Lower-Dimensional Black Hole Chemistry

TL;DR

This paper explores the thermodynamics and chemistry of lower-dimensional black holes, specifically BTZ black holes in 2+1 dimensions and limits approaching 1+1 dimensions, revealing their properties and relations like Smarr and Reverse Isoperimetric Inequality.

Contribution

It extends black hole chemistry to lower dimensions, analyzing the thermodynamic relations and inequalities, and introduces new work terms and concepts like lower bounds on mass and rotation in 1+1 dimensions.

Findings

BTZ black holes with charge violate the Reverse Isoperimetric Inequality.

Adding a thermodynamic work term maintains the inequality.

Lower bounds on mass are derived for 1+1 D black holes.

Abstract

The connection between black hole thermodynamics and chemistry is extended to the lower-dimensional regime by considering the rotating and charged BTZ metric in the $(2+1)$-D and a $(1+1)$-D limits of Einstein gravity. The Smarr relation is naturally upheld in both BTZ cases, where those with $Q \ne 0$ violate the Reverse Isoperimetric Inequality and are thus superentropic. The inequality can be maintained, however, with the addition of a new thermodynamic work term associated with the mass renormalization scale. The $D\rightarrow 0$ limit of a generic $D+2$-dimensional Einstein gravity theory is also considered to derive the Smarr and Komar relations, although the opposite sign definitions of the cosmological constant and thermodynamic pressure from the $D>2$ cases must be adopted in order to satisfy the relation. The requirement of positive entropy implies a lower bound on the mass of a $(1+1)$-D black hole. Promoting an associated constant of integration to a thermodynamic variable allows one to define a "rotation" in one spatial dimension. Neither the $D=3$ nor the $D \rightarrow 2$ black holes exhibit any interesting phase behaviour.