In this thesis we aim to develop new perspectives on the statistical mechanics of black holes using an information geometric approach (Ruppeiner and Weinhold geometry). The Ruppeiner metric is defined as a Hessian matrix on a Gibbs surface, and provides a geometric description of thermodynamic systems in equilibrium. This Ruppeiner geometry exhibits physically suggestive features; a flat Ruppeiner metric for systems with no interactions i.e. the ideal gas, and curvature singularities signaling critical behavior(s) of the system. We construct a flatness theorem based on the scaling property of the black holes, which proves to be useful in many cases. Another thermodynamic geometry known as the Weinhold geometry is defined as the Hessian of internal energy and is conformally related to the Ruppeiner metric with the system’s temperature as a conformal factor.

We investigate a number of black hole families in various gravity theories. Our findings are briefly summarized as follows: the Reissner-Nordström type, the Einstein-Maxwell-dilaton andBTZ black holes have flat Ruppeiner metrics that can be represented by a unique state space diagram. We conjecture that the state space diagram encodes extremality properties of the black hole solution. The Kerr type black holes have curved Ruppeiner metrics whose curvature singularities are meaningful in five dimensions and higher, signifying the onset of thermodynamic instabilities of the black hole in higher dimensions. All the three-parameter black hole families in our study have non-flat Ruppeiner and Weinhold metrics and their associated curvature singularities occur in the extremal limits. We also study two-dimensional black hole families whose thermodynamic geometries are dependent on parameters that determine the thermodynamics of the black hole in question. The tidal charged black hole which arises in the braneworld gravity is studied. Despite its similarity to the Reissner-Nordström type, its thermodynamic geometries are distinctive.

We study the thermodynamics of tidal charged black holes in four dimensions. Such black holes are spherically symmetric vacuum solutions of the effective Einstein equations on the brane and are characterized by the mass m and (generalizing their general relativistic counterparts) by a second parameter, the tidal charge q. The latter is an imprint of the Weyl curvature of the 5-dimensional space-time, in which the brane is embedded. The heat capacity of the tidal charged black hole diverges on a set of measure zero of the parameter space. However, there is no phase transition at those points, similarly to the Reissner-Nordstr\"om black hole. We investigate the thermodynamic geometry of such black holes by deriving the Weinhold and the Ruppeiner metrics. Whereas the Weinhold metric is flat, the Ruppeiner metric has a positive Ricci curvature, which is in sharp contrast with the Reissner-Nordstr\"om black hole, the general relativistic analogue of the tidal charged metric. The state space is conformal to the right half of the interior of the future light cone in a Minkowski plane. We find two constraints on the possible 5-dimensional extensions of the tidal-charged black hole: First, we conjecture that the 5-dimensional black object should have the same entropy as its 4-dimensional section studied here, modulo corrections which are small for large black holes. Second, for constant m any quasi-stationary evolution of the tidal charged black hole leads to a decrease of q, contributing towards the localization of gravity on the brane. This represents an important constraint on the evolution of the 5-dimensional space-time.

3. Geometry of black hole thermodynamics

Pidokrajt, Narit

et al.

Stockholm University, Faculty of Science, Department of Physics.

Åman, Jan E.

Stockholm University, Faculty of Science, Department of Physics.

Bengtsson, Ingemar

Stockholm University, Faculty of Science, Department of Physics.

Geometry of black hole thermodynamics2003In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 35, no 10, p. 1733-1743Article in journal (Refereed)

Abstract [en]

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-Nordstrom black holes, while curvature singularities occur for the Reissner-Nordstrom-anti-de Sitter and Kerr black holes.

In this talk we present the latest results from our ongoing project on geometro-thermodynamics (also known as information geometry of thermodynamics or Ruppeiner geometry) of dilaton BHs in 4D in both Einstein and string frames and a dyonic dilaton BH and at the end we report very briefly results from this approach to the 2D dilaton BHs.

5. Ruppeiner theory of black hole thermodynamics

Åman, Jan E.

et al.

Stockholm University, Faculty of Science, Department of Physics.

Bedford, James

Grumiller, Daniel

Pidokrajt, Narit

Stockholm University, Faculty of Science, Department of Physics.

The Ruppeiner metric as determined by the Hessian of the Gibbs surface provides a geometric description of thermodynamic systems in equilibrium. An interesting example is a black hole in equilibrium with its own Hawking radiation. In this article, we present results from the Ruppeiner study of various black hole families from different gravity theories e.g. 2D dilaton gravity, BTZ, general relativity and higher-dimensional Einstein-Maxwell gravity.

The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein–Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.

We investigate thermodynamic curvatures of the Kerr and Reissner-Nordström (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four-dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for a d=5 Kerr black hole is curved and divergent in the extremal limit. For a d>=6 Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For the RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In d>=5 the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in d=5 with double angular momenta.