Historical research guide · 0460 · 2013
Averaged null energy condition in a classical curved background
Eleni-Alexandra Kontou · Ken D. Olum
In one minute
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.
Connection
Chapter 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.
From the source
The 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
Theorem 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
Only 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
The 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
A 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
The 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
The way in
https://doi.org/10.1103/PhysRevD.87.064009How 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