Notation: Proper time as a parameter of a curve versus as a functional

Question

I am trying to figure out some notation issues (or at least that what I assume this is).

I assume the "proper time" could refer to a "proper time functional" of a timelike path $P(\lambda)$ with coordinates $x^\alpha(\lambda)$ $$\tau[P(\lambda)] = \int_{\lambda_A}^{\lambda_B}\sqrt{(- g_{\alpha \beta} (x^\gamma)\frac{dx^\alpha}{d \lambda}\frac{dx^\beta}{d \lambda})} d \lambda $$ and is a separate mathematical object than when we use proper time as simply the parameter along the worldline $P(\tau)$?

I assume you could also write $$ \tau[P(\tau')] = \int_{\tau'_A}^{\tau'_B} d \tau'$$ and the dummy variable $\tau'$ would be defined as the proper time parameter, $P(\tau')$, but I haven't seen any textbooks make this distinction (I assume this might just because it's mathematical/notational pedantry, but perhaps it's incorrect).

Edit to add: I suppose this is sometimes distinguished by referring to $\tau[P(\tau')] $ as the proper time interval with notation $\Delta \tau$.

Answer 1

When people wish to refer to the proper time of a curve as a functional, they will use whatever local equivalent they wish to call the length functional for a timelike curve :

\begin{equation} l_\gamma = \int_{t_1}^{t_2} \sqrt{g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))} dt \end{equation}

up to an appropriate sign under the square root to make the square root make sense. It is also called the Lorentzian length of a curve, and in the case of a geodesic is sometimes called the geodesic interval. It also relates to some other common functions in relativity such as Synge's world function.