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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?

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    $\begingroup$ As long as the theory preserves the Lorentz symmetry at least locally, you may always go to the clock's rest frame where the geometry is always the same and $d\tau^2=|ds^2| = dt_0^2$. Any construction of the clocks will evolve as the function of the time coordinate so it will of course tick at the same rate. If there's an extra perfectly massless scalar field, the rate of clock's ticking will depend on both the scalar field at that point as well as, independently of that, the construction of the clocks (because differently constructed clocks may induce a different dependence on the scalar). $\endgroup$
    – Luboš Motl
    Commented Jan 31, 2014 at 19:20
  • $\begingroup$ Thanks Luboš. But how do I know I should use $|ds^2|$ of that metric? I could consider a conformally-related metric, for example, like in scalar-tensor theories. In that metric I can also find a coordinate system, locally, where the geometry is flat, and find $d\tau^2 = dt^2_0$... but that's a different $t_0$ than I would have found by using the first metric. I believe the usual thing to do is choose the metric to which matter is minimally coupled. But why? It is intuitively true, but I don't know of a more rigorous argument. $\endgroup$
    – user1591373
    Commented Jan 31, 2014 at 19:23
  • $\begingroup$ If there's a scalar field that is not stabilized, there is no preferred metric. Any $f(\phi)g_{\mu\nu}$ is as good as any other. Different constructions of clocks may find one of the metrics more natural or relevant than others. This is an issue in string theory when the dilaton is not stabilized. One has the "Einstein frame metric" and "string frame metric" that differ multiplicatively by the factor of a function of the dilaton (scalar) field. $\endgroup$
    – Luboš Motl
    Commented Jan 31, 2014 at 19:25
  • $\begingroup$ Thanks again. So in string theory (when the dilaton isn't stabilized) you have different matter fields coupling to different effective metrics, and so the time you measure depends on which fields you use to construct your clock? That sounds reasonable. Not being a string theorist (I do modified gravity, where usually there exists a single Jordan frame for every field), do you know of any references where this is treated in some detail? e.g., if I build a clock in a certain way, the time it measures relates to the underlying geometry (in X frame) in this and that way. $\endgroup$
    – user1591373
    Commented Jan 31, 2014 at 19:30
  • $\begingroup$ Related: physics.stackexchange.com/questions/53334/… $\endgroup$
    – joshphysics
    Commented Feb 1, 2014 at 0:07

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