I want to know how to obtain the "experience of time" of an observer moving on an arbitary timelike worldline. By "experience of time" I mean the time expereinced by the observer as a function on coordinate time, at any point on it's world line. Literature says that it is the "proper time" of that path. But the mathematical definition of "proper time" just makes it a paramatrization of the path that is affinely connected to the arc length. Now in GR, time and space are in the same footing (in a sense). So, why should "proper time" or arc length be interpreted as giving the experience of time of the observer? It might as well give a measure of space or maybe an "unphysical" mixture of time and space.