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This paper investigates the behavior of photon orbits around black holes using the Eikonal equation and models of effective refractive index using the Schwarzschild metric, inspired by Lagrangian optics and the foundational work of David Nolte.Conducted as an educational exercise for advanced high school and undergraduate students, this research aims to help students combine multiple topics in physics to explore analogical relationships between different concepts such as the calculus of variations, optics, and general relativity.By employing numerical simulations to recalibrate photon trajectories, we highlight how the curvature of spacetime influences photon paths, particularly focusing on the stability of circular orbits.These results reveal significant deviations from classical predictions, emphasizing the critical role of gravitational lensing and the formation of black hole shadows.This paper is expected to provide consolidated information to understand and reproduce simulations of orbiting photons for interested high school or undergraduate students.
Tucker et al. (Thu,) studied this question.