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The geometry of the higher dimensional black hole thermodynamics in Einstein-Gauss-Bonnet theory

This paper deals with five-dimensional black hole solutions in (a) Einstein-Yang-Mills-Gauss-Bonnet theory and (b)Einstein-Maxwell-Gauss-Bonnet theory with a cosmological constant for spherically symmetric space time. The geometry of the black hole thermodynamics has been studied for both the black holes.


I. INTRODUCTION
The nice similarity between a black hole and a thermodynamical system is characterized by temperature and entropy: a black hole temperature (known as Hawking temperature) is proportional to its surface gravity on the horizon while entropy proportional to its horizon area [1,2] and they satisfy the first law of thermodynamics [3]. But the statistical origin of the black hole entropy is still a challenging problem today. Usually, a black hole is characterized by three parameters namely its mass, charge and angular momentum(known as no hair theorem) and the thermodynamical stability of the black hole is determined by the sign of its heat capacity(c v ). If c v < 0 (as Schwarzschild black hole), then black hole is thermodynamically unstable, but if c v changes sign in the parameter space such that it diverges [4] in between then from ordinary thermodynamics it indicates a second order phase transition [5]. However in black hole thermodynamics, a critical point exits also for extremal black hole and the second order phase transition takes place from an extremal black hole to its non extremal counter part. The geometrical concept into ordinary thermodynamics was first introduced by Weinhold [6](for details see the review [7]). According to him a Riemannian metric can be defined as the second derivative of the internal energy(U ) to entropy (S) and other extensive variables (y α ) of the system ,i.e., the Hessian of the energy, known as Weinhold metric as However it looses physical meaning in equilibrium thermodynamics. Subsequently, Ruppeiner [8] introduced a metric as the Hessian matrix of the thermodynamic entropy and is known as the Ruppeiner metric. It is defined on the state space as Note that the Ruppeiner metric is conformally related to the Weinhold metric with T −1 as the conformal factor ,i.e, Unlike Weinhold metric , the Ruppeiner geometry has physical relevance in the fluctuation theory of equilibrium thermodynamics. In fact, the inverse Ruppeiner metric gives the second moments of fluctuations. It was found that the Ruppeiner geometry is related to phase structure of thermodynamic system such that scalar curvature of the Ruppeiner metric diverges (R → −∞) at the phase transition and critical point while the Ruppeiner metric is flat(i.e scalar curvature is zero) for thermodynamical system having no statistical mechanical interactions.
In the present paper we study the geometry of the black hole thermodynamics in the five dimensional (a) Einstein-Yang-Mills-Gauss-Bonnet theory(EYMGB) and (b)Einstein-Maxwell-Gauss-Bonnet(EMGB) theory with a cosmological constant. With the recent development in String Theory, it is legitimate to assume the space time dimension to be more than four particularly in high energy physics. Also the classical gravitational analog of String Theory [9]demands a generalization of Einstein -Hilbert action by squares and higher powers of curvature terms, but field equations become fourth order and bring in Ghost [10]. This problem was resolved by Lovelock [11] choosing higher powered curvature terms in a particular combination so that the field equations still remain second order and the ghosts disappear. The first two terms in the Lagrangian of the Lovelock gravity are the Einstein-Hilbert (EH) and Gauss-Bonnet(GB)terms and they form the most general Lagrangian for the second-order field equations in the five dimension [12]. The GB term is consistent with the heterotic Super String Theory and play a fundamental role in Chern-Simons gravity [13]. One may note that GB term is purely topological in nature [14] in 4-D and hence it has no role in dynamics. This paper is organized a follows: section II − A describes the black hole solution in Einstein-Yang-Mills-Gauss-Bonnet theory and the corresponding geometrical aspect of black hole thermodynamics have been studied in section II − B The black hole in Einstein-Maxwell-Gauss-Bonnet theory is presented in section III − A and its thermodynamics has been examined from geometric point of view in section III − B

II. THERMODYNAMIC GEOMETRY OF EYMGB BLACK HOLES
In this section we first describe the black hole solution in Einstein-Yang-Mills-Gauss-Bonnet (EYMGB)theory and then study the geometry of the black hole thermodynamics in the next subsection:

A. SPHERICALLY SYMMETRIC YANG MILLS SOLUTION IN EINSTEIN-GAUSS-BONNET THEORY
Mazharimousavi and Halisoy [15] have recently obtained a 5-D spherically symmetric solution in EYMGB theory. The metric ansatz for 5-D spherically symmetric space-time is chosen as where they have expresed the metric on unit three sphere dΩ 2 3 in terms of Euler angles as [15] For the Yang-Mills field the energy momentum tensor is given by, where F i αβ is the Yang-Mills field 2-forms such that F i αβ F iαβ = 6Q 2 r 4 , Q the only non-zero gauge charge. The modified Einstein equations in EYMGB theory are Where G µν is the usual Einstein tensor in 5-D, T µν is the energy-momentum tensor given by equation (6), α, the GB coupling parameter is chosen to be positive in the heterotic string theory and the Lovelock tensor has the expression Now solving the non vanishing components of the field equations we have [15] U (r) = 1 + r 2 4α ± r 2 4α 2 + 1 + m 2α + Q 2 ln r α (9) with m as the constant of integration. Now as α → 0 provided negative branch is considered. The metric coefficient U (r) in (10) is identical to that of the Einstein-Yang-Mills solution and hence ′ m ′ is interpreted as the mass of the system. In the equation (9) for U the expression within the square root is positive definite for α > 0 while the geometry has a curvature singularity at the surface r = r min for α < 0 . Here r min is the minimum value of the radial coordinate such that the function under the square root is positive. Moreover, depending on the values of the parameters (m, Q, α), the singular surface can be surrounded by an event horizon with radius r h so that the space-time given by equation (1) represents a black hole. However if no event horizon exists, then there will be naked singularity. Now the metric decribed by equation (1) and (9) has a singularity at the greatest real and positive solution (r s )of the equation Note that if equation (11) has no real positive solution then the metric diverges at r = 0. However, the singularity is surrounded by the event horizon r h , which is the positive root of (the larger one if there are two positive real roots) Thus if r s < r h then the singularity will be covered by the event horizon. While the singularity will be naked for r s ≥ r h . In this connection one may note that the event horizon is independent of the coupling parameter α.

B. GEOMETRY OF THE BLACK HOLE THERMODYNAMICS IN EYMGB THEORY
We discuss the geometry of thermodynamics of the black hole described above. As the event horizon r h satisfies equation (12) so we have, The surface area of the event horizon is given by, A = 2π 2 r 3 h and hence the entropy of the black hole [16] takes the form Now choosingh = 1 and the Boltzmann constant appropriately, we have, S = r 3 h So m can be obtained as a function of S and Q in the form From the energy conservation law of the black hole the temperature (T ) and electric potential(φ) of the black hole are given by, As the Weinhold metric is the Hessian of the internal energy (mass parameter ′ m ′ here) so its explicit form is which can be diagonalized by the transformation and we obtain, which is curved and Lorentzian in nature provided Hence, using the conformal transformation (3) the Ruppeiner metric is given by, And the expression for the scalar curvature is, The region of (u, v) is restricted by the inequality, The non-flatness of Ruppeiner metric indicates that the black hole thermodynamics has statistical mechanical interactions. The expression for the heat capacity with a fixed charge is given by, Thus within the above admissible range(24) c Q will be positive provided v 2 . Further, at v u = e u 3 , c Q changes sign and the scalar curvature diverges. Therefore, there is a phase transition and corresponds to this critical point.

III. GEOMETRY OF EMGB BLACK HOLE THERMODYNAMICS
This section deals with the thermodynamics of the black hole solution in Einstein-Maxwell-Gauss-Bonnet theory with a comological constant. At first the spherically symmetric black hole solution is presented and then the thermodynamics of the black hole solution has been examined geometrically in the parameter space.

A. SPHERICALLY SYMMETRIC EINSTEIN-MAXWELL SOLUTION IN GAUSS-BONNET THEORY
The action in five dimensional space time (M, g µν ) that represents Einstein-Maxwell theory with a Gauss-Bonnet term and a cosmological constant has the expression [17][18][19] where R GB = R 2 −4R αβ R αβ +R αβγδ R αβγδ , is the Gauss-Bonnet term, α is the GB coupling parameter having dimension (length) 2 ( α −1 is related to string tension in heterotic super string theory), Λ is the cosmological constant and F µν = (∂ µ A ν − ∂ ν A µ ) is the usual electromagnetic field tensor with A µ , the vector potential.Now variation of this action with respect to the metric tensor and F µν gives the modified Einstein field equations and Maxwell's equations and where H µν is the Lovelock tensor (given by eq (8)) and is the electromagnetic stress tensor.
(Note that the modified Einstein field equations (28) do not contain any derivatives of the curvature terms and hence the field equations remain second order).
If the manifold M is chosen to be five dimensional spherically symmetric space-time having the line element with 0 ≤ θ 1 , θ 2 ≤ π , 0 ≤ θ 3 ≤ 2π, then solving the above field equations one obtains [1,19] Here in an orthonormal frame the non-null components of the electromagnetic tensors are Ftr = −Frt = Q 4πr 3 . Note that in the limit α → 0 one may recover the Einstein-Maxwell solution with a cosmological constant. Further, in the limit with Λ = 0 we have the five-dimensional Reissner-Nordström Solution and hence the parameters M (> 0) and Q can be identified as the mass and charge respectively. Moreover, for the solution (32) to be well defined, the radial coordinate r must have a minimum value (r min ) so that the expression within the square root is positive definite,i.e, the solution (32) is well defined for r > r min where r min satisfies 1 + 16mα πr 4 min − 8Q 2 α 3r 6 min + 4Λα 3 = 0 The surface r = (r min ) corresponds to a curvature singularity. However,depending on the values of the parameters this singular surface may be surrounded by the event horizon (having radius r h such that B(r h ) = 0)and the solution (32) describes a black hole solution known as EMGB black hole. On the other hand, if no event horizon exists then the above solution represents a naked singularity.
Thus, c Q > 0 if