Lower-dimensional black hole chemistry

Abstract

The connection between black hole thermodynamics and chemistry is extended to the lowerdimensional 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 = 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 behaviour.

Are “Superentropic” black holes superentropic?

“…It remains to be shown whether similar doubts could arise also for other Superentropic black holes, for example the recently studied charged BTZ black holes [57][58][59][60] have an apparently similar issue. The charged BTZ black hole is a three dimensional electrically charged solution of the Einstein-Maxwell equations [61]…”

“…This is in part the reason for maintaining the dimensionful parameter G, if we set 8πG = 1, or 1/2 as in the original paper [61], then r becomes dimensionless, and we need not add any scale inside the logarithm, however, in keeping with convention, we introduce the scale r 0 as a second integration constant. The extended thermodynamics of this black hole has been studied in [60,62], and is shown to crucially depend on the choice of the integration parameter r 0 . The conventional choice in the literature is to set r 0 = , however, in extended thermodynamics this has an important consequence:…”

“…This is precisely the term that implies the violation of the reverse isoperimetric inequality. Although preferred in [60], this option is questionable from various perspectives. First, the potential Φ in (5.4) is that of the electrostatic potential evaluated on the horizon, however this is not gauge invariant; usually one takes the potential difference between the horizon and infinity as a gauge invariant thermodynamic potential, however this is problematic in 3D as the potential at infinity obviously diverges.…”

“…This perspective leads to an alternative formulation of thermodynamics, where we identify r 0 in (5.1) as 'enclosing' the BTZ black hole in a circle of radius r 0 as in [61,63]. Upon this, the potential at 'infinity' (r = r 0 ) vanishes and we obtain the following thermodynamic quantities [60]:…”

Microscopic description of thermodynamic volume in extended black hole thermodynamics

“…From the point of view of the two dimensional CFT, Q is merely a deformation parameter, a global charge, which will be kept fixed here. The extended thermodynamics gives [35]:…”