Free massive test-particle action

Let $I\subset \mathbb{R}$ be an open interval, and let $x:I\to M$ be a $C^2$ future-directed timelike worldline on a prescribed smooth, time-oriented Lorentzian background $(M,g,T)$ of signature $(-+++)$, with $m>0$ a fixed primitive mass in kg and $c>0$ in m/s. For any compact segment $[a,b]\subset I$ with $a<b$, adopt the stationary-action model on that segment

$S_{m,[a,b]}[x]=-mc{\int }_a^b\sqrt{-g(x^{\mathrm{\prime }},x^{\mathrm{\prime }})}d\lambda \mathrm{.}$

Here $I\subset \mathbb{R}$ is an open interval, $\lambda$ is its real coordinate, and $x^{\mathrm{\prime }}$ denotes $dx\mathrm{/}d\lambda$. The future-timelike condition makes $x^{\mathrm{\prime }}$ nowhere zero. On a neighborhood of $[a,b]$, $x$ is $C^2$ (so its tangent is $C^1$). If $\tilde{\lambda }=f(\lambda )$ is an increasing $C^1$ reparametrization with $f^{\mathrm{\prime }}>0$, then $\sqrt{-g(dx\mathrm{/}d\tilde{\lambda },dx\mathrm{/}d\tilde{\lambda })}d\tilde{\lambda }=\sqrt{-g(x^{\mathrm{\prime }},x^{\mathrm{\prime }})}d\lambda$ by the chain rule. An admissible variation is a family $x_{ϵ}(\lambda )$ defined for $ϵ\in (-\delta ,\delta )$, $\delta >0$, and $\lambda$ in a common neighborhood of $[a,b]$, with $x_0=x$, that is $C^2$ jointly in $(ϵ,\lambda )$, has fixed endpoint events $x_{ϵ}(a)=x(a)$ and $x_{ϵ}(b)=x(b)$, and remains future timelike for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. The adopted motion principle is that on every such segment $[a,b]$, the physical worldline $x$ satisfies ${\frac{d}{dϵ}\mid }_{ϵ=0}{S}_{m,[a,b]}[x_{ϵ}]=0$ for every admissible variation through $x$. The integrand is the proper-time length density, so $S_{m,[a,b]}=-m{c}^2{\int }_a^{b}d\tau$; its dimension is action (J s). The adopted premise concerns this ideal free test body only. It is neither a theorem about all matter nor an empirical report, and supplies no backreaction or completeness claim. The flat ideal-clock specialization agrees with Adopted ideal-clock and ideal-vacuum-light interpretation; adopting this action on a curved background remains a separate premise.

The cotangent phase-space carrier convention assumes countable choice ACω only; full choice is not assumed.