Massless test-particle action and momentum scale

Let $a,b\in \mathbb{R}$ with $a<b$. A configuration on $[a,b]$ consists of a $C^2$ curve $x:[a,b]\to M$ and a smooth positive function $e:[a,b]\to (0,\mathrm{\infty })$ on the prescribed background of Ideal clocks on a prescribed Lorentzian background. Vary $x$ and $e$ independently: $x$ through $C^2$ variations $H:(-{ϵ}_0,{ϵ}_0)\times [a,b]\to M$ for some ${ϵ}_0>0$, with $H(0,s)=x(s)$ and $H(ϵ,a)=x(a)$, $H(ϵ,b)=x(b)$, and no causal restriction on the varied curves; and $e$ through jointly smooth positive variations $e_{ϵ}:[a,b]\to (0,\mathrm{\infty })$ whose variation field $\delta e:={\mathrm{\partial }}_{ϵ}{e}_{ϵ}{\mathrm{\mid }}_{ϵ=0}$ is compactly supported in $(a,b)$. Adopt a distinct massless test-particle sector with primitive action

$S_0[x,e]=\frac{1}{2}{\int }_a^{b}e(s{)}^{-1}{g}_{x(s)}(T(s),T(s))ds,T(s)=dx\mathrm{/}ds,$

and stationary under all such independent variations. The variational equations and their null branch follow from Positive-einbein variation and massive/null sector equations: the constraint is $g(T,T)=0$, and the physical branch requires $T$ to be nonzero and future-directed. The associated momentum $P=T\mathrm{/}e$ is nonzero future null; its positive scale is primitive initial data, not $mU$ for any $m=0$ substitution. Null curves have no proper-time parameter. With $s$ in seconds, $T$ in metres per second, and $e$ in kg${}^{-1}$, the integral has units joule-seconds. For finite intervals $[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]$ and $[a,b]$, an orientation-preserving $C^{\mathrm{\infty }}$ diffeomorphism $s=\varphi (r):[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ with ${\varphi }^{\mathrm{\prime }}>0$ transforms $T_r(r)={\varphi }^{\mathrm{\prime }}(r)T(\varphi (r))$ and $e_r(r)={\varphi }^{\mathrm{\prime }}(r)e(\varphi (r))$; this leaves the functional invariant by Positive-einbein variation and massive/null sector equations and preserves the stated configuration class. Neither the scale nor a clock reading is inferred from the ray image alone.

This is an adopted classical model formulation, not an empirical observation or a theorem. Its scope excludes mass generation, interactions, and quantum field effects.