Fixed-endpoint curved charge variation

The sets $z^{-1}(W)$, over chart domains $W$, form an open cover of $[a,b]$; by [F11] choose a finite subcover $V_1,\dots ,V_r$. There is $\delta >0$ such that every closed subinterval of length $<\delta$ lies in one $V_j$: otherwise [A1] selects closed intervals $I_n$ of length $<1\mathrm{/}n$ for each natural $n\ge 1$, none contained in any $V_j$, and their midpoints form a bounded sequence in $[a,b]$; [F17] gives a convergent subsequence with limit $t\in [a,b]$ and strictly increasing indices $n_s$. Some $V_j$ contains $t$ and, being relatively open, contains $(t-\eta ,t+\eta )\cap [a,b]$ for some $\eta >0$. Choose $N$ with $1\mathrm{/}N<\eta \mathrm{/}2$ by [F18]. Since $n_s$ is strictly increasing, it eventually exceeds $N$, so [F19] gives $\mathrm{\mid }{I}_{n_s}\mathrm{\mid }<1\mathrm{/}{n}_s<1\mathrm{/}N<\eta \mathrm{/}2$; also the midpoint is eventually within $\eta \mathrm{/}2$ of $t$. Thus $I_{n_s}\subset V_j$, a contradiction. Since $b-a>0$, [F18] applied to $\delta \mathrm{/}(b-a)$ gives $k\ge 1$ with $1\mathrm{/}k<\delta \mathrm{/}(b-a)$; hence $(b-a)\mathrm{/}k<\delta$. Divide $[a,b]$ into $k$ equal closed subintervals; each is contained in a $V_j$, so each curve subsegment lies in one chart. For each such subsegment and its chart, the inverse image of that chart under the continuous variation map is open and contains $[t_j,t_{j+1}]\times \{0\}$; a finite subcover of this compact segment by product neighborhoods supplies ${ϵ}_j>0$ keeping the whole subsegment in the chart for $\mathrm{\mid }ϵ\mathrm{\mid }<{ϵ}_j$. The minimum over finitely many segments gives a common parameter interval.

On each chart subsegment, write $x_{ϵ}^a(\tau )$ for the coordinates, $J^a={\mathrm{\partial }}_{ϵ}{x}_{ϵ}^a{\mathrm{\mid }}_0$, and $U^a={\mathrm{\partial }}_{\tau }{x}_0^a$. Differentiating the free-action integrand under the integral by [F6], and applying [F7]–[F9] to $Q=g_{ab}(x_{ϵ}){\mathrm{\partial }}_{\tau }{x}_{ϵ}^a{\mathrm{\partial }}_{\tau }{x}_{ϵ}^b$ at $Q=-c^2$, gives $\delta S_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=m\int (g_{ab}({\mathrm{\partial }}_{\tau }{J}^a)U^b+\frac{1}{2}({\mathrm{\partial }}_c{g}_{ab})J^c{U}^a{U}^b)d\tau$. The integrand is continuous, hence Riemann integrable by [F15].

The factors $J^a$ and $g_{ab}{U}^b$ are $C^1$, so their derivatives are continuous and integrable by [F15]. Integrating the ${\mathrm{\partial }}_{\tau }{J}^a$ term of step 2.1 by parts with [F10], and using the Christoffel formula and metric compatibility of [F2], gives on each subsegment $\delta S_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=m[g(J,U){]}_{t_j}^{t_{j+1}}-m{\int }_{t_j}^{t_{j+1}}g(J,{\mathrm{\nabla }}_{U}U)d\tau$. When the subsegment endpoints are fixed, the boundary term vanishes and this is the chart-local specialization of [F3]; at internal subdivision points we retain the displayed term.

On each chart subsegment, $A_a(z(\tau ))$ and $J^a$ are $C^1$, so their derivatives are continuous and integrable by [F15]. Differentiating the interaction integrand by [F6]–[F8] gives $q(({\mathrm{\partial }}_c{A}_a)J^c{U}^a+A_a{\mathrm{\partial }}_{\tau }{J}^a)$. Integration by parts [F10] yields $\delta S_{\mathrm{i}\mathrm{n}\mathrm{t}}=q[A(J){]}_{t_j}^{t_{j+1}}+q{\int }_{t_j}^{t_{j+1}}({\mathrm{\partial }}_c{A}_a-{\mathrm{\partial }}_a{A}_c)J^c{U}^{a}d\tau =q[A(J){]}_{t_j}^{t_{j+1}}+q{\int }_{t_j}^{t_{j+1}}{F}_{ca}{J}^c{U}^{a}d\tau$, with the component ordering fixed by [F4] and the notation from step 2.1.

Sum steps 3.1 and 3.2 over the finite subdivision. At each internal cut, the free boundary scalar $g(J,U)$ and interaction boundary scalar $A(J)$ agree in the adjacent charts, so their endpoint terms cancel. At $a,b$ they vanish because $J(a)=J(b)=0$. Repeated use of [F16] sums the remaining integrals to the stated first-variation formula.

Let $C$ be the continuous covector along $z$ whose chart components from step 4.1 are $C_a:=-m({\mathrm{\nabla }}_{U}U{)}_a+q{F}_{ab}{U}^b$. If $C({\tau }_{\ast })\mathrm{\ne }0$ at an interior point ${\tau }_{\ast }$, choose a chart $x:W\to x(W)\subseteq {\mathbb{R}}^4$ around $z({\tau }_{\ast })$ and a coordinate index $a_0$ with $C_{a_0}({\tau }_{\ast })\mathrm{\ne }0$. Set $\sigma =1$ when $C_{a_0}({\tau }_{\ast })>0$ and $\sigma =-1$ when $C_{a_0}({\tau }_{\ast })<0$. Continuity of $z$ and $C_{a_0}$ gives an open interval $I$ with $\bar{I}\subset (a,b)$, $z(\bar{I})\subset W\cap O$, and $\sigma C_{a_0}\ge d>0$ on $I$. Choose a nondegenerate compact interval $K=[{\tau }_{-},{\tau }_{+}]\subset I$, and use [F5] to choose $\chi \in C_c^{\mathrm{\infty }}(I)$ with $0\le \chi \le 1$ and $\chi =1$ on $K$. For $\tau \in I$ define $z_{ϵ}(\tau )$ by $x(z_{ϵ}(\tau ))=x(z(\tau ))+ϵ\sigma \chi (\tau )e_{a_0}$, where $e_{a_0}$ is the $a_0$ coordinate basis vector; set $z_{ϵ}(\tau )=z(\tau )$ outside $I$. The compact set $x(z(\mathrm{supp}\chi ))$ lies in the open set $x(W\cap O)$, so all shifted points stay in that set for sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. Since $\chi$ vanishes near $\mathrm{\partial }I$, these formulas define a $C^2$ fixed-endpoint variation, whose variation field has $J^{a_0}=\sigma \chi$ and all other chart components zero. On the compact support, $g({\mathrm{\partial }}_{\tau }{z}_{ϵ},{\mathrm{\partial }}_{\tau }{z}_{ϵ})$ and $g({\mathrm{\partial }}_{\tau }{z}_{ϵ},N)$ depend continuously on $(ϵ,\tau )$; at $ϵ=0$ the first is $-c^2$, and the second is strictly negative with a uniform negative bound by [F13] and [F14]. Shrinking the same $ϵ$-neighbourhood keeps both inequalities strict, so the varied curve remains future timelike in $O$.

Put $f(\tau ):=\sigma C_{a_0}(\tau )\chi (\tau )$ for $\tau \in I$ and $f(\tau ):=0$ for $\tau \in [a,b]\setminus I$. Then $f$ is continuous, nonnegative on $[a,b]$, and at least $d$ on $K$. By [F15] it is integrable on each subinterval; [F16] splits its integral at ${\tau }_{-}$ and ${\tau }_{+}$, and [F12] gives nonnegative outer integrals and ${\int }_{{\tau }_{-}}^{{\tau }_{+}}fd\tau \ge d({\tau }_{+}-{\tau }_{-})>0$. The first-variation formula of step 4.1 for the constructed variation is $\delta S={\int }_a^b{C}_{\mu }{J}^{\mu }d\tau ={\int }_a^{b}fd\tau >0$, contradicting stationarity. Therefore stationarity for every local chart-supported variation forces $C_a=0$ in the interior and, by continuity, at the endpoints. Conversely, $C_a=0$ makes step 4.1 vanish for every admissible variation; raising the covector index with the nondegenerate metric gives $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$.