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

TL;DR

This paper develops a covariant, gauge-invariant optical framework for electromagnetic wave propagation in Schwarzschild black hole spacetimes, introducing an effective refractive index that captures gravitational effects.

Contribution

It provides a novel, unified optical description of electromagnetic waves in static black hole backgrounds using a gauge-invariant formulation and an analytical refractive index.

Findings

Derived a closed-form refractive index for Schwarzschild spacetime.

Showed axial and polar electromagnetic perturbations obey the same master equation.

Analyzed the refractive index behavior near the horizon and at infinity.

Abstract

We present a fully covariant and gauge-invariant formulation of electromagnetic wave propagation in static, spherically symmetric black hole spacetimes, developed entirely within Schwarzschild-like coordinates. Start ing from the source-free Maxwell equations on a curved background, electromagnetic perturbations are de composed 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 parity-independent master equation, and their exact isospectrality emerges nat urally 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 grav itational 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.