There is an object in free fall above the surface of some planet. This planet has no significant atmosphere so there is no significant air resistance on the object. Also there is no significant electric force acting on this object, and any electric charge of the planet is negligible. Also the angular velocity of the planet is negligible. Also the reference frame that we are using to define the objects four velocity and spacetime coordinates is one that is on the planets surface close to the object in free fall.
Given the rest mass of the planet $M_0$, and the proper radius of the planet $r_0$, the initial four velocity of the object, and the initial spacetime coordinates of the object, and the proper time $\tau$ of the object, how would I calculate the final four velocity and final spacetime coordinates of the object using General Relativity.