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Title: Does the achronal averaged null energy condition hold in curved spacetime?
Plain: 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: 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.
Settled by: A proof of the achronal condition for interacting fields on self-consistent semiclassical backgrounds, or an explicit violating solution.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Proved in Minkowski space (2016); no curved-spacetime proof as of 2026.
Posed since: 2007