A Hamiltonian approach for the Thermodynamics of AdS black holes

Abstract

In this work we study the Thermodynamics of D-dimensional Schwarzschild-anti de Sitter (SAdS) black holes. The minimal Thermodynamics of the SAdS spacetime is briefly discussed, highlighting some of its strong points and shortcomings. The minimal SAdS Thermodynamics is extended within a Hamiltonian approach, by means of the introduction of an additional degree of freedom. We demonstrate that the cosmological constant can be introduced in the thermodynamic description of the SAdS black hole with a canonical transformation of the Schwarzschild problem, closely related to the introduction of an anti-de Sitter thermodynamic volume. The treatment presented is consistent, in the sense that it is compatible with the introduction of new thermodynamic potentials, and respects the laws of black hole Thermodynamics. By demanding homogeneity of the thermodynamic variables, we are able to construct a new equation of state that completely characterizes the Thermodynamics of SAdS black holes. The treatment naturally generates phenomenological constants that can be associated with different boundary conditions in underlying microscopic theories. A whole new set of phenomena can be expected from the proposed generalization of SAdS Thermodynamics.Comment: 13 pages, published versio

Contact geometry and thermodynamics of black holes in AdS spacetimes

“…As extensions of the geometric framework studied in this paper, it is interesting to investigate mechanical analogues for black hole systems, such as through a Lagrangian and action formulation. Some results on Hamiltonian approach to thermodynamic systems were obtained in [72,73], but the analysis was through a different route of symplectic geometry with additional degrees of freedom added by hand, and where, the Lagrangian was a total derivative and non-dynamical. The cosmological constant in these scenarios was shown to be an integration constant [73], which was also observed in the earlier works too [65], though Hamilton-Jacobi approach.…”

“…Some results on Hamiltonian approach to thermodynamic systems were obtained in [72,73], but the analysis was through a different route of symplectic geometry with additional degrees of freedom added by hand, and where, the Lagrangian was a total derivative and non-dynamical. The cosmological constant in these scenarios was shown to be an integration constant [73], which was also observed in the earlier works too [65], though Hamilton-Jacobi approach. In this paper though, we treated the cosmological constant as a dynamical variable, resulting in a pdV term in the first law of black hole mechanics and the existence of an equation of state.…”

Extended quasilocal Thermodynamics of Schwarzchild–anti de Sitter black holes

“…Homogeneity is required for extensivity [13] and for the existence of an integrating factor for the reversible heat exchange [20]. However, homogeneity requires that Λ must be introduced in the theory in a very specific manner, otherwise inconsistencies appear (for example, in the construction of the thermodynamic potentials, as pointed out in [21]). In particular, if one tries to use the cosmological constant as a thermodynamic variable in this extension, the obtained theory will present singularities in the Legendre transformation between Λ and its conjugate variable [19].…”

“…In a previous work [21], we extend the minimal setup using a Hamiltonian approach to Thermodynamics [25], where the equations of state are realized as constraints on a symplectic phase space. In this manner we are able to incorporate Λ in a consistent manner, solving the homogeneity issue and obtaining a sensible Smarr formula for the SAdS Thermodynamics.…”

“…In this manner we are able to incorporate Λ in a consistent manner, solving the homogeneity issue and obtaining a sensible Smarr formula for the SAdS Thermodynamics. However, we are not able to fix the functional form of Λ in [21], as it remains dependent on free parameters of the model. In this work we construct an extended version for the quasilocal thermodynamic description using the aforementioned Hamiltonian method.…”