Quantum inequality in spacetimes with small curvature
Eleni-Alexandra Kontou · Ken D. Olum
Abstract and summary · read the original at the source
In one page
Quantum field theory allows the energy density at a point to sit below the ambient vacuum level — the same accounting that produces the Casimir force — and how deep and how long that dip may run is what sets the engineering budget for wormholes and warp geometries. The bounds are called quantum inequalities, and almost all of them had been proved only in flat spacetime, which is exactly where a warp drive is not. Eleni-Alexandra Kontou and Ken Olum of the Tufts Institute of Cosmology close part of that gap. Starting from a general curved-spacetime bound of Fewster and Smith, they derive an explicit inequality for a massless scalar field measured along a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. The result is the familiar flat-space bound plus correction terms, and the corrections stay small whenever the sampling time is much shorter than any curvature radius. One consequence is neat: only the Ricci tensor enters, so along a path through empty space the flat-space bound applies unchanged.
Why it matters hereChapter four’s warp and wormhole designs live or die on how much energy can be held below the ambient vacuum level, and for how long. This paper is the first explicit statement of that budget in a spacetime that is actually curved — the setting any real design occupies — and it tells builders when the simpler flat-space number is safe to use. Chapter two gets the sharper picture of the vacuum floor: what may dip below it is set by curvature only through the Ricci tensor, which is to say through the matter and energy actually present.
What it claims
01Working to first order in the curvature, the flat-space quantum inequality for a minimally coupled massless scalar field survives on a geodesic in a curved spacetime, with additional terms built from the Ricci tensor and its derivatives combined with the sampling function.Abstract; Sec. VII, Eq. (128)
Published and peer-reviewed02No other part of the Riemann tensor appears in the correction, only the Ricci tensor — so in a vacuum spacetime, Schwarzschild and Kerr and pure gravitational waves included, the flat-space quantum inequality holds unmodified to first order.Sec. II, paragraph following Eq. (3); Sec. VIII, third paragraph
Published and peer-reviewed03For a Gaussian sampling function of width t-nought the bound is written explicitly, its leading term being exactly the known flat-space result and the curvature terms entering through the products of the Ricci bounds with powers of the sampling width.Sec. VII, Eqs. (129) and (130)
Published and peer-reviewed04Ford and Roman had argued that flat-space quantum inequalities can be used in curved spacetime whenever the curvature radius is large compared with the sampling time; this paper confirms that explicitly and computes the size of the deviation, adding that the curvature must also be small at every point lying in both the causal future of one point of the path and the causal past of another.Sec. VIII, fourth paragraph
Published and peer-reviewed05An ambiguity remains in every such bound, carried by the two unknown coefficients of the local curvature terms in the gravitational Lagrangian — the authors keep them explicit as the parameters a and b rather than fixing them by hand.Sec. VIII, fifth paragraph; Eq. (128)
What to watch06The result is a step toward proving the conjecture behind the authors’ earlier achronal averaged null energy condition theorem; the named next steps are to extend it to null-projected rather than timelike-projected quantum inequalities, and to slightly non-geodesic curves.Sec. VIII, final paragraph
What to watch
Read it · abstract
Abstract
Quantum inequalities bound the extent to which weighted time averages of the renormalized energy density of a quantum field can be negative. They have mostly been proved in flat spacetime, but we need curved-spacetime inequalities to disprove the existence of exotic phenomena, such as closed timelike curves. In this work we derive such an inequality for a minimally-coupled scalar field on a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. Since only the Ricci tensor enters, there are no first-order corrections to the flat-space quantum inequalities on paths which do not encounter any matter or energy.
Eleni-Alexandra Kontou and Ken D. Olum. Physical Review D 91, 104005 (2015). Author version: arXiv:1410.0665.
(Abstract only. The author version is free to read on arXiv — see the rights note for why the full text is not reproduced here, and for the two companion sheets in this library.)
The way in
https://doi.org/10.1103/PhysRevD.91.104005Published as Physical Review D 91, 104005 (2015) under the APS default licence. The author version is free to read on arXiv as 1410.0665, posted 2 October 2014 with a second version on 4 October 2014, and that record carries the arXiv non-exclusive distribution licence version 1.0 — not a Creative Commons licence, checked on the arXiv abstract page on 2026-09-08, and no Creative Commons statement appears in the text either. So this sheet carries the summary, the claims and the authors’ own abstract and sends the reader to the source. The claims below were written from the arXiv text and the locators use its section and equation numbering. Kontou and Olum write from the Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford; the work was supported by grant RFP3-1014 from the Foundational Questions Institute, and Kontou by a John F. Burlingame Graduate Fellowship. The general curved-spacetime bound they build on is Fewster and Smith’s, Annales Henri Poincaré 9, 425 (2008). Two companion sheets in this library carry the rest of this conversation: Ford and Roman’s founding quantum inequality, Physical Review D 51, 4277 (1995), at /library/stm-5e822858ce, and Ford, Helfer and Roman on why the purely spatial version does not exist in four dimensions, Physical Review D 66, 124012 (2002), at /library/stm-755a293263.
How to cite it
Eleni-Alexandra Kontou, Ken D. Olum (2015) Quantum inequality in spacetimes with small curvature. doi:10.1103/PhysRevD.91.104005
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