The Quantum Null Energy Condition in Curved Space

TL;DR

This work analyzes the quantum null energy condition (QNEC) in curved spacetime, clarifying when the bound $T_{kk} \ge \frac{1}{2\pi} S''$ remains finite and renormalization-scheme independent. It develops a detailed scheme-dependence analysis across spacetime dimensions, showing that for $d\le 3$ the QNEC is finite under minimal stationarity ($\theta|_p=0$), while for $d=4,5$ additional derivative and dominant-energy conditions are required, and for $d\ge 6$ scheme-independence generally fails due to higher-derivative counter-terms. The paper then extends holographic proofs via entanglement wedge nesting to curved backgrounds, proving a finite, renormalized QNEC for $d\le 3$ and $d=4,5$ under the stated conditions, with Killing horizons ensuring scheme-independence in arbitrary backgrounds. A corollary is a first-order semi-classical generalized second law (GSL) proof for holographic theories with $d\le 3$ at leading order in $G$, not requiring perturbations to Killing horizons. Overall, the results sharpen our understanding of QNEC in curved space, highlight the dimensional dependence of scheme-independence, and connect QNEC to holographic entropy and the GSL in new geometric settings.

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 $θ$ and shear $σ_{ab}$ of $N$ at point $p$ satisfy $θ|_p= \dotθ|_p =0$, $σ_{ab}|_p=0$. 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 $θ, σ_{ab}$ 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.