Given a metric, we find out the null and timelike geodesic which helps us conclude that how the trajectory of various particles will become in a particular curvature of spacetime.

But I don't understand that with respect to whom these geodesic are calculated. Are these geodesic observed by an observer who is sitting at infinity or are these geodesic according to a person who is actually travelling along these geodesics? Will the null geodesic be same for all the observers regardless of which frame they are in, because light should travel at speed c in every frame so is it true that light take the same path according to every observer?

How do we differentiate between the two observers by just looking at a metric and calculating the geodesic equation?

The situation should change when we consider a local inertial observer because in his frame of reference the spacetime is flat, so for him laws of special relativity would be valid. What would he see when he is falling into a black hole from a finite distance away?

• Note that a geodesic is an observer-independent notion. – Qmechanic Sep 10 '17 at 9:41

Your assertion that the speed of light is the same in every frame is not technically correct. The speed of a ray of light in a local frame is always $c$, but the speed the same ray as measured by some distant observer may not be.