Thin-shell wormholes: Linearization stability

Eric Poisson; Matt Visser (Washington University)

REVIEW

Thin-shell wormholes: Linearization stability

Eric Poisson, Matt Visser (Washington University)

General Relativity and Quantum Cosmology · Submitted 30 Jun 1995 · 9 references

Pith's verdict: A clean, correct short note that turns thin-shell wormhole stability into a one-parameter criterion on the equation of state; the algebra holds and the caveats are stated, not hidden.

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arXiv IDgr-qc/9506083 (version 1)

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CitationEric Poisson, Matt Visser (Washington University). "Thin-shell wormholes: Linearization stability." arXiv:gr-qc/9506083 (1995). https://pith.science/paper/STKG5QIK

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Editor's note

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Worth your time if you work on thin shells or wormholes, and worth citing regardless. The result is narrow but real: for a spherically symmetric thin-shell wormhole built by cut-and-paste Schwarzschild surgery, linearized radial stability reduces to the sign of V''(a0), and that sign depends on the shell's equation of state only through β0² = dp/dσ at the static solution. Eqs. (27), (29), and (31)–(36) give the criterion and map the stable region in the (β0², a0/M) plane. The punchline in Section IV is the useful part: for a0 > 3M you need β0² < 0, so a shell with a conventional sound speed is always unstable. That is a quotable no-go within the model class, and it is more general than the earlier equation-of-state-by-equation-of-state treatments in Visser's papers [2,5]. The paper says so plainly and does not oversell it (abstract:s2). I re-derived the chain from the junction conditions (8)–(12) through the conservation law (15) to (26) and (27); it comes out. The quadratic (33), the discriminant condition β0² ≥ 3/2 + √3 or ≤ 3/2 − √3, and the two stability regions all check numerically. No data, no fitting, no circularity, and the debt to Brady–Louko–Poisson [6] is methodological, not load-bearing. Soft spots are modeling choices, and the authors own most of them. The barotropic assumption p = p(σ) with fixed β² is assumed, not derived, and it is what lets σ be written as σ(a); drop it and the criterion does not apply. Spherical symmetry and Schwarzschild exteriors are likewise assumed. Section IV is candid that β0 need not be a literal sound speed, and the Casimir and false-vacuum examples (β² = −1) are cited correctly. Two small things I would flag to the authors rather than to you: the 'if and only if' in Eq. (29) skips the marginal case V''(a0) = 0, and the roots a0± in (34) are not ordered consistently between regions I and II, so a reader has to sort out which root is which. Both are presentational. The figure is referenced only by caption, which is a bit thin. None of this touches the central claim. Send it out; it is a short, honest paper that does exactly what it says.

Referee report

Significance

The paper gives a clean, self-contained derivation from the Einstein field equations and the Israel junction conditions, with no fitted parameters and no data. The result is a falsifiable, general criterion: Eq. (29) is a sharp inequality in the two dimensionless quantities β0² and a0/M, and the stability regions of Eqs. (35)–(36) and Fig. 1 are explicit. The discussion of β0² as a would-be sound speed, and of the Casimir and false-vacuum examples with β0² = −1, is a useful and honest attempt to interpret an unusual result. The paper also correctly scopes its claim to linearized, spherically symmetric, barotropic perturbations, and it explicitly notes that a microphysical model for the exotic matter is needed to settle whether β0² < 0 is physically realizable. These are genuine strengths.

Minor objections (7)

[Section IV, first paragraph] The text writes 'β² = ∂σ/∂p = −1' for the Casimir and false-vacuum examples, but Eq. (23) defines β² ≡ ∂p/∂σ. For these two examples the two expressions coincide numerically, so the physics is unaffected, but the notation should be made consistent with Eq. (23) (i.e., write β² = ∂p/∂σ = −1). It would also help to state explicitly the sign convention for σ in S_ij = diag(−σ, p, p), since the sign of β² for the Casimir and false-vacuum examples depends on it.
[Section III, after Eq. (28)] The statement 'the wormhole is stable if and only if [Eq. (29)]' should be qualified for the marginal case V''(a0) = 0. The expansion in Eq. (28) is to quadratic order, and when V''(a0) vanishes the leading behaviour is cubic (or higher), so the criterion as derived applies only for V''(a0) ≠ 0. A sentence noting that the boundary of the stability region is marginal would make the 'if and only if' precise.
[Eq. (6)] The factor δ_ij multiplying [K] in Eq. (6) should be the induced metric on the throat (h_ij or γ_ij), not the Kronecker delta; as written the equation is not consistent with the spherical-symmetry reduction in Eq. (8). This appears to be a typographical slip, but it should be corrected.
[Eqs. (35)–(36)] The roots a0^± defined in Eq. (34) are not ordered consistently across the two branches: for β0² > −1/4 (region I) one has a0^- < a0^+, while for β0² < −1/4 (region II) the ordering is reversed, a0^- > a0^+. The formulas in Eqs. (35)–(36) are correct once this is taken into account, but a sentence stating the ordering explicitly would help the reader apply them without having to re-sort the roots.
[Section IV, first paragraph] The sentence 'stability implies that large traversable wormhole (a0 > 3M) are stable only for β0² < 0' has a grammatical error: 'wormhole' should be 'wormholes' (or the sentence should read 'a large traversable wormhole ... is stable only for β0² < 0').
[Fig. 1 caption] The caption identifies the two dashed lines but does not state which side of each line is stable. Adding a phrase such as 'the shaded regions are stable' and marking the a0 = 3M line would make the figure self-contained.
[Manuscript throughout] The submitted text contains several broken words and symbols ('Mis souri', 'Schw arzs child', 'p articularly', 'diﬀerent', and the '12' in Eq. (3) for 1/2). These should be corrected in the final version.

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The paper

The paper claims

Spherically symmetric thin-shell wormholes are linearly stable only if the throat matter has an unusual equation of state: for throats larger than three times the mass, stability requires the squared sound-speed parameter to be negative.

The reading

A thin-shell wormhole is a toy model of a traversable wormhole built by cutting two copies of the Schwarzschild black-hole spacetime and gluing them together along a spherical surface. The seam is the wormhole throat, and it must carry exotic matter with negative energy density to hold the geometry open. Once you specify how that surface matter responds to compression, its equation of state, the Einstein equations determine how the throat radius moves. This paper asks a narrower question: if the wormhole sits at rest at some radius, is that rest state stable? The authors perturb the throat radius slightly, expand the effective potential to second order, and find that stability is controlled by a single number, beta-squared, defined as the derivative of surface pressure with respect to surface energy density. The answer is a clean inequality relating beta-squared to the throat radius in units of the wormhole mass. The result is counterintuitive. If beta is interpreted as a sound speed and restricted to the ordinary range between zero and the speed of light, the inequality can never be satisfied: every such wormhole is unstable. Stability is possible only in two pockets of parameter space, one requiring a large negative beta-squared at or below minus one half, and one requiring a very large positive beta-squared above about 3.23 with the throat radius pinned near twice the mass. The authors point out that known exotic sources such as the Casimir vacuum and false vacuum have beta-squared equal to minus one, so the negative pocket is not empty. They also caution that beta need not literally be a sound speed, since no microphysical model of the exotic matter exists.

Extended reading notes

Core claim

The paper's load-bearing assertion is Eq. (29): a spherically symmetric thin-shell wormhole is linearly stable against radial perturbations if and only if 2M/a0 + (M^2/a0^2)/(1-2M/a0) + (1+2*beta0^2)(1-3M/a0) < 0, where beta0^2 = dp/dsigma evaluated at the static solution. The consequence stated in Section IV is that for a0 > 3M stability requires beta0^2 < 0, so a wormhole whose surface matter has a conventional sound speed (0 < beta0 <= 1) is always unstable. If correct, this is a general no-go for stabilizing thin-shell wormholes with ordinary matter.

Load-bearing premise

The derivation assumes the perturbed shell still obeys a barotropic equation of state p = p(sigma) with a fixed beta^2 = dp/dsigma, so that sigma can be written as a function of the throat radius alone (Section III, Eq. 16 and the paragraph beginning 'If we choose a particular equation of state'). It also assumes spherical symmetry and that the throat radius is the only dynamical degree of freedom, with Schwarzschild geometry unchanged on either side. If the surface matter is not barotropic, or if non-spherical or radiative modes grow faster, the stability conclusion need not hold.

Original abstract

The class of spherically-symmetric thin-shell wormholes provides a particularly elegant collection of exemplars for the study of traversable Lorentzian wormholes. In the present paper we consider linearized (spherically symmetric) perturbations around some assumed static solution of the Einstein field equations. This permits us to relate stability issues to the (linearized) equation of state of the exotic matter which is located at the wormhole throat.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The class of spherically-symmetric thin-shell wormholes provides a particularly elegant collection of exemplars for the study of traversable Lorentzian wormholes. In the present paper we consider linearized (spherically symmetric) perturbations around some assumed static solution of the Einstein field equations. This permits us to relate stability issues to the (linearized) equation of state of the exotic matter which is located at the wormhole throat.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The class of spherically-symmetric thin-shell wormholes provides a particularly elegant collection of exemplars for the study of traversable Lorentzian wormholes.	background 1.00	0.06	0.03	asserted 1.00
s1	In the present paper we consider linearized (spherically symmetric) perturbations around some assumed static solution of the Einstein field equations.	contribution 0.68	0.90	0.04	asserted 0.94
s2	This permits us to relate stability issues to the (linearized) equation of state of the exotic matter which is located at the wormhole throat.	interpretation 0.31	0.77	0.16	asserted 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** FIG. 1. Regions of stability: Traversable wormholes in

Figures reproduced from arXiv:gr-qc/9506083 by the authors.

References

Works cited (9)

[6]

M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61 , 1446 (1988)

View on Pith 1988
[1]

Electronic mail: poisson@wuphys.wustl.edu

View on Pith
[2]

Electronic mail: visser@kiwi.wustl.edu

View on Pith
[3]

M. Visser, Phys. Rev. D 39 , 3182 (1989)

View on Pith 1989
[4]

M. Visser, Nucl. Phys. B328 , 203 (1989)

View on Pith 1989
[5]

M.S. Morris and K.S. Thorne, Am. J. Phys. 56 , 395 (1988)

View on Pith 1988
[7]

M. Visser, Lorentzian wormholes --- from Einstein to Hawking\/ (American Institute of Physics, New York, 1995)

View on Pith 1995
[8]

P.R. Brady, J. Louko, and E. Poisson, Phys. Rev. D 44 , 1891 (1991)

View on Pith 1991
[9]

N. Sen, Ann. Phys. (Leipzig) 73 , 365 (1924); K. Lanczos, Ann. Phys. (Leipzig) 74 , 518 (1924); W. Israel, Nuovo Cimento 44B , 1 (1966); erratum--- ibid. 48B , 463 (1967). references figure [] Regions of stability: Traversable wormholes in the indicated regions of the ^2_0 versus a_0/M plane are stable against radial perturbations. The higher dashed line ...

View on Pith 1924

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How later papers used it

69 later papers cite it

Evidence: substantial. 58 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 121 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool and 1 compared its own results with it. 1 of them appears under more than one label. 1 more paper by the same authors is listed: 1 built on it.

Builds on it 1 of 2 papers with this label · 1 by the same authors

By the same authors

Generalized previous work in which the case of M_-=M_+ was studied by this paper.

Generic spherically symmetric dynamic thin-shell traversable wormholes in standard general relativity (2011) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

Uses its method or tool 4 of 9 papers with this label

Adopted this paper's linear relation for the matter to study the stability of the thin-shell wormhole under a radial perturbation of the throat radius.

Traversable Thin-shell Wormhole in the 4D Einstein-Gauss-Bonnet Theory (2020) · 0.99 · paraphrased: the licence does not allow quoting
Followed the Poisson-Visser reasoning of this paper, putting quite generally P=P(σ) rather than choosing a specific equation of state.

Plane symmetric thin-shell wormholes: solutions and stability (2008) · 0.96 · paraphrased: the licence does not allow quoting
Followed the other path provided by Poisson and Visser in this paper.

Thin-shell wormholes in d-dimensional general relativity: Solutions, properties, and stability (2010) · 0.99 · paraphrased: the licence does not allow quoting
We also discuss the potential function by the method of Poisson $&$ Visser as discussed in their work [ref] .

Exact Gravastar Solution (2026) · Surface Energy Density and Pressure · 1.00

Compares with it 1 of 8 papers with this label

Noted that when α→0 its own result is the same as that in this paper.

Traversable Thin-shell Wormhole in the 4D Einstein-Gauss-Bonnet Theory (2020) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

56 arXiv papers cite this in 104 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 51 sentences without a confident label

Builds on it 2 papers

This analysis generalizes previous work, where the case of $M_-=M_+$ was studied [ref] .

Generic spherically symmetric dynamic thin-shell traversable wormholes in standard general relativity · Thin-shell Schwarzschild wormhole: $b_±=2M_±$ and $Φ_±=0$ · 0.97
The dynamical analysis in [ref] was generalized by considering solutions with electrical charge [ref] , solutions in the presence of a cosmological constant [ref] , and spherically symmetric dynamical solutions [ref] .

Plane symmetric thin-shell wormholes: solutions and stability · Introduction · 0.94

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To study the stability of the thin-shell wormhole under a radial perturbation of the throat radius, we adopt following linear relation [ref] for the matter.

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In this subsection, rather than choosing a specific equation of state, we shall follow the Poisson-Visser reasoning [ref] , and put quite generally $P=P(σ)$, see ([ref]).

Plane symmetric thin-shell wormholes: solutions and stability · General considerations with a general equation of state · 0.96
Writing $K=η^îĵK_îĵ$ for the orthonormal trace and $[Q]=Q^(+)-Q^(-)$ for the jump of any junction quantity, the Israel–Lanczos equation in the same basis is [ref] [equation]

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Note that when $α→0$, we get [equation] which is the same as that in [ref] .

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For the specific symmetric case, $m_+=m_-$, and for vanishing bounce parameters $a_±=0$, the analysis reduces to that of reference [ref] .

Dynamic thin-shell black-bounce traversable wormholes · Lanczos equations and surface stress-energy · 0.85
Comparing our results in this work and those in literatures [ref] , we see that, {$H(r_0)=0$, $H'(r_0)=0$, $H”(r_0)<0$} are non necessary conditions for getting stable wormholes by the cut-and-paste method.

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That $η$ may exceed unity in this region is physically admissible for thin shells, as discussed by Poisson and Visser [ref] and Lobo [ref] .

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Linearized stability analyses of thin-shell wormholes have since been carried out for a variety of background spacetimes [ref] .

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Lorentzian wormholes were studied in General Relativity (GR) [ref] , in which conditions for traversable wormholes were found introducing a static spherically symmetric metric.

Quasinormal modes and bound states of massive scalar fields in wormhole spacetimes · Introduction · 0.97
Various strategies extensively explored in the literature include rotating solutions [ref] , evolving wormhole spacetimes [ref] , thin-shell wormholes employing the cut-and-paste procedure [ref] - [ref] , and modifications to theories of gravity [ref] - [ref] (see Chapter [ref] for more details).

Wormhole geometries in modified gravity · Energy conditions · 0.95

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [6]

M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61 , 1446 (1988)

View on Pith 1988

[2] [1]

Electronic mail: poisson@wuphys.wustl.edu

View on Pith

[3] [2]

Electronic mail: visser@kiwi.wustl.edu

View on Pith

[4] [3]

M. Visser, Phys. Rev. D 39 , 3182 (1989)

View on Pith 1989

[5] [4]

M. Visser, Nucl. Phys. B328 , 203 (1989)

View on Pith 1989

[6] [5]

M.S. Morris and K.S. Thorne, Am. J. Phys. 56 , 395 (1988)

View on Pith 1988

[7] [7]

M. Visser, Lorentzian wormholes --- from Einstein to Hawking\/ (American Institute of Physics, New York, 1995)

View on Pith 1995

[8] [8]

P.R. Brady, J. Louko, and E. Poisson, Phys. Rev. D 44 , 1891 (1991)

View on Pith 1991

[9] [9]

N. Sen, Ann. Phys. (Leipzig) 73 , 365 (1924); K. Lanczos, Ann. Phys. (Leipzig) 74 , 518 (1924); W. Israel, Nuovo Cimento 44B , 1 (1966); erratum--- ibid. 48B , 463 (1967). references figure [] Regions of stability: Traversable wormholes in the indicated regions of the ^2_0 versus a_0/M plane are stable against radial perturbations. The higher dashed line ...

View on Pith 1924