Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been "good", or not?

Question

In MTW, "Gravitation", there is a brief section (p. 26 and Fig. 1.9) discussing that clocks may be characterized, trial by trial, as having been "good" in the trial under consideration, or having been "bad" (i.e., presumably, not "good") in that trial:

Let $t$ be time on a "good" clock (time coordinate of a local inertial frame); it makes the tracks of free particles through the local region of space time look straight. Let $T[~t~]$ be the reading of the "bad" clock; it makes the world lines of free particles through the local region of space time look curved (Fig. 1.9). [...]

This description doesn't seem quite rigorous (e.g.: What does it mean for a track or a world line to be "made to look" in some particular geometric sense at all ??), it seems to be restricted to clocks in regions admitting inertial frames, and instead of being stated in terms of explicit geometric notions it refers to "free particles" (i.e. a notion which would require explicit, comprehensible definition in the first place).

Therefore:

Is there a rigorous, explicitly geometric, general characterization for whether a given clock had been "good", or not, in some particular trial ?

For definiteness and to fix the notation, consider clock $~\mathcal C := (~C, t_{\mathcal C}~)$,
consisting of "clock face $C$" and "clock readings $t_{\mathcal C}$",
in the trial from $C$ having taken part in coincidence event $\varepsilon_{C J}$ until $C$ having taken part in coincidence event $\varepsilon_{C Q}$,
with $\mathcal S_C := \{ C_J, ... C_K, ... C_P, ... C_Q \}$ the ordered set of indications of clock face $C$ corresponding to the ordered set of events $\{ \varepsilon_{C J}, ... \varepsilon_{C K}, ... \varepsilon_{C P}, ... \varepsilon_{C Q} \}$ in which $C$ took part in the course of the trial under consideration,
and the corresponding "readings" of clock $\mathcal C$ expressed as (monotonous) function $t_{\mathcal C}~:~ \mathcal S_C \rightarrow \mathbb R$.

Answer 1

Writing $\mathcal X_C^{n} := \{ \varepsilon_{ C \mathcal X }^1, ~..., \varepsilon_{ C \mathcal X }^n \}$ for any (variable) suitable ordered subset of $n \ge 1$ events in which $C$ took part, and abbreviating

$${\! \large \tilde\tau} C_K^P := \! \! \mathop{ \bf \text{ infimum } }_{ \large {\mathcal X_C^{n} } \subseteq \{ \large{ \varepsilon_{C K}, ~... ~ \varepsilon_{C P} \}} } \! \! \small{ \left[ \sqrt{ \frac{s^2[~\varepsilon_{C K}, \varepsilon_{ C \mathcal X }^1~]}{s^2[~\varepsilon_{C K}, \varepsilon_{C P}~]} } \! + \! \! \left( \sum_{ m = 1 }^{n - 1} \sqrt{ \frac{s^2[~\varepsilon_{ C \mathcal X }^m, \varepsilon_{ C \mathcal X }^{(m + 1)}~]}{s^2[~\varepsilon_{C K}, \varepsilon_{C P}~]} } \right) \! \! + \! \sqrt{ \frac{s^2[~\varepsilon_{ C \mathcal X }^n, \varepsilon_{C P}~]}{s^2[~\varepsilon_{C K}, \varepsilon_{C P}~]} } \right] },$$

a clock $\mathcal C := (~C, t_{\mathcal C}~)$ is called having been good in the trial from $C$ having taken part in coincidence event $\varepsilon_{C J}$ until $C$ having taken part in coincidence event $\varepsilon_{C Q}$ if and only if for any two indications $C_K, C_P \in \mathcal S_C$ of the clock in the course of the trial the corresponding "readings" satisfy:

$$ \begin{array}{rl} t_{\mathcal C}[~C_P~] - t_{\mathcal C}[~C_K~] & = \left( t_{\mathcal C}[~C_P~] - t_{\mathcal C}[~C_J~] \right) \left( \frac{ {\! \large \tilde\tau} C_K^P }{ {\! \large \tilde\tau} C_J^P } \right) \sqrt{ \frac{s^2[~\varepsilon_{ C K }, \varepsilon_{C P}~]}{s^2[~\varepsilon_{C J}, \varepsilon_{C P}~]} } \cr ~ & = \left( t_{\mathcal C}[~C_Q~] - t_{\mathcal C}[~C_K~] \right) \left( \frac{ {\! \large \tilde\tau} C_K^P }{ {\! \large \tilde\tau} C_K^Q } \right) \sqrt{ \frac{s^2[~\varepsilon_{ C K }, \varepsilon_{C P}~]}{s^2[~\varepsilon_{C K}, \varepsilon_{C Q}~]} }. \end{array}$$

Whether this definition is rigorous, as requested, depends of course on whether the required values of interval ratios are definite in the first place, before and without in turn requiring knowledge or presumptions about whether given clocks were good, or not.