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Thermodynamics and Phase Transition of a Gauss-Bonnet Black Hole in a Cavity

Considering a canonical ensemble, in which the temperature and the charge on a wall of the cavity are fixed, we investigate the thermodynamics of a D-dimensional Gauss-Bonnet black hole in a finite spherical cavity. Moreover, it shows that the first law of thermodynamics is still satisfied. We then discuss the phase structure and transition in both five and six dimensions. Specifically, we show that there always exist two regions in the parameter space. In one region, the system possesses one single phase. However in the other region, there could coexist three phases and a van der Waals-like phase transition occurs. Finally, we find that there is a fairly close resemblance in thermodynamic properties and phase structure of a Gauss-Bonnet-Maxwell black hole, either in a cavity or in anti-de Sitter space.


Introduction
The thermodynamics of black holes has been a subject of intensive study for several decades since J. Bekenstein and S. Hawking discovered that the black hole entropy was proportional to the area of its event horizon [1,2]. A Schwarzschild black hole in asymptotically flat space has a negative specific heat and hence become unstable as a thermodynamic system. That being said, the Schwarzschild black hole radiates more when it becomes smaller.
To make a black hole thermally stable, it is necessary to put the black hole in a closed system. One of the most popular way to approach this aim is to place a black hole in anti-de Sitter (AdS) space with the negative cosmological constant, in which the timelike boundary can reflected radiation back into the bulk. Hawking and Page in [3] first studied the thermodynamic properties of a Schwarzschild black hole in AdS space and discovered the Hawking-Page phase transition. Since then, the thermodynamic properties and phase transition of various black holes in AdS space have been discussed in [4][5][6][7][8][9][10].
On the other hand, another popular choice is considering a cavity in an asymptotic flat space, where the Dirichlet boundary condition is imposed on the wall of the cavity. It was discovered by York in [11] that a Schwarzschild black hole in a cavity is thermally stable and a Hawking-Page-like transition can occur as the temperature decreases, which is quite similar to the behavior of a Schwarzschild-AdS black hole. Later, a Reissner-Nordstrom (RN) black hole in a cavity was studied in a grand canonical ensemble [12] and a canonical ensemble [13]. It showed that a Hawking-Page-like phase transition and a van der Waals-like one occur in the canonical ensemble and grand canonical ensemble, respectively, which is similar to the AdS case [14]. In the following papers [15][16][17][18][19][20], the phase structure of various black holes in the cavity is studied, where Hawking-Page-like or van der Waals-like phase transitions were found except for some special cases. Considering charged scalars, boson stars and hairy black holes in a cavity in [21][22][23][24], it showed that the phase structure of the gravity system in a cavity is strikingly similar to that of holographic superconductors in the AdS gravity. In [25][26][27][28][29][30][31][32], the stabilities of solitons, stars and black holes in a cavity were also investigated. It is interesting to note that most studies in the literature have been consider in the framework of the Eisenstein-Maxwell theory. On the other hand, we recently found that the phase structure of a nonlinear electrodynamics black hole in a cavity can be different from that in AdS space [33,34]. It naturally raises a question whether there exists other theory beyond the Eisenstein-Maxwell theory, in which thermodynamics of a black hole can be dependent on boundary conditions. The Gauss-Bonnet gravity is the simplest case of Lovelock theories, which extends the general relativity theory through adding higher derivative terms into the Einstein-Hilbert action. In [35,36], it showed that the Gauss-Bonnet term is naturally consistent with the first-order α ′ correction of closed string low energy effective action. The Gauss-Bonnet AdS black solution was first obtained in [8]. After that, the thermodynamic properties and phase structure of a Gauss-Bonnet black hole in AdS space are discussed in various scenarios [37][38][39][40]. It is worth noting that the Gauss-Bonnet term is a topological invariant in four dimensions, and hence thermodynamics of a Gauss-Bonnet black hole is always analyzed in higher dimensions.
In this paper, we study the thermodynamic properties and phase structure of a Gauss-Bonnet-Maxwell black hole in a cavity in a canonical ensemble. We find that thermodynamics and phase structure of a Gauss-Bonnet-Maxwell black hole in a cavity bear striking resemblance to that in AdS space. This paper is organized as follows. In section 2, we first review the Gauss-Bonnet-Maxwell black hole solution and obtain the Euclidean action of the Gauss-Bonnet-Maxwell black hole in a cavity. In section 3, we then discuss thermodynamics of the Gauss-Bonnet-Maxwell black hole in a cavity and show that the first law of thermodynamics is satisfied. In section 4, the phase structure of the black hole in cavity is discussed in five and six dimensions. We summarize our results in section 5. Finally, we discuss the phase structure of a Gauss-Bonnet-Maxwell AdS black hole in the appendix. We take G = = c = k B = 1 for simplicity in this paper.

Gauss-Bonnet Black Hole in a Cavity
In this section, we briefly review the Gauss-Bonnet-Maxwell black hole solution and obtain the Euclidean action.
It is worth noting that in this paper, we study the black hole thermodynamics in a canonical ensemble, in which the temperature and charge are fixed on the boundary.
Considering the Gauss-Bonnet gravity coupled to Maxwell theory on a D-dimensional spacetime manifold M with a time-like boundary ∂M, we can write the action as Here, the bulk action is given by where α is the Gauss-Bonnet coupling constant. Generally, α is positive since it is associated with string length's square of string theory [35]. Furthermore, F µν is the electromagnetic field strength tensor, which is defined as On the boundary ∂M, the surface terms are where γ is the determinant of the induced metric on ∂M, K µν is the external curvature of ∂M, K is the trace of the external curvature, J is the trace of

4)
G αβ is the D − 1 dimensional Einstein tensor on ∂M corresponding to induced metric γ ab , and K 0 , J 0 , G αβ 0 are the correlative quantities when boundary ∂M embedded in flat spacetime [41]. Note that the Gauss-Bonnet term is a topological invariant in four dimensions, so we will consider D ≥ 5 in what follows.
By varying the action (2.1), we find the equations of motion We consider a static spherically symmetric black hole solution with the metric The equations of motion then reduce to for simplicity. Integrating eqns. (2.8) and (2.10), we obtain the solution where M is the ADM mass and Q is the charge of the black hole, and ω D−2 is the volume of the unit D − 2 sphere [35]. The outer event horizon radius r + of the black hole satisfies f (r + ) = 0. Therefore, the metric function f (r) can be rewritten in terms of r + : The Euclidean action S E can be related to the action S (2.1): S E = iS. Using the analytic continuation which gives F rτ = iF rt . Suppose that the black hole lives in a spherical cavity, where the boundary ∂M is at Since the temperature T is fixed on the boundary of the cavity, we can impose the boundary condition at r = r B in terms of the reciprocal temperature: , and hence the period of τ is . Integrating the Euclidean action and using eqn. (2.12), the Euclidean action is rewritten as

Thermodynamics
In the semi-classical approximation, the on-shell Euclidean action is related to the free energy F : From eqn. (2.15), we can express the the free energy F in terms of the temperature T , the charge Q, the Gauss-Bonnet parameter α ( α) , the cavity radius r B and the horizon radius r + : where T , Q , α ( α) and r B are parameters of the canonical ensemble and the horizon radius r + is the only variable, By extremizing the free energy F (r + ; T, Q, α, r B ) with respect to r + , we can determine the only variable r + :
Since the Hawking temperature of the black hole is defined as T h = f ′ (r + ) /4π, eqn. (3.19) can be written as where the Hawking temperature is (3.21) So for the observer on the wall, the temperature T on the cavity is blueshifted from Hawking temperature T h .
At the locally stationary point r + = r + (T, Q, α, r B ), the free energy F (r + ; T, Q, α, r B ) can be express only in terms of T , Q, α ( α) and r B : For later convenience, F (r + ; T, Q, α, r B ) and F (T, Q, α, r B ) can be abbreviated to F (r + ) and F , respectively.
Furthermore, the thermal energy of the black hole in the cavity is where the thermal energy E is expressed in terms of the entropy S, the charge Q and the cavity radius r B .
Moreover, we can define an electric potential and thermodynamic surface pressure as (3.24) It is easy to verify that the differential E with respect to S, Q and area A is satisfied: is the surface area of the cavity. Using eqns. To discuss the thermodynamic stability of the Gauss-Bonnet-Maxwell black hole in the cavity, we consider the specific heat at constant electric charge (3.27) Since the system is thermally stable with C Q > 0, the black holes is thermally stable when ∂r + (T, Q, r B ) /∂T > 0. From ∂ 2 F/∂ 2 T = −C Q , the thermally stable/unstable phases have concave downward/upward F -T curves.
The black hole phase is thermally stable/unstable when r + (T, Q, α, r B ) is a local minimum/maximum of F (r + ).
Note that the physical space of r + has boundaries, such as r e ≤ r + ≤ r B , (3.28) where r e is the horizon radius of the extremal black hole.

Phase Transition
In this section, we will discuss the phase transition of a Gauss-Bonnet-Maxwell black hole in a cavity for D = 5 and D = 6. For convenience, we express the variables in terms of r B : (4.29) From eqns. (3.17) and (2.12), we obtain the free energy as a function of x: where the metric function becomes as (4.31) Moreover, the Hawking temperature in eqn. (3.21) is rewritten as (4.32)

Five dimensions
When D = 5, the thermodynamic expressions are rewritten as: , (4.34) where the metric function is simplified as When it comes to the phase structure, we need to consider the locally stationary points r + = r + (T, Q, α, r B ), which can be multivalued and lead to more than one phase. The globally stable phase and phase transitions can be determined by calculating the free energy.

Six dimensions
In six dimensions, the thermodynamic expressions are simplified as follow: It is worth noting that, unlike the D = 5 case, the extremal temperature is dependent onᾱ in the D = 6 case.

Conclusion
In this paper, we first calculated the Euclidean action of a Gauss-Bonnet-Maxwell black hole in a finite spherical cavity and obtained the corresponding free energy in a canonical ensemble by semi-classical approximation.
Moreover, the first law of thermodynamics was found to be satisfied. In the rest of this paper, we mainly discussed the phase structure and transition of a Gauss-Bonnet-Maxwell black hole in a cavity for D = 5 and D = 6. In five dimensions, there are two regions in theᾱ-Q phase space in FIG. 1

A Gauss-Bonnet black hole in AdS space
In this appendix, we briefly discuss phase structure of a Gauss-Bonnet-Maxwell black hole in AdS space. In [8], the metric function was obtained as where l is the radius of the AdS space. Using eqn. (A.39), we can express ADM mass M of the black hole in terms of the horizon radius r + , .
Furthermore, the black hole temperature and entropy are