Contact and metric structures in black hole chemistry

TL;DR

This paper reviews the application of contact and thermodynamic geometry to black holes in AdS spacetimes, exploring their phase structure, ensemble behavior, and microscopic interactions within the extended thermodynamics framework.

Contribution

It introduces a comprehensive analysis of thermodynamic geometries for black holes, linking geometric features to phase transitions and microscopic properties, and extends the approach to holographic gauge theories.

Findings

Thermodynamic curvature relates to critical points in black hole phase transitions.

Ensemble non-equivalence is characterized by different geometric behaviors.

Black hole thermodynamics exhibits features analogous to classical systems like van der Waals fluids.

Abstract

We review recent studies of contact and thermodynamic geometry for black holes in AdS spacetimes in the extended thermodynamics framework. The cosmological constant gives rise to the notion of pressure $P = -\Lambda / 8 \pi$ and, subsequently a conjugate volume $V$, thereby leading to a close analogy with hydrostatic thermodynamic systems. To begin with, we review the contact geometry approach to thermodynamics in general and then consider thermodynamic metrics constructed as the Hessians of various thermodynamic potentials. We then study their correspondence to statistical ensembles for systems with two-dimensional spaces of equilibrium states. From the zeroes and divergences of the curvature scalar obtained from the metric, we carefully analyze the issue of ensemble non-equivalence and show certain complimentary behaviors in the description of a thermodynamic system. Following a thorough analysis of the familiar van der Waals system, we turn our attention to black holes in extended phase space. Considering the example of charged AdS black holes, we discuss the generic features of their thermodynamic geometry in detail. The relationship of the thermodynamic curvature(s) with critical points as well as microscopic interactions in black holes is also briefly explored. We finally set up the thermodynamic geometry for finite temperature gauge theories dual to black holes in AdS via holographic correspondence and comment on recent progress.