Historical research guide · 0527 · 2003
Can a wormhole supported by only small amounts of exotic matter really be traversable?
Peter K. F. Kuhfittig
In one minute
A traversable wormhole is a shortcut between two distant places, and holding one open calls for matter whose energy density some observers would measure as negative — what Morris and Thorne named exotic matter. Two published results had made that sound almost affordable: quantum effects may be enough to support a throat, and the total amount of exotic matter can be made as small as you like. Peter Kuhfittig, a mathematician at the Milwaukee School of Engineering, asks the practical follow-up. Squeezing the exotic matter into a vanishingly thin shell makes the tunnel flare out so slowly that it can become enormous, and it pushes the throat towards being an event horizon, which would trap a traveller rather than pass them through. He builds a model that avoids both. With one choice of the two shape functions the tidal forces stay inside human limits, the wormhole mouth sits about four astronomical units out, a ship at one gravity reaches the throat in roughly six days — and the exotic matter required is still vanishingly small.
Connection
Chapter 4 is where the metric stops being scenery and becomes something you engineer, and this is the practical end of that argument: it takes the two published escape routes — quantum support for a throat, and arbitrarily little exotic matter — and works out what the geometry has to look like before a person could actually go through it, with tidal limits, a mouth distance and a transit time attached.
From the source
To hold a traversable wormhole open, violations of certain energy conditions are unavoidable, so the energy density of the supporting matter may be seen as negative by some observers; all classical forms of matter obey the weak energy condition, but quantum fields can generate locally negative energy densities, and wormhole spacetimes also violate the averaged null energy condition.Section 1, Introduction
Recent studies show that quantum effects may be sufficient to support a wormhole throat, and that the total amount of exotic matter can be made arbitrarily small — the latter following from the assumed averaged-null-energy-condition violation, provided the redshift factor at the throat tends to zero over a sequence of wormholes.Abstract, points (a) and (b); Section 1
Using only small amounts of exotic matter carries two costs: the shape function’s derivative sits close to unity near the throat, so the embedding diagram flares out very slowly and the wormhole can become too large to traverse in a reasonable length of time, and combined with the Ford-Roman constraints the geometry may come close to having an event horizon at the throat.Abstract; Section 2, discussion following Eq. (10)
The model takes the metric functions as inverse powers of the distance from the throat and from an interior point, with the redshift constant fixed by matching the two slopes at the outer edge of the shell; the weak energy condition is then violated only inside the shell and satisfied outside it, the Ford-Roman constraints are met, and no event horizon forms because the redshift function stays finite at the throat.Section 2, Eq. (8) and the inequalities following it
For the simplest case the radial tidal constraint is met with a scale constant of five times ten to the minus nine light years; placing the space stations at about 6.6 times ten to the eighth kilometres, roughly four astronomical units, satisfies the redshift-gradient constraint and keeps the shape ratio within one per cent of unity, and a spaceship accelerating at one Earth gravity halfway and decelerating the rest of the way reaches the throat in about six days.Section 3, Traversability conditions, Eq. (11) and the numerical estimates
The integral measuring the total amount of energy-condition-violating matter in the shell is smaller than ten to the minus one hundred even for the simplest case, and the conclusion does not depend on the particular functions chosen, because the redshift function stays finite at the throat while the other function diverges there, driving both limiting quantities to zero.Section 3, Eq. (12) and the limit argument; Section 4, Conclusion
The way in
https://doi.org/10.1103/PhysRevD.68.067502How to cite it
Peter K. F. Kuhfittig (2003) Can a wormhole supported by only small amounts of exotic matter really be traversable? doi:10.1103/PhysRevD.68.067502