A note on the averaged null energy condition in quantum field theory

TL;DR

This paper proposes a generalized averaged null energy condition (ANEC) in quantum field theory that includes a finite lower bound, potentially preserving its validity in curved spacetimes and impacting the understanding of wormholes.

Contribution

It introduces a modified ANEC with a finite lower bound, addressing the limitations of the original ANEC in curved spacetimes and exploring its implications for energy conditions and wormholes.

Findings

Generalized ANEC can be valid in curved spacetimes.

If generalized ANEC holds, macroscopic wormholes are ruled out.

Implications for total energy positivity and singularity theorems.

Abstract

The averaged null energy condition has been recently shown to hold for linear quantum fields in a large class of spacetimes. Nevertheless, it is easy to show by using a simple scaling argument that ANEC as stated cannot hold generically in curved four-dimensional spacetime, and this scaling argument has been widely interpreted as a death-blow for averaged energy conditions in quantum field theory. In this note I propose a simple generalization of ANEC, in which the right-hand-side of the ANEC inequality is replaced by a finite (but in general negative) state-independent lower bound. As long as attention is focused on asymptotically well-behaved spacetimes, this generalized version of ANEC is safe from the threat of the scaling argument, and thus stands a chance of being generally valid in four-dimensional curved spacetime. I argue that when generalized ANEC holds, it has implications for the non-negativity of total energy and for singularity theorems similar to the implications of ANEC. In particular, I show that if generalized ANEC is satisfied in static traversable wormhole spacetimes (which is likely but remains to be shown), then macroscopic wormholes (but not necessarily microscopic, Planck-size wormholes) are ruled out by quantum field theory.