Gravitational Thermodynamics: From Black Holes to Holography

Abstract

The subject of gravitational thermodynamics lies at the center of numerous fields of study, many of which may seem disconnected, yet have proven to be deeply entwined. This thesis examines two primary facets of this subject, the study of black hole thermodynamics, and the principle of bulk/boundary duality (or `holography') as applied to gravitating systems.

In Part I of this thesis we explore thermodynamic aspects of a wide variety of black hole spacetimes. We focus on asymptotically de Sitter black holes, in an extended phase space where the cosmological constant is interpreted as a thermodynamic pressure. We begin with the prototypical classes, examining general relativistic Schwarzschild- and Reissner-Nordstr\"{o}m-de Sitter black holes. We demonstrate the consistent formulation of their thermodynamics in the extended phase space using a Euclidean path integral approach, and uncover novel compact small-large black hole transitions not seen in asymptotically AdS spacetimes. We also consider a number of extensions of Einstein-Maxwell theory: Born-Infeld electrodynamics, conformally coupled scalar fields, and Gauss-Bonnet gravity. We study the thermodynamic properties and phase structure of black hole solutions in these theories, uncovering (among other things) a unique reentrant phase transition in the grand canonical ensemble, compact reentrant phase transitions, and isolated critical points. We also examine the analogy these systems make with ordinary fluid systems, showing that in contrast to asymptotically anti-de Sitter black holes, de Sitter black holes have nonlinear equations of state which forbid such an interpretation.

Part II of this thesis represents an attempt to understand the thermodynamic nature of gravity from a broader perspective. Here, we take a `holographic' approach, promoting the gravitational screen formalism to a fully covariant mapping between bulk geometric quantities and those of a relativistic dissipative fluid system on the (arbitrary, timelike) boundary. We demonstrate the projection of the field equations onto the screen boundary, derive the corresponding fluid conservation equations, and explicitly construct the dictionary relating the two systems. We show how entropy production in the fluid is tied to gravitational wave propagation in the bulk, and discuss the role of the equation of state of the fluid in the correspondence. Finally, we explicitly construct several gravitational screens in spherically symmetric spacetimes. We determine the properties of the resulting holographic fluids, and use thermodynamical laws governing the fluid to assign a notion of temperature and entropy to the bulk geometry.