# Geodesic equation in different coordinate systems

Suppose I have a Schwarzschild metric in Schwarzschild-coordinates and I obtain the geodesic equations for this metric.

Suppose I transform the Schwarzschild metric into a different coordinate system, say Painleve-Gullstrand, or proper distance coordinates, and after the transformation I also derive the geodesic equations.

If I solve the geodesic equations numerically, for the same initial conditions, will the one written in Schwarzschild coordinates give the same implications as one that is written in different coordinates? If yes, then will it be advisable to find a coordinate transformation where the Christoffel symbols look simpler before numerically solving the geodesic equations?

• What do you mean by ‘same implications’? The initial conditions won’t be the same in different coordinates. – Rumplestillskin Apr 4 at 7:24