Recently I've been trying to learn General and Special Relativity by myself. There is an specific thing I do not understand perfectly, proper time in the metric of the space-time.
Take the case of an empty space-time:
$$-c^2 \mathrm d \tau^2 = -c^2 \mathrm d t^2 + \mathrm d x^2 + \mathrm d y^2 + \mathrm d z^2$$ where $\tau$ is the proper time of an object.
I don't understand when $c^2 \mathrm d \tau^2$ is used as $ds^2$. Why is this possible? It is related to the worldline traced by an object? Could you calculate the line integral in order to find the length of the path the object? This line integral would give the path length of a geodesic?
Could you get the speed of an object by doing this (correct me if I'm wrong): $$-c^2 \mathrm d \tau^2 = -c^2 \mathrm d t^2 + \mathrm d x^2 + \mathrm d y^2 + \mathrm d z^2 \rightarrow \\ -c^2 \left( \frac{\mathrm d \tau}{\mathrm d t} \right)^2 = -c^2 + \left( \frac{\mathrm d x}{\mathrm d t} \right)^2 + \left( \frac{\mathrm d y}{\mathrm d t} \right)^2 + \left( \frac{\mathrm d z}{\mathrm d t} \right)^2 $$ If you assume that: $\dot{x}^2 + \dot{y}^2 + \dot{z}^2 = ||v||^2$, then by rearranging a little bit the equation, you could get the velocity of an object.