Spacetime averaged null energy condition

The averaged null energy condition has known violations for quantum fields in curved space, even when one considers only achronal geodesics. Many such examples involve rapid variation in the stress-energy tensor in the vicinity of the geodesic under consideration, giving rise to the possibility that averaging in additional dimensions would yield a principle universally obeyed by quantum fields. However, after discussing various procedures for additional averaging, including integrating over all dimensions of the manifold, we give here a class of examples that violate any such averaged condition.