Extrinsic and intrinsic curvatures in thermodynamic geometry

TL;DR

This paper explores how intrinsic and extrinsic curvatures in thermodynamic geometry reveal phase transition information in black hole systems, showing extrinsic curvature correlates with heat capacity signs and diverges at critical points.

Contribution

It demonstrates the relationship between extrinsic curvature and thermodynamic phase transitions in black holes, providing a geometric perspective that can be generalized to other thermodynamic systems.

Findings

Extrinsic curvature sign matches heat capacity around phase transitions.

Extrinsic curvature diverges at phase transition points.

Intrinsic curvature diverges at critical points but does not indicate heat capacity sign.

Abstract

We investigate the intrinsic and extrinsic curvatures of certain hypersurfaces in the thermodynamic geometry of a physical system and show that they contain useful thermodynamic information. For an anti-Reissner-Nordstr\"{o}m-(A)de Sitter black hole (Phantom), the extrinsic curvature of a constant $Q$ hypersurface has the same sign as the heat capacity around the phase transition points. For a Kerr-Newmann-AdS (KN-AdS) black hole, the extrinsic curvature of $Q \to 0$ hypersurface (Kerr black hole) or $J \to 0$ hypersurface (RN black black hole) has the same sign as the heat capacity around the phase transition points. The extrinsic curvature also diverges at the phase transition points. The intrinsic curvature of the hypersurfaces diverges at the critical points but has no information about the sign of the heat capacity. Our study explains the consistent relationship holding between the thermodynamic geometry of the KN-AdS black holes and those of the RN and Kerr ones \cite{ref1}. This approach can be easily generalized to an arbitrary thermodynamic system.