In GR, it's usually taken for granted - or as a definition - that the time measured by an observer's clock is related to the geometry in a very simple way, $d\tau^2 = |ds^2|$. This is easy enough to see in some simple contexts, like special relativity, but it might not be obvious in general spacetimes.

In principle I should be able to construct a model of a clock using the matter sector alone, and then show that, assuming that matter is minimally coupled to the metric, the time this clock measures is related to the spacetime interval in the usual way.

Is that possible, and if so does anyone know where I could find a derivation like that?

In GR, it's usually taken for granted - or as a definition - that the time measured by an observer's clock is related to the geometry in a very simple way, $d\tau^2 = |ds^2|$. This is easy enough to see in some simple contexts, like special relativity, but it might not be obvious in general spacetimes.

In principle I should be able to construct a model of a clock using the matter sector alone, and then show that, assuming that matter is minimally coupled to the metric, the time this clock measures is related to the spacetime interval in the usual way.

Is that possible, and if so does anyone know where I could find a derivation like that?

EDIT: We can think about the question like this. I might consider having a second metric related to the first, e.g., by a conformal factor depending on a scalar field like in many modified gravity theories. So for which metric do you assume that $d\tau^2 = |ds^2|$ measures an observer's clock time? The standard answer is you use the metric to which matter is minimally coupled. But I would be very surprised if this were just an assumption; minimal coupling should be able to tell us the relation between proper time and geometry. But how?