Historical research guide · 0563 · 1998
Quantum Inequality Restrictions on Negative Energy Densities in Curved Spacetimes
Michael John Pfenning · L. H. Ford
In one minute
Negative energy density is a real feature of quantum field theory — arrange a field the right way and a detector reads below the ambient vacuum level — and both warp bubbles and traversable wormholes need it. Michael Pfenning’s Tufts dissertation, written with Lawrence Ford, works out how much of it nature will lend you and for how long. The tool is the quantum inequality, an uncertainty-principle-style relation tying the depth of a negative energy dip to its duration. Pfenning and Ford derive the general form in static curved spacetime, where it becomes the Euclidean wave operator acting on the Euclidean Green’s function, and then solve it exactly for Robertson-Walker universes, flat space with mirrors, Rindler and de Sitter space, and the region outside a black hole. Then they price the Alcubierre warp bubble. The inequality forces the bubble wall to be only hundreds of Planck lengths thick, and for a bubble a hundred metres across the negative energy needed comes out far above the mass of the visible universe.
Connection
This dissertation is where chapter 4 gets its price tag: the negative-energy requirement of the Alcubierre metric turned into a number, with the derivation shown. Every warp design that came after — Van Den Broeck’s bubble, White’s toroidal reshaping, the Bobrick and Martire shells — is an attempt to cut that bill, and chapter 6 needs the same inequalities because they are the rule that governs how far below the ambient vacuum level any device can push.
From the source
A general quantum inequality on the energy density can be derived for both the quantized scalar field and the electromagnetic field in static curved spacetimes; for the scalar field it can be written as the Euclidean wave operator acting on the Euclidean Green’s function.Chapter 3, Sections 3.1 and 3.4; Abstract
In the limit of short sampling times, a small-distance expansion of the Green’s function reduces the curved-space inequality to the flat-space form plus subdominant correction terms that depend on the local spacetime geometry — so the flat-space bound is the right one whenever the sampling time is short compared with the smallest local radius of curvature.Section 3.3
Exact quantum inequalities are obtained for the three- and four-dimensional static Robertson-Walker spacetimes, flat space with perfectly reflecting mirrors, Rindler and static de Sitter space, and the spacetime outside a black hole; in every case the bound limits how much negative energy can be observed relative to the vacuum energy density of that spacetime, and outside a black hole it is measured relative to the Boulware vacuum.Chapter 4; dissertation abstract, third paragraph
Applied to the Alcubierre metric, the quantum inequality constrains the thickness of the warp bubble wall: taking the sampling time as a tenth of the local radius of curvature, the wall thickness is bounded by about one hundred times the bubble velocity times the Planck length, so the walls are typically only hundreds or thousands of Planck lengths thick and the shape function behaves as a step function.Section 5.3, equations 5.22 and 5.23; Section 5.5
Integrating the local energy density over proper volume for a bubble of 100 metre radius gives a total negative energy bounded by about 6.2 times ten to the sixty-fifth grams multiplied by the bubble velocity, which is roughly three times ten to the twentieth galaxy masses per unit velocity, or about ten orders of magnitude greater than the total mass of the visible universe.Section 5.4, equations 5.25 to 5.30
The price is extremely sensitive to the constraints: a wall one metre thick, if the inequality could be evaded, would bring a 100 metre bubble down to about a quarter of a solar mass, and a bubble the size of one electron Compton wavelength needs about 400 solar masses; the derivation assumes a quantized massless scalar field, and the corresponding inequalities for massive scalar fields and for the electromagnetic field are more restrictive still.Section 5.4, closing paragraph; Section 5.5
The way in
https://arxiv.org/abs/gr-qc/9805037How to cite it
Michael John Pfenning, L. H. Ford (1998) Quantum Inequality Restrictions on Negative Energy Densities in Curved Spacetimes arXiv:gr-qc/9805037