Applications of Geometrothermodynamics in Non–Standard Theories of Gravity

Abstract

Black holes are one of the most interesting objects of study in physics. Classically, they trap everything including light. They are characterized by an event horizon which encloses a curvature singularity. A study of black hole physics shows that a black hole behaves as a thermodynamic system, with the area of the event horizon playing the role of the entropy and a geometric quantity called surface gravity as the temperature of the black hole. It is possible to introduce differential geometric concepts in ordinary thermodynamics. The most known structures were postulated by Weinhold and Ruppeiner who introduced Riemannian metrics in the space of equilibrium states of a thermodynamic system. These geometric structures can obviously be applied in black hole thermodynamics. Unfortunately, the results are not in agreement with ordinary thermodynamics which is manifestly Legendre invariant. To overcome this inconsistency, the theory of geometrothermodynamics was proposed. It incorporates arbitrary Legendre transformations into the geometric structure of the equilibrium space in an invariant manner. InthisthesisIhavestudiedthethermodynamicsofvariousblackholesindifferentgravity theories by means of thermodynamic Riemannian curvatures. The thermodynamics of black holes is reformulated within the context of the formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations and to allow several equivalent representations. Legendre invariance allows to explain a series of contradictory results known in the literature from the use of Weinhold’s and Ruppeiner’s thermodynamic metrics for black holes. I present a brief review of classical and black hole thermodynamics and the basic mathematical elements of geometrothermodynamics in the first chapter. Then I present a systematic and consistent construction of geometrothermodynamics by using Riemannian contact geometry for the phase manifold and harmonic maps for the equilibrium manifold in section 1.11. In chapter two, using the formalism of geometrothermodynamics, I investigate the geometric properties of the equilibrium manifold for diverse thermodynamic systems. Starting from Legendre invariant metrics of the phase manifold, I derive thermodynamic metrics for the equilibrium manifold whose curvature becomes singular at those points where phase transitions of first and second order occur.I present the thermodynamics and the thermodynamic geometries of charged rotating BTZ black holes in the third chapter. The thermodynamics of these black holes is investigated within the context of the Weinhold and Ruppeiner geometries and the formalism of geometrothermodynamics. Considering the behavior of the heat capacity and the Hawking temperature, I show that Weinhold and Ruppeiner geometries cannot describe completely the thermodynamics of these black holes and of their limiting case of vanishing electric charge. In contrast, the Legendre invariance imposed on the metric in geometrothermodynamics allows one to describe these black holes and their limiting cases in a consistent and invariant manner. In the fourth chapter, the thermodynamic properties of five-dimensional static and spherically symmetric black holes in Einstein-Gauss-Bonnet theory are investigated. To formulate the thermodynamics of these black holes I use the Bekenstein-Hawking entropy relation and, alternatively, a modified entropy formula which follows from the first law of thermodynamics of black holes. The results of both approaches are not equivalent. Chapter five is devoted to the study of thermodynamic geometries of the most general static, spherically symmetric, topological black holes of the Hoˇrava–Lifshitz gravity. In particular, I show that a Legendre invariant metric derived in the context of geometrothermodynamics for the equilibrium manifold reproduces correctly the phase transition structure of these black holes. Moreover, the limiting cases in which the mass, the entropy or the Hawking temperature vanish are also accompanied by curvature singularities which indicate the limit of applicability of the thermodynamics and the geometrothermodynamics of black holes. The Einstein limit and the case of a black hole with flat horizon are also investigated.