Metamaterial analog of a black hole shadow: An exact ray-tracing simulation based on the spacetime index of refraction

Abstract: In this letter first we show that the equation of null geodesics in spherically symmetric spacetimes in isotropic coordinates is identical to the equation of light ray trajectories in isotropic media in flat spacetime. Based on this analogy we introduce an exact simulation of the light ray trajectories both in these spacetimes and in their metamaterial analogs in terms of the spacetime index of refraction. As unstable light trajectories, the photon spheres form in these metamaterial analogs at {\it exactly} the same radial distances as expected from the corresponding black hole geometries. Using the same ray-tracing simulation we find the analog of a simple black hole shadow formed by the metamaterial analog of a Schwarzschild black hole, eclipsing a line of light sources near its analog horizon.