Abstract
In this paper we discuss a formulation of extended phase space thermodynamics of black holes in Anti de Sitter (AdS) spacetimes from the contact geometry point of view. Thermodynamics of black holes can be understood within the framework of contact geometry as flows of vector fields generated by Hamiltonian functions on equilibrium submanifolds in the extended phase space that naturally incorporates the structure of a contact manifold. Deformations induced by the contact vector fields are used to construct various maps among thermodynamic quantities. Thermodynamic variables and equations of state of Schwarzschild black holes are mapped to that of Reissner-Nordstr\"{o}m black holes in AdS, with charge as the deformation parameter. In addition, the equations of state of general black holes in AdS are shown to emerge from the high-temperature ideal gas limit equations via suitable deformations induced by contact vector fields. The Hamilton-Jacobi formalism analogous to mechanics is set up, and the corresponding characteristic curves of contact vector fields are explicitly obtained to model thermodynamic processes of black holes. Extension to thermodynamic cycles in this framework is also discussed.
Highlights
Formulation of the precise laws of black hole mechanics, in analogy with the laws of thermodynamics long ago [1], has paved the way for exciting proposals in general relativity [2,3,4,5,6]
For black holes in anti–de Sitter (AdS) spacetimes in particular, there has been a lot of activity, as they show interesting phase transitions, such as the Hawking-Page transition [7] and novel thermodynamic structure due to the presence of a cosmological constant [8,9,10]
We focus our attention on the contact structure rather than the metric structure of the extended thermodynamic phase space and uncover novel connections between various black holes in AdS
Summary
Formulation of the precise laws of black hole mechanics, in analogy with the laws of thermodynamics long ago [1], has paved the way for exciting proposals in general relativity [2,3,4,5,6]. We focus our attention on the contact structure rather than the metric structure of the extended thermodynamic phase space and uncover novel connections between various black holes in AdS. One should note an important difference with classical mechanics: In mechanics, the phase space is always even dimensional, whereas in the thermodynamic case, the phase space turns out to be odd dimensional, which can be seen from the expression dU − TdS þ PdV 1⁄4 0: It is clear that U has no conjugate variable, and the thermodynamic phase space in this case is five dimensional with local coordinates fP; V; T; S; Ug. in the case of black holes in extended phase space, enthalpy takes center stage, and we work with the following relation: dH − TdS − VdP 1⁄4 0: The thermodynamic phase space assumes the structure of a contact manifold. The thermodynamic potential assumes the role of Hamilton’s principal function
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