Averaged null energy condition in spacetimes with boundaries

TL;DR

This paper proves that the Averaged Null Energy Condition (ANEC) cannot be uniformly violated by a quantized, minimally coupled real scalar field along a complete null geodesic when the geodesic sits inside a flat tubular neighborhood whose causal structure matches Minkowski space. The authors build on Quantum Null Energy Inequalities (QNEI) and impose precise geometric and quantum-field-theoretic assumptions, including Hadamard states and a tubular Minkowski-like region, to bridge from local QNEIs to the global ANEC integral $A(t)=\int_{-\infty}^{\infty} d\lambda\, T_ω(\Phi_0(\lambda,t))$. They prove Theorem II.1, ruling out uniform negative convergence of $A(t)$ near the geodesic, and provide a variant (Theorem III.1) under uniform continuity of $T_ω$ that yields a liminf bound for the weighted null-energy integral on large scales. An illustrative cylinder-spacetime example shows that non-achronal windings can violate ANEC, while complete geodesics with appropriate geometric conditions retain ANEC. Overall, the work reinforces the robustness of ANEC against distant boundaries or curvature effects in the semiclassical regime and informs ongoing efforts to rule out exotic spacetimes such as traversable wormholes in these settings.

Abstract

The Averaged Null Energy Condition (ANEC) requires that the average along a complete null geodesic of the projection of the stress-energy tensor onto the geodesic tangent vector can never be negative. It is sufficient to rule out many exotic phenomena in general relativity. Subject to certain conditions, we show that the ANEC can never be violated by a quantized minimally coupled free scalar field along a complete null geodesic surrounded by a tubular neighborhood in which the geometry is flat and whose intrinsic causal structure coincides with that induced from the full spacetime. In particular, the ANEC holds in flat space with boundaries, as in the Casimir effect, for geodesics which stay a finite distance away from the boundary

Figures (1)

• Figure 1: The parallelogram $\Phi_v(\eta,\tau)$, shown shaded, can be considered as a set of timelike geodesics parameterized by $\tau$ with $\eta$ fixed (short dashes), or of null geodesics parameterized by $\eta$ with $\tau$ fixed (long dashes).