Seeing relativity-I: Ray tracing in a Schwarzschild metric to explore the maximal analytic extension of the metric and making a proper rendering of the stars

We present an implementation of a ray tracing code in the Schwarzschild metric. We aim at building a numerical code with a correct implementation of both special (aberration, amplification, Doppler) and general (deflection of light, lensing, gravitational redshift) relativistic effects so as to simulate what an observer with arbitrary velocity would see near, or possibly within, the black hole. We also pay some specific attention to perform a satisfactory rendering of stars. Using this code, we then show several unexplored features of the maximal analytical extension of the metric. In particular, we study the aspect of the second asymptotic region of the metric as seen by an observer crossing the horizon. We also address several aspects related to the white hole region (i.e., past singularity) seen both from outside the black hole, inside the future horizon and inside the past horizon, which gives rise to the most counter-intuitive effects.