# 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

## 1 Answer

In general relativity a timelike/lightlike geodesic describes the path of a massive/massless free-falling particle in spacetime. In a different coordinate system the initial conditions and the picture of the path will look different, but the physical occurrence is identical.

However different coordinate systems not necessarily cover the same physical regions of spacetime. For instance Schwarzschild coordinates arrive till the horizon. To cross the horizon you need Eddington–Finkelstein coordinates or Kruskal–Szekeres coordinates or other.

The choice of a convenient coordinate system is constrained by the regions of spacetime you want to explore.