left(+\mu +(1-f)\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{r^2{\Lambda }^2}}\left[{\Lambda }^3-2(2-3f){\Lambda }^2+6(1-3f)(1-f)\Lambda +18f{(1-f)}^2\right],\hfill \end{array}$$

(86)

and the source term in Equation (85) is given by

$$\begin{array}{ccc}\hfill S_{\left({\Phi }_{\left(e\right)}\right)}& :=& -{\partial }_t{S}_{\left(ec\right)t}-f^2{\partial }_r{S}_{\left(ec\right)r}+2f{\displaystyle \frac{f-1}{r}}{S}_{\left(ec\right)r}+{\displaystyle \frac{r}{2}}f{\partial }_r{S}_{\left(F\right)}+{\displaystyle \frac{1}{2}}{S}_{\left(F\right)}-{\displaystyle \frac{1}{4}}\Lambda S_{\left(F\right)}\hfill \\ & & +{\displaystyle \frac{3(1-f)}{\Lambda }}\left[-S_{\left(\mathbb{F}\right)tt}-{\partial }_t{S}_{\left(ec\right)t}-f^2{\partial }_r{S}_{\left(ec\right)r}-{\displaystyle \frac{4}{r}}{f}^2{S}_{\left(ec\right)r}+{\displaystyle \frac{f}{2}}{S}_{\left(F\right)}\right]\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Lambda }{2f}}-1\right){\tilde{T}}_{tt}+{\displaystyle \frac{1}{2}}\left((2-f)-{\displaystyle \frac{1}{2}}\Lambda \right)f{\tilde{T}}_{rr}-{\displaystyle \frac{1}{2}}r{\partial }_r{\tilde{T}}_{tt}+{\displaystyle \frac{1}{2}}{f}^{2}r{\partial }_r{\tilde{T}}_{rr}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{f}{2}}{\tilde{T}}_{\left(e0\right)}-l(l+1)f{\tilde{T}}_{\left(e1\right)r}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{2}}{r}^2{\partial }_t^2{\tilde{T}}_{\left(e2\right)}-{\displaystyle \frac{1}{2}}{f}^2{r}^2{\partial }_r^2{\tilde{T}}_{\left(e2\right)}-{\displaystyle \frac{1}{2}}3(1+f)rf{\partial }_r{\tilde{T}}_{\left(e2\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{1}{2}}(7-3f)f{\tilde{T}}_{\left(e2\right)}+{\displaystyle \frac{1}{4}}(l(l+1)-1-f)(l(l+1)+2){\tilde{T}}_{\left(e2\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{3(1-f)}{\Lambda }}\left[{\tilde{T}}_{tt}+r{f}^2{\partial }_r{\tilde{T}}_{\left(e2\right)}+{\displaystyle \frac{1}{2}}(1+7f)f{\tilde{T}}_{\left(e2\right)}\right].\hfill \end{array}$$

(88)

To solve the master Equation (85), we have to impose appropriate boundary conditions and solve it as the Cauchy problem. In the book [18], it is shown that the Zerilli Equation (85) without the source term, i.e., $S_{\left({\Phi }_{\left(e\right)}\right)}=0$, can be transformed to the Regge–Wheeler equation. This transformation is called the Chandrasekhar transformation. Since the Regge–Wheeler equation can be solved by the MST (Mano Suzuki Takasugi) formulation [45,46,47], we may say that the solution to the Zerilli Equation (85) without the source term is also obtained through the MST formulation.

Finally, we note that the solutions ${\Phi }_{\left(e\right)}$ and $\tilde{F}$ satisfy Equation (69) due to the consistency of the linearized Einstein equation. Here, the source term $S_{\left(F\right)}$ is explicitly given by Equation (48). Here, we check this consistency of the initial value constraint (80) and the evolution Equation (69). From Equations (69) and (85), we obtain

$$\begin{array}{ccc}\hfill 0& =& r^2\Lambda {\partial }_t^2{S}_{(\Lambda \tilde{F})}-\left[(5-3f)\Lambda +3(1-f)(1+f)+18{\displaystyle \frac{1}{\Lambda }}f{(1-f)}^2\right]f{S}_{(\Lambda \tilde{F})}\hfill \end{array}$$

$$\begin{array}{ccc}& & -2\left[3(1-f)+2\Lambda \right]f^{2}r{\partial }_r{S}_{(\Lambda \tilde{F})}-\Lambda r^{2}f{\partial }_r\left[f{\partial }_r{S}_{(\Lambda \tilde{F})}\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{4}}\left[(1-3f)-\Lambda \right]{\Lambda }^{2}f{S}_{\left(F\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & -2r{f}^2\Lambda {\partial }_r{S}_{\left({\Phi }_{\left(e\right)}\right)}-\left[\Lambda +(1+3f)\right]\Lambda f{S}_{\left({\Phi }_{\left(e\right)}\right)}.\hfill \end{array}$$

(89)

This is an identity of the source terms. We confirm that Equation (89) is an identity due to the definitions (48), (82), and (88) and the continuity Equations (53)–(55) of the perturbative energy–momentum tensor. This means that the evolution Equation (69) is trivial when $l\ne 0$. Thus, we confirm that the above strategy for $l\ne 0$ modes is consistent.

Of course, this strategy is valid only when $l\ne 0$. In the $l=0$ case, we have to consider a different strategy to obtain the variables $X_{\left(e\right)}$, $Y_{\left(e\right)}$ and $\tilde{F}$. This will be discussed in Section 4.

Before going on to the discussion on the strategy to solve $l=0$-mode Einstein equations, we comment on the original equation derived by Zerilli [13,14] for $l\ge 2$. We consider the original time derivative of the Moncrief master variable (77) as

$$\begin{array}{ccc}\hfill {\partial }_t{\Phi }_{\left(e\right)}& =& {\displaystyle \frac{r}{\Lambda }}\left[f{\partial }_t{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}\Lambda {\partial }_t\tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_t{\partial }_r\tilde{F}\right].\hfill \end{array}$$

(90)

On the other hand, Equation (79) is given by

$$\begin{array}{ccc}\hfill {\partial }_t{X}_{\left(e\right)}& =& {\displaystyle \frac{l(l+1)}{2r}}{Y}_{\left(e\right)}-{\displaystyle \frac{r}{2}}{\partial }_t{\partial }_r\tilde{F}-{\displaystyle \frac{3f-1}{4f}}{\partial }_t\tilde{F}-8\pi r{S}_{\left(Y_{\left(e\right)}\right)}.\hfill \end{array}$$

(91)

Substituting Equation (91) into Equation (90), for $l\ne 0$ modes, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{l(l+1)}}{\partial }_t{\Phi }_{\left(e\right)}& =& {\displaystyle \frac{1}{2\Lambda }}\left[f{Y}_{\left(e\right)}-{\displaystyle \frac{r}{2}}{\partial }_t\tilde{F}\right]-8\pi r^{2}f{\displaystyle \frac{1}{l(l+1)\Lambda }}{S}_{\left(Y_{\left(e\right)}\right)}.\hfill \end{array}$$

(92)

Here, if we define the variable ${\Psi }_{\left(e\right)}$ by

$$\begin{array}{ccc}\hfill {\Psi }_{\left(e\right)}& :=& {\displaystyle \frac{1}{2\Lambda }}\left[f{Y}_{\left(e\right)}-{\displaystyle \frac{r}{2}}{\partial }_t\tilde{F}\right]\hfill \end{array}$$

(93)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{l(l+1)}}{\partial }_t{\Phi }_{\left(e\right)}+8\pi r^{2}f{\displaystyle \frac{1}{l(l+1)\Lambda }}{S}_{\left(Y_{\left(e\right)}\right)},\hfill \end{array}$$

(94)

the variable ${\Psi }_{\left(e\right)}$ corresponds to the original Zerilli master variable. Roughly speaking, the variable ${\Psi }_{\left(e\right)}$ corresponds to the time derivative of the variable ${\Phi }_{\left(e\right)}$ with additional source terms from the matter fields. Therefore, it is trivial that ${\Psi }_{\left(e\right)}$ also satisfies the Zerilli equation with different source terms. In other words, the Zerilli equation for ${\Psi }_{\left(e\right)}$ is derived by the time derivative of the Zerilli equation for ${\Phi }_{\left(e\right)}$. This means that the solution to the Zerilli equation for ${\Psi }_{\left(e\right)}$ may include an additional arbitrary function of r as an “integration constant”. This “integration constant” is not included in the solution ${\Phi }_{\left(e\right)}$ for the Zerilli Equation (85). In this sense, the restriction of the initial value of Equation (85) for ${\Phi }_{\left(e\right)}$ is stronger than that of Equation (85) for ${\Psi }_{\left(e\right)}$.

4. $l=0$-Mode Perturbations on the Schwarzschild Background

Here, we consider $l=0$-mode perturbations based on the perturbation equations for the even mode on the Schwarzschild background which are summarized in Section 3. Since Proposal 1 enables us to carry out mode-by-mode analyses including $l=0,1$ modes, all equations in Section 3 except for Equations (92) and (94) are valid even in the $l=0$ mode. However, the strategy to solve these equations is different from that for $l\ne 0$ modes, because Equations (79) and (80) do not directly give the components $(\tilde{F},Y_{\left(e\right)})$ of the metric perturbation for the $l=0$ mode.

Before showing the strategy to solve even-mode Einstein equations for the $l=0$ mode, we note that

$$\begin{array}{c}\hfill {\widehat{D}}_p{k}_{\left(\widehat{\Delta }\right)}=0={\widehat{D}}_p{\widehat{D}}_q{k}_{\left(\widehat{\Delta }\right)}\end{array}$$

(95)

if we impose the regularity $\delta =0$ on the harmonic function $k_{\left(\widehat{\Delta }\right)}$. In this case, the only remaining components of the linearized energy–momentum tensor are

$$\begin{array}{c}\hfill {\mathcal{T}}_{ab}={\tilde{T}}_{AB}{k}_{\left(\widehat{\Delta }\right)}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b+{\displaystyle \frac{1}{2}}{r}^2{\gamma }_{pq}{\tilde{T}}_{\left(e0\right)}{k}_{\left(\widehat{\Delta }\right)}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\end{array}$$

(96)

Therefore, we can safely regard that

$$\begin{array}{c}\hfill {\tilde{T}}_{\left(e2\right)}=0,{\tilde{T}}_{\left(e1\right)A}=0.\end{array}$$

(97)

Owing to Equation (97), the trace of the perturbation ${\tilde{F}}_{AB}$ is determined by the Einstein Equation (34), i.e.,

$$\begin{array}{c}\hfill {\tilde{F}}_D^D=0.\end{array}$$

(98)

In the case of the $l=0$ mode, $\Lambda$, defined by Equation (75), is given by

$$\begin{array}{c}\hfill \Lambda =1-3f.\end{array}$$

(99)

Then, the Moncrief master variable ${\Phi }_{\left(e\right)}$ is given by Equation (77), i.e.,

$$\begin{array}{c}\hfill {\Phi }_{\left(e\right)}:={\displaystyle \frac{r}{1-3f}}\left[f{X}_{\left(e\right)}-{\displaystyle \frac{1}{4}}(1-3f)\tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\right].\end{array}$$

(100)

This is equivalent to Equation (78) with $l=0$ as

$$\begin{array}{c}\hfill f{X}_{\left(e\right)}={\displaystyle \frac{1-3f}{r}}{\Phi }_{\left(e\right)}+{\displaystyle \frac{1-3f}{4}}\tilde{F}-{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}.\end{array}$$

(101)

As in the case of the $l\ne 0$ mode, this equation yields that the component $X_{\left(e\right)}$ of the metric perturbation is determined by $(\tilde{F},{\Phi }_{\left(e\right)})$.

The crucial difference between the $l=0$ mode and $l\ne 0$ modes is Equations (79) and (80). In the $l=0$ case, these equations yield

$$\begin{array}{ccc}& & {\partial }_r\left[(1-3f){\Phi }_{\left(e\right)}\right]=-{\displaystyle \frac{8\pi r^2}{f}}{\tilde{T}}_{tt},\hfill \end{array}$$

(102)

$$\begin{array}{ccc}& & {\partial }_t\left[(1-3f){\Phi }_{\left(e\right)}\right]=-8\pi r^{2}f{\tilde{T}}_{tr},\hfill \end{array}$$

(103)

where we use Equation (101) to derive Equation (103).

The components of the divergence of the energy–momentum tensor are summarized as

$$\begin{array}{ccc}& & {\partial }_t{\tilde{T}}_{tt}-f^2{\partial }_r{\tilde{T}}_{rt}-{\displaystyle \frac{(1+f)f}{r}}{\tilde{T}}_{rt}=0,\hfill \end{array}$$

(104)

$$\begin{array}{ccc}& & {\partial }_t{\tilde{T}}_{tr}-{\displaystyle \frac{1-f}{2rf}}{\tilde{T}}_{tt}-f^2{\partial }_r{\tilde{T}}_{rr}-{\displaystyle \frac{(3+f)f}{2r}}{\tilde{T}}_{rr}=0,\hfill \end{array}$$

(105)

$$\begin{array}{ccc}& & {\tilde{T}}_{\left(e0\right)}=0.\hfill \end{array}$$

(106)

Here, we check the integrability condition of Equations (102) and (103). Differentiating Equation (102) with respect to t and differentiating Equation (103) with respect to r, we obtain the integrability condition of Equations (102) and (103) as follows:

$$\begin{array}{ccc}\hfill 0& =& {\partial }_t\left(-8\pi {\displaystyle \frac{r^2}{f}}{\tilde{T}}_{tt}\right)-{\partial }_r\left(-8\pi r^{2}f{\tilde{T}}_{tr}\right)\hfill \\ & =& -8\pi r^2{\displaystyle \frac{1}{f}}\left[{\partial }_t{\tilde{T}}_{tt}-f^2{\partial }_r{\tilde{T}}_{tr}-{\displaystyle \frac{(1+f)f}{r}}{\tilde{T}}_{tr}\right].\hfill \end{array}$$

(107)

This coincides with the component (104) of the continuity equation of the matter field. Thus, Equations (102) and (103) are integrable and there exists the solution ${\Phi }_{\left(e\right)}={\Phi }_{\left(e\right)}[T_{tt},T_{tr}]$ to these equations.

In the case of the $l=0$ mode, the evolution Equation (85) has the same form, but the potential $V_{even}$, defined by Equation (86), with $l=0$, is given by

$$\begin{array}{c}\hfill V_{even}={\displaystyle \frac{3(1-f)(1+3f^2)}{r^2{(1-3f)}^2}}\end{array}$$

(108)

and the source term in Equation (88) is given by

$$\begin{array}{c}\hfill S_{\left({\Phi }_{\left(e\right)}\right)}=-{\tilde{T}}_{tt}-{\displaystyle \frac{r}{2}}{\partial }_r{\tilde{T}}_{tt}+{\displaystyle \frac{r}{2}}{\partial }_t{\tilde{T}}_{tr}-{\displaystyle \frac{3(1-f)}{1-3f}}{\tilde{T}}_{tt}.\end{array}$$

(109)

Through Equations (102) and (103), we obtain

$$\begin{array}{ccc}& & -{\displaystyle \frac{1}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+{\partial }_r\left(f{\partial }_r{\Phi }_{\left(e\right)}\right)-V_{even}{\Phi }_{\left(e\right)}\hfill \\ & =& {\displaystyle \frac{16\pi r}{1-3f}}\left[-{\tilde{T}}_{tt}-{\displaystyle \frac{r}{2}}{\partial }_r{\tilde{T}}_{tt}+{\displaystyle \frac{r}{2}}{\partial }_t{\tilde{T}}_{tr}-{\displaystyle \frac{3(1-f)}{1-3f}}{\tilde{T}}_{tt}\right].\hfill \end{array}$$

(110)

This coincides with the master Equation (85) with $l=0$. Thus, the master Equation (85) does not give us any information other than that of Equations (102) and (103).

As in the case of $l\ne 0$ modes, the metric component $X_{\left(e\right)}$ is determined by the variables $(\tilde{F},{\Phi }_{\left(e\right)})$, as seen in Equation (101). Although $\tilde{F}$ is determined by Equation (80) in the $l\ne 0$ case, this is impossible in the $l=0$ case. Therefore, we have to consider Equation (69) for the variable $\tilde{F}$ which is trivial in the $l\ne 0$ case.

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2\tilde{F}+{\partial }_r\left(f{\partial }_r\tilde{F}\right)+{\displaystyle \frac{1}{r^2}}3(1-f)\tilde{F}+{\displaystyle \frac{4}{r^3}}(1-3f){\Phi }_{\left(e\right)}=16\pi \left(-{\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}\right).\end{array}$$

(111)

This equation has the same form as the inhomogeneous version of the Regge–Wheeler equation with $l=0$, while the original Regge–Wheeler equation is valid only for $l\ge 2$ modes. If we solve Equation (111), we can determine the variable $\tilde{F}$ which depends on the variable ${\Phi }_{\left(e\right)}$ and the matter fields ${\tilde{T}}_{tt}$ and ${\tilde{T}}_{rr}$. Then, through the solution $\tilde{F}=\tilde{F}[{\Phi }_{\left(e\right)},{\overline{T}}_{tt},{\overline{T}}_{rr}]$ and the solution to Equations (102) and (103), we can obtain the variable $X_{\left(e\right)}$ through Equation (101) as a solution to the linearized Einstein equation for the $l=0$ mode.

The remaining component to be obtained is the component $Y_{\left(e\right)}$ of the metric perturbation. To obtain the variable $Y_{\left(e\right)}$, we recall the original initial value constraints (43) and (44). In the $l=0$-mode case, the source term $S_{\left(ec\right)t}$ and $S_{\left(ec\right)r}$ are given by $S_{\left(ec\right)t}=S_{\left(ec\right)r}=0$ from Equations (45) and (97). Then, the initial value constraints (43) and (44) are given by

$$\begin{array}{ccc}& & {\partial }_r\left(f{Y}_{\left(e\right)}\right)={\displaystyle \frac{1}{2}}{\partial }_t\tilde{F}-{\partial }_t\left(X_{\left(e\right)}\right),\hfill \end{array}$$

(112)

$$\begin{array}{ccc}& & {\partial }_t\left(f{Y}_{\left(e\right)}\right)=-f{\partial }_r\left(f{X}_{\left(e\right)}\right)-{\displaystyle \frac{1}{2}}{f}^2{\partial }_r\tilde{F}.\hfill \end{array}$$

(113)

We may regard that Equations (112) and (113) are equations to obtain the variable $Y_{\left(e\right)}$. Actually, the integrability of these equations is guaranteed by Equations (101)–(103), (105) and (111). Then, we can obtain the component $Y_{\left(e\right)}$ of the metric perturbation by the direct integration of Equations (112) and (113).

We may carry out the above strategy to obtain the $l=0$-mode solution to the linearized Einstein equations, but it is instructive to consider the vacuum case where all components of the linearized energy–momentum tensor ${}^{\left(1\right)}{\mathcal{T}}_{ab}$ vanish before the derivation of the non-vacuum case.

4.1. $l=0$-Mode Vacuum Case

Here, we consider the vacuum case of the above equations for $l=0$-mode perturbations. First, we consider Equations (102) and (103) with the vacuum condition ${\tilde{T}}_{tt}={\tilde{T}}_{tr}=0$ as

$$\begin{array}{ccc}& & {\partial }_r\left[(1-3f){\Phi }_{\left(e\right)}\right]=0,\hfill \end{array}$$

(114)

$$\begin{array}{ccc}& & {\partial }_t\left[(1-3f){\Phi }_{\left(e\right)}\right]=0.\hfill \end{array}$$

(115)

These equations are easily integrated as

$$\begin{array}{c}\hfill (1-3f){\Phi }_{\left(e\right)}=-2M_1,M_1\in \mathbb{R}.\end{array}$$

(116)

Furthermore, the variable $\tilde{F}$ is determined by Equation (111) with the vacuum condition ${\tilde{T}}_{tt}={\tilde{T}}_{rr}=0$ as

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2\tilde{F}+{\partial }_r\left(f{\partial }_r\tilde{F}\right)+{\displaystyle \frac{1}{r^2}}3(1-f)\tilde{F}-{\displaystyle \frac{8M_1}{r^3}}=0.\end{array}$$

(117)

From Equations (101) and (116), we obtain the component $X_{\left(e\right)}$ of the metric perturbation as follows:

$$\begin{array}{c}\hfill f{X}_{\left(e\right)}=-{\displaystyle \frac{2M_1}{r}}+{\displaystyle \frac{1-3f}{4}}\tilde{F}-{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}.\end{array}$$

(118)

Moreover, the components $Y_{\left(e\right)}$ obtain the direct integration of Equations (112) and (113), because the integrability is already guaranteed. Substituting Equation (118) into Equations (112) and (113), we obtain

$$\begin{array}{ccc}& & f{\partial }_r\left(f{Y}_{\left(e\right)}\right)=-{\displaystyle \frac{1}{4}}(1-5f){\partial }_t\tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t\tilde{F},\hfill \end{array}$$

(119)

$$\begin{array}{ccc}& & {\partial }_t\left(f{Y}_{\left(e\right)}\right)={\displaystyle \frac{2M_{1}f}{r^2}}-{\displaystyle \frac{3}{4r}}f(1-f)\tilde{F}-{\displaystyle \frac{1}{4}}f(1-3f){\partial }_r\tilde{F}+{\displaystyle \frac{1}{2}}r{\partial }_t^2\tilde{F},\hfill \end{array}$$

(120)

where we use Equation (117).

Here, we assume the existence of the solution to Equation (117) and we denote this solution by

$$\begin{array}{c}\hfill \tilde{F}=:{\partial }_t{\rm Y},\end{array}$$

(121)

for our convention. Substituting Equation (121) into Equation (117) and integrating by t, we obtain

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2{\rm Y}+{\partial }_r(f{\partial }_r{\rm Y})+{\displaystyle \frac{1}{r^2}}3(1-f){\rm Y}-{\displaystyle \frac{8M_1}{r^3}}t+\zeta \left(r\right)=0.\end{array}$$

(122)

where $\zeta \left(r\right)$ is an arbitrary function of r. Using Equation (121) and integrating by t, Equation (120) yields

$$\begin{array}{c}\hfill f{Y}_{\left(e\right)}={\displaystyle \frac{2M_{1}f}{r^2}}t-{\displaystyle \frac{3}{4r}}f(1-f){\rm Y}-{\displaystyle \frac{1}{4}}f(1-3f){\partial }_r{\rm Y}+{\displaystyle \frac{1}{2}}r{\partial }_t^2{\rm Y}+\Xi \left(r\right),\end{array}$$

(123)

where $\Xi \left(r\right)$ is an arbitrary function of r. Substituting Equation (123) into Equation (119) and using Equation (122), we obtain

$$\begin{array}{c}\hfill \zeta \left(r\right)=-{\displaystyle \frac{4}{1-3f}}{\partial }_r\Xi \left(r\right).\end{array}$$

(124)

In summary, we obtain the components of $X_{\left(e\right)}$, $Y_{\left(e\right)}$, and $\tilde{F}$ of the metric perturbations as follows:

$$\begin{array}{ccc}\hfill f{X}_{\left(e\right)}& =& -{\displaystyle \frac{2M_1}{r}}+{\displaystyle \frac{1-3f}{4}}{\partial }_t{\rm Y}-{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t{\rm Y},\hfill \end{array}$$

(125)

$$\begin{array}{ccc}\hfill f{Y}_{\left(e\right)}& =& {\displaystyle \frac{2M_{1}f}{r^2}}t-{\displaystyle \frac{3}{4r}}f(1-f){\rm Y}-{\displaystyle \frac{1}{4}}f(1-3f){\partial }_r{\rm Y}+{\displaystyle \frac{1}{2}}r{\partial }_t^2{\rm Y}+\Xi \left(r\right),\hfill \end{array}$$

(126)

and

$$\begin{array}{c}\hfill \tilde{F}={\partial }_t{\rm Y},-{\displaystyle \frac{1}{f}}{\partial }_t^2{\rm Y}+{\partial }_r(f{\partial }_r{\rm Y})+{\displaystyle \frac{1}{r^2}}3(1-f){\rm Y}-{\displaystyle \frac{8M_1}{r^3}}t-{\displaystyle \frac{4}{1-3f}}{\partial }_r\Xi \left(r\right)=0,\end{array}$$

(127)

and $\Xi \left(r\right)$ is an arbitrary function of r.

Here, we consider the covariant form ${\mathcal{F}}_{ab}$ of the $l=0$-mode metric perturbation. According to Proposal 1, we impose the regularity of $S^2$ on the harmonic function $k_{(\Delta )}$ so that

$$\begin{array}{c}\hfill k_{(\Delta )}=1.\end{array}$$

(128)

Since ${\tilde{F}}_{AB}$ is traceless according to Equation (98) for $l=0$-mode perturbations, the gauge-invariant metric perturbation ${\mathcal{F}}_{ab}$ for the $l=0$ mode is given by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& {\tilde{F}}_{AB}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b+{\displaystyle \frac{1}{2}}{\gamma }_{pq}{r}^2\tilde{F}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b\hfill \end{array}$$

$$\begin{array}{ccc}& =& -\left(f{X}_{\left(e\right)}\right)\left\{{\left(dt\right)}_a{\left(dt\right)}_b+f^{-2}{\left(dr\right)}_a{\left(dr\right)}_b\right\}+2\left(f{Y}_{\left(e\right)}\right)f^{-1}{\left(dt\right)}_{(A}{\left(dr\right)}_{B)}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{2}}{\gamma }_{pq}{r}^2\tilde{F}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\hfill \end{array}$$

(129)

As in the case of the $l=1$ odd-mode perturbation in the Part I paper [29], the solutions (125)–(127) may include the terms in the form of ${\pounds }_V{g}_{ab}$ for the vector field $V^a$. To find the term ${\pounds }_V{g}_{ab}$, we consider the generator $V_a$ whose components are given by

$$\begin{array}{c}\hfill V_a=V_t(t,r){\left(dt\right)}_a+V_r(r,t){\left(dr\right)}_a.\end{array}$$

(130)

Then, the nonvanishing components of ${\pounds }_V{g}_{ab}$ are given by

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tt}=2{\partial }_t{V}_t-f{f}^{\prime }{V}_r,\hfill \end{array}$$

(131)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tr}={\partial }_t{V}_r+{\partial }_r{V}_t-{\displaystyle \frac{f^{\prime }}{f}}{V}_t,\hfill \end{array}$$

(132)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{rr}=2{\partial }_r{V}_r+{\displaystyle \frac{f^{\prime }}{f}}{V}_r,\hfill \end{array}$$

(133)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\theta \theta }=2rf{V}_r,\hfill \end{array}$$

(134)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\varphi \varphi }=2rf{sin}^2\theta V_r.\hfill \end{array}$$

(135)

From Equations (129), (134), and (135), we choose

$$\begin{array}{c}\hfill V_r={\displaystyle \frac{1}{4f}}r\tilde{F}={\displaystyle \frac{1}{4f}}r{\partial }_t{\rm Y},{\pounds }_V{g}_{\theta \theta }={\displaystyle \frac{1}{{sin}^2\theta }}{\pounds }_V{g}_{\varphi \varphi }={\displaystyle \frac{1}{2}}{r}^2\tilde{F}={\displaystyle \frac{1}{2}}{r}^2{\partial }_t{\rm Y}.\end{array}$$

(136)

Substituting Equation (136) into Equations (131)–(133), we obtain

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tt}=2{\partial }_t{V}_t-{\displaystyle \frac{1}{4}}(1-f){\partial }_t{\rm Y},\hfill \end{array}$$

(137)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tr}={\displaystyle \frac{1}{4f}}r{\partial }_t^2{\rm Y}+{\partial }_r{V}_t-{\displaystyle \frac{1}{fr}}(1-f)V_t,\hfill \end{array}$$

(138)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{rr}=-{\displaystyle \frac{1}{4f^2}}(1-3f){\partial }_t{\rm Y}+{\displaystyle \frac{1}{2f}}r{\partial }_r{\partial }_t{\rm Y}.\hfill \end{array}$$

(139)

To identify the degree of freedom, which is expressed as ${\pounds }_V{g}_{ab}$ in $X_{\left(e\right)}$, we choose

$$\begin{array}{c}\hfill {\partial }_t{V}_t={\displaystyle \frac{1}{4}}f{\partial }_t{\rm Y}+{\displaystyle \frac{1}{4}}rf{\partial }_r{\partial }_t{\rm Y}\end{array}$$

(140)

so that

$$\begin{array}{c}\hfill {\pounds }_V{g}_{tt}=-{\displaystyle \frac{1}{4}}(1-3f){\partial }_t{\rm Y}+{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t{\rm Y}.\end{array}$$

(141)

Then, we obtain

$$\begin{array}{c}\hfill V_t={\displaystyle \frac{1}{4}}f{\rm Y}+{\displaystyle \frac{1}{4}}rf{\partial }_r{\rm Y}+\gamma \left(r\right),\end{array}$$

(142)

where $\gamma \left(r\right)$ is an arbitrary function of r. Substituting Equation (142) into Equation (138) and using the Equation (127) for ${\rm Y}$, we obtain

$$\begin{array}{ccc}\hfill {\pounds }_V{g}_{tr}& =& {\displaystyle \frac{2M_1}{r^2}}t+{\displaystyle \frac{r}{2f}}{\partial }_t^2{\rm Y}-{\displaystyle \frac{1}{4}}(1-3f){\partial }_r{\rm Y}-{\displaystyle \frac{3}{4r}}(1-f){\rm Y}\hfill \\ & & +{\displaystyle \frac{r}{1-3f}}{\partial }_r\Xi \left(r\right)+{\partial }_r\gamma \left(r\right)-{\displaystyle \frac{1}{fr}}(1-f)\gamma \left(r\right).\hfill \end{array}$$

(143)

From solutions (125)–(127) and Expression (129) of the gauge-invariant part of the metric perturbation and Components (136), (139), (141), and (143) of ${\pounds }_V{g}_{ab}$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& {\displaystyle \frac{2M_1}{r}}\left({\left(dt\right)}_a{\left(dt\right)}_b+f^{-2}{\left(dr\right)}_a{\left(dr\right)}_b\right)+{\pounds }_V{g}_{ab}\hfill \\ & & +2\left({\displaystyle \frac{1}{f}}\Xi \left(r\right)-{\displaystyle \frac{r}{1-3f}}{\partial }_r\Xi \left(r\right)-{\partial }_r\gamma \left(r\right)+{\displaystyle \frac{1}{fr}}(1-f)\gamma \left(r\right)\right){\left(dt\right)}_{(a}{\left(dr\right)}_{b)}.\hfill \end{array}$$

(144)

As a choice of the generator $V_a$, we choose the arbitrary function $\gamma \left(r\right)$ in $V_a$ such that

$$\begin{array}{c}\hfill \gamma \left(r\right)=-{\displaystyle \frac{r}{(1-3f)}}\Xi \left(r\right)+f\int dr{\displaystyle \frac{2}{f{(1-3f)}^2}}\Xi \left(r\right).\end{array}$$

(145)

Then, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& {\displaystyle \frac{2M_1}{r}}\left({\left(dt\right)}_a{\left(dt\right)}_b+f^{-2}{\left(dr\right)}_a{\left(dr\right)}_b\right)+{\pounds }_V{g}_{ab},\hfill \end{array}$$

(146)

where

$$\begin{array}{c}\hfill V_a=\left({\displaystyle \frac{f}{4}}{\rm Y}+{\displaystyle \frac{rf}{4}}{\partial }_r{\rm Y}-{\displaystyle \frac{r}{1-3f}}\Xi \left(r\right)+f\int dr{\displaystyle \frac{2}{f{(1-3f)}^2}}\Xi \left(r\right)\right){\left(dt\right)}_a+{\displaystyle \frac{r}{4f}}{\partial }_t{\rm Y}{\left(dr\right)}_a.\end{array}$$

(147)

The function ${\rm Y}(t,r)$ is the solution to the second equation in Equation (127).

The solution (146) is the $O\left(ϵ\right)$-mass parameter perturbation $M+ϵM_1$ of the Schwarzschild spacetime, apart from the term of the Lie derivative of the background metric $g_{ab}$. Since the $l=0$ mode is for spherically symmetric perturbations, Solution (146) is the realization of the linearized gauge-invariant version of Birkhoff’s theorem [36]. We also note that the vector field $V_a$ is also gauge-invariant in the sense of the second kind. Here, we have to emphasize that the generator (147) with the second equation in Equation (127) is necessary if we include the perturbative Schwarzschild mass parameter $M_1$ as the solution to the linearized Einstein equation in our framework. This can be seen from the second equation in Equation (127). This equation indicates that $M_1=0$ if we choose ${\rm Y}=0$ for the arbitrary time t.

4.2. $l=0$-Mode Non-Vacuum Case

Inspecting the above vacuum case, we apply the method of variational constants. In Equation (116), the Schwarzschild mass parameter perturbation $M_1$ is an integration constant. Then, we choose the function $m_1(t,r)$ so that

$$\begin{array}{c}\hfill m_1(t,r):=-{\displaystyle \frac{1}{2}}(1-3f){\Phi }_{\left(e\right)}.\end{array}$$

(148)

The integrability of Equations (102) and (103) is already confirmed in Equation (107). Then, we obtain

$$\begin{array}{c}\hfill m_1(t,r)=4\pi \int dr{\displaystyle \frac{r^2}{f}}{\tilde{T}}_{tt}+M_1=4\pi \int dt{r}^{2}f{\tilde{T}}_{tr}+M_1.\end{array}$$

(149)

Equation (101) yields the component $X_{\left(e\right)}$ of the metric perturbation as follows:

$$\begin{array}{c}\hfill f{X}_{\left(e\right)}=-{\displaystyle \frac{2m_1(t,r)}{r}}+{\displaystyle \frac{1-3f}{4}}\tilde{F}-{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}.\end{array}$$

(150)

As discussed above, the variable $\tilde{F}$ is determined by Equation (111). As in the vacuum case in Section 4.1, we introduce the function ${\rm Y}$ such that

$$\begin{array}{ccc}& & \tilde{F}=:{\partial }_t{\rm Y},\hfill \\ & & -{\displaystyle \frac{1}{f}}{\partial }_t^2{\rm Y}+{\partial }_r(f{\partial }_r{\rm Y})+{\displaystyle \frac{3(1-f)}{r^2}}{\rm Y}-{\displaystyle \frac{8}{r^3}}\int dt{m}_1(t,r)+\zeta \left(r\right)\hfill \end{array}$$

(151)

$$\begin{array}{ccc}& =& 16\pi \int dt\left(-{\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}\right),\hfill \end{array}$$

(152)

where $\zeta \left(r\right)$ is an arbitrary function of r. Through the variable ${\rm Y}$ and Equation (150), Equation (113) is integrated as follows:

$$\begin{array}{ccc}\hfill f{Y}_{\left(e\right)}& =& {\displaystyle \frac{2f}{r^2}}\int dt{m}_1(t,r)+8\pi r{f}^2\int dt{\tilde{T}}_{rr}\hfill \\ & & -{\displaystyle \frac{3f(1-f)}{4r}}{\rm Y}-{\displaystyle \frac{f(1-3f)}{4}}{\partial }_r{\rm Y}+{\displaystyle \frac{r}{2}}{\partial }_t^2{\rm Y}+\Xi \left(r\right),\hfill \end{array}$$

(153)

where $\Xi \left(r\right)$ is a different arbitrary function of r from $\zeta \left(r\right)$. Substituting Equation (153) into Equation (112), and using Equations (149), (150), and (152) and the component (105) of the continuity equation, we obtain

$$\begin{array}{ccc}\hfill \zeta \left(r\right)& =& -{\displaystyle \frac{4}{1-3f}}{\partial }_r\Xi \left(r\right)\hfill \end{array}$$

(154)

as expected from the vacuum case in Section 4.1.

In summary, we obtain the solution to the components of the metric perturbations $X_{\left(e\right)}$, $Y_{\left(e\right)}$, and $\tilde{F}$ as follows:

$$\begin{array}{ccc}\hfill f{X}_{\left(e\right)}& =& -{\displaystyle \frac{2m_1(t,r)}{r}}+{\displaystyle \frac{1-3f}{4}}{\partial }_t{\rm Y}-{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t{\rm Y},\hfill \end{array}$$

(155)

$$\begin{array}{ccc}\hfill f{Y}_{\left(e\right)}& =& {\displaystyle \frac{2f}{r^2}}\int dt{m}_1(t,r)+8\pi r{f}^2\int dt{\tilde{T}}_{rr}\hfill \end{array}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{3f(1-f)}{4r}}{\rm Y}-{\displaystyle \frac{f(1-3f)}{4}}{\partial }_r{\rm Y}+{\displaystyle \frac{r}{2}}{\partial }_t^2{\rm Y}+\Xi \left(r\right),\hfill \end{array}$$

(156)

$$\begin{array}{ccc}\hfill \tilde{F}& =:& {\partial }_t{\rm Y},\hfill \end{array}$$

(157)

$$\begin{array}{ccc}& & {\partial }_t^2{\rm Y}-f{\partial }_r(f{\partial }_r{\rm Y})-{\displaystyle \frac{3f(1-f)}{r^2}}{\rm Y}+{\displaystyle \frac{8f}{r^3}}\int dt{m}_1(t,r)+{\displaystyle \frac{4f{\partial }_r\Xi \left(r\right)}{1-3f}}\hfill \end{array}$$

$$\begin{array}{ccc}& =& 16\pi \int dt\left({\tilde{T}}_{tt}-f^2{\tilde{T}}_{rr}\right).\hfill \end{array}$$

(158)

Here, we consider the covariant form of the above $l=0$-mode non-vacuum solutions. As in the vacuum case in Section 4.1, we show expression (129) of the above non-vacuum solution

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& -\left(f{X}_{\left(e\right)}\right)\left[{\left(dt\right)}_a{\left(dt\right)}_b+f^{-2}{\left(dr\right)}_a{\left(dr\right)}_b\right]+2\left(f{Y}_{\left(e\right)}\right)f^{-1}{\left(dt\right)}_{(A}{\left(dr\right)}_{B)}\hfill \\ & & +{\displaystyle \frac{1}{2}}{\gamma }_{pq}{r}^2{\partial }_t{\rm Y}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\hfill \end{array}$$

(159)

The components of ${\mathcal{F}}_{ab}$ are given by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tt}& =& {\displaystyle \frac{2m_1(t,r)}{r}}-{\displaystyle \frac{1-3f}{4}}{\partial }_t{\rm Y}+{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t{\rm Y},\hfill \end{array}$$

(160)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tr}& =& {\displaystyle \frac{2}{r^2}}\int dt{m}_1(t,r)+8\pi rf\int dt{\tilde{T}}_{rr}\hfill \\ & & -{\displaystyle \frac{3(1-f)}{4r}}{\rm Y}-{\displaystyle \frac{(1-3f)}{4}}{\partial }_r{\rm Y}+{\displaystyle \frac{r}{2f}}{\partial }_t^2{\rm Y}+{\displaystyle \frac{1}{f}}\Xi \left(r\right),\hfill \end{array}$$

(161)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{rr}& =& {\displaystyle \frac{2m_1(t,r)}{r{f}^2}}-{\displaystyle \frac{1-3f}{4f^2}}{\partial }_t{\rm Y}+{\displaystyle \frac{r}{2f}}{\partial }_r{\partial }_t{\rm Y},\hfill \end{array}$$

(162)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\theta \theta }& =& {\displaystyle \frac{r^2}{2}}{\partial }_t{\rm Y}={\displaystyle \frac{1}{{sin}^2\theta }}{\mathcal{F}}_{\varphi \varphi }.\hfill \end{array}$$

(163)

As in the vacuum case, we consider the term in the form ${\pounds }_V{g}_{ab}$ with the generator

$$\begin{array}{c}\hfill V_a=V_t(t,r){\left(dt\right)}_a+V_r(r,t){\left(dr\right)}_a.\end{array}$$

(164)

Then, we obtain Equations (131)–(135). Comparing Equations (134), (135), and (163), we choose $V_r$ so that

$$\begin{array}{c}\hfill V_r={\displaystyle \frac{1}{4f}}r{\partial }_t{\rm Y},{\pounds }_V{g}_{\theta \theta }={\displaystyle \frac{1}{{sin}^2\theta }}{\pounds }_V{g}_{\varphi \varphi }={\displaystyle \frac{1}{2}}{r}^2{\partial }_t{\rm Y},\end{array}$$

(165)

and we have

$$\begin{array}{c}\hfill {\mathcal{F}}_{\theta \theta }={\pounds }_V{g}_{\theta \theta },{\mathcal{F}}_{\varphi \varphi }={\pounds }_V{g}_{\varphi \varphi }.\end{array}$$

(166)

Substituting the choice (165) into Equation (133) and comparing it with Equation (162), we obtain

$$\begin{array}{c}\hfill {\pounds }_V{g}_{rr}=-{\displaystyle \frac{1-3f}{4f^2}}{\partial }_t{\rm Y}+{\displaystyle \frac{1}{2f}}r{\partial }_r{\partial }_t{\rm Y},{\mathcal{F}}_{rr}={\displaystyle \frac{2m_1(t,r)}{r{f}^2}}+{\pounds }_V{g}_{rr}.\end{array}$$

(167)

Substituting the choice $V_r$ in Equation (165) into Equation (131) and comparing it with Equation (160), we choose

$$\begin{array}{c}\hfill V_t={\displaystyle \frac{1}{4}}f{\rm Y}+{\displaystyle \frac{1}{4}}rf{\partial }_r{\rm Y}+\gamma \left(r\right),\end{array}$$

(168)

and obtain

$$\begin{array}{c}\hfill {\pounds }_V{g}_{tt}=-{\displaystyle \frac{1-3f}{4}}{\partial }_t{\rm Y}+{\displaystyle \frac{1}{2}}rf{\partial }_r{\partial }_t{\rm Y},{\mathcal{F}}_{tt}={\displaystyle \frac{2m_1(t,r)}{r}}+{\pounds }_V{g}_{tt}.\end{array}$$

(169)

Finally, from Equation (132) with the choice (168) of $V_t$ and the choice (165) of $V_r$, we obtain

$$\begin{array}{ccc}\hfill {\pounds }_V{g}_{tr}& =& {\displaystyle \frac{1}{4f}}r{\partial }_t^2{\rm Y}-{\displaystyle \frac{1-3f}{4}}{\partial }_r{\rm Y}+{\displaystyle \frac{1}{4}}r{\partial }_r(f{\partial }_r{\rm Y})+{\partial }_r\gamma \left(r\right)-{\displaystyle \frac{1-f}{fr}}\gamma \left(r\right).\hfill \end{array}$$

(170)

Furthermore, using Equation (158), we have

$$\begin{array}{ccc}\hfill {\pounds }_V{g}_{tr}& =& {\displaystyle \frac{2}{r^2}}\int dt{m}_1(t,r)-4\pi {\displaystyle \frac{r}{f}}\int dt{\tilde{T}}_{tt}+4\pi rf\int dt{\tilde{T}}_{rr}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{2f}}r{\partial }_t^2{\rm Y}-{\displaystyle \frac{1-3f}{4}}{\partial }_r{\rm Y}-{\displaystyle \frac{3(1-f)}{4r}}{\rm Y}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\partial }_r\gamma \left(r\right)-{\displaystyle \frac{1-f}{fr}}\gamma \left(r\right)+{\displaystyle \frac{r{\partial }_r\Xi \left(r\right)}{1-3f}}.\hfill \end{array}$$

(171)

Through Equation (161), we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tr}& =& 4\pi r\int dt\left({\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}\right)+{\pounds }_V{g}_{tr}\hfill \\ & & +f\left({\displaystyle \frac{2}{f{(1-3f)}^2}}\Xi \left(r\right)-{\partial }_r\left({\displaystyle \frac{r}{f(1-3f)}}\Xi \left(r\right)\right)-{\partial }_r\left({\displaystyle \frac{1}{f}}\gamma \left(r\right)\right)\right).\hfill \end{array}$$

(172)

The same choice of $\gamma \left(r\right)$ in the generator $V_a$ as in Equation (145) yields

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tr}& =& 4\pi r\int dt\left({\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}\right)+{\pounds }_V{g}_{tr}.\hfill \end{array}$$

(173)

Thus, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& {\displaystyle \frac{2}{r}}\left(M_1+4\pi \int dr{\displaystyle \frac{r^2}{f}}{T}_{tt}\right)\left({\left(dt\right)}_a{\left(dt\right)}_a+{\displaystyle \frac{1}{f^2}}{\left(dr\right)}_a{\left(dr\right)}_a\right)\hfill \\ & & +2\left[4\pi r\int dt\left({\displaystyle \frac{1}{f}}{\tilde{T}}_{tt}+f{\tilde{T}}_{rr}\right)\right]{\left(dt\right)}_{(a}{\left(dr\right)}_{b)}+{\pounds }_V{g}_{ab},\hfill \end{array}$$

(174)

where

$$\begin{array}{c}\hfill V_a=\left({\displaystyle \frac{f}{4}}{\rm Y}+{\displaystyle \frac{rf}{4}}{\partial }_r{\rm Y}-{\displaystyle \frac{r\Xi \left(r\right)}{(1-3f)}}+f\int dr{\displaystyle \frac{2\Xi \left(r\right)}{f{(1-3f)}^2}}\right){\left(dt\right)}_a+{\displaystyle \frac{1}{4f}}r{\partial }_t{\rm Y}{\left(dr\right)}_a.\end{array}$$

(175)

The variable ${\rm Y}$ must satisfy Equation (158). We also note that the expression of ${\mathcal{F}}_{ab}$ is not unique, since we may choose a different vector field, $V_a$. We can also choose the time component $V_t$ of the vector field $V_a$ so that ${\mathcal{F}}_{tr}={\pounds }_V{g}_{tr}$. In this case, the additional terms appear in the component ${\mathcal{F}}_{tt}$.

We also note that the term ${\pounds }_V{g}_{ab}$ in Equation (174) is gauge-invariant of the second kind. Furthermore, unlike the vacuum case, the variable ${\rm Y}$ in this term includes information of the matter field through Equation (158). In this sense, the term ${\pounds }_V{g}_{ab}$ in Equation (174) is physical.

5. $l=1$-Mode Non-Vacuum Perturbations on the Schwarzschild Background

In this section, we consider $l=1$-mode perturbations based on the variables defined in Section 2 and Section 3. Even in the case of the $l=1$ mode, the gauge-invariant variables given by Equations (28)–(30) are valid. Since mode-by-mode analyses are possible in our formulation, we can consider $l=1$ modes separately. For $l=1$ even-mode perturbations, the component ${\mathcal{F}}_{Ap}$ of the gauge-invariant part of the metric perturbation vanishes and the other components are given by

$$\begin{array}{c}\hfill {\mathcal{F}}_{AB}:=\sum _{m=-1}^1{\tilde{F}}_{AB}{k}_{\left(\widehat{\Delta +2}\right)m},{\mathcal{F}}_{pq}:={\displaystyle \frac{1}{2}}{\gamma }_{pq}{r}^2\sum _{m=-1}^1\tilde{F}{k}_{(\widehat{\Delta }+2)m}.\end{array}$$

(176)

We can also separate the trace part ${\tilde{F}}_D^D$ and the traceless part ${\tilde{\mathbb{F}}}_{AB}$ for the metric perturbation ${\tilde{F}}_{AB}$ as Equation (36). We also consider the components of the traceless part ${\mathbb{F}}_{AB}$ as Equation (42).

Following Proposal 1, we impose the regularity on the harmonic function $k_{(\widehat{\Delta }+2)m}$. Then, we have

$$\begin{array}{c}\hfill \left({\widehat{D}}_p{\widehat{D}}_q-{\displaystyle \frac{1}{2}}{\gamma }_{pq}\widehat{\Delta }\right)k_{(\widehat{\Delta }+2)m}={ϵ}_{r(p}{\widehat{D}}_{q)}{\widehat{D}}^r{k}_{(\widehat{\Delta }+2)m}=0.\end{array}$$

(177)

In this case, the only remaining components of the linearized energy–momentum tensor ${}^{\left(1\right)}{\mathcal{T}}_{ab}$ are given by

$$\begin{array}{ccc}\hfill {}^{\left(1\right)}{\mathcal{T}}_{ab}& =& \sum _{m=-1}^1{\tilde{T}}_{AB}{k}_{\left(\widehat{\Delta +2}\right)}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b\hfill \end{array}$$

$$\begin{array}{ccc}& & +2r\sum _{m=-1}^1\left\{{\tilde{T}}_{\left(e1\right)A}{\widehat{D}}_p{k}_{(\widehat{\Delta }+2)m}+{\tilde{T}}_{\left(o1\right)A}{ϵ}_{pr}{\widehat{D}}^r{k}_{(\widehat{\Delta }+2)m}\right\}{\left(d{x}^A\right)}_{(a}{\left(d{x}^p\right)}_{b)}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{2}}{r}^2{\gamma }_{pq}\sum _{m=-1}^1{\tilde{T}}_{\left(e0\right)}{k}_{(\widehat{\Delta }+2)m}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\hfill \end{array}$$

(178)

Therefore, for even-mode perturbations, we can safely regard that

$$\begin{array}{c}\hfill {\tilde{T}}_{\left(e2\right)}=0.\end{array}$$

(179)

From Equations (34) and (179), the component ${\tilde{F}}_{AB}$ is traceless. Then, we may concentrate on the components $X_{\left(e\right)}$ and $Y_{\left(e\right)}$, defined by Equation (42), and the component $\tilde{F}$ as the metric perturbations. Furthermore, all arguments in Section 3 are valid even in the case of $l=1$ modes. Therefore, we may use Equations (77)–(89) when we derive the $l=1$-mode solutions to the linearized Einstein equations.

From Definition (75) of $\Lambda$, we obtain

$$\begin{array}{c}\hfill \Lambda =3(1-f).\end{array}$$

(180)

Then, the Moncrief variable ${\Phi }_{\left(e\right)}$, defined by Equation (77), is given by

$$\begin{array}{c}\hfill {\Phi }_{\left(e\right)}:={\displaystyle \frac{r}{3(1-f)}}\left[f{X}_{\left(e\right)}-{\displaystyle \frac{3(1-f)}{4}}\tilde{F}+{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\right].\end{array}$$

(181)

In other words, the component $X_{\left(e\right)}$ is given by

$$\begin{array}{c}\hfill f{X}_{\left(e\right)}={\displaystyle \frac{3(1-f)}{r}}{\Phi }_{\left(e\right)}+{\displaystyle \frac{3(1-f)}{4}}\tilde{F}-{\displaystyle \frac{1}{2}}rf{\partial }_r\tilde{F}\end{array}$$

(182)

as a solution to the linearized Einstein equation if the variables ${\Phi }_{\left(e\right)}$ and $\tilde{F}$ are given as solutions to the linearized Einstein equation. Furthermore, from Equations (79) and (80), we obtain

$$\begin{array}{ccc}\hfill \tilde{F}& =& -4f{\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{4(1-f)}{r}}{\Phi }_{\left(e\right)}-{\displaystyle \frac{32\pi r^2}{3(1-f)}}{\tilde{T}}_{tt},\hfill \end{array}$$

(183)

$$\begin{array}{ccc}\hfill f{Y}_{\left(e\right)}& =& rf{\partial }_t\left(X_{\left(e\right)}+{\displaystyle \frac{r}{2}}{\partial }_r\tilde{F}\right)+{\displaystyle \frac{3f-1}{4}}r{\partial }_t\tilde{F}+8\pi r^{2}f{\tilde{T}}_{tr}\hfill \end{array}$$

$$\begin{array}{ccc}& =& (1-f){\partial }_t{\Phi }_{\left(e\right)}-2rf{\partial }_t{\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{16\pi r^3}{3(1-f)}}{\partial }_t{\tilde{T}}_{tt}+8\pi r^{2}f{\tilde{T}}_{tr},\hfill \end{array}$$

(184)

where we use Equations (182) and (183) in the derivation of Equation (184). Under the given components ${\tilde{T}}_{tt}$ and ${\tilde{T}}_{tr}$ of the linearized energy–momentum tensor, Equations (183) and (184) yield that the components $\tilde{F}$ and $Y_{\left(e\right)}$ are determined by ${\Phi }_{\left(e\right)}$. Furthermore, substituting Equation (183) into Equation (182), we obtain

$$\begin{array}{ccc}\hfill f{X}_{\left(e\right)}& =& -{\displaystyle \frac{f(1-f)}{r}}{\Phi }_{\left(e\right)}-f(1-f){\partial }_r{\Phi }_{\left(e\right)}+2rf{\partial }_r\left(f{\partial }_r{\Phi }_{\left(e\right)}\right)\hfill \end{array}$$

$$\begin{array}{c}-8\pi r^2{\tilde{T}}_{tt}+{\displaystyle \frac{16\pi r^{2}f}{(1-f)}}{\tilde{T}}_{tt}+{\displaystyle \frac{16\pi r^{3}f}{3(1-f)}}{\partial }_r{\tilde{T}}_{tt}.\hfill \end{array}$$

(185)

This also yields that the component $X_{\left(e\right)}$ is determined by ${\Phi }_{\left(e\right)}$ under the given components of the linearized energy–momentum tensor. Thus, the components $X_{\left(e\right)}$, $Y_{\left(e\right)}$, and $\tilde{F}$ are determined by the single variable ${\Phi }_{\left(e\right)}$, apart from the contribution from the components of the linearized energy–momentum tensor.

The determination of the Moncrief variable ${\Phi }_{\left(e\right)}$ is accomplished by solving the master Equation (85):

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]-{\displaystyle \frac{1-f}{r^2}}{\Phi }_{\left(e\right)}=16\pi {\displaystyle \frac{r}{\Lambda }}{S}_{\left({\Phi }_{\left(e\right)}\right)}.\end{array}$$

(186)

And, the source term in Equation (85) is given by

$$\begin{array}{ccc}\hfill S_{\left({\Phi }_{\left(e\right)}\right)}& =& {\displaystyle \frac{1}{2}}r{\partial }_t{\tilde{T}}_{tr}-{\displaystyle \frac{1}{2}}r{\partial }_r{\tilde{T}}_{tt}+{\displaystyle \frac{1-4f}{2f}}{\tilde{T}}_{tt}-{\displaystyle \frac{f}{2}}{\tilde{T}}_{rr}-f{\tilde{T}}_{\left(e1\right)r}.\hfill \end{array}$$

(187)

The master variable ${\Phi }_{\left(e\right)}$ is determined through the master Equation (186) with appropriate initial conditions.

Furthermore, we have to take into account the perturbation of the divergence of the energy–momentum tensor, which is summarized as follows:

$$\begin{array}{ccc}& & -{\partial }_t{\tilde{T}}_{tt}+f^2{\partial }_r{\tilde{T}}_{rt}+{\displaystyle \frac{(1+f)f}{r}}{\tilde{T}}_{rt}-{\displaystyle \frac{2f}{r}}{\tilde{T}}_{\left(e1\right)t}=0,\hfill \end{array}$$

(188)

$$\begin{array}{ccc}& & -{\partial }_t{\tilde{T}}_{tr}+{\displaystyle \frac{1-f}{2rf}}{\tilde{T}}_{tt}+f^2{\partial }_r{\tilde{T}}_{rr}+{\displaystyle \frac{(3+f)f}{2r}}{\tilde{T}}_{rr}-{\displaystyle \frac{2f}{r}}{\tilde{T}}_{\left(e1\right)r}-{\displaystyle \frac{f}{r}}{\tilde{T}}_{\left(e0\right)}=0,\hfill \end{array}$$

(189)

$$\begin{array}{ccc}& & -{\partial }_t{\tilde{T}}_{\left(e1\right)t}+f^2{\partial }_r{\tilde{T}}_{\left(e1\right)r}+{\displaystyle \frac{(1+2f)f}{r}}{\tilde{T}}_{\left(e1\right)r}+{\displaystyle \frac{f}{2r}}{\tilde{T}}_{\left(e0\right)}=0.\hfill \end{array}$$

(190)

The expression of (187) for the source term $S_{\left({\Phi }_{\left(e\right)}\right)}$ in Equation (186) is derived by using Equation (189).

5.1. $l=1$-Mode Vacuum Case

As in the case of $l=0$ modes, it is instructive to consider the vacuum case where all components of the linearized energy–momentum tensor vanish before the derivation of the non-vacuum case.

Here, we consider the covariant form ${\mathcal{F}}_{ab}$ of the $l=1$-mode metric perturbation as follows:

$$\begin{array}{c}\hfill {\mathcal{F}}_{ab}=\sum _{m=-1}^1{\tilde{F}}_{AB}{k}_{(\Delta +2)m}{\left(d{x}^A\right)}_a{\left(d{x}^B\right)}_b+{\displaystyle \frac{1}{2}}\sum _{m=-1}^1{\gamma }_{pq}{r}^2\tilde{F}{k}_{(\Delta +2)m}{\left(d{x}^p\right)}_a{\left(d{x}^q\right)}_b.\end{array}$$

(191)

The harmonic function $k_{(\Delta +2)m}$ is explicitly given by Equations (26) and (27). If we impose regularity on these harmonics by the choice $\delta =0$, these harmonics are given by the spherical harmonics $Y_{l=1,m}$ with $l=1$:

$$\begin{array}{c}\hfill Y_{l=1,m=0}\propto cos\theta ,Y_{l=1,m=1}\propto sin\theta e^{i\varphi },Y_{l=1,m=-1}\propto sin\theta e^{-i\varphi }.\end{array}$$

(192)

Since the extension of our arguments to $m=\pm 1$ modes is straightforward, we concentrate only on $m=0$ modes.

For the $m=0$ mode, the gauge-invariant part ${\mathcal{F}}_{ab}$ of the metric perturbation is given by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& \left(f{X}_{\left(e\right)}\right)cos\theta \left\{-{\left(dt\right)}_a{\left(dt\right)}_b-f^{-2}{\left(dr\right)}_a{\left(dr\right)}_b\right\}+{\displaystyle \frac{2}{f}}\left(f{Y}_{\left(e\right)}\right)cos\theta {\left(dt\right)}_{(a}{\left(dr\right)}_{b)}\hfill \end{array}$$

$$\begin{array}{c}+{\displaystyle \frac{1}{2}}{r}^2\tilde{F}cos\theta \left\{{\left(d\theta \right)}_a{\left(d\theta \right)}_b+{sin}^2\theta {\left(d\varphi \right)}_a{\left(d\varphi \right)}_b\right\}.\hfill \end{array}$$

(193)

By choosing ${\tilde{T}}_{tt}={\tilde{T}}_{tr}=0$ in Equations (183)–(185), we obtain the vacuum solutions $\tilde{F}$, $Y_{\left(e\right)}$, and $X_{\left(e\right)}$ of the metric perturbation as follows:

$$\begin{array}{ccc}\hfill X_{\left(e\right)}& =& -{\displaystyle \frac{1}{r}}(1-f){\Phi }_{\left(e\right)}-(1-f){\partial }_r{\Phi }_{\left(e\right)}+2r{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right],\hfill \end{array}$$

(194)

$$\begin{array}{ccc}\hfill Y_{\left(e\right)}& =& {\partial }_t\left[+{\displaystyle \frac{1}{f}}(1-f){\Phi }_{\left(e\right)}-2r{\partial }_r{\Phi }_{\left(e\right)}\right],\hfill \end{array}$$

(195)

$$\begin{array}{ccc}\hfill \tilde{F}& =& -4f{\partial }_r{\Phi }_{\left(e\right)}-4{\displaystyle \frac{1-f}{r}}{\Phi }_{\left(e\right)}.\hfill \end{array}$$

(196)

Here, ${\Phi }_{\left(e\right)}$ is a solution to the equation

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]-{\displaystyle \frac{1-f}{r^2}}{\Phi }_{\left(e\right)}=0.\end{array}$$

(197)

As in the case of the $l=0$ mode, we consider the problem of whether Solution (193) with Equations (194)–(196) is described by ${\pounds }_V{g}_{ab}$ for the appropriate vector field $V_a$ or not. From the symmetry of the above solution, we consider the case where the vector field $V_a$ is given by

$$\begin{array}{c}\hfill V_a=V_t{\left(dt\right)}_a+V_r{\left(dr\right)}_a+V_{\theta }{\left(d\theta \right)}_a,{\partial }_{\varphi }{V}_t={\partial }_{\varphi }{V}_r={\partial }_{\varphi }{V}_{\theta }=0\end{array}$$

(198)

and calculate all components of ${\pounds }_V{g}_{ab}$. We note that all components of ${\mathcal{F}}_{ab}$ given by Equation (193) are proportional to $cos\theta$. Therefore, if we may identify some components of ${\mathcal{F}}_{ab}$ with ${\pounds }_V{g}_{ab}$, the $\theta$-dependence of the components in Equation (198) should be given by

$$\begin{array}{c}\hfill V_a=v_t(t,r)cos\theta {\left(dt\right)}_a+v_{r}cos\theta {\left(dr\right)}_a+v_{\theta }sin\theta {\left(d\theta \right)}_a.\end{array}$$

(199)

Then, the non-trivial components of ${\pounds }_V{g}_{ab}$ are given by

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tt}=\left(2{\partial }_t{v}_t-f{f}^{\prime }{v}_r\right)cos\theta \ne 0,\hfill \end{array}$$

(200)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tr}=\left({\partial }_t{v}_r+{\partial }_r{v}_t-{\displaystyle \frac{f^{\prime }}{f}}{v}_t\right)cos\theta \ne 0,\hfill \end{array}$$

(201)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{t\theta }=\left({\partial }_t{v}_{\theta }-v_t\right)sin\theta =0,\hfill \end{array}$$

(202)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{rr}=2f^{-1/2}{\partial }_r\left(f^{1/2}{v}_r\right)cos\theta \ne 0,\hfill \end{array}$$

(203)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{r\theta }=\left(r^2{\partial }_r\left({\displaystyle \frac{1}{r^2}}{v}_{\theta }\right)-v_r\right)sin\theta =0,\hfill \end{array}$$

(204)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\theta \theta }=2\left(v_{\theta }+rf{v}_r\right)cos\theta \ne 0,\hfill \end{array}$$

(205)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\varphi \varphi }=2\left(rf{v}_r+v_{\theta }\right){sin}^2\theta cos\theta \ne 0.\hfill \end{array}$$

(206)

From Equations (202) and (204), we obtain

$$\begin{array}{c}\hfill r^{2}v(t,r):=v_{\theta },v_t={\partial }_t{v}_{\theta }=r^2{\partial }_{t}v,v_r=r^2{\partial }_r\left({\displaystyle \frac{1}{r^2}}{v}_{\theta }\right)=r^2{\partial }_{r}v,\end{array}$$

(207)

i.e.,

$$\begin{array}{c}\hfill V_a=r^2{\partial }_{t}vcos\theta {\left(dt\right)}_a+r^2{\partial }_{r}vcos\theta {\left(dr\right)}_a+r^{2}vsin\theta {\left(d\theta \right)}_a.\end{array}$$

(208)

Then, Equations (200)–(206) are summarized as

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tt}=r^2\left(2{\partial }_t^{2}v-{\displaystyle \frac{f(1-f)}{r}}{\partial }_{r}v\right)cos\theta ,\hfill \end{array}$$

(209)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tr}={\partial }_t\left(2r^2{\partial }_{r}v-{\displaystyle \frac{1-3f}{f}}rv\right)cos\theta ,\hfill \end{array}$$

(210)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{rr}=2f^{-1/2}{\partial }_r\left(f^{1/2}{r}^2{\partial }_{r}v\right)cos\theta ,\hfill \end{array}$$

(211)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\theta \theta }=2r^2\left(rf{\partial }_{r}v+v\right)cos\theta .\hfill \end{array}$$

(212)

As the first trial, we consider the correspondence

$$\begin{array}{c}\hfill {\pounds }_V{g}_{\theta \theta }={\mathcal{F}}_{\theta \theta },\end{array}$$

(213)

i.e.,

$$\begin{array}{c}\hfill rf{\partial }_{r}v+v=-f{\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{1-f}{r}}{\Phi }_{\left(e\right)}.\end{array}$$

(214)

As the second trial, we consider the correspondence

$$\begin{array}{c}\hfill {\pounds }_V{g}_{rr}={\mathcal{F}}_{rr},\end{array}$$

(215)

i.e.,

$$\begin{array}{c}\hfill -{\displaystyle \frac{1-5f}{f}}r{\partial }_{r}v+{\displaystyle \frac{2r^2}{f}}{\partial }_r\left(f{\partial }_{r}v\right)={\displaystyle \frac{1-f}{rf}}{\Phi }_{\left(e\right)}+{\displaystyle \frac{1-f}{f}}{\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{2r}{f}}{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right].\end{array}$$

(216)

From Equations (214) and (216), we obtain

$$\begin{array}{c}\hfill v=-{\displaystyle \frac{1}{r}}{\Phi }_{\left(e\right)}.\end{array}$$

(217)

Substituting Equation (217) into Equation (201), we obtain

$$\begin{array}{c}\hfill {\pounds }_V{g}_{tr}={\partial }_t\left(-2r{\partial }_r{\Phi }_{\left(e\right)}+{\displaystyle \frac{1}{f}}(1-f){\Phi }_{\left(e\right)}\right)cos\theta ={\mathcal{F}}_{tr}.\end{array}$$

(218)

Furthermore, substituting Equation (217) into Equation (200), we obtain

$$\begin{array}{ccc}\hfill {\pounds }_V{g}_{tt}& =& \left(-2r{\partial }_t^2{\Phi }_{\left(e\right)}-{\displaystyle \frac{f(1-f)}{r}}{\Phi }_{\left(e\right)}+f(1-f){\partial }_r{\Phi }_{\left(e\right)}\right)cos\theta \hfill \end{array}$$

$$\begin{array}{ccc}& =& -f\left(-{\displaystyle \frac{1}{r}}(1-f){\Phi }_{\left(e\right)}-(1-f){\partial }_r{\Phi }_{\left(e\right)}+2r{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]\right)cos\theta \hfill \end{array}$$

$$\begin{array}{ccc}& =& {\mathcal{F}}_{tt},\hfill \end{array}$$

(219)

where we use Equation (197).

Then, we have shown that

$$\begin{array}{c}\hfill {\mathcal{F}}_{ab}={\pounds }_V{g}_{ab},\end{array}$$

(220)

where

$$\begin{array}{c}\hfill V_a=-r{\partial }_t{\Phi }_{\left(e\right)}cos\theta {\left(dt\right)}_a+\left({\Phi }_{\left(e\right)}-r{\partial }_r{\Phi }_{\left(e\right)}\right)cos\theta {\left(dr\right)}_a-r{\Phi }_{\left(e\right)}sin\theta {\left(d\theta \right)}_a.\end{array}$$

(221)

Thus, the vacuum solution of $l=1$-mode perturbations is described by the Lie derivative of the background metric through the master Equation (197).

5.2. $l=1$-Mode Non-Vacuum Case

Here, we consider the non-vacuum solution to the $l=1$ even-mode linearized Einstein equations. In this non-vacuum case, we concentrate only on $m=0$ mode perturbations, as in the vacuum case, because the extension of our arguments to $m=\pm 1$ modes is straightforward. The solution is given by the covariant form (193), as in the case of the vacuum case. The non-vacuum solutions for the variable $\tilde{F}$, $Y_{\left(e\right)}$, and $X_{\left(e\right)}$ are given by Equations (183)–(185), respectively. The master variable ${\Phi }_{\left(e\right)}$ must satisfy the master Equation (186) with the source term (187). We have to emphasize that the components of the linear perturbation of the energy–momentum tensor satisfy the continuity Equations (188)–(190). Then, the components of the gauge-invariant part ${\mathcal{F}}_{ab}$ for $l=1$ even-mode non-vacuum perturbations are summarized as follows:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tt}& =& f\left[{\displaystyle \frac{1}{r}}(1-f){\Phi }_{\left(e\right)}+(1-f){\partial }_r{\Phi }_{\left(e\right)}-2r{\partial }_r\left[f{\partial }_r{\Phi }_{\left(e\right)}\right]\right]cos\theta \hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{8\pi r^2}{3(1-f)}}\left[3(1-3f){\tilde{T}}_{tt}-2rf{\partial }_r{\tilde{T}}_{tt}\right]cos\theta ,\hfill \end{array}$$

(222)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tr}& =& r{\partial }_t\left[{\displaystyle \frac{1-f}{rf}}{\Phi }_{\left(e\right)}-2{\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{16\pi r^2}{3f(1-f)}}{\tilde{T}}_{tt}\right]cos\theta +8\pi r^2{\tilde{T}}_{tr}cos\theta ,\hfill \end{array}$$

(223)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{rr}& =& {\displaystyle \frac{1}{f}}\left[{\displaystyle \frac{1-f}{r}}{\Phi }_{\left(e\right)}+(1-f){\partial }_r{\Phi }_{\left(e\right)}-2r{\partial }_r\left(f{\partial }_r{\Phi }_{\left(e\right)}\right)\right]cos\theta \hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{8\pi r^2}{3f^2(1-f)}}\left[3(1-3f){\tilde{T}}_{tt}-2rf{\partial }_r{\tilde{T}}_{tt}\right]cos\theta ,\hfill \end{array}$$

(224)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\theta \theta }& =& -2r\left[rf{\partial }_r{\Phi }_{\left(e\right)}+(1-f){\Phi }_{\left(e\right)}+{\displaystyle \frac{8\pi r^3}{3(1-f)}}{\tilde{T}}_{tt}\right]cos\theta ,\hfill \end{array}$$

(225)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\varphi \varphi }& =& -2r\left[rf{\partial }_r{\Phi }_{\left(e\right)}+(1-f){\Phi }_{\left(e\right)}+{\displaystyle \frac{8\pi r^3}{3(1-f)}}{\tilde{T}}_{tt}\right]{sin}^2\theta cos\theta .\hfill \end{array}$$

(226)

As seen in the vacuum case, if we choose the generator $V_a$ as Equation (221), i.e.,

$$\begin{array}{ccc}\hfill V_a=V_{\left(vac\right)a}& :=& -r{\partial }_t{\Phi }_{\left(e\right)}cos\theta {\left(dt\right)}_a+\left({\Phi }_{\left(e\right)}-r{\partial }_r{\Phi }_{\left(e\right)}\right)cos\theta {\left(dr\right)}_a\hfill \\ & & -r{\Phi }_{\left(e\right)}sin\theta {\left(d\theta \right)}_a,\hfill \end{array}$$

(227)

we obtain

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tt}=f\left[-{\displaystyle \frac{2r}{f}}{\partial }_t^2{\Phi }_{\left(e\right)}+(1-f){\partial }_r{\Phi }_{\left(e\right)}-{\displaystyle \frac{1-f}{r}}{\Phi }_{\left(e\right)}\right]cos\theta ,\hfill \end{array}$$

(228)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{tr}=r{\partial }_t\left({\displaystyle \frac{1-f}{rf}}{\Phi }_{\left(e\right)}-2{\partial }_r{\Phi }_{\left(e\right)}\right)cos\theta ,\hfill \end{array}$$

(229)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{rr}={\displaystyle \frac{1}{f}}\left[{\displaystyle \frac{1-f}{r}}{\Phi }_{\left(e\right)}+(1-f){\partial }_r{\Phi }_{\left(e\right)}-2r{\partial }_r\left(f{\partial }_r{\Phi }_{\left(e\right)}\right)\right]cos\theta ,\hfill \end{array}$$

(230)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\theta \theta }=-2r\left[rf{\partial }_r{\Phi }_{\left(e\right)}+(1-f){\Phi }_{\left(e\right)}\right]cos\theta ,\hfill \end{array}$$

(231)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{\varphi \varphi }=-2r\left(rf{\partial }_r{\Phi }_{\left(e\right)}+(1-f){\Phi }_{\left(e\right)}\right){sin}^2\theta cos\theta ,\hfill \end{array}$$

(232)

$$\begin{array}{ccc}\hfill & & {\pounds }_V{g}_{t\theta }={\pounds }_V{g}_{t\varphi }={\pounds }_V{g}_{r\theta }={\pounds }_V{g}_{r\varphi }={\pounds }_V{g}_{\theta \varphi }=0.\hfill \end{array}$$

(233)

Through these formulae of the components ${\pounds }_V{g}_{ab}$ and Equations (222)–(226) for the components of ${\mathcal{F}}_{ab}$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tt}& =& {\pounds }_V{g}_{tt}-{\displaystyle \frac{16\pi r^2{f}^2}{3(1-f)}}\left[{\displaystyle \frac{1+f}{2}}{\tilde{T}}_{rr}+rf{\partial }_r{\tilde{T}}_{rr}-{\tilde{T}}_{\left(e0\right)}-4{\tilde{T}}_{\left(e1\right)r}\right]cos\theta ,\hfill \end{array}$$

(234)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{tr}& =& {\pounds }_V{g}_{tr}-{\displaystyle \frac{16\pi r^3}{3f(1-f)}}\left[{\partial }_t{\tilde{T}}_{tt}-{\displaystyle \frac{3f(1-f)}{2r}}{\tilde{T}}_{tr}\right]cos\theta ,\hfill \end{array}$$

(235)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{rr}& =& {\pounds }_V{g}_{rr}-{\displaystyle \frac{16\pi r^3}{3f(1-f)}}\left[{\partial }_r{\tilde{T}}_{tt}-{\displaystyle \frac{3(1-3f)}{2rf}}{\tilde{T}}_{tt}\right]cos\theta ,\hfill \end{array}$$

(236)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\theta \theta }& =& {\pounds }_V{g}_{\theta \theta }-{\displaystyle \frac{16\pi r^4}{3(1-f)}}{\tilde{T}}_{tt}cos\theta ,\hfill \end{array}$$

(237)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\varphi \varphi }& =& {\pounds }_V{g}_{\varphi \varphi }-{\displaystyle \frac{16\pi r^4}{3(1-f)}}{\tilde{T}}_{tt}{sin}^2\theta cos\theta ,\hfill \end{array}$$

(238)

where we use Equation (186) with the source term (187) and the component (189) of the continuity equation in Equation (234). Equations (234)–(238) are summarized as

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{ab}& =& {\pounds }_V{g}_{ab}-{\displaystyle \frac{16\pi r^2}{3(1-f)}}\left[f^2\left\{{\displaystyle \frac{1+f}{2}}{\tilde{T}}_{rr}+rf{\partial }_r{\tilde{T}}_{rr}-{\tilde{T}}_{\left(e0\right)}-4{\tilde{T}}_{\left(e1\right)r}\right\}{\left(dt\right)}_a{\left(dt\right)}_b\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{2r}{f}}\left\{{\partial }_t{\tilde{T}}_{tt}-{\displaystyle \frac{3f(1-f)}{2r}}{\tilde{T}}_{tr}\right\}{\left(dt\right)}_{(a}{\left(dr\right)}_{b)}\hfill \end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{r}{f}}\left\{{\partial }_r{\tilde{T}}_{tt}-{\displaystyle \frac{3(1-3f)}{2rf}}{\tilde{T}}_{tt}\right\}{\left(dr\right)}_a{\left(dr\right)}_b\hfill \end{array}$$

$$\begin{array}{ccc}& & \left.+r^2{\tilde{T}}_{tt}{\gamma }_{ab}\right]cos\theta .\hfill \end{array}$$

(239)

We note that there may exist the term ${\pounds }_W{g}_{ab}$ in the right-hand side of Equation (239) in addition to the term ${\pounds }_V{g}_{ab}$ discussed above. Such a term depends on the equation of the state of the matter field. This situation can be seen in the Part III paper [37]. Even if we consider such terms, we will not have a simple expression of the metric perturbation, in general. Therefore, we will not carry out such further considerations here.

6. Summary and Discussion

In summary, after reviewing our general framework for the general relativistic gauge-invariant perturbation theory and our strategy for linear perturbations on the Schwarzschild background spacetime proposed in refs. [28,29], we developed the component treatments of the even-mode linearized Einstein equations. Our proposal in refs. [28,29] was on the gauge-invariant treatments of $l=0,1$ mode perturbations on the Schwarzschild background spacetime. Since we used singular harmonic functions at once in our proposal, we had to confirm whether our proposal is physically reasonable or not.

To confirm this, in the Part I paper [29], we carefully discussed the solutions to the Einstein equations for odd-mode perturbations. We obtained the Kerr parameter perturbations in the vacuum case, which are physically reasonable. In this paper, we carefully discussed the solutions to the even-mode perturbations. Due to Proposal 1, we could treat $l=0,1$-mode perturbations through an equivalent manner to $l\ge 2$-mode perturbations. For this reason, we derived the equations for even-mode perturbations without making a distinction among $l\ge 0$ modes for even-mode perturbations.

To derive even-mode perturbations, it was convenient to introduce the Moncrief variable. In this paper, we explained the introduction of the Moncrief variable through an initial value constraint (61) which was regarded as an equation for the component $\tilde{F}$ of the metric perturbation and the Moncrief variable ${\Phi }_{\left(e\right)}$. This consideration led to the well-known definition of the Moncrief variable ${\Phi }_{\left(e\right)}$. Furthermore, from the evolution Equation (49), we obtained the well-known master Equation (85) for the Moncrief variable ${\Phi }_{\left(e\right)}$.

Moreover, we obtained the constraint Equations (79) and (80) together with the definition (78) of the Moncrief variable. From their derivations, we showed that these equations are valid not only for $l\ge 2$ but also for $l=0,1$ modes. We also checked the consistency of these equations, and we derived the identity of the source terms which are given by the components of the linear perturbation of the energy–momentum tensor. This identity was confirmed by the components of the linear perturbation of the energy–momentum tensor.

In this paper, we also carefully discussed the $l=0,1$ mode solutions to the linearized Einstein equations for even-mode perturbations to check that Proposal 1 is physically reasonable.

The $l=0$-mode solutions were discussed in Section 4. After summarizing the linearized Einstein equations and the linearized continuity equations for the generic matter field for the $l=0$ mode, we first considered the vacuum solution of $l=0$-mode perturbations following Proposal 1. Then, we showed that the additional mass parameter perturbation of the Schwarzschild spacetime is the only solution apart from the terms of the Lie derivative of the background metric $g_{ab}$ in the vacuum case. This is the gauge-invariant realization of the linearized version of the Birkhoff theorem [36].

In the non-vacuum case, we used the method of the variational constant with the Schwarzschild mass constant parameter in the vacuum case. Then, we obtained the general non-vacuum solution to the linearized Einstein equation for the $l=0$ mode. As the result, we obtained the linearized metric (174). Solution (174) includes the additional mass parameter perturbation $M_1$ of the Schwarzschild mass and the integration of the energy density. Furthermore, in Solution (174), we have the $2{\left(dt\right)}_{(a}{\left(dr\right)}_{b)}$ term due to the integration of the components of the energy–momentum tensor. In the solution (174), we also have the term which has the form of the Lie derivative of the background metric $g_{ab}$. The off-diagonal term of $2{\left(dt\right)}_{(a}{\left(dr\right)}_{b)}$ can be eliminated by the replacement of the generator $V_a$ of the Lie derivative of the background metric $g_{ab}$. However, if we eliminate the off-diagonal term of $2{\left(dt\right)}_{(a}{\left(dr\right)}_{b)}$ through the replacement of the generator $V_a$, we have an additional term in the diagonal components of the linearized metric perturbation (174). Since these diagonal components have complicated forms, we did not carry out this displacement.

We also discussed $l=1$-mode perturbations in Section 5. In this paper, we concentrated only on the $m=0$ mode, since the extension to $m=\pm 1$ modes is straightforward. The solution of the $l=1$ mode was obtained through a similar strategy to the case of $l\ge 2$ modes which were discussed in Section 3. As in the case of $l=0$-mode perturbations, we first discussed the vacuum solution for $l=1$-mode perturbations. As a result, $l=1$-mode vacuum metric perturbations are described by the Lie derivative of the background metric $g_{ab}$ with an appropriate operator. On the other hand, in non-vacuum $l=1$-mode perturbations, the $l=1$ mode metric perturbation have a contribution from the components of the energy–momentum tensor of the matter field in addition to the term of the Lie derivative of the background metric $g_{ab}$ which was derived as the above vacuum solution.

As the odd-mode solutions in the Part I paper [29], we also have the terms of the Lie derivative of the background metric $g_{ab}$ in the derived solutions in the $l=0,1$ even-mode solutions. We have to recall that our definition of gauge-invariant variables is not unique, and we may always add the term of the Lie derivative of the background metric $g_{ab}$ with a gauge-invariant generator, as emphasized in Section 2.1. In other words, we may have such terms in derived solutions at any time, and we may say that the appearance of such terms is a natural consequence due to the symmetry in the definition of gauge-invariant variables. Furthermore, since our formulation completely excludes the second-kind gauge through Proposal 1, these terms of the Lie derivative should be regarded as the degree of freedom of the first-kind gauge, i.e., the coordinate transformation of the physical spacetime ${\mathcal{M}}_{ϵ}$, as emphasized in the Part I paper [29]. This discussion is the consequence of our distinction of the first- and second-kind gauges and the complete exclusion of the degree of freedom of the second-kind gauge, as emphasized in the Part I paper [29].

We also note that the existence of the additional mass parameter perturbation $M_1$ requires the perturbations of $\tilde{F}$ due to the linearized Einstein equations. In this sense, the term described by the Lie derivative of the background spacetime is necessary. The solutions derived in this paper and the Part I paper [29] are local perturbative solutions. If we construct the global solution, we have to use the solutions obtained in this paper and in the Part I paper [29] as local solutions and have to match these local solutions. We expect that the term of the Lie derivative derived here will play important roles in this case.

Besides the term of the Lie derivative of the background metric $g_{ab}$, we realized the Birkhoff theorem for $l=0$ even-mode solutions and the Kerr parameter perturbations in $l=1$ odd-mode solutions. These solutions are physically reasonable. This also implies that Proposal 1 is physically reasonable. Nevertheless, we used singular mode functions at once to construct gauge-invariant variables and imposed the regular boundary condition for the functions on $S^2$ when we solved the linearized Einstein equations, while the conventional treatment through the decomposition through the spherical harmonics $Y_{lm}$ corresponded to the imposition of the regular boundary condition from the starting point.