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 \(d\ge6\) 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.