The intrinsic curvature of thermodynamic potentials for black holes with critical points

TL;DR

This paper investigates the geometric structure of thermodynamic potentials for black holes in anti-de Sitter space, analyzing their curvature and critical points, and finds that curvature singularities do not always indicate phase transitions.

Contribution

It provides a detailed analysis of the intrinsic curvature of thermodynamic metrics for black holes, including extended phase space, revealing that singularities do not necessarily correspond to critical phenomena.

Findings

Curvature diverges when heat capacities diverge.

Ricci scalar singularities do not always reflect critical behavior.

Extended thermodynamics introduces curvature singularities at specific thermodynamic conditions.

Abstract

The geometry of thermodynamic state space is studied for asymptotically anti-de Sitter black holes in D-dimensional space times. Convexity of thermodynamic potentials and the analytic structure of the response functions is analysed. The thermodynamic potentials can be used to define a metric on the space of thermodynamic variables and two commonly used such metrics are the Weinhold metric, derived from the internal energy, and the Ruppeiner metric, derived from the entropy. The intrinsic curvature of these metrics is calculated for charged and for rotating black holes and it is shown that the curvature diverges when heat capacities diverge but, contrary to general expectations, the singularities in the Ricci scalars do not reflect the critical behaviour. When a cosmological constant is included as a state space variable it can be interpreted as a pressure and the thermodynamically conjugate variable as a thermodynamic volume. The geometry of the resulting extended thermodynamic state space is also studied, in the context of rotating black holes, and there are curvature singularities when the heat capacity at constant angular velocity diverges and when the black hole is incompressible. Again the critical behaviour is not visible in the singularities of the thermodynamic Ricci scalar.