Ideal clocks on a prescribed Lorentzian background

Adopted convention. On a prescribed background $(M,g,N)$ from Particle spacetime, observers and worldlines, let $x:[a,b]\to M$, $a<b$, be a continuous piecewise-$C^1$ future timelike curve in the sense of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold: it has a finite partition $a=s_0<\cdots <s_k=b$, each segment is $C^1$ up to its endpoints with future timelike one-sided tangents there, and the two tangents at a corner need not agree. If $s$ denotes its parameter and $T_i=dx\mathrm{/}ds$ on the $i$th segment, an ideal clock carried along it records the increment

$\mathrm{\Delta }\tau =\frac{1}{c}{\sum }_{i=1}^k{\int }_{s_{i-1}}^{s_i}\sqrt{-g(T_i,T_i)}ds\mathrm{.}$

The increment is independent of the admissible finite partition, by additivity of the integral, and is additive across corners. On each smooth segment this is the proper-time parameter with $d\tau =c^{-1}\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds$. The flat-spacetime specialization is inherited from Adopted ideal-clock and ideal-vacuum-light interpretation; extending that convention to the curved prescribed background is a separate adoption of this framework.

When the parameter $s$ is measured in seconds, $x^0=ct$ is in metres, $c$ is in metres per second, and $\tau$ is in seconds. This is a model postulate, not a theorem or a reported measurement. It does not assert that a physical clock realizes the idealization exactly, that the background solves gravitational field equations, or that the test body affects the background.