Wave Propagation and Effective Refraction in Lorentz-Violating Wormhole Geometries

TL;DR

The paper addresses how Lorentz-violating wormhole spacetimes modify massless scalar-wave propagation by developing a geometric-optical framework that derives a radial Helmholtz equation and a geometry-induced refractive index $n(\omega,x)$. It shows that divergences of $n$ align with Killing horizons while turning points defined by $n^2=0$ govern reflection, transmission, and confinement, with the behavior depending on the lapse profile. By analyzing constant, linear, and quadratic lapse functions, the work demonstrates horizonless, asymmetric, and multi-horizon regimes, respectively, revealing dispersive, graded-index-like wave dynamics in curved vacua. These results provide a unifying optical interpretation of wave phenomena in Lorentz-violating gravities and point toward observational signatures through wave scattering and resonances in such backgrounds.

Abstract

We study the propagation of massless scalar waves in static, spherically symmetric Lorentz-violating wormhole spacetimes within a geometric-optical framework. Starting from a general metric characterized by an arbitrary lapse function and areal radius, we derive curvature invariants, establish regularity conditions at the wormhole throat, and reduce the Klein-Gordon equation to a Helmholtz-type radial wave equation. This formulation naturally leads to a position- and frequency-dependent effective refractive index determined by the underlying spacetime geometry and Lorentz-violating structure, resulting in effective frequency-dependent wave-optical behavior. We show that divergences of the refractive index coincide with Killing horizons, while curvature-induced turning points control reflection, transmission, and confinement of scalar waves. By analyzing constant, linear, and quadratic lapse profiles, we identify horizonless transmission regimes, asymmetric wave propagation, and multi-horizon trapping structures. Our results reveal that Lorentz violation can significantly modify wave-optical properties of curved spacetime, generating graded-index analogues and geometric confinement of modes without curvature singularities. This unified optical perspective provides a robust framework for investigating wave scattering, resonances, and potential observational signatures in Lorentz-violating gravitational backgrounds.

Figures (3)

• Figure 1: Plots of $n(\omega,x)^2$ versus $x$ for $\ell=a=1$ and different Lorentz-violating parameters $\eta=0,2/3,0.9$ at $\omega=1,2,4$. Table \ref{['tab:turning_points1']} reports the corresponding turning point locations.
• Figure 2: Plots of $n(\omega,x)^2$ versus $x$ for $\ell = a = 1$, $\eta=2/3$, and $\omega=1,2,4$, for different Rindler-type acceleration values $\chi=0.02,0.2,1$. Table \ref{['tab:turning_points2']} reports the corresponding turning points and Killing horizon locations.
• Figure 3: Effective refractive index squared $n(\omega,x)^2$ for the quadratic-lapse geometry, with $\ell = a = 1$. When $n(\omega,x)^2<0$ within the plotted ranges, indicating evanescent modes. Table \ref{['tab:turning_points3']} summarizes the turning points (a sample is shown on each panel) and Killing horizons for $\omega=0.1,1,2,4$.