Simulating an orbit around a spherical mass
The curvature of space-time around a spherical mass is described by the Schwarzschild metric.
We provide an Excel file simulating the orbit of a test particle around the spherical mass using a straightforward forward Euler simulation.
Download link: 20161106_geodesic_line_simulation_riemann_schwarzschild_2d_v3
The simulation is performed in a 2-dimensional space where and meaning the metric reduces to:
where the right hand side uses Einstein notation.
Given coördinates and first order derivatives at time , the 2nd order derivatives feeding the simulation can be derived using the geodesic equation:
where is derived from
again using Einstein notation and with the inverse matrix of the metric .
The exact matrices are provided in the Excel.
An example is provided below and can be simulated in the Excel by setting the starting values to and and the simulation step size to .