Particle trajectory near a rotating black hole

In this study, we explore the trajectory of a particle near a rotating black hole using the framework of General Relativity, with a focus on the Kerr-Schild metric in Cartesian coordinates. Our approach involves developing Python code to symbolically compute the Christoffel symbols associated with the Kerr-Schild metric. These Christoffel symbols are essential for describing the curvature of spacetime and are complicated in the context of rotating black holes. This requires significant computational effort for these all-symbolic calculations. After the Christoffel symbols are obtained, we generate a Fortran implementation to numerically solve the geodesic equations as Ordinary differential equations (ODEs) that govern the motion of particles in curved spacetime. To integrate these geodesic equations, we are using Fourth-order Runge-Kutta (RK4) numerical method with specific initial conditions to simulate the trajectory of particle interacting with the gravitational field of a rotating black hole. We compare our results with angular momentum a= 0, the well-studied case of a non-rotating (Schwarzschild) black hole to study how the rotation alters the particle's trajectory.