Schwarzschild Geodesics

By: Gijs Bellaard

The last few months I have been extremely interested in the theory of General Relativity. To hone my understanding I have already made a Kerr-Newman Metric Black Hole simulation, but the solution method is rough, to say the least. It simply applies the forward Euler method directly to the geodesic equations expressed in their Hamiltonian form. This time I did it differently: I just asked Mathematica to integrate the geodesics equations in their Euler-Lagrange form! The result is the following Mathematica Notebook and animation:

Schwarzschild Geodesic

What is visualized above is the orbit of a test particle within the Schwarzschild metric, which corresponds to the gravitational field of an uncharged, non-rotating, spherically symmetric body of a certain mass. The Schwarzschild metric can be expressed in an infinitude of coordinates, but in this case I have used the isotropic coordinates system. One notable effect that can be seen in the animation is the apsidal precession of the orbit. This effect is special to Einstein's general theory of relativity, and does not appear in Newton's theory of gravitation.