The quantum null energy condition in curved space. (31st October 2017)

• Record Type: Journal Article
• Authors: Fu, Zicao; Koeller, Jason; Marolf, Donald
• Is Part Of: Classical and quantum gravity. Volume 34:Number 22(2017:Nov.)
• Publication Date: 2017-10-31
• Subjects: quantum null energy condition -- holographic entanglement entropy -- quantum fields in curved space; Quantum gravity -- Periodicals; Gravitation -- Periodicals; Relativity (Physics) -- Periodicals; Space and time -- Periodicals; Periodicals; 521.1
• DOI: 10.1088/1361-6382/aa8f2c
• Languages: English
• ISSNs: 0264-9381

Abstract

Abstract: The quantum null energy condition (QNEC) is a conjectured bound on components ( T k k = T a b 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 ⩽ 3, for arbitrary backgroud metric we show that the QNEC is naturally finite and independent of renormalization scheme when the expansion θ of N at the point p vanishes. This is consistent with the original QNEC conjecture which required θ and the shear σ a b to satisfy θ | p = θ ˙ | p = 0, σ a b | p = 0 . But for d = 4, 5 more conditions than even these 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 ⩾ 6 we argue these properties to fail even for weakly isolated horizons (where all derivatives of θ, σ a b 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 ⩽ 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.