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

Abstract

We consider the universal sector of a d > 2 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) + A(H_(B, perturbed))/4G_d is non-decreasing, where A(H_(B_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_(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) – TS_(CFT) – ΩJ, with T, Ω constants set by H_B. This latter corollary constitutes a 2nd law for appropriate non-compact AdS event horizons.

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