Averaged null energy condition in a classical curved background
Eleni-Alexandra Kontou · Ken D. Olum
Abstract and summary · read the original at the source
In one page
A wormhole, a warp bubble or a time machine can always be written down as a geometry; what decides whether one can exist is the matter it would take to hold it open. The sharpest version of that rulebook is the achronal averaged null energy condition: add up the energy a light ray meets along its whole path, and the total must not come out negative. Eleni-Alexandra Kontou and Ken Olum, at Tufts, extend that result off flat space and into curved space, which is where any real design would sit. Their theorem says that a free, minimally coupled quantum scalar field cannot violate the condition along a light ray that travels inside a tube of ordinary, classically sourced curvature. The proof rests on one stated conjecture — that the flat-space limit on how deep and how long energy may dip below the ambient vacuum level survives, slightly corrected, when curvature is small. Their closing section then maps exactly where the remaining room is.
Why it matters hereChapter 4 scores every warp and wormhole design on how much below-ambient-vacuum energy it needs and along what path, and this paper narrows that specification: with an ordinary quantum scalar field, the light ray you are trying to bend cannot pay for itself inside a region of ordinary curvature. Chapter 2 gets the sharper statement of what the vacuum will and will not lend, and the two places Kontou and Olum say the room actually is — fields coupled to curvature, and a second field working in the spacetime a first one has already made. Read it beside their curved-spacetime quantum inequality at /library/stm-efbb5f52fe and the full rulebook at /library/stm-1bcd336342.
What it claims
01The achronal averaged null energy condition is the useful form of the rule: strong enough to exclude wormholes, superluminal travel and time machines, yet weak enough that no violation is known for minimally coupled scalar fields. It requires only that the stress-energy projected along a complete null geodesic integrate to a non-negative value, and only for geodesics that are achronal — no two points of which can be joined by a timelike curve.Sec. I, Introduction, Eq. 1 and the paragraph following it
Settled physics02Theorem 1: for a free, minimally coupled quantum scalar field in a Hadamard state, the averaged null energy integral cannot converge uniformly to negative values on all geodesics of a tubular neighbourhood of a null geodesic, provided that neighbourhood obeys the null convergence condition, has bounded and smooth curvature, is flat in the distant past and future, and has a causal structure unaffected by conditions elsewhere.Sec. III, The theorem, and Eq. 7
Published and peer-reviewed03Only a neighbourhood of the light ray has to be well behaved, so the result covers any geodesic that never encounters a material source, even where such sources exist elsewhere in the same spacetime.Sec. II C, and the paragraph following Eq. 3
Published and peer-reviewed04The proof rests on one stated conjecture: that the flat-space, null-contracted and timelike-averaged quantum inequality of Fewster and Roman continues to hold, with a small correction, in spacetimes of small curvature — where small means every component of the Riemann tensor in the observer’s parallel-transported frame, multiplied by the square of the sampling time, stays much less than one.Sec. IV, Quantum inequality, Eqs. 8 and 9
What to watch05A single effect cannot both violate the averaged null energy condition and supply the curvature that would license the violation. Writing the exotic stress-energy against the curvature it induces gives a scaling in which the tensor is bounded by itself times the square of the Planck length over a much larger characteristic length — which cannot hold.Sec. VI, Discussion, Eqs. 67 to 69
Published and peer-reviewed06The authors name what their theorem leaves open. Scalar fields with non-minimal curvature coupling can violate the condition even classically once field values approach the Planck scale, no quantum inequalities of the usual kind are known for them, and a two-stage construction — one field that breaks the pointwise condition while obeying the averaged one, and a second, weaker effect that breaks the averaged condition in the spacetime the first has made — is not yet ruled out.Sec. VI, Discussion, closing paragraphs
What to watch
Read it · abstract
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. Exotic spacetimes, such as those allow wormholes or the construction of time machines are possible in general relativity only if ANEC is violated along achronal geodesics. Starting from a conjecture that flat-space quantum inequalities apply with small corrections in spacetimes with small curvature, we prove that ANEC is obeyed by a minimally coupled, free quantum scalar field on any achronal null geodesic surrounded by a tubular neighborhood whose curvature is produced by a classical source.
The way in
https://doi.org/10.1103/PhysRevD.87.064009LICENCE. Published as Physical Review D 87, 064009 (2013) under the APS default licence; the publisher PDF at link.aps.org refused every fetch on 2026-09-08 with HTTP 403. The author manuscript is free to read on arXiv as 1212.2290v1, posted 11 December 2012, and that record carries the arXiv non-exclusive distribution licence version 1.0 — arXiv distribution rights only, no re-use — checked on the arXiv abstract page on 2026-09-08, and no Creative Commons statement appears in the manuscript text either. So this sheet carries the summary, the claims and the authors’ own abstract, and sends the reader to the full text at the source. TEXT. The claims below were written from the complete arXiv manuscript and the locators use its own section, theorem and equation numbering; the abstract reproduced here is the published one. Kontou and Olum wrote from the Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford. Two companion sheets carry the rest of this conversation: the same pair’s curved-spacetime quantum inequality at /library/stm-efbb5f52fe, and the Kontou and Sanders review of the whole energy-condition rulebook at /library/stm-1bcd336342.
How to cite it
Eleni-Alexandra Kontou, Ken D. Olum (2013) Averaged null energy condition in a classical curved background. doi:10.1103/PhysRevD.87.064009
Where it sits in the curriculum