Scalar fields, energy conditions and traversable wormholes
Carlos Barceló · Matt Visser
Abstract and summary · read the original at the source · none found
In one page
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.
Why it matters hereChapter 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.
What it claims
01The 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
Published and peer-reviewed02The 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
Published and peer-reviewed03Solving 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
Published and peer-reviewed04Every 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
Published and peer-reviewed05Ordinary 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
What to watch06These 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
Settled physics
Read it · abstract
Abstract
We describe the different possibilities that a simple and apparently quite harmless classical scalar field theory provides to violate the energy conditions. We demonstrate that a non-minimally coupled scalar field with a positive curvature coupling — xi greater than zero — can easily violate all the standard energy conditions, up to and including the averaged null energy condition (ANEC). Indeed this violation of the ANEC suggests the possible existence of traversable wormholes supported by non-minimally coupled scalars. To investigate this possibility we derive the classical solutions for gravity plus a general (arbitrary xi) massless non-minimally coupled scalar field, restricting attention to the static and spherically symmetric configurations. Among these classical solutions we find an entire branch of traversable wormholes for every positive xi. (This includes and generalizes the case of conformal coupling, xi equal to one sixth, we considered in Phys. Lett. B466 (1999) 127–134.) For these traversable wormholes to exist we demonstrate that the scalar field must reach trans-Planckian values somewhere in the geometry. We discuss how this can be accommodated within the current state of the art regarding scalar fields in modern theoretical physics. We emphasize that these scalar field theories, and the traversable wormhole solutions we derive, are compatible with all known experimental constraints from both particle physics and gravity.
The way in
https://doi.org/10.1088/0264-9381/17/18/318Published as Classical and Quantum Gravity 17 (2000) 3843, by Carlos Barceló and Matt Visser, both then at the Physics Department of Washington University in Saint Louis. Licence checked directly: the IOP version of record carries the publisher’s own terms, and the author version, arXiv:gr-qc/0003025v2, dated 14 July 2000 and marked accepted for publication in Classical and Quantum Gravity, carries arXiv’s assumed licence for 1991 to 2003 postings. Neither is a Creative Commons grant, so this sheet carries the summary, the claims and the authors’ own abstract and sends the reader to the source; the author version is free to read on arXiv and the IOP page is bot-walled to automated readers. The summary and claims below were written from the complete twenty-five-page author version, and the locators cite its sections and equation numbers. The abstract is the authors’ own, with the inequality signs written out in words because the site’s renderer does not carry mathematical markup. Three companion sheets in this library carry the rest of this conversation: Teo’s rotating traversable wormholes at /library/stm-a473e80f36, Sushkov’s wormholes supported by phantom energy at /library/stm-35489e8f11, and Kuhfittig on how little exotic material a traversable throat really needs at /library/stm-d54d953175.
How to cite it
Carlos Barceló, Matt Visser (2000) Scalar fields, energy conditions and traversable wormholes. doi:10.1088/0264-9381/17/18/318
Where it sits in the curriculum
The metric, warp drives and wormholesThe vacuum as a quantum fluid