[Paper Review] General parametrization of wormhole spacetimes and its application to shadows and quasinormal modes

К. А. Бронников, R. A. Konoplya+1 more|arXiv (Cornell University)|Feb 21, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy75 references93 citations

TL;DR

This paper introduces a general parametrization for spherically symmetric, traversable, asymptotically flat wormhole spacetimes using continued-fraction expansions in a compactified radial coordinate, valid across the entire spacetime. The method achieves high accuracy—relative errors below 1%—with just first-order expansion for key observables like shadows and quasinormal modes, demonstrating that only a few parameters dominate observable physics in wormhole backgrounds.

ABSTRACT

The general parametrization for spacetimes of spherically symmetric Lorentzian, traversable wormholes in an arbitrary metric theory of gravity is presented. The parametrization is similar in spirit to the post-Newtonian parametrized formalism, but with validity that extends beyond the weak field region and covers the whole space. Our method is based on a continued-fraction expansion in terms of a compactified radial coordinate. Calculations of shadows and quasinormal modes for various examples of parametrization of known wormhole metrics that we have performed show that, for most cases, the parametrization provides excellent accuracy already at the first order. Therefore, only a few parameters are dominant and important for finding potentially observable quantities in a wormhole background. We have also extended the analysis to the regime of slow rotation.

Motivation & Objective

• To develop a general, gravity-independent parametrization of static, spherically symmetric, traversable wormholes valid across the full spacetime, extending beyond weak-field approximations.
• To adapt the Rezzolla-Zhidenko black hole parametrization framework to wormholes by replacing the event horizon with the throat radius as the natural length scale.
• To test the accuracy of the parametrization on known analytic wormhole solutions, focusing on observable quantities such as shadows and quasinormal modes.
• To extend the formalism to slow rotation via axial symmetry, enabling broader astrophysical applicability.
• To demonstrate that only a few dominant parameters in the parametrization are sufficient to accurately describe potentially observable phenomena in wormhole spacetimes.

Proposed method

• Employ a continued-fraction expansion of metric functions in terms of a compactified radial coordinate, ensuring superior convergence across the full radial range from throat to infinity.
• Use the wormhole throat radius r₀ as the fundamental length scale, replacing the black hole horizon radius used in the original Rezzolla-Zhidenko formalism.
• Construct parametrized metrics in the Morris-Thorne frame and in non-Morris-Thorne frames, optimizing coordinate choices based on throat behavior.
• Apply the parametrization to specific wormhole solutions (e.g., Bronnikov-Kim, Simpson-Visser, Casadio-Fabbri-Mazzacurati) and compare with exact solutions.
• Calculate shadow radii and quasinormal modes using numerical methods (e.g., Prony method) to assess parametrization accuracy.
• Use gauge-invariant error metrics to compare parametrized results with exact solutions across multiple examples and orders of expansion.

Experimental results

Research questions

• RQ1 Can a general, gravity-independent parametrization of wormhole spacetimes be constructed that is valid across the entire radial domain, not just in the weak-field limit?
• RQ2 How accurately can the parametrization reproduce key observable quantities—such as shadow radii and quasinormal modes—for known analytic wormhole solutions?
• RQ3 What is the convergence behavior of the parametrization, and how many parameters are needed to achieve high accuracy in observable predictions?
• RQ4 How does the parametrization perform in the regime near the black hole–wormhole transition, where spacetime geometry becomes highly sensitive?
• RQ5 Can the formalism be extended to slowly rotating wormholes while preserving accuracy and interpretability?

Key findings

• The parametrization achieves relative errors of less than 1% in shadow radii and quasinormal mode frequencies for most wormhole solutions already at first-order expansion.
• For the Bronnikov-Kim II braneworld wormhole, the first-order parametrization yields a relative error of approximately 1% in quasinormal mode frequencies, improving to sub-1% at second order.
• The Simpson-Visser wormhole shadow radius is reproduced with a relative error of about 0.5% at first order, with further improvement at second order.
• The method remains convergent and accurate even near the black hole–wormhole transition point, where spacetime geometry is highly sensitive.
• The Prony method applied to quasinormal modes confirms that the first-order parametrization captures the dominant oscillatory behavior with sufficient accuracy for observational comparison.
• The parametrization demonstrates that only a few parameters—specifically, the throat parameters h₀, f₀ and deformation parameters ϵ, a₁, b₁—are dominant for observable quantities, enabling efficient phenomenological modeling.