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

TL;DR

This work develops two parallel routes to thermodynamics in curved spacetimes: (i) a first-law based treatment of electrically charged self-gravitating thin shells in arbitrary dimensions, elucidating entropy and stability and revealing a black-hole limit with Smarr-like relations; and (ii) a Euclidean path-integral formulation that constructs canonical/grand-canonical ensembles of black holes and self-gravitating matter in curved backgrounds (finite or infinite cavities, AdS and flat spaces). By applying the York/Gibbons-Hawking formalism and zero-loop semiclassical approximations, the study derives thermodynamic potentials, phase diagrams, and stability criteria for RN and RN-AdS black holes, as well as for self-gravitating shells, and demonstrates the consistency between the Euclidean-quantum gravity approach and Davies’ thermodynamic theory in various dimensions. Key results include entropy expressions for shells that reproduce Bekenstein-Hawking values in the black-hole limit, detailed intrinsic stability conditions for shells under multifluctuation modes, and comprehensive mappings of phase structure (hot space vs black holes) across AdS/cavity setups, including Hawking-Page-type transitions and planar AdS limits. The findings illuminate how semiclassical gravity encodes thermodynamic behavior of curved spacetimes and offers a framework to probe gravity–thermodynamics links at microscale scales, with implications for quantum gravity and holographic contexts. $S$ and $A_+$ relations, horizon radii $r_+$, inverse temperatures $\beta$, and other thermodynamic quantities are consistently treated with $($A$)$, $($S$)$, and $($T$)$ relations expressed in $ $Delimiters$.

Abstract

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 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 descrip-tions 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 and choose equations of state which allow 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 phase transitions between hot matter and black holes. We 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 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.

Figures (26)

• Figure 1: Region of thermodynamic stability of the shell for mass fluctuations only for $d=5$ with the curves of marginal stability $a(x,y)$ plotted in function $x= r_+^{2}/R^{2}$, and $y=r_-^{2}/r_+^{2}$: (a) certain values of $y$; (b) certain values of $y$;. The regions below the curves describe the stable configurations.
• Figure 2: Region of thermodynamic stability of the shell for mass fluctuations only and $a=1$: (a) in stripes in terms of $\mu M/R^2$ and $\sqrt{\mu}Q/R^2$; (b) above the curves for different $d$ in terms of $x=r_+^{d-3}/R^{d-3}$ and $y=r_-^{d-3}/r_+^{d-3}$.
• Figure 3: Region of thermodynamic stability of the shell for area fluctuations only for $d=5$ with the curves of marginal stability $a(x,y)$ plotted in function $x=r_+^{2}/R^{2}$, and $y=r_-^{2}/r_+^{2}$: (a) certain values of $y$; (b) certain values of $x$. The regions below the curves describe the stable configurations.
• Figure 4: Region of thermodynamic stability of the shell for charge fluctuations only for $d=5$ with the curves of marginal stability $a(x,y)$ plotted in function $x=r_+^{2}/R^{2}$, and $y=r_-^{2}/r_+^{2}$: (a) certain values of $y$; (b) certain values of $x$. The regions below the curves describe the stable configurations.
• Figure 5: Region of thermodynamic stability of the shell for mass and area fluctuations together for $d=5$ with the curves of marginal stability $a(x,y)$ plotted in function $x=r_+^{2}/R^{2}$, and $y=r_-^{2}/r_+^{2}$: (a) certain values of $y$; (b) certain values of $x$. The regions below the curves describe the stable configurations.