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 Bañados, Teitelboim, and Zanelli (BTZ) metric in the (2 þ 1)-dimensional and (1 þ 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 þ 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 ð1 þ 1Þ-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 ¼ 3 nor the D → 2 black holes exhibit any interesting phase behavior.