Proper time in general relativity 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?
 A: Yes, the proper time along a timelike curve in relativity is very much analogous to the length of a curve.  Just as the length of a curve is invariant under rotations, the proper time along a curve is invariant under Lorentz transformations.
One difference with conventional length is that although a straight line is the shortest length between two points, a straight line in spacetime (constant-velocity trajectory in flat space, or geodesic in curved space) is the longest proper time between two events.  That happens because when calculating the proper time, the space components enter with the opposite sign to the time component.
The additional minus sign in that equation is purely a matter of convention, though. It's needed in the (−, +, +, +) metric signature, but would be absent in the (+, −, −, −) signature.
As for the physical interpretation, proper time is just the time that would be measured by a clock traveling along that timelike curve as its world-line.  Or equivalently, the amount of time a traveler would experience if they flew that trajectory in their spaceship.  (This is true whether the curve is a geodesic or not.  A geodesic corresponds to the ship just floating along freely and not accelerating, while a general timelike curve could include acceleration.)
A: All curves can be parameterised by an affine parameter (commonly written as $\lambda$ in the GR books I have). However only for timelike curves does the parameter $\lambda$ have a physical meaning i.e. it's the elapsed time $\tau$ shown on a clock by the observer following the curve.
So there's no mathematical difference between parameterising a timelike curve using $\tau$ and a general curve using $\lambda$, only one of interpretation.
