# (2+1)-dimensional Static Cyclic Symmetric Traversable Wormhole: Quasinormal Modes and Causality

## TL;DR

This paper investigates a (2+1)-dimensional static cyclic symmetric traversable wormhole, analyzing its causality structure and calculating quasinormal modes for a scalar field, revealing a discrete spectrum due to the effective potential.

## Contribution

It provides an exact solution for a (2+1)-D wormhole with nonlinear electrodynamics and explores its causal structure and quasinormal modes, which is a novel combination.

## Key findings

- Causality analysis with Kruskal-Szekeres and Penrose diagrams.
- Exact calculation of scalar quasinormal modes showing a discrete spectrum.
- Effective potential resembles a shifted harmonic oscillator.

## Abstract

In this paper we study a static cyclic symmetric traversable wormhole in $(2+1)-$dimensional gravity coupled to nonlinear electrodynamics in anti-de Sitter spacetime. The solution is characterized by three parameters: mass $M$, cosmological constant $\Lambda$ and one electromagnetic parameter, $q_{\alpha}$. The causality of this spacetime is studied, determining its maximal extension and constructing then the corresponding Kruskal-Szekeres and Penrose diagrams. The quasinormal modes (QNMs) that result from considering a massive scalar test field in the wormhole background are determined by solving in exact form the Klein-Gordon equation; the effective potential resembles the one of a harmonic oscillator shifted from its equilibrium position and, consequently, the QNMs have a pure point spectrum.