Electromagnetic wave propagation in static black hole spacetimes: an effective refractive index description in Schwarzschild geometry

Abstract

We investigate electromagnetic wave propagation in static, spherically symmetric black hole spacetimes using a covariant and gauge-invariant framework based on the established Maxwell perturbation formalism. Building upon known parity decompositions and gauge-invariant master equations, we reformulate the resulting radial dynamics entirely within Schwarzschild coordinates and introduce an effective refractive-index description of electromagnetic propagation in curved spacetime. Starting from the source-free Maxwell equations on a curved background, electromagnetic perturbations are decomposed according to parity and systematically reduced to gauge-invariant dynamical variables without introducing auxiliary coordinate transformations or horizon-regular variables. Both axial and polar sectors are shown to obey the same parityindependent master equation, and their exact isospectrality emerges naturally as a direct consequence of Maxwell theory in four dimensions. By eliminating first-derivative terms through an appropriate field redefinition, the radial dynamics is cast into a Helmholtz-type equation, which motivates the introduction of an effective, position- and frequency-dependent refractive index encoding gravitational redshift, curvature effects, and angular momentum within a unified optical framework. Specializing to the Schwarzschild geometry, we obtain the refractive index in closed analytical form and analyze its behavior in the near-horizon, intermediate, and asymptotic regimes. The resulting description provides a transparent and physically intuitive interpretation of electromagnetic evanescence, and propagation in black hole spacetimes, and establishes a robust foundation for wave-optical, semiclassical, and numerical studies in more general static gravitational backgrounds.