well, The geodedic equations are different. please look at the link that I send you, how is the calculation of the geodetic equations. you start with the metric ,input, so for different metric you get different geodetic equations.
In minkowski space you need the so call signature this is the eta matrix in my program input.
You get also different geodetic line for sphere, torus, zylinder , hyperbola ......, what so ever.
With this program that I put hear you can calculate the geodetic , equation of motions very easy.
Try to get the geodetic with maple programm with(tensor) ....., so you can see how easy is it to use my programm.
If you have problem with it, I can send you the maple calculations my examples.