Revisiting Einstein's analogy: black holes as gradient-index lenses

Abstract: According to Albert Einstein, gravitation is analogous to an optical medium. Building on this idea, various definitions of the gradient-index (GRIN) medium representing curved spacetime have been proposed; often, these approaches demand advanced knowledge of General Relativity (GR) and its associated mathematical methods. This paper introduces an alternative approach for generating the form of GRIN media that reproduces the exact behavior of photon trajectories in the presence of a Schwarzschild black hole or about the equatorial plane of a Kerr black hole. Our approach is based on Noether's theorem that leads to the optical Fermat principle, leveraging the system's symmetry, resulting in an unrestricted and robust method for finding GRIN media in a flat spacetime that reproduces the behavior of null geodesics in Schwarzschild-like spacetimes, employing only photon trajectories as input. This approach is also valid for massive particles and provides a complementary understanding of GR. For this reason, the method is applied to generate the GRIN medium from the S2 star's trajectory around SgrA*, demonstrating that sufficient precision can be achieved to reproduce the orbit's subtle apsidal precession.