Static and dynamic traversable wormholes

What it claims

1. 01The energy conditions are not part of general relativity. They are additional limitations imposed on the stress-energy tensor to assure the physical reasonability of solutions to the Einstein field equations — which is why violating them is a licence question rather than a law of physics, and why the whole wormhole problem reduces to how much violation is needed.Chapter 3 Energy Conditions, section 3.1; restated in Chapter 6 Summary, second paragraph

   Published and peer-reviewed

2. 02Known physics already violates them. Between two Casimir plates the energy density is negative — minus pi squared times the reduced Planck constant, over seven hundred and twenty times the plate separation to the fourth power — so the weak and dominant conditions fail there, and because energy density plus the pressure along the plate normal is negative the null and strong conditions fail too. Adamiak adds the topological version, in which periodic boundary conditions imposed on the universe itself violate all of them, and a non-zero cosmological constant, whose term is identified with the vacuum energy density and includes the zero-point fluctuations of every field-theory degree of freedom.Section 3.2 Violations, equations 3.28 to 3.33

   Settled physics

3. 03The nine requirements a traversable wormhole has to meet, as Adamiak lists them after Morris and Thorne: the metric spherically symmetric and static; the solution obeying the Einstein field equations everywhere; a throat connecting two asymptotically flat regions; no horizon, since a horizon would prohibit two-way travel; tidal gravitational forces reasonably small; a crossing time acceptable from the point of view of both the traveller and an outside observer; a physically reasonable stress-energy tensor; stability; and the requirement that it must be possible to assemble the thing.Section 2.1 Desired Properties of Traversable Wormhole, the numbered list of nine

   Designed, not yet built

4. 04How much exotic matter is measured, not assumed. Adamiak reviews the volume-integral quantifiers of averaged-null-energy-condition-violating material, and the three conditions a proper integration measure must satisfy: it must be natural, so that the resulting total mass is the same in any coordinate system; it must reproduce, at least to first order, the scalar charge of a solution where a scalar field plays the role of exotic matter; and the mass must obey a conservation principle. His verdict is that there is a strong indication the amount of exotic matter needed to support a wormhole may not be as huge as was thought a decade earlier, and that introducing dynamics usually makes that amount smaller still.Section 3.3 Quantification of exotic matter, equations 3.49 and 3.50; Chapter 6 Summary, numbered point 3

   Published and peer-reviewed

5. 05The evolving-wormhole result, stated in his own terms: writing the metric with a time-dependent conformal factor and imposing the weak energy condition gives three inequalities, and the flare-out condition at the throat forces the combination of the conformal factor and its first two time derivatives to stay positive. That constraint cannot be satisfied over the whole domain for a conformal factor that is positive and bounded. So although wormholes can be created without exotic matter, their lifetime is limited, because the controlling function sooner or later becomes negative.Section 4.2 Conformal Approach, equations 4.5 to 4.10 with Figure 4.2

   Published and peer-reviewed

6. 06The rotating-wormhole result, in two parts. Rotation does not remove the need for exotic matter — an energy-condition violation is always present near the throat. But the exotic matter can be moved around the throat, so that some class of infalling observers would never encounter it. In Teo’s solution the throat radius picks up a term in the angular momentum and the cosine of the polar angle, giving the throat a dumbbell-like shape, and at substantial rotation speed the time-time component of the metric can turn positive outside the throat — an ergoregion.Chapter 5 Rotating Wormholes, the two numbered results after equation 5.13, with the Teo solution at equation 5.14

   Published and peer-reviewed

The way in

https://doi.org/10.1142/9789812834300_0357ATTRIBUTION, CHECKED AT CROSSREF 2026-09-09. A previous reader suspected this record of being mis-attributed. It is not. The Crossref deposit for DOI 10.1142/9789812834300_0357 names one author, Jaroslaw P. Adamiak, with the affiliation Department of Mathematical Sciences, University of South Africa, PO Box 392, Unisa 0003, South Africa; the container is The Eleventh Marcel Grossmann Meeting, the event is the Proceedings of the MG11 Meeting on General Relativity held in Berlin, Germany, the publisher is World Scientific, the pages are 2187 to 2189 and the print date is September 2008. OpenAlex agrees and adds no other author. The registry title was recorded with a broken word, ‘TRA VERSABLE’, which is a line-break artefact of the source page and not a different work; the title is given here in normal case. The author’s full name as printed on his own title page is Jaroslaw Pawel Adamiak. WHAT THE DOI NAMES, AND WHAT WAS READ. The DOI names a three-page contribution to the MG11 proceedings, which is closed at World Scientific and was not reached. OpenAlex lists a second location for the same record, the University of South Africa institutional repository under handle 10500/2058, and that is the master’s dissertation the contribution summarises: ‘Static and dynamic traversable wormholes’, submitted in part fulfilment of the degree of Master of Science in Applied Mathematics, University of South Africa, January 2005, supervisor Professor N. T. Bishop, joint supervisor Professor W. M. Lesame, six plus one hundred leaves, deposited 25 August 2009. The UnisaIR web front answered ‘Service Temporarily Unavailable, deliberate shutdown’ during this work, so the repository record and the dissertation PDF were read through the Internet Archive captures of 14 October 2018 and 16 August 2017 respectively; the PDF is the complete dissertation with its six chapters, five appendices and one hundred and fourteen references, and it was read in full on 2026-09-09. EVERY CLAIM BELOW IS LOCATED IN THAT DISSERTATION, by its own chapter, section or equation number, and never in the three-page proceedings contribution, which was not read. LICENCE. No Creative Commons statement appears on the proceedings record or on the repository record, so nothing beyond the author’s own abstract is reproduced here. THE ABSTRACT. The abstract below is the one printed on page i of the dissertation, and it is word for word the abstract OpenAlex carries for this DOI with one exception: the repository’s catalogue record contains a typing slip, ‘attempts to exocit matter quantification’, where the dissertation itself prints ‘attempts to exotic matter quantification’. The dissertation’s own wording is the one reproduced. Equations are described in words, since the extracted text carries the original typesetting only as characters.

How to cite it

Jarosław P. Adamiak (2008) Static and dynamic traversable wormholes. doi:10.1142/9789812834300_0357

Where it sits in the curriculum

Wormholes, energy conditions and the negative-energy budget