Fixed-endpoint variation of the relativistic particle functional

Fix an inertial coordinate frame; the continuous function $s\mapsto -\eta (\dot{X},\dot{X})$ is strictly positive on the compact $[a,b]$ and attains a positive minimum $m_0>0$ by [F3], and ${\dot{X}}^0$ likewise has a positive minimum by future directedness [F1]; the variation $h$ and its derivative are bounded on $[a,b]$ with bounds $H_1,H_2$, and the compact image of $X$ lies in the open domain of $A_{\mu }$ with a positive Lebesgue number for a finite coordinate-ball cover.

Put $M:={\max }_{[a,b]}\mathrm{\parallel }\dot{X}{\mathrm{\parallel }}_2$ and $H:={\max }_{[a,b]}\mathrm{\parallel }\dot{h}{\mathrm{\parallel }}_2$, both finite. Since the diagonal metric has entries of modulus one, the estimate of [F5], $\mathrm{\mid }\eta (v,w)\mathrm{\mid }\le \mathrm{\parallel }v{\mathrm{\parallel }}_2\mathrm{\parallel }w{\mathrm{\parallel }}_2$ gives $-\eta (\dot{X}+\epsilon \dot{h},\dot{X}+\epsilon \dot{h})\ge m_0-2\mathrm{\mid }\epsilon \mathrm{\mid }MH-{\epsilon }^2{H}^2$. Choose ${\epsilon }_0>0$ so that $2{\epsilon }_{0}MH+{\epsilon }_0^2{H}^2<m_0\mathrm{/}2$ and ${\epsilon }_{0}H<\frac{1}{2}{\min }_{[a,b]}{\dot{X}}^0$ (when $H=0$ these restrictions are automatic). Then for $\mathrm{\mid }\epsilon \mathrm{\mid }<{\epsilon }_0$ the squared timelike margin is at least $m_0\mathrm{/}2$ and the time component is positive. Thus $X+\epsilon h$ is again a $C^2$ future timelike curve with the same endpoints; shrinking ${\epsilon }_0$ further using the boundedness of $h$ and the compact image's positive distance from the complement of the potential domain keeps every $X+\epsilon h$ inside that neighbourhood.

With the parameter rectangle $[a,b]\times (-{\epsilon }_0,{\epsilon }_0)$ the integrands of $S$ and their $\epsilon$-derivatives are continuous and the square-root argument is bounded away from zero by step 2.1, so [F3] applies and the first variation is $\delta S=-mc{\int }_a^b\frac{-\eta (\dot{h},\dot{X})}{\sqrt{-\eta (\dot{X},\dot{X})}}ds+q{\int }_a^b[({\mathrm{\partial }}_{\nu }{A}_{\mu })(X)h^{\nu }{\dot{X}}^{\mu }+A_{\nu }(X){\dot{h}}^{\nu }]ds$; by [F2] the first integrand is $m{U}_{\nu }{\dot{h}}^{\nu }$.

Integration by parts [F4] with $h(a)=h(b)=0$ gives ${\int }_a^{b}m{U}_{\nu }{\dot{h}}^{\nu }ds=-{\int }_a^{b}m\frac{d{U}_{\nu }}{ds}{h}^{\nu }ds$ and ${\int }_a^b{A}_{\nu }(X){\dot{h}}^{\nu }ds=-{\int }_a^b({\mathrm{\partial }}_{\mu }{A}_{\nu })(X){\dot{X}}^{\mu }{h}^{\nu }ds$, since all factors are $C^1$ on $[a,b]$; hence $\delta S={\int }_a^b{C}_{\nu }(s)h^{\nu }(s)ds$ with $C_{\nu }=-m\frac{d{U}_{\nu }}{ds}+q({\mathrm{\partial }}_{\nu }{A}_{\mu }-{\mathrm{\partial }}_{\mu }{A}_{\nu })(X){\dot{X}}^{\mu }=-m\frac{d{U}_{\nu }}{ds}+q{F}_{\nu \mu }{\dot{X}}^{\mu }$, a continuous coefficient on $[a,b]$.

Choice-free coefficient vanishing: if $C_{\nu }(s_0)\mathrm{\ne }0$ at some interior point, choose the sign $\sigma \in \{-1,1\}$ with $\sigma C_{\nu }(s_0)>0$ and a small open interval $J\subset (a,b)$ on which $\sigma C_{\nu }\ge d>0$, then a nondegenerate closed subinterval $K\subset J$; the one-dimensional ZF cutoff [F4] gives $\zeta \in C_c^{\mathrm{\infty }}(J)$ with $0\le \zeta \le 1$ and $\zeta =1$ near $K$, and $h^{\nu }:=\sigma \zeta$ (other components zero) is an admitted fixed-endpoint $C^2$ variation, for which monotonicity and additivity [F4] give $\delta S=\int \sigma C_{\nu }\zeta ds\ge d\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}(K)>0$, contradicting stationarity.

Hence $C_{\nu }=0$ on $(a,b)$ and, by continuity, on $[a,b]$; conversely $C_{\nu }\equiv 0$ makes the first variation $\delta S=\int C_{\nu }{h}^{\nu }ds$ vanish for every fixed-endpoint $C^2$ variation, so stationarity is indeed equivalent to $C_{\nu }\equiv 0$.

Dividing $m\frac{d{U}_{\nu }}{ds}=q{F}_{\nu \mu }{\dot{X}}^{\mu }$ by the positive factor ${\tau }^{\mathrm{\prime }}=d\tau \mathrm{/}ds=\sqrt{-\eta (\dot{X},\dot{X})}\mathrm{/}c$ of [F2] and using ${\dot{X}}^{\mu }={\tau }^{\mathrm{\prime }}{U}^{\mu }$ gives $m\frac{d{U}_{\nu }}{d\tau }=q{F}_{\nu \mu }{U}^{\mu }$; raising the index with the constant metric gives the equivalent vector form. The proof uses only finite choices, finite-interval Riemann calculus and the explicit ZF cutoff, and asserts no existence or coupling theory for singular particles and fields; adopting $S$ as a physical action is an additional model assumption.