> My understanding is that in GR, massive observers move along timelike curves xμ(λ)
... where xμ:EO→Rn appears to be some coordinate assignment to the set of events EO in which the observer under consideration (O) took part
(perhaps not even "just any" coordinate assignment, but rather an assignment which represents the events EO in the order in which observer O took part, as a "path Rn");
and λ:(subset of) R→(subset of) Rn is a real-valued parametrization of those coordinates
(perhaps not even "just any" parametrization, but again rather a parametrization which represents the order of observer O's indications monotonously) ...
> and if an observer moves from point xμ(λa) to xμ(λb)
... or focussing on the physical: if observer O moves from one particular event (which we happened to denote by coordinates xμ(λa)) to another particular event (denoted by xμ(λb)) ...
> then his clock will measure that an amount of time tba given by the curve's arc length;
This seems to indicate a misunderstanding, either of terminology and conventions of notation, or even more profoundly:
in the theory of relativity, "time" means
- foremost an indication of an observer
(such as the indication OA of observer O of having been in coincidence with observer A, having jointly taken part in coincidence event εAO),
- and in a derived sense a (real) number value t assigned as a clock reading to an indication of an observer (or also assigned to the entire event, of which the observer indicated its participance), such as any function t:EO→R
(or perhaps not even "just any" such function, but rather only such functions which are monotonous wrt. the order in which observer O took part in the events of set EO).
Obviously, there is a tremendously large number of distinct such clock reading functions t;
any two of which are not necessarily continuous wrt. each other, much less differentiable, or even smooth (differentiable to any order), or even affine (proportional to each other).
On the other hand, the arc length of the timelike curve of a particular observer (between two particular events, such as between the two events εAO and εBO in which observer O took part, or between two particular indications of the observer, such as correspondingly between the two indications OA and OB of observer O),
a.k.a. the duration of observer O between these two indications,
a.k.a. the proper time of observer O between these two indications
(as far as the term "proper" is permissible at all, since it suggests the possibility of "improper" notions as well),
is usually denoted by the letter τ;
for the above case explicitly as τO[_A,_B].
Ratios of durations are unambiguos real numbers, i.e. quantities to be measured.
Further, the readings t assigned to indications of a given clock provide a measure of its coresponding durations τ as far as for any three distinct indications OH,OJ,OK holds:
(t[OK]−t[OH])τO[_H,_J]=(t[OJ]−t[OH])τO[_H,_K].
If (and only if) this was satisfied, the clock under consideration (incl. the clock reading assignment t) is said to have been "good", or having "run evenly".
> Why is this so?
As a matter of definition;
in particular of the notions "duration τ (of an observer, between two of its indications)", and of whether a given clock was "good" or to which extent it was not.