A coarse-grained generalized second law for holographic conformal field theories

Abstract

We consider the universal sector of a $d$-dimensional large-$N$ strongly-interacting holographic CFT on a black hole spacetime background $B$. When our CFT${}_d$ is coupled to dynamical Einstein-Hilbert gravity with Newton constant $G_d$, the combined system can be shown to satisfy a version of the thermodynamic Generalized Second Law (GSL) at leading order in $G_d$. The quantity $S_{CFT}+\frac{A(H_{B,\text{perturbed}})}{4G_d}$ is non-decreasing, where $A(H_{B,\text{perturbed}})$ is the (time-dependent) area of the new event horizon in the coupled theory. Our $S_{CFT}$ is the notion of (coarse-grained) CFT entropy outside the black hole given by causal holographic information -- a quantity in turn defined in the AdS${}_{d+1}$ dual by the renormalized area $A_{ren}(H_{\mathrm{bulk}})$ of a corresponding bulk causal horizon. A corollary is that the fine-grained GSL must hold for finite processes taken as a whole, though local decreases of the fine-grained generalized entropy are not obviously forbidden. Another corollary, given by setting $G_d=0$, states that no finite process taken as a whole can increase the renormalized free energy $F=E_{out}-T{S}_{CFT}-\hat{I}\text{c◯}J-\hat{I}¦Q$, with $T,\hat{I}\text{c◯},\hat{I}¦$ constants set by $H_B$. This latter corollary constitutes a 2nd law for appropriate non-compact AdS event horizons.

minor corrections, 18 pages