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

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          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.

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          Author and article information

          Journal
          2015-08-31
          2015-11-05
          Article
          10.1088/0264-9381/33/5/055008
          1509.00074
          f2a4074c-78c1-467c-9969-5770f53bb93a

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          minor corrections, 18 pages
          hep-th

          High energy & Particle physics
          High energy & Particle physics

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