Thanks in advance - this problem has been bothering me for a while! I'm working with an unpowered spaceship orbiting a large mass M. The orbit is circular and it is following the geodesic freely. It has an orbit radies of r = R. My question is this. The metric of the space-time curvature is the Schwarszchild metric: ds^2 = - (1 - 2GM/(r*c^2) )*(c^2)dt^2 + [(1 - 2GM/(r*c^2) )^-1]*dr^2 + (r^2)dθ^2 + (r^2)*(sinθ^2)d∅^2 -(1) But I keep seeing references to ds^2 = - (c^2)*d(tau)^2 -(2) I understand the angular terms and dr disappear as the d(tau) means we are observing the orbit from the reference frame stationary with the satellite. But I can only how equation (1) = equation (2) in a Schwarszchild curved space if r (distance between mass and satellite) tends to infinity. But, in the case of the satellite's reference frame r=R. I will be eternally grateful to anyone that can shed light on my error of understanding. Many thanks.