Historical research guide · 0888 · 2015
Quantum inequality in spacetimes with small curvature
Eleni-Alexandra Kontou · Ken D. Olum
In one minute
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.
Connection
Chapter 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.
From the source
Working 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)
No 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
For 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)
Ford 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
An 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)
The 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
The way in
https://doi.org/10.1103/PhysRevD.91.104005How to cite it
Eleni-Alexandra Kontou, Ken D. Olum (2015) Quantum inequality in spacetimes with small curvature doi:10.1103/PhysRevD.91.104005