Does the achronal averaged null energy condition hold in curved spacetime?
In plain words
Quantum fields can have negative energy locally, which could in principle hold open wormholes or build time machines. A weaker condition, energy averaged along light rays that never come back to their own past, would exclude these, but it is unproved in curved spacetime.
Precise statement
The averaged null energy condition, integral of $\langle T_{ab}\rangle k^{a} k^{b} d\lambda \ge 0$ along complete null geodesics with tangent $k^{a}$ and affine parameter $\lambda$, is proved for quantum field theory in Minkowski space (Faulkner, Leigh, Parrikar, Wang 2016) but fails on some curved backgrounds. Prove or refute the self-consistent achronal version (Graham and Olum 2007): for every achronal complete null geodesic in a solution of semiclassical gravity, the integral is non-negative. An answer is a proof for a stated class of field theories and spacetimes, or a counterexample that solves the semiclassical equations.
What would settle it
A proof of the achronal condition for interacting fields on self-consistent semiclassical backgrounds, or an explicit violating solution.
Status in the literature
Unverified note
Proved in Minkowski space (2016); no curved-spacetime proof as of 2026.