Black hole geometric thermodynamics

The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudo-Riemannian manifold (M, g) and a smooth function f : M → R, whose Hessian with respect to g is non-degenerate, one can define on M the associated pseudo-Riemannian Hessian metric h = Hess g f . In the following, we apply this methodology for describing the geometrical properties of some interesting mathematical objects like the higher dimensional Reissner-Nördstrom black holes. The paper is organized as follows. Section 1 reviews the history of pseudo-Riemannian Geometry and points out which Hessian is more convenient for physical problems. Section 2 gives the Christoffel symbols and the system of geodesics of pseudo-Riemannian manifold (M, h = Hess g f ), establishes the relation between the components of the curvature tensors field of (M, h) and (M, g), and determines the PDEs representing the coincidence between the Christoffel symbols of (M, h) and the Christoffel symbols of (M, g). The last section presents the comparison of null-length curves trajectories obtained with the two metrics for a 5-dimensional RN black hole.

Motivated by the energy representation of Riemannian metric, in this paper we will study different approaches toward the geometrical concept of black hole thermodynamics. We investigate thermodynamical Ricci scalar of Weinhold, Ruppeiner and Quevedo metrics and show that their number and location of divergences do not coincide with phase transition points arisen from heat capacity. Next, we introduce a new metric to solving these problems. The denominator of the this thermodynamical metric whose Ricci scalar only contains terms that match to the phase transition of heat capacity.

General Relativity and Gravitation

The Hessian of the entropy function can be thought of as a metric tensor on the state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study this geometry for some families of black holes. It is flat for the BTZ and Reissner-Nordström black holes, while curvature singularities occur for the Reissner-Nordström-anti-de Sitter and Kerr black holes.

Entropy, 2014

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.

Entropy, 2015

We give a brief survey of thermodynamic metrics, in particular the Hessian of the entropy function, and how they apply to black hole thermodynamics. We then provide a detailed discussion of the Gibbs surface of Kerr black holes. In particular we analyze its global properties, and extend it to take the entropy of the inner horizon into account. A brief discussion of Kerr-Newman black holes is included.

General Relativity and Gravitation, 2018

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald's method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald's method. The new method is applied to Schwarzschild, Kerr, and Kerr-Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.

The Eleventh Marcel Grossmann Meeting - On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories - Proceedings of the MG11 Meeting on General Relativity, 2008

We report outcomes from the geometrical studies (Ruppeiner and Weinhold geometries) of thermodynamics of certain black hole families in various spacetime dimensions e.g. BTZ, general relativity and higher-dimensional Einstein-Maxwell gravity. So far the most physically suggestive result occurs for the Kerr black hole in D ≥ 5 in which thermodynamic stability is found to be encoded in the curvature of the Ruppeiner metric. This is consistent with the known result in the literature.

The European Physical Journal C, 2014

We study the thermodynamics and thermodynamic geometry of Park black hole in Hořava gravity. By incorporating the ideas of differential geometry, we have investigated the thermodynamics using Weinhold geometry and Ruppeiner geometry. We have also analyzed it in the context of newly developed geometrothermodynamics(GTD). Divergence of specific heat is associated with the second order phase transition of black hole. Here in the context of Park black hole, both Weinhold's metric and Ruppeiner's metric well explain this phase transition. But these explanations depend on the choice of potential. Hence the Legendre invariant GTD is used, and with the true singularities in the curvature scalar, GTD well explain the second order phase transition. All these methods together give an exact idea of all the behaviors of the Park black hole thermodynamics.

Thermodynamic properties of locally anisotropic (2 + 1)-black holes are studied by applying geometric methods. We consider a new class of black holes with a constant in time elliptical event horizon which is imbedded in a generalized Finsler like spacetime geometry induced from Einstein gravity. The corresponding thermodynamic systems are three dimensional with entropy S being a hypersurface function on mass M, anisotropy angle θ and eccentricity of elliptic deformations ε. Two-dimensional curved thermodynamic geometries for locally anistropic deformed black holes are constructed after integration on anisotropic parameter θ. Two approaches, the first one based on two-dimensional hypersurface parametric geometry and the second one developed in a Ruppeiner-Mrugala-Janyszek fashion, are analyzed. The thermodynamic curvatures are computed and the critical points of curvature vanishing are defined.

Physical Review D, 1998

We consider the thermodynamic properties of the constant curvature black hole solution recently found by Bañados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are not isotherms so the first law of thermodynamics exists only in an integral form. These properties arise from the unusual topology of the Euclidean black hole instanton. 04.70.Dy, 04.20.Ha, 04.70.Bw * An examination of all identifications in four dimensional anti-de Sitter spacetime has been presented by Holst and Peldan [3], but the global properties of their black hole solution differ from those of the solution found by Bañados.