Wave Propagation and Effective Refraction in Lorentz-Violating Wormhole Geometries

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.

💡 Research Summary

The paper investigates the propagation of mass‑less scalar waves in static, spherically symmetric wormhole spacetimes that incorporate Lorentz‑violating effects. The authors begin by writing the most general line element for such a wormhole in terms of two arbitrary functions: a lapse (or redshift) function A(x) and an areal radius r(x), where the radial coordinate x runs from –∞ to +∞ and smoothly covers both asymptotic regions. By computing the Christoffel symbols, Ricci tensor, Ricci scalar and Kretschmann scalar, they obtain explicit curvature invariants that depend only on A, r and their first and second derivatives. The regularity conditions at the throat are identified as r′(x₀)=0 and r′′(x₀)>0, guaranteeing a smooth minimum of the areal radius.

Next, the covariant Klein‑Gordon equation for a scalar field Φ is separated using the standard ansatz Φ(t,x,θ,φ)=e^{-iωt}Y_{ℓm}(θ,φ)R(x). This yields a second‑order radial equation containing a first‑derivative term that reflects the non‑trivial radial dependence of the background. By redefining the radial function as ψ(x)=R(x) r(x) A(x)^{p} (with an appropriate power p), the authors cast the equation into a Schrödinger‑like form
ψ″(x)+