Motion equations in a Kerr–Newman–de Sitter spacetime: some methods of integration and application to black holes shadowing in Scilab

Abstract

In this paper, we recall some basic facts about the Kerr-Newman-(anti) de Sitter (KNdS) spacetime and review several formulations and integration methods for the geodesic equation of a test particle in such a spacetime. In particular, we introduce some basic general symplectic integrators in the Hamiltonian formalism and we re-derive the separated motion equations using Carter's method.

After this theoretical background, we explain how to ray-trace a KNdS black hole, equipped with a thin accretion disk, using Scilab. We compare the accuracy and execution time of the previous methods, concluding that the Carter equations is the best one. Then, inspired by Hagihara, we apply Weierstrass' elliptic functions to the non-rotating case, yielding a fairly fast shadowing program for such a spacetime.

We provide some illustrations of the code, including a depiction of the effects of the cosmological constant on shadows and accretion disk, as well as a simulation of M87*.

Influences of tilted thin accretion disks on the observational appearance of hairy black holes in Horndeski gravity

“…These images can be simulated by tracing the light rays backward in time from the observer's image screen to the light source near the BH. To date, a number of ray tracing algorithms have been developed and efficiently applied to various astrophysical problems [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89]. Here, we have chosen Odyssey [73] as a robust tool for tracking the evolution of light rays.…”