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

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 $$\Delta \tau$$.

• It is simply not very fun to be working with physically meaningless/arbitrary path parameters $\lambda$ when we have the physically meaningful proper time $\tau=\tau[P(\lambda)]$, which can be seen as a relationship between $\tau$ and $\lambda$. Now, the path parametrisation $\lambda$ should be one-to-one and monotonic, and it should then be obvious that $\tau$ is too; and that they are in one-to-one correspondence with each other. Then you can just invert the relationship and use $\tau$ as the parametrisation by considering $P[\lambda(\tau)]$ Commented Aug 17, 2023 at 5:12
• It seems like there is overloading of notation when you write $\tau = \tau[P(\lambda)]$; e.g., I'm not sure it makes sense to extremise a variable $\tau$ (or perhaps it does?), but it makes sense to extremise a function(al) $\tau[P(\lambda)]$. I'm not sure I understand this change when you invert the relationship to get $\tau$ as the parametrisation. Commented Aug 17, 2023 at 5:30
• You are correct that, before the extremisation of the functional $\tau[P(\lambda)]$, even the numerical value of the extremised function itself is not defined. It is only after the extremisation that, everywhere along the extremised path, do you have a well-defined proper time to numerically associate with each position in spacetime. Only then can you invert the relationship, only for the extremised path. What you are arguing is indeed very pedantic, but it will turn up later when thinking of quantum theory, because the non-extremal paths also contribute. i.e. you are onto something. Commented Aug 17, 2023 at 5:39
• How are $P$ and $x$ related? Commented Aug 17, 2023 at 6:19
• @Qmechanic, I should have specified x refers to the coordinates of the path P in a chosen basis. Commented Aug 17, 2023 at 6:22

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