The Quantum Null Energy Condition in Curved Space

Abstract

The quantum null energy condition (QNEC) is a conjectured bound on components $(T_{kk}=T_{ab}{k}^a{k}^b$) of the stress tensor along a null vector $k^a$ at a point $p$ in terms of a second $k$-derivative of the von Neumann entropy $S$ on one side of a null congruence $N$ through $p$ generated by $k^a$. The conjecture has been established for super-renormalizeable field theories at points $p$ that lie on a bifurcate Killing horizon with null tangent $k^a$ and for large-N holographic theories on flat space. While the Koeller-Leichenauer holographic argument clearly yields an inequality for general $(p,k^a)$, more conditions are generally required for this inequality to be a useful QNEC. For $d\le 3$, for arbitrary backgroud metric satisfying the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$, we show that the QNEC is naturally finite and independent of renormalization scheme when the expansion $\theta$ and shear ${\sigma }_{ab}$ of $N$ at point $p$ satisfy $\theta {|}_p=\dot{\theta }{|}_p=0$, ${\sigma }_{ab}{|}_p=0$. This is consistent with the original QNEC conjecture. But for $d=4,5$ more conditions are required. In particular, we also require the vanishing of additional derivatives and a dominant energy condition. In the above cases the holographic argument does indeed yield a finite QNEC, though for $d\ge 6$ we argue these properties to fail even for weakly isolated horizons (where all derivatives of $\theta ,{\sigma }_{ab}$ vanish) that also satisfy a dominant energy condition. On the positive side, a corrollary to our work is that, when coupled to Einstein-Hilbert gravity, $d\le 3$ holographic theories at large $N$ satisfy the generalized second law (GSL) of thermodynamics at leading order in Newton's constant $G$. This is the first GSL proof which does not require the quantum fields to be perturbations to a Killing horizon.

arXiv comment31 pages, no figure; v2: signs corrected and new comments for d=3; v3: minor corrections; v4: minor modifications to address referee comments; v5: paragraph containing equation (2.13) modified

References in corpus (39)

• A Covariant Holographic Entanglement Entropy Proposal
• Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence
• A Covariant Entropy Conjecture
• Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
• Quantum corrections to holographic entanglement entropy
• Causality & holographic entanglement entropy
• Extended Gravity Theories and the Einstein-Hilbert Action
• Holographic Entanglement Entropy for General Higher Derivative Gravity
• A Quantum Focussing Conjecture
• Averaged Energy Conditions and Quantum Inequalities
• Deriving covariant holographic entanglement
• Geometry of Generic Isolated Horizons
• Proof of Classical Versions of the Bousso Entropy Bound and of the Generalized Second Law
• Proof of the Quantum Null Energy Condition
• Causal Holographic Information
• Entanglement Entropy, Trace Anomalies and Holography
• Topological Censorship
• Restrictions on Negative Energy Density in Flat Spacetime
• Achronal averaged null energy condition
• The Quantum Interest Conjecture
• A Second Law for Higher Curvature Gravity
• Holographic Proof of the Quantum Null Energy Condition
• Holographic Entanglement Entropy for the Most General Higher Derivative Gravity
• Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions
• Some Calculable Contributions to Holographic Entanglement Entropy
• On Spacetime Entanglement
• Semiclassical energy conditions for quantum vacuum states
• A holographic proof of the averaged null energy condition
• Classical and quantum flux energy conditions for quantum vacuum states
• Renormalized entanglement entropy
• Thermalization of Causal Holographic Information
• Generalized second law at linear order for actions that are functions of Lovelock densities
• Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality
• Entropy increase during physical processes for black holes in Lanczos-Lovelock gravity
• Coarse-grained entropy and causal holographic information in AdS/CFT
• A Lower Bound on the Energy Density in Classical and Quantum Field Theories
• Properties of Causal Holographic Information
• Averaged null energy condition in a classical curved background
• A coarse-grained generalized second law for holographic conformal field theories

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• Violation of the quantum null-energy condition in a holographic wormhole and infrared effects
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