Scalars on asymptotically locally AdS wormholes with $\mathcal{R}^{2}$ terms

Abstract

In this paper we study the propagation of a probe scalar on an asymptotically locally AdS wormhole solution of Einstein-Gauss-Bonnet theory in five dimensions. The radial coordinate $\rho$ connects both asymptotic regions located at $\rho\rightarrow\pm\infty$. The metric is characterized by a single integration constant $\rho_{0}$ and the wormhole throat is located at $\rho=0$. In the region $0<\rho<\rho _{0}$, both the gravitational pull as well as the centrifugal contributions to the geodesic motion point in the same direction and therefore they cannot balance. We explore the consequences of the existence of this region on the propagation of a scalar probe. The cases with $\rho_{0}=0$ as well as the limit $\rho_{0}\rightarrow +\infty$ lead to exactly solvable differential eigenvalue problems, with shape-invariant potentials of the Rosen-Morse and Scarf family, respectively. Here, we numerically obtain the normal modes of a scalar field when $\rho_0\neq 0$, with reflecting boundary conditions at both asymptotic regions. We also explore the effect of a non-minimal coupling between the scalar curvature and the scalar field. Remarkably, there is a particular value of the non-minimal coupling parameter that leads to fully resonant spectra in the limit of vanishing $\rho_0$ as well as when $\rho_0\rightarrow+\infty$, for purely radial modes.Comment: 16 pages, many figures. V2: typos corrected and new appendix on the wormhole obtained in the limit $\rho_0\rightarrow+\infty