[Paper Review] Conformal continuations and wormhole instability in scalar-tensor gravity

К. А. Бронников, S.V. Grinyok|arXiv (Cornell University)|Nov 12, 2004

Cosmology and Gravitation Theories7 references19 citations

TL;DR

This paper investigates the stability of static, spherically symmetric traversable wormholes in scalar-tensor gravity theories with conformal continuations and zero scalar potential. Using analytical methods based on self-adjoint operator theory and numerical verification, it proves that all such wormholes are unstable under spherically symmetric perturbations due to at least one exponentially growing mode, with a finite growth rate rather than catastrophic divergence.

ABSTRACT

We study the stability of static, spherically symmetric, traversable wormholes existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Fisher's well-known scalar-vacuum solution holds in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically with the aid of the theory of self-adjoint operators in Hilbert space and is confirmed by a numerical computation.

Motivation & Objective

• To analyze the stability of static, spherically symmetric wormholes arising from conformal continuations in scalar-tensor gravity with zero scalar potential.
• To account for physically meaningful boundary conditions on both scalar and metric perturbations, ensuring regularity across the transition sphere.
• To determine whether such wormholes exhibit instability under spherically symmetric perturbations, particularly in the context of sign-changing gravitational coupling.
• To establish the existence of at least one growing mode of perturbation, confirming instability, using rigorous analytical and numerical methods.

Proposed method

• Formulates the problem in the Einstein conformal frame, where Fisher’s scalar-vacuum solution applies, enabling a well-defined wave equation for perturbations.
• Reduces the perturbation equations to a Schrödinger-type eigenvalue problem with a potential derived from the background geometry and scalar field coupling.
• Applies the theory of self-adjoint operators in Hilbert space to prove the existence of negative eigenvalues, implying exponentially growing solutions.
• Uses the minimax principle with a trial function (k=1/2) to estimate the ground state energy, yielding an upper bound below zero.
• Employs numerical computation via the Fortran SLEIG package after coordinate transformation to solve the boundary-value problem and confirm the existence of a discrete eigenvalue.
• Verifies that metric perturbations remain regular at the transition sphere by checking the finiteness of first and second derivatives in the Gaussian radial coordinate.

Experimental results

Research questions

• RQ1 Do static, spherically symmetric wormholes in scalar-tensor gravity with conformal continuations and zero potential remain stable under spherically symmetric perturbations?
• RQ2 How do physical boundary conditions on scalar and metric perturbations constrain the stability analysis?
• RQ3 Does the presence of a sign change in the effective gravitational coupling lead to instability, and if so, what is the nature of the growth rate?
• RQ4 Can the existence of a growing mode be rigorously proven using self-adjoint operator theory in the Einstein frame?
• RQ5 What is the characteristic decay time of such unstable wormholes, and how does it scale with physical parameters like wormhole radius?

Key findings

• All static, spherically symmetric wormholes in the considered class of scalar-tensor theories are unstable under spherically symmetric perturbations due to the existence of at least one exponentially growing mode.
• The instability is not catastrophic; the perturbation growth rate is finite, with a characteristic decay time τ ≈ 5m, where m is a parameter related to the wormhole size.
• Numerical computation confirms a single discrete eigenvalue at E ≈ −0.048, consistent with the analytical estimate of μ′₀ ≈ −0.039, proving the existence of negative eigenvalues in the spectrum.
• Metric perturbations remain regular at the transition sphere, as their first derivatives vanish and second derivatives are finite, confirming physical validity of the perturbation modes.
• For a wormhole of stellar size (~10⁶ km radius), the decay time is estimated at a few seconds, consistent with light-crossing timescales.
• The instability persists across different scalar-tensor models, with the decay time scaling linearly with the wormhole radius for fixed geometry parameters.