[Paper Review] Electrically charged and neutral wormhole instability in scalar-tensor gravity

К. А. Бронников, S.V. Grinyok|ArXiv.org|Sep 15, 2005

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper investigates the stability of electrically charged and neutral traversable wormholes in scalar-tensor gravity theories with zero scalar potential, using self-adjoint operator theory and numerical methods. It analytically proves that all such wormholes with zero or small electric charge are unstable under spherically symmetric perturbations, with instability growth timescales on the order of the light-crossing time of the throat radius.

ABSTRACT

We study the stability of static, spherically symmetric, traversable wormholes with or without an electric charge, existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Penney's or Fisher's well-known solutions hold 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 with zero or small electric charge 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 numerical computations.

Motivation & Objective

• To investigate the stability of static, spherically symmetric traversable wormholes in scalar-tensor gravity theories with zero scalar field potential.
• To extend previous instability results for neutral wormholes to the case of electrically charged wormholes in the same class of theories.
• To account for physical regularity constraints on metric and scalar field perturbations, which impose nontrivial boundary conditions.
• To determine whether small electric charges can stabilize otherwise unstable wormhole solutions under spherically symmetric perturbations.

Proposed method

• Formulates the action for a general class of scalar-tensor theories with metric, scalar field, and electromagnetic field as sources.
• Applies conformal continuation to map solutions from the Einstein frame (where Penney’s or Fisher’s solutions hold) to the Jordan frame, enabling wormhole geometry.
• Derives the linearized perturbation equations for metric and scalar fields, transforming them into a self-adjoint Sturm-Liouville form.
• Transforms the Sturm-Liouville equation into a Schrödinger-type equation with a potential exhibiting specific asymptotic behaviors at spatial infinity and the transition sphere.
• Applies self-adjoint operator theory to analyze the spectrum, focusing on the existence of negative eigenvalues corresponding to exponentially growing perturbations.
• Uses linear operator perturbation theory and numerical computations to confirm the presence of unstable modes for small charges.

Experimental results

Research questions

• RQ1 Are electrically charged traversable wormholes in scalar-tensor gravity with zero potential unstable under spherically symmetric perturbations?
• RQ2 How do boundary conditions—particularly regularity at the transition sphere and asymptotic behavior—constrain the perturbation spectrum?
• RQ3 Does the presence of small electric charge alter the instability properties of previously known neutral wormhole solutions?
• RQ4 Can the instability be analytically proven using self-adjoint operator theory in Hilbert space?
• RQ5 What is the quantitative behavior of the instability growth rate as a function of electric charge?

Key findings

• All electrically charged and neutral traversable wormholes in the considered scalar-tensor theories with zero potential are unstable under spherically symmetric perturbations when the electric charge is zero or small.
• The instability is analytically proven using the theory of self-adjoint operators in Hilbert space, which confirms the existence of at least one negative eigenvalue in the perturbation spectrum.
• Numerical computations confirm the analytical results, showing that the instability increment for small charges is close to that of the neutral case.
• The characteristic timescale for instability growth is approximately τ ≈ 5h, where h is the throat radius in proper units, implying decay times on the order of a few seconds for stellar-sized wormholes.
• As the electric charge increases, the instability increment diminishes, suggesting possible stabilization at large charge values, though numerical reliability decreases for large charges.
• The results are general and apply to all scalar-tensor theories admitting wormhole solutions via conformal continuation, not just specific models.