[Paper Review] On the stability of thin-shell wormholes

Peter K. F. Kuhfittig|arXiv (Cornell University)|Sep 30, 2014

Black Holes and Theoretical Physics10 references3 citations

TL;DR

This paper investigates the stability of thin-shell wormholes by modeling them as arbitrarily thin spherical shells of exotic matter, using a continuous line element instead of the standard cut-and-paste construction. It shows that such wormholes are unstable to linearized radial perturbations due to negative second derivative of the effective potential, regardless of the interpretation of the speed-of-sound parameter β.

ABSTRACT

A thin-shell wormhole is theoretically constructible by surgically grafting together two Schwarzschild spacetimes using the so-called cut-and-paste technique. By describing such a wormhole as the limiting case of a spherical shell, it is shown that the structure must be unstable to linearized radial perturbations. Some earlier studies by the author et al. have shown, however, that under certain conditions, thin-shell wormholes can be stable.

Motivation & Objective

• To examine the stability of thin-shell wormholes by modeling them as arbitrarily thin spherical shells with continuous spacetime geometry.
• To challenge the assumption of stability in earlier cut-and-paste thin-shell models by analyzing radial perturbations in a physically continuous framework.
• To determine whether the instability persists when the wormhole is approximated as a thin shell with nonzero surface stresses.
• To clarify the role of surface energy density σ and surface pressure 𝒫 in determining stability.
• To reconcile conflicting results on stability by analyzing the second derivative of the effective potential V(a) around equilibrium.

Proposed method

• Constructs a wormhole solution using a constant energy density ρ₀ confined in a spherical shell between r₀ and a, with b(r) derived from the Einstein field equations.
• Matches the interior solution to the exterior Schwarzschild metric at r = a, ensuring continuity of metric coefficients.
• Applies the Lanczos junction conditions to compute surface energy density σ and surface pressure 𝒫 from the discontinuity in extrinsic curvature.
• Introduces time dependence via a(t) = a(τ), where τ is proper time on the junction surface, to model dynamic throat evolution.
• Derives the effective potential V(a) from the equation of motion, using σ and 𝒫 as functions of da/dτ.
• Performs linear stability analysis by expanding V(a) around static equilibrium a = a₀, computing V''(a₀) to determine stability.

Experimental results

Research questions

• RQ1 Is a thin-shell wormhole constructed from a continuous spherical shell of exotic matter stable under linearized radial perturbations?
• RQ2 How does the stability criterion derived from the effective potential V(a) behave when the shell is made arbitrarily thin?
• RQ3 Does the presence of nonzero surface stresses (σ and 𝒫) lead to instability even when the cut-and-paste construction suggests possible stability?
• RQ4 Can the parameter β (interpreted as speed of sound) influence the stability outcome, or is the instability robust regardless of β's value?
• RQ5 Under what conditions might a thin-shell wormhole remain stable, and how does this compare to results from noncommutative geometry or other modified gravity models?

Key findings

• The second derivative of the effective potential, V''(a₀), is negative for the thin-shell wormhole model, indicating instability to linearized radial perturbations.
• The instability arises primarily from the negative contributions of terms involving M, b(a₀), and mₛ, especially when mₛ is treated as constant.
• Even in the limit where the shell thickness approaches zero (a → r₀), the surface energy density σ remains negative and the surface pressure 𝒫 remains positive, preserving instability.
• The result holds regardless of the interpretation of β², which is not constrained to (0,1] due to exotic matter properties.
• The instability contradicts earlier claims of stability in the cut-and-paste model, suggesting that the thin-shell approximation may not preserve stability under dynamic perturbations.
• Stability is only possible under alternative physical frameworks, such as noncommutative geometry or generalized dilaton-axion gravity, as shown in prior works cited.