Historical research guide · 0801 · 2000
Scalar fields, energy conditions and traversable wormholes
Carlos Barceló · Matt Visser
In one minute
Carlos Barceló and Matt Visser ask what ordinary classical matter is actually obliged to do, and find the obligation is weaker than the textbooks suggest. The energy conditions of general relativity — the rules that would forbid a wormhole you could travel through — are assumptions about matter, not theorems, and a plain classical scalar field breaks them as soon as the field is coupled to spacetime curvature rather than sitting inertly on top of it. The authors solve gravity plus such a field exactly, for static spherically symmetric geometries, and find a whole branch of traversable wormhole solutions for every positive value of the curvature coupling, generalising the conformally coupled case they had published the year before. The price is specific and stated plainly: somewhere in the geometry the scalar field has to reach values above the Planck scale. They then show these theories sit comfortably inside every solar-system measurement made of gravity, and note that string-inspired physics is full of exactly such fields.
Connection
Chapter 4 needs geometries a craft could actually cross, and this paper moves the wormhole out of the quantum-exotic-matter corner and into classical field theory: the source is a scalar field of the kind modern physics expects to exist anyway, and the authors name the one extreme it demands — a trans-Planckian field value — as the thing to work on next.
From the source
The energy conditions of general relativity are assumptions about what classical matter does, not theorems about it, and a simple classical scalar field coupled to the spacetime curvature violates all of the pointwise conditions and can violate the averaged null energy condition too — which is the condition the topological censorship theorem uses to rule wormholes out.Section 1, opening paragraphs; Section 2
The averaged null energy integral shows exactly where the violation lives: for negative curvature coupling, and for positive coupling while the field stays below the square root of the ratio of the inverse Newton constant to the coupling, the integrand is positive and the condition holds — violation needs positive coupling and places along the geodesic where the field exceeds that value, which is where the boundary terms can no longer be discarded and can contribute negatively without bound.Section 2.2, Eq. 2.11 with the three cases that follow it
Solving gravity plus a massless non-minimally coupled scalar field exactly, in the static and spherically symmetric class, yields an entire branch of traversable wormhole solutions for every positive curvature coupling — which includes and generalises the conformally coupled case, coupling one sixth, that the same authors published in Physics Letters B 466, 127, in 1999.Sections 3 and 4; Abstract
Every one of those wormhole solutions demands the same extreme: the scalar field has to reach absolute values above roughly the Planck mass divided by the square root of the curvature coupling, so either the field goes trans-Planckian somewhere in the geometry or the coupling constant has to be made very large.Section 4, paragraph following Eq. 4.44
Ordinary matter placed in these geometries feels an effective Newton constant set by the local scalar field, and in the solutions as derived that constant changes sign between the two asymptotic regions unless the matter coupling function is chosen to cancel the factor — the authors point to thin-shell surgery as the way to confine that behaviour to a thin region around the throat, and flag the stability of the solutions as untouched work.Section 4, Eqs. 4.45 to 4.47 with footnotes 4 and 5
These scalar theories are compatible with everything gravity has been measured to do: very long baseline interferometry of light deflection by the Sun fixes the parametrized post-Newtonian parameter gamma at one minus 0.6 plus or minus 3.1 parts in ten thousand, which puts the Brans-Dicke-like parameter above 3000 in magnitude, a limit easily met when the scalar field is a small fraction of the Planck scale in the solar neighbourhood — and membrane-inspired low-energy theories supply such fields automatically, the dilaton and the moduli among them.Section 5, Eqs. 5.52 and 5.53 and the paragraphs that follow
The way in
https://doi.org/10.1088/0264-9381/17/18/318How to cite it
Carlos Barceló, Matt Visser (2000) Scalar fields, energy conditions and traversable wormholes doi:10.1088/0264-9381/17/18/318