Well, do you know what the Schwarzschild metric is? In Schwarzschild (spherical) coordinates, if a particle starts at the point ##r, \theta, \phi## at time ##t## and travels a small distance to the point ##r+\delta r, \theta + \delta \theta, \phi + \delta \phi## by time ##t+\delta t##, then the the change in proper time ##\delta \tau## satisfies: