Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

Journal of High Energy Physics, Feb 2017

In this paper we revisit the question that in what sense empty AdS5 black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic c-function using causal horizon in the black brane geometry. The UV value of the c-function is a UV and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the c-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature. In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs at finite temperature the renormalized entanglement negativity of an interval has UV (Low-T) value c UV and IR (High-T) value zero. So this is a potential candidate for our c-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic c-function is sensitive to the physics of the interior. Correspondingly the field theory c-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate c-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be done in specific cases only. The existence of this c-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the c-function reaches zero at the curvature singularity correlates the two facts: loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity.

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Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

Received: September Black hole singularity, generalized (holographic) c-theorem and entanglement negativity Shamik Banerjee 0 1 3 4 Partha Paul 0 1 2 4 Open Access 0 1 4 c The Authors. 0 1 4 0 Bhubaneshwar-751005 , Odisha , India 1 5-1-5 Kashiwa-no-Ha , Kashiwa City, Chiba 277-8568 , Japan 2 Institute of Physics , Sachivalaya Marg 3 Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo 4 [46] E. Perlmutter, M. Rangamani and M. Rota, Central Charges and the Sign of Entanglement In this paper we revisit the question that in what sense empty AdS5 black brane geometry can be thought of as RG- ow. We do this by c-function using causal horizon in the black brane geometry. The UV value of the cfunction is aUV and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the c-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are e ectively massive with a gap set by the temperature. In eld theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs at the renormalized entanglement negativity of an interval has UV (Low-T) value cUV and IR (High-T) value zero. So this is a potential candidate for our c-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic c-function is sensitive to the physics of the interior. Correspondingly the eld theory c-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate c-function) may be a way to probe part of the physics near the singularity. Negativity at nite temperature depends on the full operator content of the theory and so perhaps this can be done in speci c cases only. AdS-CFT Correspondence; Black Holes in String Theory; Renormalization - Calculation and results Towards a physical interpretation 3.1 Is nite temperature entanglement negativity a generalized c-function? Black hole singularity from loss of quantum correlation An infalling observer? 1 Introduction 2 3 3.2 A Perturbation near In AdS-CFT black brane is a thermal state of the boundary conformal eld theory living on the Minkowski space-time. This is not a relevant deformation of the CFT Hamiltonian and there is no renormalization group ow in the ordinary sense. Therefore the question of the existence of a c-function, in the sense of Zamolodchikov [1{11], does not naturally arise in this situation. Moreover Zamolodchikov c-function is constant at a and independent of the state of the CFT. The purpose of this note is to point out that AdS-CFT duality and the thermodynamic nature of classical gravity allows us to introduce a generalized notion of c-function, at least for large-N theories with classical gravity dual. This generalized c-function cannot be interpreted as an o -shell central charge. Rather it can be interpreted as a measure of quantum entanglement that exists at di erent energy scales in the given state. We will construct this c-function holographically when the CFT is in thermal state and the gravity dual is an empty black brane geometry. We focus on four dimensional eld theories only. Our choice of the thermal state is motivated by the fact that the gravity dual has a curvature singularity and the Lorentz invariance is broken everywhere except near the UV boundary of AdS. So it can teach us some lessons about RG- ow interpretation of more general geometries. Throughout this paper we will assume that the bulk theory is Einstein gravity coupled minimally to a set of matter elds. Holographic view The holographic picture is based on the fact that the gravity dual of c-theorem is the second law of causal horizon thermodynamics in asymptotically AdS spaces [20, 21]. In a nutshell, second law for causal horizons say that if we consider the future bulk light-cone of a boundary point then the expansion of the null geodesic generators of the light-cone is negative [26, 27]. Now one can assign Bekenstein-Hawking entropy to the causal horizon. The fact that the expansion is negative then implies that as we move away from the boundary the entropy density decreases monotonically. This is essentially holographic c-theorem [14{19] if we specialize to a domain-wall geometry. The bulk future light-cone interpolates between the UV-AdS and the IR-AdS and the monotonically decreasing Bekenstein-Hawking entropy density gives the holographic c-function [20, 21]. If we focus on domain-wall geometry then the second law has the interpretation of holographic c-theorem. But what about other asymptotically AdS (AAdS) geometries? Second law of causal horizon thermodynamics holds in any AAdS geometry and in fact holographic RG [17] applies to any such setup. It has been argued that the holographic RG in the bulk is dual to th (...truncated)


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Shamik Banerjee, Partha Paul. Black hole singularity, generalized (holographic) c-theorem and entanglement negativity, Journal of High Energy Physics, 2017, pp. 43, Volume 2017, Issue 2, DOI: 10.1007/JHEP02(2017)043