Is proper time a definition in GR?

Question

Where does come from in GR that the proper time between two events measured by an observer is the length of his worldline calculated using the metric? Is it assumed or postulated in analogy with special relativity?

I thought that it could be proved by saying that there is a certain chart in which the worldline is parametrized with constant spatial coordinates (I would like to know if this is always possible, mine is just a guess). So the length of the worldline in this chart is equal to the difference of the temporal coordinates of the two events. But in GR all the coordinates are arbitrary so we can't assume that the first of the four coordinates equals the time of the observer. Is this true or is it possible to distinguish the temporal coordinate from the spacial ones in some way and assign to it a physical meaning?

Answer 1

In an observer's rest frame their four-velocity is always $(1, 0, 0, 0)$, so in a time $\tau$ recorded by their clock the length of their trajectory is just $\tau$.

Answer 2

In order to have a scientific theory you need a mathematical framework and a mapping between terms in the mathematical framework and experimental measurements. This mapping is sometimes called the minimal interpretation. The minimal interpretation is an essential part of the scientific theory, without it a pure mathematical framework cannot be tested using the scientific method.

In relativity, there are two possibilities for clocks. One is to simply declare, as part of the minimal interpretation, that the reading on a clock is the integral of the metric along the clock’s worldline. Then this fact is a part of the definition of the theory itself. This is my preference.

The other possibility is to say that the reading on a clock is the coordinate time in an inertial frame (using the Einstein synchronization convention) where the clock is at rest. With this definition one can derive the relationship between a clock and the metric. Then this fact is a derived result of the theory.

I prefer the first approach because it places a greater focus on invariants and reduces the focus on reference frames. However, this is merely my personal preference and is not shared by all. Some prefer the other approach because historically it came first.