Ray Tracing Through Absorbing Dielectric Media in the Schwarzschild Spacetime

TL;DR

This paper extends general relativistic ray tracing to include absorption in dielectric media near massive objects, analyzing how absorption affects light propagation and observable signals in curved spacetime.

Contribution

It introduces a complex Hamiltonian approach to incorporate dispersion and absorption in ray tracing within Schwarzschild spacetime, extending previous models to include complex refractive indices.

Findings

Transmission requires the power-law index h > 1.

Exponential absorption occurs for h ≤ 1.

Results reproduce known literature in specific limits.

Abstract

General Relativity describes the trajectories of light-rays through curved spacetime near a massive object. In addition to gravitational lensing, we include an absorbing dielectric medium given by a complex refractive index known as the Drude model. When absorption is included the eikonal becomes complex, with the imaginary part related to the absorption along a ray between emission and observation points. We extend results from the literature to include dispersion in the index of refraction. The complex Hamiltonian splits into a real part that describes the equations of motion and a constraint equation that governs the momentum loss in the system. We work in coordinates which are fully real, with a real metric in physical spacetime. We assume the dust and plasma distributions of the Drude matter to coincide and vary as a power-law $1/r^h$. We find that transmission requires $h>1$, otherwise exponential absorption occurs along ray paths. We use ray-tracing through strongly absorbing matter near the surface of the compact star, as well as specializing to a point-lens in the weak-field limit with weakly absorbing matter to generate potentially observable light curves for distant observers. In the appropriate limits, our theory reproduces results from the literature.