Entropy and temperatures of Nariai black hole

The statistical entropy of the Nariai black hole in a thermal equilibrium is calculated by using the brick-wall method. Even if the temperature depends on the choice of the time-like Killing vector, the entropy can be written by the ordinary area law which agrees with the Wald entropy. We discuss some physical consequences of this result and the properties of the temperatures.


I. INTRODUCTION
It has been claimed that the entropy of a black hole is proportional to the surface area at the event horizon [1], and then the Schwarzschild black hole has been studied through the quantum field theoretic calculation [2]. One of the convenient methods to get the entropy is to use the brick-wall method, which gives the statistical entropy satisfying the area law of the black hole [3]. Then, there have been extensive applications to various black holes .
In fact, another way to obtain the entropy is to regard the entropy of black holes as the conserved Noether charge corresponding to the symmetry of time translation [27]. For the Einstein gravity, the Wald entropy is always given by the A H /(4G), where A H and G are the surface area at the event horizon and the Newton's gravitational constant, respectively.
Actually, there are many extended studies for the entropy as the Noether charge in the general theory of gravity including the higher power of the curvature [28][29][30][31][32][33].
The fact that the cosmological constant seems to be positive in our universe deserves to study the Schwarzschild black hole on the de Sitter background, which can be easily realized in the form of the Schwarzschild-de Sitter (SdS) spacetime. It has the black hole horizon and the cosmological horizon, and the observer lives between them. In this spacetime, the temperature of the black hole is different from the temperature due to the cosmological horizon  [39].
In this paper, we would like to study the statistical entropy of the Nariai black hole by using the brick-wall method. In section II, we introduce the SdS spacetime and the Nariai spacetime, and define two kinds of temperatures based on the different normalizations of the Killing vectors. We will also apply the Wald formula tor the Nariai black hole in order to get the entropy without resort to normalizations of the Killing vector. In section III, the entropy will be calculated by using the brick-wall method. Although the energy and the temperature depend on the normalization of the time-like Killing vector, the normalization-independent statistical entropy can be obtained, which is compatible with the Wald entropy. Finally, summary and discussion are given in section IV.

II. TEMPERATURES AND WALD ENTROPY IN NARIAI BLACK HOLE
Let us start with the four-dimensional Einstein-Hilbert action with the cosmological constant Λ, which is given by The equation of motion obtained from the action (1) becomes The static and spherically symmetric solution of Eq. (2) is written as with Hereafter, we will consider only the Schwarzschild-de Sitter spacetime with Λ > 0. For , it has two horizons of the black hole horizon r b and the cosmological horizon r c . In this case, the metric function (4) can be neatly written as For M = 0, it has only the cosmological horizon with r c = 3/Λ.

The symmetry of time translation in the SdS spacetime can be described by a timelike
Killing vector, which is written as where γ t is a normalization constant. In the standard normalization, γ t is obtained from the condition to satisfy ξ µ ξ µ = −1 at the asymptotically flat Minkowski spacetime. For instance, its value usually becomes γ t = 1 for a Schwarzschild metric. In the SdS spacetime, there is no asymptotically flat region, so that we should consider the reference point r g where the gravitational acceleration vanishes due to the balance between the forces of the black hole by the mass and the cosmological horizon by the cosmological constant. Thus, we can choose the normalization constant in Eq. (5) to satisfy ξ µ ξ µ = −1 at that reference point r g , which yields where the reference point can be found from f ′ (r g ) = 0 and is explicitly given by r g = (3M/Λ) 1/3 . Now, the surface gravities κ b and κ c on the black hole horizon and the cosmological horizon are written as respectively. Then, the temperatures along with the normalization (6) are calculated as which are called the Bousso-Hawking temperatures [39]. This temperature can be also On the other hand, in the Euclidean geometry, the Hawking temperature agrees with the inverse of the period of the Euclidean time to avoid a conical singularity at the horizon.
Setting the Euclidean time τ to τ = it, the Euclidean line element of Eq. (3) is written as From Eq. (9), the Hawking temperatures for the black hole horizon and the cosmological horizon become respectively, which agree with the temperatures obtained from the Killing vector (5)  In this degenerate case with r b = r c , the metric (3) should be transformed to an appropriate coordinate system because it has the coordinate singularity and becomes inappropriate.
Near the degenerate case, the mass can be written as [38][39][40] where the degenerate case can be obtained by taking ǫ = 0. One can define the new time and the radial coordinate ψ and χ by In terms of the new coordinates (12), the line element (3) is written in the form of up to the first order in ǫ. For the case of ǫ = 0, Eq. (13) is called the Nariai metric, which is given by In this coordinate system, the back hole horizon and the cosmological horizon correspond to χ = 0 and χ = π, respectively, where the proper distance between the two horizons is given by π/ √ Λ which is not zero. From now on, we will study this Nariai black hole which is actually real geometry to describe thermal equilibrium since the horizon temperature is the same with the cosmological temperature. However, there are two kinds of temperatures depending on the definitions of the normalization of the Killing vector.
With Eqs. (11) and (12), the Killing vector (5) becomes to the leading order in ǫ. Using Eq. (7), the Bousso-Hawking temperature is calculated as This can be also obtained from the Killing vectorξ = ∂/∂ψ at the coordinate system with the rescaled timeψ = ψ/ √ Λ. As expected, the temperature of the black hole horizon is the same with that of the cosmological horizon. On the other hand, one can also get the Hawking temperature from the Euclidean metric (14) by setting the Euclidean time as ψ E = iψ. Then, the Euclidean Nariai metric can be written as In order to avoid a conical singularity at the two horizons, the period of the Euclidean time for the black hole horizon or the cosmological horizon are chosen as 2π, respectively. Then the Hawking temperatures are given by which corresponds to the surface gravity obtained from the Killing vector ∂/∂ψ using Eq. (7).
It is interesting to note that the Hawking temperature is constant as long as the Nariai condition M = 1/(3 √ Λ) is met. Moreover, it can be easily checked that the Bousso-Hawking temperature (16) is obtained from the condition to avoid a conical singularity at the horizons for the scaled Euclidean timeψ E = ψ E / √ Λ.
In order to find the Wald entropy of the Nariai spacetime, one should consider a diffeomorphism invariance with the Killing vector ξ µ which is associated with the conservation law of ∇ µ J µ = 0 [27][28][29][30][31][32], for which the Noether potential J µν can be defined by J µ = ∇ ν J µν .
If a Lagrangian is written in the form of L = L(g µν , R µνρσ ), then the Noether potential is given by [31,32] where For a timelike Killing vector, the Wald entropy [27] is expressed by where κ and h µν are the surface gravity and the induced metric on the hypersurface Σ of a horizon, respectively. And ǫ µν is defined by where n µ is the outward unit normal vector of Σ. The proper velocity u µ of a fiducial observer moving along the orbit of ξ µ is given by For the Nariai metric (14), the Killing vector is given by where γ is a normalization constant, which will be not specified in this section. From the norm of the Killing vector, we obtain α = γ sin χ/ √ Λ and The outward unit normal vectors of the black hole horizon and the cosmological horizon are calculated as n µ = (1/ √ Λ)δ χ µ and n µ = −(1/ √ Λ)δ χ µ , respectively. Then, the nonzero components of Eq. (22) are ǫ ψχ = −ǫ χψ = ± sin χ/(2Λ), where the upper sign and the lower sign correspond to the black hole horizon and the cosmological horizon, respectively. Now, for the action (1), we obtain which leads to Inserting Eq. (23) into Eq. (7), we can obtain κ b,c = γ. Then, from Eq. (21), the Wald entropy is given by where A b and A c are the areas of the black hole horizon and the cosmological horizon, respectively. The total area given by the two horizons becomes A = 8π/Λ since A b = A c = 4π/Λ. Eventually, the entropy (26) can be rewritten as which also agrees with the Bekenstein-Hawking entropy. After all, we obtained the Wald entropy expressed by the expected area law, which is independent of the normalization of the Killing vector.

III. ENTROPY FROM BRICK-WALL METHOD
In the Nariai black hole governed by the line element (14), the black hole temperature is the same with the cosmological temperature as seen from Eqs. (16) and (18), which imply that the net flux is in fact zero. Thus, the thermal equilibrium can be realized in this special configuration, which is different from the non-equilibrium SdS black hole. In order to calculate the statistical entropy in this thermal background [3], we will consider a quantum scalar field in a box surrounded by the two horizons. The Klein-Gordon equation for the scalar field is written as where m is the mass of the scalar field. By using the WKB approximation with Φ ∼ exp[−iωψ + iS(χ, θ, φ)] under the Nariai metric (14), the square module of the momentum is obtained as where k χ = ∂S/∂χ, k θ = ∂S/∂θ, and k φ = ∂S/∂φ. Then, the number of quantum states with the energy less than ω is calculated as where V p denotes the volume of the phase space satisfying k 2 + m 2 ≤ 0. For simplicity, we take the massless limit of m 2 = 0. As seen from (30), the number of states diverges at the horizons of χ = 0, π, so that we need the UV cutoff at χ = h b and χ = π − h c . The UV cutoff parameters h b and h c are assumed to be very small. Then, the free energy is given by Then, the entropy becomes The proper lengths for the UV parameters are given bȳ which leads to h b,c = √ Λh b,c . Then, Eq. (32) is written as within the leading order ofh b,c .
When we perform the WKB approximation with the line element (14), the coordinate ψ plays a role of the time. The corresponding Killing vector is given by ξ = ∂/∂ψ and β in Eq. (35) should be taken as the inverse of the Hawking temperature (18). Then, the entropy is obtained as where ℓ P ≡ G /c 3 is the Plank length. If the cutoff is chosen ash b,c = ℓ P / √ 90π like the case of the Schwarzschild black hole [3], the entropy (36) is remarkably written as where the total area is defined by A = A b + A c for convenience. Then, it agrees with one quarter of the horizon area of the Bekenstein-Hawking entropy.
From the viewpoint of the renormalization [42], the total entropy can be written as the sum of the Wald entropy (27)  show that β H ω = β BHω . In the calculation of the free energy (31), ω in the integrand should be replaced byω/ √ Λ and the integration should be performed forω. Then, we can obtain the same entropy with Eq. (36) based on the Bousso-Hawking temperature. Therefore, the entropy is always written as the area law of the Wald entropy, whereas the temperature and the energy depend on the choice of the time, that is, the normalization of the timelike Killing vector.
The final comment is in order. As for the Bousso-Hawking temperature, it can be regarded as a Tolman temperature [43]. It was defined at the vanishing surface gravity where it is the counterpart of the asymptotically Minkowski space in the asymptotically flat black holes. The Bousso-Hawking temperature can be derived from the definition of the Tolman temperature of T loc = T H / √ g ψψ = √ Λ/(2π sin χ) where T H = 1/(2π). If we move the observer, for instance, to the black hole horizon of χ = 0 or to the cosmological horizon of χ = π, the temperature goes to infinity. In particular, at the middle point of χ = π 2 , it produces the Bousso-Hawking temperature. So the Bousso-Hawking normalization of Killing vector is compatible with the Tolman temperature. So, we can identify the Bousso-Hawking temperature with the Tolman temperature at the reference point.