Averaged null energy condition and quantum inequalities in curved spacetime

TL;DR

This paper proves that the Averaged Null Energy Condition (ANEC) holds for quantum scalar fields in curved spacetime with small curvature, using quantum inequalities to rule out exotic phenomena.

Contribution

It introduces a null-projected quantum inequality and applies it to demonstrate ANEC in curved spacetime, extending previous flat spacetime results.

Findings

ANEC is obeyed by quantum scalar fields in small curvature spacetimes.

Derived a null-projected quantum inequality applicable to curved spacetime.

Established constraints on negative energy densities in curved backgrounds.

Abstract

The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. ANEC can be used to rule out spacetimes with exotic phenomena, such as closed timelike curves, superluminal travel and wormholes. We prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal null geodesic (not two points can be connected with a timelike curve) surrounded by a tubular neighborhood whose curvature is produced by a classical source. To prove ANEC we use a null-projected quantum inequality, which provides constraints on how negative the weighted average of the renormalized stress-energy tensor of a quantum field can be. Starting with a general result of Fewster and Smith, we first derive a timelike projected quantum inequality for a minimally-coupled scalar field on flat spacetime with a background potential. Using that result we proceed to find the bound of a quantum inequality on a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. The last step is to derive a bound for the null-projected quantum inequality on a general timelike path. Finally we use that result to prove achronal ANEC in spacetimes with small curvature.