STM-D-0764Paper1998Published and peer-reviewed

Rotating traversable wormholes

Edward Teo

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Morris and Thorne showed in 1988 how to write down a wormhole you could actually walk through, and then read off from Einstein’s equations what kind of matter would be needed to hold it open. Their wormhole was static and perfectly spherical. Edward Teo, then at Cambridge and the National University of Singapore, does the same job for a wormhole that spins. He writes the most general stationary, axially symmetric metric a traversable wormhole can have, so any rotating wormhole can be described inside one framework. Two results follow. The throat still needs matter that violates the null energy condition — no rotation removes that bill. But rotation lets you move that matter around the throat, and Teo gives an explicit wormhole in which fast enough infalling travellers cross without ever passing through it. Spin the wormhole hard enough and it also grows an ergoregion, a tube around its equator where nothing can stand still and rotational energy can be extracted.

Why it matters hereChapter 4 is about engineering the metric rather than pushing through it, and this is the paper that gives the rotating case its general form. It matters for two practical reasons: it tells a designer exactly where the exotic matter has to sit, and it shows that the traveller’s own path can be routed around it — the objection that a wormhole traveller must swim through negative-energy material is answered here, for rotating wormholes, with an explicit metric and an explicit energy threshold. The ergoregion result adds the other half of chapter 4’s interest: a rotating throat is not only a door, it is a place from which energy can be taken.

What it claims

01The most general stationary, axially symmetric traversable wormhole can be written in a canonical form with four functions of radius and polar angle — the analogues of Morris and Thorne’s redshift and shape functions, plus an angular velocity term for the dragging of inertial frames. This gives an explicit class of rotating wormholes generalising the static, spherically symmetric ones.Abstract; Sect. 2, Eq. (19)
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02Such a wormhole generically violates the null energy condition at the throat. Teo proves it directly rather than by the general defocusing argument, choosing a null vector that acquires an angular velocity as the throat is approached because of frame dragging, and showing the resulting quantity is negative somewhere on the throat.Sect. 3, Eqs. (20) to (25)
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03The exotic matter supporting the wormhole can be moved around the throat by choosing the metric functions appropriately, so that some class of infalling observers never encounters it — in contrast with the static, spherically symmetric case, where every such observer does. The key point stated alongside it is that one can never avoid the use of exotic matter altogether.Sect. 3, final paragraph
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04In the worked example, with angular momentum parameter equal to one quarter, null geodesics falling in without angular momentum give a strictly positive Ricci contraction, and time-like ones do too provided the conserved energy per unit rest mass is greater than about 1.75 — so those travellers traverse the wormhole without meeting any matter of negative energy density.Sect. 4, Eqs. (26), (34) and (35)
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05If the rotation is fast enough — in the worked example, an angular momentum parameter whose absolute value is greater than one half — an ergoregion appears outside the throat, where particles can no longer remain stationary with respect to infinity. It forms a tube around the equator rather than a shell, because with no event horizon it cannot reach the poles, and rotational energy can be extracted from it by the Penrose process.Sect. 4, final two paragraphs; Fig. 1
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06Teo names the semi-classical regime as the most promising application of the framework: quantum fields, unlike classical matter, do violate the energy conditions, and self-consistent wormhole solutions supported by a quantised scalar field already exist with throats of order the Planck length — candidate models for spacetime foam, whose generalisation to slowly rotating wormholes was being taken up when he wrote.Sect. 5, Concluding remarks
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Abstract

The general form of a stationary, axially symmetric traversable wormhole is discussed. This provides an explicit class of rotating wormholes that generalize the static, spherically symmetric ones first considered by Morris and Thorne. In agreement with general analyses, it is verified that such a wormhole generically violates the null energy condition at the throat. However, for suitable model wormholes, there can be classes of geodesics falling through it which do not encounter any energy-condition-violating matter. The possible presence of an ergoregion surrounding the throat is also noted.

The way in

https://doi.org/10.1103/PhysRevD.58.024014LICENCE CHECK. The version of record is Physical Review D volume 58, article 024014 (1998), under the APS default licence, and the author copy arXiv:gr-qc/9803098v2, dated 27 April 1998, carries the arXiv assumed licence for 1991 to 2003 postings. Neither is a Creative Commons grant, so no full text is reproduced here and the sheet stays abstract-only. The abstract below is the author’s own, as printed on the arXiv posting and the published paper; the preprint carries the Cambridge DAMTP report number R98/17.

How to cite it

Edward Teo (1998) Rotating traversable wormholes. doi:10.1103/PhysRevD.58.024014

Where it sits in the curriculum

The metric, warp drives and wormholes