Lower-Dimensional Black Hole Chemistry

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.