Historical research guide · 0559 · 1997
Restrictions on negative energy density in flat spacetime
L. H. Ford · Thomas A. Roman
In one minute
Lawrence Ford and Thomas Roman, at the Tufts Institute of Cosmology, ask the question any vacuum engineer eventually has to ask: how far below the ambient vacuum level can a field be pushed, and for how long? Quantum field theory allows the energy density at a single point to sit below the flat-space vacuum floor without limit — the Casimir effect and squeezed light both show it — but Ford and Roman show that nature keeps the books over time. Fold the energy density a moving observer measures against a bell-shaped sampling function of width t-nought, and the average cannot fall below a number that scales as one over t-nought to the fourth power. Deeper means briefer, exactly like an uncertainty relation. The contribution here is a far simpler derivation, done with plane waves instead of spherical ones, extended for the first time to the electromagnetic field and to massive scalar fields. Mass tightens the bound. Stretch the sampling time to infinity and the averaged weak energy condition falls out of it.
Connection
Chapter 4’s warp and wormhole geometries and chapter 6’s vacuum-energy devices both turn on the same quantity — how far a field can be held below the ambient vacuum level, and for how long — and this is the paper that turns that question into a number with a name attached. It is also the specification every later positive-energy warp shell is designed to satisfy rather than dodge.
From the source
Quantum field theory permits the energy density at a spacetime point to sit below the flat-space vacuum level without bound, violating the classical pointwise energy conditions, and two instances of it have observational support: the Casimir effect and squeezed states of light.Section 1, first paragraph
For the free quantized massless scalar field in four-dimensional Minkowski spacetime, the energy density an arbitrary inertial observer measures, folded against a Lorentzian sampling function of characteristic time t-nought, is bounded below by minus three divided by thirty-two pi squared times t-nought to the fourth power — a bound the authors here re-derive in a plane-wave mode expansion far simpler than their original spherical-wave argument.Section 1, Eq. 1; Section 2.1, Eqs. 14 to 17
Giving the scalar field a mass multiplies that bound by a function G of the dimensionless combination twice the mass times the sampling time; G tends to one as the mass goes to zero, recovering the massless result, and falls to zero as the mass or the sampling time grows — so a massive field must hold its below-vacuum region more tightly localized than a massless one, because the rest mass energy has to be overcome as well.Section 2.1, Eqs. 15 and 16 and the discussion following Eq. 17; Figure 1
For the free quantized electromagnetic field in four-dimensional Minkowski spacetime the same method gives a bound of minus three divided by sixteen pi squared times t-nought to the fourth power — weaker than the massless scalar bound by exactly the factor of two the field’s two polarization states would lead you to expect.Section 3, Eqs. 45 to 48
Taking the sampling time to infinity samples the observer’s entire timelike geodesic, and every one of these bounds then returns the averaged weak energy condition — so in Minkowski spacetime the averaged weak energy condition is a consequence of the quantum inequality rather than a separate assumption.Section 4, Eq. 49
Applied to Morris-Thorne wormhole geometries in the short-sampling-time limit, the bound says the throat must be close to Planck size or the below-vacuum energy must sit in an extraordinarily thin band — for a throat one metre across, a band about a millionth of a proton radius thick — and the authors name the two ways out they can see: on the order of ten to the sixty-second power of independent fields, or a field whose numerical constant on the right-hand side is many orders of magnitude larger than any yet examined.Section 4, third and fourth paragraphs
The way in
https://doi.org/10.1103/PhysRevD.55.2082How to cite it
L. H. Ford, Thomas A. Roman (1997) Restrictions on negative energy density in flat spacetime doi:10.1103/PhysRevD.55.2082