Historical research guide · 1005 · 2018

Note on a thin-shell wormhole in extremal Reissner–Nordström geometry

S. Habib Mazharimousavi · Mustafa Halilsoy

In one minute

A wormhole needs something unusual holding its throat open: matter whose energy, measured along a light ray, dips below the ordinary vacuum level. The thin-shell method, due to Matt Visser, pushes all of that unusual material into a single thin surface so the bulk of spacetime stays perfectly ordinary. Habib Mazharimousavi and Mustafa Halilsoy ask what happens when the surrounding geometry is an extremal Reissner–Nordström black hole — a charged hole carrying exactly as much charge as mass, whose horizon is cold, at zero Hawking temperature, and sits at a doubled root of the metric function. Place the throat at that horizon, they show, and the energy density on the shell falls to zero while the angular pressure stays positive and finite, equal to two divided by the hole’s mass. Then they shake it. Perturbed radially, the shell moves in an effective potential that has a well at the equilibrium radius for a range of the fluid parameter, so it oscillates instead of collapsing.

Connection

Chapter 4 treats wormholes as engineering problems in the metric, and the engineering has always come down to one line item: how much exotic material the throat needs and whether the thing holds still once built. This Note answers both in the same construction, and chapter 2 owns the answer, because the surface fluid it needs is ordinary — positive pressure, vanishing energy density — rather than the negative-energy stuff the older models required. Chapter 13 gets the closing remark, that the same argument runs for any cold or ultracold extremal horizon.

From the source

The construction: take the bulk to be an extremal black hole, one whose metric function is a perfect square so that both the function and its first derivative vanish at the horizon, and place the static throat of a thin-shell wormhole exactly at that double root. The surface energy density then vanishes and the angular pressure equals twice the first derivative of the square root of the metric function at the horizon — positive and finite.Section II A, equation 7, arXiv preprint 1704.04435
For the extremal Reissner–Nordström hole, where mass equals charge and the double-root horizon sits at a radius equal to the mass, the throat therefore carries zero energy density and an angular pressure of two divided by the mass — inversely proportional to the mass or charge of the hole. A bigger hole needs a gentler shell.Section II A, the paragraph following equation 7, arXiv preprint 1704.04435
The extremal case is special for a reason the paper states before it uses it: for an ordinary black hole bulk the throat radius must be strictly larger than the horizon radius, because the surface pressure diverges at the horizon and a throat there is not physical. Extremality removes that divergence, which is what makes putting the throat at the horizon legitimate at all.Section II, the paragraph beginning ‘Next, if we assume that the bulk spacetime is a black hole’, arXiv preprint 1704.04435
On the energy condition, the preprint states the strong form: with vanishing energy density and positive angular pressure at static equilibrium the shell satisfies the energy conditions, and under radial perturbation the sum of energy density and pressure stays positive in the neighbourhood of the equilibrium radius. The published version of record states the same result more carefully, as a throat at the cold near horizon carrying arbitrarily small exotic matter rather than exactly none — and that is the wording this library follows.Published abstract, and Section II B with Figure 5 of arXiv preprint 1704.04435
Stability is tested rather than assumed. Perturbing the throat radially and adopting a variable equation of state with a single constant parameter, the conservation equation gives the energy density in closed form and the motion of the shell reduces to a one-dimensional problem in an effective potential. That potential has a local minimum at the equilibrium radius for values of the parameter below one quarter, which means the shell is stable against small radial perturbations; at one quarter and above the minimum disappears and the shell is unstable. Stability is strongest near a parameter value of zero, where the potential barriers are highest.Section II B, equations 13 to 15 with Figures 1 and 2, arXiv preprint 1704.04435
Two open ends the authors name themselves. Under a strong rather than a small perturbation the shell can still fail, and the way it fails depends on the sign of the fluid parameter: for positive values both collapse to the centre and indefinite expansion are possible, while for negative values collapse cannot take place. And the whole argument is not tied to this one metric — what has been shown for the cold, zero Hawking temperature extremal Reissner–Nordström hole holds for other cold and ultracold extremal black holes too.Section II B and the closing paragraph of Section III Conclusion, arXiv preprint 1704.04435

The way in

https://doi.org/10.1142/s0218271818500281

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

PUBLICATION. International Journal of Modern Physics D, volume 27, number 3, article 1850028, February 2018; published online 13 February 2018. Both authors are at the Department of Physics, Eastern Mediterranean University, Gazimagusa, North Cyprus. LICENCE, CHECKED 2026-09-08. The Crossref record registers no licence at all for this DOI and no Creative Commons statement appears on the article or on the preprint, so nothing beyond the work’s own abstract is reproduced here. SOURCE READ. Unpaywall records the work as green open access through arXiv 1704.04435, version 1, dated 12 April 2017 — the only version arXiv holds — and that preprint, the complete four-page Note with its five figures and nine references, was retrieved and read in full on 2026-09-08. TWO VERSIONS, ONE DIFFERENCE WORTH KNOWING. The abstract reproduced below is the published one, taken from the Crossref deposit for the version of record. It differs from the 2017 preprint’s abstract in two places: the preprint says the cold horizon where the published version says the cold near horizon, and the preprint says zero total exotic matter where the published version says arbitrarily small exotic matter. The preprint text places the static throat exactly at the double-root horizon and obtains a surface energy density of exactly zero. The claims below are drawn from the preprint and say so; where the published wording is the more careful one, that is noted in the claim. Equations are described in words, since the extracted text carries the original typesetting only as characters. RELATED PAGES. The energy-condition requirement this construction is designed around is at /library/stm-a2430e29f0 and /library/stm-b8b6ab9b94; the question of whether an arbitrarily small amount of exotic matter is enough is treated head-on at /library/stm-d54d953175; the thin-shell method itself comes from Visser, at /library/stm-1e6a0b75bc, and its static and dynamic cases are worked through at /library/stm-e4138028fb (Adamiak); and the Casimir-supported and phantom-supported alternatives are at /library/stm-87af684cc9 and /library/stm-35489e8f11.

How to cite it

S. Habib Mazharimousavi, Mustafa Halilsoy (2018) Note on a thin-shell wormhole in extremal Reissner–Nordström geometry doi:10.1142/s0218271818500281

Where it sits in the curriculum

Wormholes, energy conditions and the negative-energy budget