Stability of the Schwarzschild-de Sitter black hole in the dRGT massive gravity theory

The Schwarzschild-de Sitter solution in the Einstein theory with a positive cosmological constant $\Lambda=m^2/\alpha$ becomes an exact solution to the dRGT non-linear massive gravity theory with the mass parameter $m$ when the theory parameters $\alpha$ and $\beta$ satisfy the relation $\beta=\alpha^2$. We study the perturbative behaviour of this black hole solution in the non-linear dRGT theory with $\beta=\alpha^2$. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of the gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the St\"uckelberg field sector at the linear perturbation level in this background. Physical significance of this degenercy depends on how the St\"uckelberg fields couple observable fields.


Introduction
One of the biggest problems in cosmology is to explain the current accelerated expansion of the universe. In the standard theory of gravity, i.e. general relativity, this reduces to the cosmological constant (Λ) problem or the dark energy problem [1,2] if we require the spatial homogeneity(Cf. [3][4][5]). Beside this standard approach, many alternative theories have been suggested in order to solve this problem. Among the most populars, we have modified gravity theories (MOG) [6], non-localities [7] and massive gravity theories [8], which are just large scale modifications of gravity.
In order for such a theory to be a real theory of nature, it must be consistent with all the observed features. In particular, it must be consistent with the 'observed' existence of astrophysical black holes. In many case, this requirement leads to non-trivial constraints. For example, it was recently claimed that the bi-Schwarzschild solution is unstable against a spherically symmetric perturbation in the bi-metric theory of gravity [9]. Motivated by this, the stability of the Schwarzschild-de Sitter black hole was analyzed in the framework of the linear massive gravity theory by Brito, Cardoso and Pani [10,11]. They found that the black hole is unstable generically, but becomes stable when the mass of the graviton takes the particular value m 2 = 2Λ/3. In this case, the theory is inside the regime of partially massless gravity, where the Vainshtein mechanism seems to be unnecessary since the DVZ discontinuity does not appear anymore [12]. However, it has been demonstrated that the partially massless theories of gravity have several problems of consistency [13].
In the present paper, we analyze the stability of the Schwarzschild-de Sitter solution in the framework of the non-linear dRGT massive theory of gravity. We do not introduce the cosmological constant as an extra parameter of the theory, but instead, we utilize the fact that the Schwarzschild-de Sitter black hole is an exact solution to the non-linear dRGT theory if the parameters α = 1 + 3α 3 and β = 3(α 3 + 4α 4 ) of the theory satisfy the relation β = α 2 . For this parameter choice, the mass term of the theory behaves exactly as the cosmological constant term in the Einstein theory for a spherically symmetric geometry as pointed out by Berezhiani et al [14]. We exhaust all Schwarzschild-de Sitter-type solutions to the non-linear dRGT theory in the unitary gauge for the Stückelberg fields assuming β = α 2 . We find a family of solutions that are gauge equivalent to the standard Schwarzschildde Sitter solution if we neglect the non-trivial transformation of the Stückelberg fields. In the massive gravity theory, they should be regarded as different solutions because if the metric are put into the standard Schwarzschild-de Sitter form, the Stückelberg fields behave differently.The solution obtained in [14] is one solution in this family that is regular at the future horizon. There exists no solution that is regular both at the future and the past horizons.
We consider linear perturbations of this background solution in the framework of the nonlinear dRGT theory only assuming the parameter relation β = α 2 . Hence, we generally expect to obtain perturbation equations that are different from those in the Einstein theory with the cosmological constant. In fact, we do if we do not impose the constraint coming from the Bianchi identity on the mass term. However, when we impose that constraint, the extra terms are required to vanish. Hence, we obtain the perturbation equations that are identical to those in the Einstein theory with a cosmological constant and some additional constraints on the metric perturbation variables that correspond to the gauge-dependent parts in the Einstein theory. From this result and the Birkhoff theorem for the Einstein theory, we can easily find the general solution to the perturbation equations and deduce the stability of the black hole against linear perturbations concerning the spacetime structure. However, we also find that this general solution contains an arbitrary function of the spacetime coordinates that reduces to a part of the gauge transformation freedom in the absense of the Stückelberg fiels. In the gauge in which the background metric takes the standard Schwarzschild-de Sitter form, this freedom goes to the Stückelberg fields. Hence, we cannot determine the behavior of the fields only by initial data. Along with this general argument, we point out that the general solution to the vector-type perturbation equations contains a family of stationary modes that correspond the rotation of a black hole in the Einstein theory.
The paper is organized as follows. In Section 2, we summarize the basic part of the dRGT non-linear massive gravity formalism that is relevant to the present paper. In Section 3, we show that the mass term in the field equation of the dRGT theory becomes identical to the cosmological constant term for an arbitrary spherically symmetric metric in the unitary gauge for the Stückelberg fields when the theory parameters satisfy the relation β = α 2 , and that as a consequence the Schwarzschild-de Sitter spacetime becomes an exact solution in the dRGT theory for this parameter relation. We also develop details of the Schwarzschild-de Sitter solution in the dRGT theory. In Section 4, we make a brief review of the gaugeinvariant formulation for perturbations of a black hole. In Section 5, we derive perturbation equations for the Schwarzschild-de Sitter type background in the dRGT theory, and then in Section 6, we introduce gauge-invariant variables for the present system by treating the Stückelberg fields to be dynamical and express the perturbation equations in terms of them. In Section 7, we summarize and conclude. In Appendix A, we show that there exist other parameter choices for which the dRGT theory admits a Schwarzschild-de Sitter type solution and exhaust all possibilities.

The dRGT theory
In the standard formalism of the dRGT theory, the action is given by [8] with the effective potential depending on two free parameters as The dependence of each term U n on the metric g and the Stückelberg field φ a is determined in terms of the matrix Q = (Q µ ν ) defined by as The potential U is unique. It is impossible to add polynomial terms without introducing a ghost [8,14]. By taking a variation of the action with respect to the metric, we obtain the field equation where Its mixed components X = (X µ ν ) = g µλ X λν can be explicitly expressed in the matrix form in terms of the matrix Q as with Throughout the present paper, we use α and β instead of α 3 and α 4 . In this generally covariant formulation, we can regard the Stückelberg fields either to be dynamical or to be non-dynamical. This is because the dynamical equation for φ a obtained from the action by a variation with respect to φ a is practically equivalent to the consistency equation obtained from (6) by the Bianchi identity.
To see this, we use the diffeomorphism invariance of the mass term of the action, For an infinitesimal coordinate transformation this equation leads to Because ζ µ is an arbitrary vector field, we obtain Therefore, if the field equation (6) holds, the left-hand side of this equation should vanish due to the Bianchi identity ∇ ν G ν µ ≡ 0. Because ∂ µ φ a is a regular matrix, this constraint is equivalent to the Euler equation for the Stückelberg field,

The Schwarzschild-de Sitter solution
If the Schwarzschild-de Sitter solution satisfies the field equations in massive gravity, the tensor X µν becomes a constant multiple of g µν for that metric [14]: Conversely, if a solution to the field equations (6) satisfies this relation, it must be a solution to the vacuum Einstein equations with Λ. Hence, if it is spherically symmetric, the solution must be diffeomorphic to the Schwarzschild-de Sitter solution. Note that this does not implies the uniqueness of the solution because although the matrices of two solutions are related by a coordinate transformation, the Stückelberg fields may not be related by the same transformation.
In this section, we examine under what conditions (16) holds for spherically symmetric spacetimes. In particular, we show that if the parameters α 3 and α 4 satisfy the relation any spherically symmetric metric of the form ds 2 = g tt (t, r)dt 2 + 2g tr (t, r)dtdr + g rr (t, r)dr 2 + r 2 S(t, r) 2 dΩ 2 2 (18) satisfies the condition (16) with if S(t, r) is a constant given by Note that the cosmological constant Λ is different from zero for any finite value of α if m 2 = 0.
To prove this, we work in the unitary gauge in which the Stückelberg fields φ a are given by in the Cartesian Minkowski coordinates. In this gauge, the reference metric f µν in the spherical coordinates is given by Hence, for the metric (18), the matrix M 2 defined by (3a) is given by From this, we find that the matrix Q can be expressed in the form where a, b and c are expressed in terms of the metric coefficients as with M 1 = (−g (2) ) −1/2 −g tt + g rr + 2(−g (2) ) 1/2 1/2 , (26) We can also express g µν in terms of the components of Q as In particular, If we substitute the expression for Q in terms of a, b, c and S into (8), we get where F 1 , F 2 and F 3 are functions of S defined by Now, it is easy to see that all of F 1 , F 2 and F 3 vanish if the relations (17) and (20) hold. This means that X = (X µ ν ) becomes a multiple of the unit matrix: Note that this holds independent of the functional dependences of a(t, r), b(t, r) and c(t, r).
If we require that the metric (18) be a solution of the field equations (6) with (17), owing to the Birkhoff theorem for the Einstein vacuum system, it must be isomorphic to the Schwarzschild-de Sitter solution in the standard form for which g tt = −f (r), g tr = 0 and g rr = 1/f (r) with f (r) = 1 − 2M/r − Λr 2 /3. The above result means that g tt , g tr and g rr obtained from this standard form by arbitrary change of time coordinate t → T (t, r) also satisfies the field equations (6). Because we have already fixed the spacetime coordinates by the unitary gauge condition (21), these solutions obtained from the standard form by fixing 6/26 the Stückelberg fields and applying the coordinate transformation only to the metric should be regarded to be inequivalent mutually.
Finally, we notice that the above parameter relation is not the only case in which a metric isomorphic to the Schwarzschild-de Sitter solution satisfies the field equation (6). In the Appendix A, we exhaust all such possibilities.

Gauge invariant formulation for Black Hole perturbations
In this section, we introduce some notions to describe perturbations of a black hole spacetime and its gauge-invariant treatment formulated previously in [15,16]. We start from a general spherically symmetric background metric given by where g ab is the metric of a two-dimensional spacetime N 2 and is the metric of a unit two-sphere S 2 , whose Ricci tensor is given byR ij = γ ij . We denote the covariant derivative, connection coefficients and curvature tensors as for the four-dimensional whole spacetime, for the two-dimensional spacetime N 2 , and for the 2-sphere S 2 . The spherical symmetry of the background requires the background energy-momentum tensor to be given by

Tensorial decomposition of perturbations
We classify perturbation variables into two different types according to their tensorial behavior on S 2 so that we get a decoupled closed set of differential equations for each type of perturbations. For that purpose, we decompose the tensors h ab (y), h ai (y) and h ij (y) on S 2 defined by the metric perturbation h µν = δg µν as into these irreducible tensorial components as follows.
First, h ab are scalar with respect to transformations over S 2 . Next, the vector h ai on S 2 can be uniquely decomposed into the scalar h a and the divergence-free vector h (1) ai as up to the addition of arbitrary functions only of y to h a , which correspond to the exceptional l = 0 mode (S-mode) in the harmonic expansion explained later. This implies that this exceptional mode for h a is a spurious mode and should be discarded.

7/26
Finally, the 2-tensor h ij on S 2 can be decomposed into three parts as For this decomposition, h T is uniquely determined up to functions belonging to the kernel of the operatorL ij , which consists of the S-mode (l = 0) and the l = 1 modes in the harmonic expansion. Similarly, h T i is unique up to a combination of the Killing vector of S 2 with arbitrary functions of y as coefficients. This corresponds to the exceptional mode with l = 1 in the harmonic expansion. These exceptional modes are spurious as the S-mode for h a and should be discarded in physical arguments. With this understanding, the scalar components (h ab , h a , h L , h T ) of the metric perturbation h µν describe the scalar perturbation, and the vector components (h In a similar way, we can decompose the energy-momentum perturbations as where Hence, the scalar and vector components of the perturbation of the energy-momentum tensor consist of (δT ab , δT a , δP, δT

Gauge invariant variables
The Einstein equations are invariant under the diffeomorphism generated by any vector field ζ M . The perturbation variable h µν and its image h µν − £ ζ g µν obtained by an infinitesimal diffeomorphism should represent the same physical situation. Then, we have an ambiguity since there are infinite varieties of values for the perturbation variables representing the same physical situation. One way to remove this redundancy is to construct gauge-invariant variables and express the perturbation equations in terms of them. This automatically extracts the physical degrees of freedom related to the perturbations. We start from the gauge transformation laws for perturbation variables. First, for the infinitesimal gauge transformation δx µ = ζ µ , the metric perturbation h µν transforms as Next, the perturbation of the energy-momentum tensor, δT µν , transforms as These transformation laws can be translated to those for the perturbation variables describing each type of perturbations by decomposing the vector field ζ µ into vector and scalar components as Now, we execute this translation and construct gauge-invariant variables for each type of perturbations.
4.2.1. Vector perturbations. For vector perturbations, the above gauge transformation law for the metric perturbation can be translated into the irreducible vector components as From this, it follows that the combination is gauge invariant for generic modes. On the other hand, for the exceptional mode, h T i does not exist, and only the combination In contrast to the metric perturbation, δT a i and δT i j for a vector perturbation of the energy-momentum tensor become gauge invariant by themselves : For the exceptional perturbations, τ (1)i j does not exist. Note that any gauge-invariant variable for a generic vector perturbation can be expressed as a linear combination of (F T i just transforms like ζ i . Hence, if we express this variable in terms of the gauge-invariant variables, gauge is automatically specified. The exceptional perturbations should be treated with more care.

Scalar perturbations.
For scalar perturbations, the scalar components of the metric perturbation transform as If we define X µ = (X a , X i =D i X L ) as 9/26 X µ just transforms like X µ → X µ + ζ µ : (X a , X L ) → (X a + T a , X L + S).
Hence, we can define the following set of gauge-invariant variables for a generic metric perturbation: For the exceptional modes, these are not gauge invariant.
Similarly, for generic matter perturbations, we can construct the following basic gaugeinvariants: For the exceptional modes, all or some of these are not gauge invariant. Further, for the S-modes, Σ ai and Π ij do not exist, and for the exceptional modes with l = 1, Π ij does not exist.
As in the vector case, any gauge invariant for generic scalar perturbations can be expressed as a combination of the variables (F L , Π (0) ) and their derivatives. Further, when we express the metric and matter perturbation variables in terms of these gauge invariants and X µ , we can fix gauge by specifying the X µ as a linear function of the gauge-invariant variables. In the next section, we work in the unitary gauge to derive perturbation equations for the Schwarzschild-de Sitter black hole in the dRGT theory, and then in Section 6, we will express the perturbation equations obtained there in the gaugeinvariant form using the formulation explained here.

Harmonic expansions
In practical arguments, it is often more convenient to use the harmonic expansions for perturbation variables and their gauge-invariant combinations. We also use it in the subsequent sections. So, we here give some expressions for scalar and vector harmonic expansions relevant to the analysis in our paper, but more details can be found in [15,16].
First, in order to expand vector perturbations, we use the irreducible harmonic vectors defined by the eigenvalue problem For S 2 , the eigenvalue k 2 v is given by Note that V i is proportional to ǫ ijD j S where S is some scalar harmonics with the same l.
The lowest mode with l = 1 is exceptional because it can be shown to be a Killing vector 10/26 field on S 2 and satisfies The basic variables for vector perturbations can be expanded in terms of the vector-type harmonic basis as and correspondingly, the gauge-invariant variables are expanded as for the case of generic modes satisfying m V := k 2 v − 1 = (l + 2)(l − 1) > 0, where the indices of the harmonic tensors are lowered and raised by γ ij . Here and in the following, we omit the index for the harmonic basis and the corresponding summation symbols for simplicity.
For the exceptional modes with m V = 0, i.e. l = 1, there is only one gauge-invariant: For scalar perturbations, we use a basis for the scalar harmonic functions satisfying the eigenvalue problem△ and the associated vector and tensors defined by In terms of these harmonic tensors, the perturbation variables for scalar perturbations can be expanded as and the corresponding gauge-invariant variables are For exceptional modes, τ T does not exist for the l = 0 and l = 1 modes, and Σ a does not exist for the l = 0 modes.

Perturbation analysis in the dRGT formalism
In this section, we derive perturbation equations for the Schwarzschild-de Sitter solution in the dRGT theory with non-linear mass terms. 11/26

Background solution
As we have shown in Section 3, when the theory parameters α and β satisfy the relation (17), the Schwarzschild-de Sitter solution in the form (18) with S = S 0 becomes an exact solution to the field equations of the dRGT theory in the unitary gauge (21) for the Stückelberg fields φ a . In this form of the solution, the extra constant factor S 0 appears in front of the angular part of the metric. In studying perturbations of this background, we remove this constant factor by the coordinate transformation S 0 r → r so that we can use various formula for perturbations in the literature: where the index a and b run over 0 and 1 with y 0 = t and y 1 = r. This coordinate transformation transforms the unitary gauge condition (21) on the Stückelberg field to in the Cartesian Minkowski coordinates, and the reference metric f µν to The metric (67) should be obtained from the standard form for the Schwarzschild-de Sitter solution by a coordinate transformation t → T 0 (t, r) where T 0 (t, r) is an arbitrary function of t and r with ∂ t T 0 = 0. Hence, Thus, the background solution has a degeneracy represented by an arbitrary function of t and r even under the spherical symmetry requirement. This degeneracy cannot be gauged away because of the existence of the Stückelberg fields. This implies that the dRGT theory is dynamically pathological at this background. We will see that this degeneracy extends to freedom represented by an arbitrary function of full coordinates in the linear perturbation level. The above r-coordinate rescaling also affects the Q matrix. Because the dRGT theory has general covariance, (g * f * ) = (g µα f αν ) transforms as Because the mixed tensor Q should behave exactly as g * f * under a coordinate transformation, the r-rescaling transforms Q from the old value Q ′ to 12/26 Note that due to the r-rescaling, the expression for g ab in terms of a, b, and c is modified as follows: Similarly, a, b and c are expressed in terms of the new metric g ab as withM 1 = (−g (2) ) −1/2 −g tt + S 2 0 g rr + 2S 0 (−g (2) g (2) = g tt g rr − g 2 tr . (77)

Perturbation of X
Now, we calculate the perturbation of the tensor X = (X µ ν ) corresponding to the metric perturbation where Y , Y i and Y ij represents the corresponding tensors for either the scalar or vector harmonics. For vector perturbations, the terms in proportion to Y do not exist. First, from (8), a perturbation of the matrix X is determined by δQ as Here, δχ n is a linear combination of δQ n , which is given by where h * * is the matrix notation for the mixed tensor h µ ν . 13/26 In general, δQ is determined as the solution to In solving this, it is important that the background metric g and the matrix M = g * f * are the direct sum of two-dimensional submatrices, because the calculations of δQ a b , δQ a i and δQ i j decouple from each other except for the calculation of δQ n , which can be directly calculated by the above formula. The results for δQ n are given in the Appendix B.
First, the angular part δQ i j can be easily calculated because 1 − Q (2) = (1/S 0 )I 2 : The corresponding components of δX are expressed in terms of this as The result of the calculation is where w(r) = 1 + α α β(c 2 + ab) + α(a + b) + 1 .
Next, for the t − r part, solving the matrix equation we obtain Inserting this into we find Finally, because (Q n ) a i = 0 for the background Q, we have 14/26 Here, Hence, Now, (81) for δQ a i reduces to Hence, we obtain By inserting the above background value for Q (1) , we find this vanishes identically!!:

Vector perturbations
For vector perturbations, the metric perturbation h µν = δg µν has the harmonic expansion Similarly, a vector perturbation of the energy-momentum tensor has the harmonic expansion where τ a and τ T are gauge-invariant. From the calculations in the previous section, we obtain These source terms have to satisfy the Bianchi identities, which for a vector perturbation reduce to [15,16] D a (r 3 τ a ) + (l + 2)(l − 1) 2[l(l + 1) − 1] 1/2 r 2 τ T = 0 ⇒ (l − 1)w(r)H T = 0.
Because w(r) = 0 for β = α 2 , it follows that H T = 0 for l ≥ 2. Hence, the perturbation equations are identical to those for the vacuum Einstein system, and for l ≥ 2, we obtain 15/26 the additional constraint H T = 0. This implies that the general solution to the perturbation equation is given by where F a is the gauge-invariant variable for vector perturbations satisfying the perturbed vacuum Einstein equations In particular, we can conclude that the system is stable for vector perturbations.
For the exceptional mode with l = 1 for which H T does not exist, F a is not gauge-invariant and transforms for ζ a = 0, ζ i = LV i as We know that the general solution for l = 1 in the Einstein case is a linear combination of this gauge mode and the rotational perturbation corresponding to the angular momentum component in the Kerr metric [16]. Hence, the general solution in the present case is given by In particular, this shows that the dRGT theory admit a rotational black hole solution in the linear perturbation level.

Scalar perturbations
For scalar perturbations, Hence, the perturbation of the effective energy-momentum tensor is given by The corresponding standard gauge-invariant variables are and τ T . These should satisfy the conservation laws [15,16] 1 1 16/26 where k 2 s = l(l + 1). These reduce to −2l(l + 1)H L = (l + 2)(l − 1)H T (l ≥ 1), (110a) Hence, for all modes including the case l = 0, 1 for which H T does not exist, we obtain the constraint H L = H T = 0, and the perturbation equations are identical to those for the vacuum Einstein system with Λ, which has the structure where E ab , E a and E L are tensors written as differential linear combinations of the gaugeinvariants F ab and F . In particular, no instability occurs. The general solution for l ≥ 2 is expressed in terms of the gauge-invariant quantities satisfying the perturbations equations for the vacuum Einstein system with Λ as where f t (t, r) is left as an arbitrary function. This corresponds to the freedom associated with the infinitesimal coordinate transformation, δt = T t S, δr = 0, δz i = 0: The exceptional modes with l = 0, 1 should be treated with care. First, for the S-mode with l = 0, the variables f a and H T do not exists. Hence, Now, F ab is not gauge invariant, and transforms as The residual gauge freedom is represented by δt = T t (t, r). This result is consistence with the existence of the degeneracy represented by the single function T 0 (t, r) in the background solution.
Because the solution satisfies the Einstein equations, from the Birkhoff theorem, we know that the general solution is a linear combination of the above gauge transformation from the background solution and the perturbation corresponding to the variation of the mass 17/26 parameter in the background metric, Next, for the l = 1 mode, there exists no H T again, but now we have f a . However, due to the absence of H T , F and F ab are not gauge invariant, and transforms under δy a = T a S and δz i = L(t.r)S i as L is restricted by the condition H L = 0 as Because we know that the corresponding solutions with l = 1 to the vacuum Einstein system is exhausted by (F, F ab ) obtained from the trivial solution (0, 0) by the above gauge transformation [15], the general solution to our perturbation equations with l = 1 is given by where

Gauge-invariant formulation for perturbations in the dRGT theory
Because the dRGT theory is a completely general covariant theory if the Stückelberg field is treated as a dynamical one, the perturbation equations can be also written in the gauge-invariant form by introducing gauge-invariant variables for the perturbation of the Stückelberg field φ α . Let us denote a perturbation of φ α as then, from the general theory, its gauge transformation under the coordinate transformation δ g x µ = ζ µ is given by In the unitary gauge, the background value of φ α is 18/26 Hence, for δ g y a = T a , δ g z i = LY i , σ a transforms as where

Vector perturbations
For vector perturbations, we have From we can construct a gauge-invariant variablê for generic modes with l ≥ 2, in addition to the standard gauge-invariant variable for the metric, Then, the source term for the massive gravity equation can be expressed in terms of it as τ a = 0, κ 2 τ T = m 2 w(r)kS 0σT .
Hence, in terms of the gauge-invariantσ T , our result is expressed aŝ This implies that the dynamical degree of freedom of the Stückelberg field is completely suppressed, and the perturbation of the metric behaves exactly in the same way as for the Einstein gravity. For the exceptional modes with l = 1, we only have a single gauge-invariant quantitŷ Our analysis showed that for l = 1, the general solution for F a is given bŷ where L(t, r) is an arbitrary function and α is an arbitrary constant corresponding to the angular momentum parameter. Thus, a functional degeneracy appears. 19/26

Scalar perturbations
For generic modes (l ≥ 2) of scalar perturbations, we adopt the gauge-invariant variables for σ α defined byσ In terms of these, the source terms corresponding to δX are expressed as We have found that all of these gauge-invariant source terms vanish, hencê butσ t (t, r) can be an arbitrary function. Hence, the functional degeneracy appears even for generic modes.
For the exceptional modes with l = 1, F , F ab ,σ a andσ T are not gauge invariant because we have to set H T = 0 in their definitions and transform as However, we can construct the following basic gauge invariants from these: The perturbation equations for these variables are obtained by the replacements F →F , F ab →F ab ,σ a →σ a ,σ T → 0.
Hence, the general solution for this case is expressed in terms of these variables aŝ where L(t, r) is an arbitrary function. Finally, for the exceptional modes with l = 0, from the gauge transformation formula we can construct the following gauge invariants from f ab , H L and σ a : We have shown that the general solution for l = 0 can be expressed in terms of these gauge invariants asF where T t is an arbitrary function of t and r, and δM is an arbitrary constant corresponding to the mass variation.

Summary and Conclusions
In the present paper, we first looked for the parameter relation for which the non-linear massive gravity theory admits the Schwarzschild-de Sitter black hole as an exact solution systematically. We found that when the parameters satisfies the relation β = α 2 , there exists a family of solutions parameterized by an arbitrary function T 0 (t, r), which are isomorphic to the Schwarzschild-de Sitter spacetime but are not equivalent if the configuration of the Stückelberg fields are taken into account. We next investigated the perturbative stability of this family of Schwarzschild-de Sittertype black holes in the framework of the dRGT formulation of the non-linear massive gravity with β = α 2 . We found that the perturbative equations derived from the field equations of the dRGT theory becomes identical to the perturbations equation for the vacuum Einstein theory with cosmological constant if we take into account the consistency condition obtained from the field equations by the Bianchi identity. This consistency condition is essentially equivalent to the field equation for the Stückelberg field. This implies that the Schwarzschildde Sitter black hole solution is stable in the non-linear massive gravity theory as far as the spacetime structure is concerned at least in the linear perturbation level, in contrast to the bi-Schwarzschild solution in the bi-metric theory.
In spite of this stability result, we found a pathological feature of the black hole solution in the dRGT theory with the parameter relation β = α 2 ; the general solution to the perturbation equations contains an arbitrary function of the spacetime coordinates. This implies that the predictability of dynamics is lost at least in the linear perturbation level around this black hole solution. This degeneracy can be removed by coordinate transformations if we neglect the Stückelberg fields. Hence, the pathology appears to come from the dynamics of the Stückelberg fields. Because the Schwarzschild-de Sitter black hole becomes an exact solution only when the higher-order mass terms exist, there is a possibility that this pathology might be removed in the non-linear level of perturbations. as S = α + β ± α 2 − β 1 + 2α + β , (A12) and the corresponding cosmological constant is given by The condition Λ = 0 is given by Note that (A11) has a solution for T 0 locally with respect to r at most in general. One exception is the solution Interestingly, this corresponds to a Finkelstein-type time coordinate which is regular at the future horizon or the past horizon.
Finally, as the summary of this appendix, we give an exhaustive list of the spherically symmetric solutions isomorphic to the Schwarzschild-de Sitter solution and the corresponding parameter constraints in the dRGT massive gravity theory: • Solution F: The solution with a flat metric. The metric form should be that of (A3) with S given by one of the values in (A4). No constraint on the parameters α and β is required. • Solution SdS-I: The Schwarzschild-de Sitter type solution discussed in Section 3. The cosmological constant is given by Λ = m 2 /α, and the metric is given by (A9) with S = α/(1 + α). The parameters are constrained as β = α 2 , but the function T 0 (t, r) can be arbitrary. • Solution SdS-II: The Schwarzschild-de Sitter type solution whose metric is given by (A9) with constant S given by (A12) and the cosmological constant (A13). The parameters α and β are weakly constrained as β < α 2 , but the function T 0 is constrained to those satisfying (A11).