Separate massless shell action

On a prescribed background $(M,g,T)$, choose an open parameter interval $I$ and a compact segment $[a,b]\subset I$ with $a<b$. In a spacetime chart containing the curve and sufficiently small variations, let $x:I\to M$ be smooth near $[a,b]$, let $p(\lambda )\in T_{x(\lambda )}^{\ast }M$ be a smooth covector field along $x$, and let $e$ be smooth and positive near $[a,b]$. In this chart, write $g_{ab}=g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ for $a,b\in \{0,1,2,3\}$, let $(g^{ab})$ be the inverse matrix, and define $p^{\mathrm{♯}}={\sum }_{a,b=0}^3{g}^{ab}{p}_b{\mathrm{\partial }}_a$; equivalently, $g(p^{\mathrm{♯}},w)=p(w)$ for every tangent vector $w$. Adopt the distinct massless classical particle action on this segment

$S_{0,[a,b]}[x,p,e]={\int }_a^b[p_a{x}^{\mathrm{\prime }a}-e{C}_0(x,p)]d\lambda ,C_0=\frac{1}{2}{g}^{ab}{p}_a{p}_b\mathrm{.}$

Admissible variations are smooth one-parameter families of chart fields $(x_{ϵ},p_{ϵ},e_{ϵ})$ on a common neighborhood of $[a,b]$, with $x_{ϵ}(a)=x(a)$ and $x_{ϵ}(b)=x(b)$; the $p_{ϵ}$ and $e_{ϵ}$ variations are independent and have no endpoint constraints, while $e_{ϵ}>0$ for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. Impose the physical branch only on stationary solutions after these variations: $p\mathrm{\ne }0$ and $p^{\mathrm{♯}}$ is future null. With all spacetime coordinates measured in metres and dimensionless $\lambda$, $p_a$ has units kg m/s, $C_0$ has units kg²m²/s² and $e$ has units s/kg, making the action J s. Other parameter units scale $e$ accordingly. This is a separate adopted model: null curves have no proper-time parameter, the model does not arise by setting $m=0$ in $P=mU$, and its positive momentum scale is additional initial data. On the nonzero branch, the vertical differential is $d_p{C}_0=p^{\mathrm{♯}}\mathrm{\ne }0$: nondegeneracy of $g^{-1}$ and $p\mathrm{\ne }0$ give $p^{\mathrm{♯}}\mathrm{\ne }0$. Hence the shell is a regular primary constraint in the sense of Regular primary constraints and total generator. This postulate is not an empirical observation or a deduction from the massive model.

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