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 Ba\~nados, Teitelboim, and Zanelli (BTZ) metric in the ($2+1$)-dimensional and ($1+1$)-dimensional limits of Einstein gravity. The Smarr relation is naturally upheld in both BTZ cases, where those with Q≠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→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 an upper 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→2 black holes exhibit any interesting phase behavior.