For general relativity, Wald's GR states that timelike curves, with the norm $g_{ab}T^{a}T^{b} < 0$, can be parameterized by the "proper time" $$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$ This looks awfully similar to the formula for the length of a curve in any arbitrary Riemannian space with positive signature, $$\ell = \int (g_{ab}T^aT^b)^{1/2} dt,$$ where $t$ is the parameter of the curve. The length formula makes sense; it is a generalization of the usual length formula in flat Euclidean space. My question is: what is the relation between the proper time definition and the length formula? What is it about that additional minus sign that turns length into "proper time"?
I'm also finding that Wald is short on explanation of the physics. In special relativity, I can say in ordinary language that the proper time is the time between two events measured by an inertial frame that is located at the same position as the two events. What, then, is the intuitive meaning of that "proper time" defined by Wald's formula?