Note on thermodynamics method of black hole/CFT correspondence

Abstract

In the paper we further refine the thermodynamics method of black hole/CFT correspondence. We show that one can derive the central charges of different holographic pictures directly from the entropy product S+S− if it is mass-independent, for a black hole in the Einstein gravity or the gravity without diffeomorphism anomaly. For a general black hole in the Einstein gravity that admits holographic descriptions, we show that the thermodynamics method and asymptotic symmetry group (ASG) analysis can always give consistent results in the extreme limit. Furthermore, we discuss the relation between black hole thermodynamics and the hidden conformal symmetry. We show that the condition T+A+ = T−A−, with A± being the outer and inner horizon areas, is the necessary, but not sufficient, condition for a black hole to have the hidden conformal symmetry. In particular, for the Einstein(-Maxwell) gravity T+A+ = T−A− is just the condition T+S+ = T−S−, with S± being the outer and inner horizon entropies, which is the condition for the entropy product S+S− being mass-dependent. When there exists the hidden conformal symmetry in the low-frequency scattering off the generic non-extremal black hole, it always leads to the same temperatures of dual CFT as the ones got from the thermodynamics method. *

Hot attractors

“…Crucially, the temperatures of the left movers and the right movers in the CFT are both positive. These relations apply to the BTZ black hole [41,42] and have been discussed in the literature [31][32][33][34][35][36][37][38]. To briefly summarize a few of the other known results, let us note that (1.1) encapsulates a statement about the thermodynamic entropy:…”

“…While it remains unclear what quantum mechanical degrees of freedom the area of the inner horizon counts, a first law of thermodynamics applies to this surface [31][32][33][34][35][36][37][38]. For comparison, we write…”

Entropy relations and bounds of horizons in modified gravity

“…When one find the UER for general black holes, one can built the left/right-moving models in someway, and hence lead to the physical picture for the statistical origin of black holes. The black hole/CFT (BH/CFT) correspondence [17,[69][70][71][72][73][74] open a window for this viewpoint, where the mass-independent entropy product is useful to built the holographic descriptions of black hole entropy, for black hole with two horizons,.…”

Aspects of warped AdS3/CFT2 correspondence

“…From the thermodynamics laws of both horizons, it is straightforward to read the thermodynamics of left-and right-mover and the corresponding dual temperatures. This method has been tested and applied to many cases [27][28][29][30][31]. Especially in [31], the relation between the thermodynamics method and other conventional methods has been clarified.…”

“…This method has been tested and applied to many cases [27][28][29][30][31]. Especially in [31], the relation between the thermodynamics method and other conventional methods has been clarified. It turns out that they are consistent with each other for general black holes in Einstein(-Maxwell) theory and in some cases the thermodynamics method is even more powerful.…”