Question

Can we derive the relation between proper time and the spacetime interval?

Answer 1

In GR, it's usually taken for granted - or as a definition - that thetime measured by an observer's clock is related to the geometry in a very simple way, d s 2.

Answer 2

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 dτ2=|ds2|=dt20dτ2=|ds2|=dt02. 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).