Lorentz-Violating Wormhole Optics

Abstract

We study massless spin-1 field propagation in a static, circularly symmetric $(2+1)$-dimensional wormhole with spatial Lorentz-violating anisotropy characterized by the throat radius $a$ and deformation parameter $η$. The geometry is horizon-free, geodesically complete, and asymptotically flat, with negative Gaussian curvature localized near the throat. Using the fully covariant vector boson formalism and covariant Maxwell theory, we derive an exact Schrödinger-type radial equation with a curvature-induced effective potential. Recasting the dynamics in Helmholtz form yields an effective refractive-index profile, showing that the wormhole acts as an inhomogeneous optical medium with position-dependent refractive index and frequency-dependent confinement, where low-frequency modes are strongly trapped while high-frequency modes propagate almost freely. A differential-geometric correspondence with helicoidal surfaces is established via $1/[a^2(1-η)] \leftrightarrow w^2$, demonstrating that Lorentz-violation-induced curvature is mathematically equivalent to curvature generated by geometric twist and linking the model to twisted graphene nanoribbons as analog-gravity platforms. These results provide a geometric framework for curvature-driven localization, dispersion, and anisotropic wave propagation in topologically nontrivial $(2+1)$-dimensional backgrounds.

Figures (4)

• Figure 1: Gaussian curvature $\mathcal{K}(x)$ of a $(2+1)$-dimensional Lorentz-violating wormhole for varying Lorentz-violation parameters $\eta \in [0,0.9]$ and three different throat radii $a = 0.5, 1.0, 1.5$. Each subplot corresponds to a fixed wormhole throat, while the curves within each subplot represent increasing values of $\eta$. The curvature is strictly negative throughout, indicating a hyperbolic optical geometry, and the magnitude of $K$ near the throat increases with both decreasing throat radius and increasing Lorentz-violation parameter, reflecting the enhancement of geometric flaring due to Lorentz-symmetry breaking.
• Figure 2: 3D embedding diagrams of Lorentz-violating wormholes (with $x_{\text{max}}= 8$) for different values of the Lorentz-violation parameter $\eta$ at fixed throat radius $a=1$. The surfaces correspond to $\eta = 0$, $\eta = 0.45$, and $\eta = 0.9$. Both the upper ($z>0$) and lower ($z<0$) embedding branches are displayed to show the full wormhole geometry. Increasing $\eta$ significantly modifies the radial stretching of the embedding surface, indicating a strong deformation of the wormhole geometry induced by Lorentz-violating effects.
• Figure 3: Effective potential $V_{\rm eff}(x)$\ref{['eq:Veff-explicit']} for massless spin-1 vector bosons in the Lorentz-violating wormhole. The potential is plotted for $a=0.5, 1.0, 1.5$ and $\eta \in [0,0.9]$, showing the deformation induced by Lorentz violation.
• Figure 4: Refractive index squared $n^2(x)$\ref{['R-Index']} for massless spin-1 vector boson propagation in the Lorentz-violating wormhole geometry for four representative electromagnetic frequencies: IR ($\omega = 3 \times 10^{14}$ Hz), visible light ($\omega = 5.5 \times 10^{14}$ Hz), X-ray ($\omega = 3 \times 10^{18}$ Hz), and gamma-ray ($\omega = 1 \times 10^{20}$ Hz). The wormhole throat radius is taken as $a = 5$ nm, and the spin of the propagating mode is $s=1$. The curves correspond to ten values of the Lorentz-violation parameter $\eta$ in the range $[0,0.9]$, with $\alpha = 1-\eta$, showing how Lorentz-violation modifies the effective refractive index profile along the longitudinal coordinate $x \in [-20,20]$ nm. The black dashed line indicates $n^2(x)=0$, separating propagating regions ($n^2>0$) from evanescent or exponentially decaying regions ($n^2<0$). The spatial variations are localized near the wormhole throat, while far from the throat, $n^2(x)$ asymptotically approaches unity.