Thermodynamics and statistical mechanical ensembles of black holes and self-gravitating matter

Black holes exist all over our Universe, possessing a very wide range of masses. At the moment, they serve as a probe to test general relativity at astrophysical scales, but in the future they may also give us information about gravity at the microscale. Black holes seem to have thermodynamic properties, such as the Bekenstein-Hawking entropy, which are important when considering black holes with size of a few centimeters or smaller. Since entropy in statistical mechanics is related to the number of possible microstates of a system, several questions arise: what gives rise to the black hole entropy? Can it be explained by a quantum description of gravity?

In order to further study these questions, the connection between thermodynamics and gravity must be explored at the microscale. In this doctoral thesis, we aim to understand this connection using two descriptions that yield the thermodynamics of curved spacetimes. We start by imposing the first law of thermodynamics to a charged self-gravitating matter thin shell in higher dimensions. The fundamental pressure equation of state can be used for the shell, which is given solely by general relativity.

An equation of state for temperature of the shell is also chosen, so that it allows the study of the black hole limit and the recovery of black hole thermodynamics. Furthermore, we use the Euclidean path integral approach to quantum gravity to construct statistical ensembles of black hole spacetimes and self-gravitating matter, in order to study semiclassically the possible phase transitions between hot matter and black holes.

We also show the power of the formalism in obtaining the thermodynamic properties of curved spacetimes. Namely, we study the canonical and grand canonical ensemble of charged black holes inside a cavity, which may have a finite or infinite radius. We also construct ensembles of a self-gravitating matter thin shell, both in anti-de Sitter and in asymptotically flat spaces, in order to understand the thermodynamic features of the shell and the possible phase transitions to black hole configurations.