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

TL;DR

This paper establishes a generalized second law for holographic conformal field theories coupled to gravity, showing that a combined entropy involving CFT and horizon area is non-decreasing, with implications for the thermodynamics of black holes in AdS spaces.

Contribution

It introduces a coarse-grained GSL for holographic CFTs coupled to gravity, linking CFT entropy with horizon area and deriving a second law for non-compact AdS horizons.

Findings

The combined entropy S_{CFT} + area term is non-decreasing in the coupled system.

Finite processes cannot increase the renormalized free energy F in the dual gravitational description.

The coarse-grained GSL implies the fine-grained GSL holds for entire processes.

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 $A_{ren}(H_{\rm bulk})$ 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 $F = E_{out} - T S_{CFT} - \Omega J - \Phi Q$, with $T, \Omega, \Phi$ constants set by ${H}_B$. This latter corollary constitutes a 2nd law for appropriate non-compact AdS event horizons.