Echoes from bounded universes

TL;DR

The paper constructs a class of modified Ellis wormholes in which one asymptotic region is a bounded 2-sphere core with finite areal radius, characterized by a throat at $x=0$ and an inner region ending at $R$ (with $R$ possibly zero). It demonstrates geodesic completeness despite $r(x)$ possibly tending to zero and analyzes scalar perturbations to reveal time-domain echoes arising from a cavity-like inner region, where a stable photon sphere can trap modes. The study shows that the global, bounded inner geometry qualitatively alters geodesic structure, effective potentials, and late-time perturbation signals, offering potentially observable signatures that distinguish these objects from standard black holes or conventional Ellis wormholes. The work highlights how the parameters $c$ and $R$ control the cavity width and barrier height, respectively, thus governing echo strength and quasi-resonant behavior, and points to future explorations of boundary conditions and Schwarzschild-like generalizations to broaden phenomenological implications.

Abstract

We construct a general class of modified Ellis wormholes, where one asymptotic Minkowski region is replaced by a bounded 2-sphere core, characterized by asymptotic finite areal radius. We pursue an in-depth analysis of the resulting geometry, outlining that geodesic completeness is guaranteed also when the radial function asymptotically shrinks to zero. Then, we study the evolution of scalar perturbations, bringing out how these geometric configurations can in principle affect the time-domain profiles of quasinormal modes, pointing out the distinctive features with respect to other black holes or wormholes geometries.

Figures (9)

• Figure 1: Modified areal radius \ref{['eq:areal_radius']}. Top panel: $r(x)^2$ for fixed value of $c$ and different asymptotic 2-sphere radius $R$. Bottom panel: $r(x)^2$ for fixed $R$ and some choices of $c$. We normalized the plots with the throat radius $a$.
• Figure 2: Embedding diagrams of finite 2-sphere radius EWHs with fixed value of $c$. In the top-left panel we exhibit a wormhole-like object in which the 2-sphere radius shrinks to zero, creating a sort of "bubble" in the internal region. The other panels correspond to asymptotic 2-sphere radius $R$ smaller (top-right), equal (bottom-left) and bigger (bottom-right) than the wormhole throat $a$ (represented by the red line).
• Figure 3: Effective potential of four modified EWHs with fixed $c$ and four choices of $R$, namely $R/a=0,0.5,1$ and 5. We also plot the inverse of the impact parameter squared of some photons that enter in the inner region of the modified EWH.
• Figure 4: Null geodesics in the modified EWH. Solid lines correspond to light rays propagating in the outer universe, while dashed lines represent geodesics in the inner universe. We are considering photons with the same values of impact parameter in the outer universe, and showing how these photons are scattered or absorbed depending on the modified EWH configuration. The circle with radius 1 corresponds to the throat of the wormhole; the outermost circle corresponds to the local maximum of the areal radius $r(x)$ in the inner universe. The other circles are the asymptotic 2-sphere radius of each configuration.
• Figure 5: Representation of the numerical grid used for the integration of \ref{['eq:uvdiffeq']}. The evaluation points of \ref{['eq:discretizedPhi']} are represented with $S$, $W$, $E$ and $N$. The stepsize of the grid can be visualized by the distance between to consecutive points in the same axis $h=u_E-u_S=v_N-v_E$.