In the contest of the newtonian limit in general relativity, if I consider a timelike geodesic that can represent the motion of a free falling particle under the influence of the gravitational force of the perturbed metric, I can't understand why the proper time of the particle is such that: $$-c^2d\tau^2=-c^2dt^2+dx^2+dy^2+dz^2$$ I know that a geodesic is parametrized at proper time if $g(\dot \gamma, \dot \gamma)=-1$, but from here I don't know how deduce the expression above. Really it is an expression for the proper time that I have encountered also many other times but I can't understand it:
$\textbf{EDIT:}$ I mean why in general can I write $d\tau=\sqrt{-g_{\alpha\beta}dx^{\alpha}dx^\beta}$?
I have thoguht that if $\gamma=(t(\tau), x(\tau), y(\tau), z(\tau))$ then $$g(\dot \gamma, \dot \gamma)=-1\iff g(\frac{d\gamma}{d\tau}, \frac{d\gamma}{d\tau})=-1\iff -(\frac{dt}{d\tau})^2+(\frac{dx}{d\tau})^2+(\frac{dy}{d\tau})^2+(\frac{dz}{d\tau})^2=-1\iff -dt^2+dx^2+dy^2+dz^2=-d\tau^2$$ where for $g$ I have used the perturbed metric at first order that is minkowski metric.
Instead in non relativitusc measure units the condition $g(\dot \gamma, \dot \gamma)=-1$ becomes $g(\dot \gamma, \dot \gamma)=-c^2$, right?
Can you help me?
