I understand what geodesics are and how to calculate them from Christoffel symbols and all that. But I've just come across a question I have no idea about. I've been given the dust filled Friedmann solution: ds^2 = -dt^2 + a(t)^2 (dX^2 + X^2 dO^2) (O=omega) And been told to show that radial geodesics obey: a^2 (dX/dT) = k where k constant and T is proper time. And hence (dt/dT)^2 = 1 + k/a^2 How do I introduce proper time? And where to I get an expression for dX/dT from? I'm completely stumped...and help would be much appreciated!