Thermodynamic geometry and locally anisotropic black holes

We study new classes three dimensional black hole solutions of Einstein equations written in two holonomic and one anholonomic variables with respect to anholonomic frames. Thermodynamic properties of such (2 + 1)-black holes with generic local anisotropy (having elliptic horizons) are studied by applying geometric methods. 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. *

We study new classes three dimensional black hole solutions of Einstein equations written in two holonomic and one anholonomic variables with respect to anholonomic frames. Thermodynamic properties of such (2 + 1)-black holes with generic local anisotropy (having elliptic horizons) are studied by applying geometric methods. 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. *

Section 1 reviews the history of Differential Geometry derived from a fundamental tensor of type (0, 2) and points out which Hessian is more convenient to be fundamental tensor. Section 2 starts with the fundamental tensor h = Hess g f , gives the Christoffel symbols and the system of geodesics, establishes the relation between the components of the curvature tensors field of h and (R n , g), and determines the PDEs representing the coincidence between the Christoffel symbols of h and the Christoffel symbols of g. Section 3 analyses the Reissner-Nordstrom black hole from Pidokrajt point of view and from our point of view, underlying the physical characteristics of the geometrical models via fundamental tensor, Christoffel symbols, geodesics and curvature. The physical geometric models are total different, the most important differences being the degeneration curves, the null length curves and the sign of sectional curvature.

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 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.

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.

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.

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.

2000

By applying the method of moving frames modelling one and two dimensional local anisotropies we construct new solutions of Einstein equations on pseudo-Riemannian spacetimes. The first class of solutions describes non-trivial deformations of static spherically symmetric black holes to locally anisotropic ones which have elliptic (in three dimensions) and ellipsoidal, toroidal and elliptic and another forms of cylinder symmetries (in four dimensions). The second class consists from black holes with oscillating elliptic horizons.

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.