Massive spinless test-particle action

Let $a,b\in \mathbb{R}$ with $a<b$, and let $x:[a,b]\to M$ be a $C^2$ future timelike curve on the prescribed background of Ideal clocks on a prescribed Lorentzian background. Let $H:(-{ϵ}_0,{ϵ}_0)\times [a,b]\to M$ be a $C^2$ variation, for some ${ϵ}_0>0$, such that each $H(ϵ,\cdot )$ is future timelike, $H(0,s)=x(s)$, and $H(ϵ,a)=x(a)$ and $H(ϵ,b)=x(b)$. For a spinless, structureless massive test body of constant primitive mass $m>0$, measured in kilograms, and the SI conversion constant $c>0$ metres per second, adopt the action on this finite segment

$S_m[x]=-mc{\int }_a^b\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds=-m{c}^2\mathrm{\Delta }\tau ,$

where $\mathrm{\Delta }\tau :=c^{-1}{\int }_a^b\sqrt{-g(dx\mathrm{/}ds,dx\mathrm{/}ds)}ds$ is the ideal-clock increment specified by Ideal clocks on a prescribed Lorentzian background on this smooth segment. The model's free trajectories are precisely the curves stationary under the fixed-endpoint $C^2$ variations just specified; this is the local variation class in Timelike length action: arbitrary-parameter first variation. When $s$ is measured in seconds, $T$ has units metres per second, and the action has units joule-seconds. This is an adopted model premise, not a consequence of the metric or clock postulate. It excludes self-gravity, radiation reaction, spin, finite-size structure, particle creation, and quantum effects; it does not assert empirical adequacy for any particular body.