Geometric and Wave Optics in a BTZ Optical Metric-Based Wormhole

Abstract

We investigate the geometric and wave optical properties of a (2 + 1)-dimensional ultra-static spacetime conformally related to the static BTZ black hole, characterized by constant negative Gaussian curvature. The associated optical metric defines a hyperbolic wormhole geometry, wherein null geodesics experience a P & ouml;schl-Teller-type repulsive effective potential that suppresses circular photon orbits and directs all trajectories toward the optical origin. In the wave regime, we reformulate the Helmholtz equation into a Schr & ouml;dinger-like form, revealing a spatially localized effective potential that encodes curvature and angular momentum effects. The resulting refractive index n(p,w)) is both spatially and spectrally dispersive, leading to a position-dependent critical frequency wc(P) that delineates the boundary between propagating and evanescent modes. At high frequencies, the medium becomes asymptotically transparent, while for w<wc(P), waves undergo exponential attenuation. These results demonstrate intrinsic curvature-induced spectral filtering and provide a geometrically tunable framework for analog gravity systems and graphene-based photonic platforms.