Timelike length action: arbitrary-parameter first variation

For a $C^1$ field $V$ along a $C^1$ curve $y$, define $D_{s}V$ in coordinates by $(D_{s}V{)}^a=(V^a{)}^{\mathrm{\prime }}+{\mathrm{\Gamma }}^a{}_{bc}(y)(y^b{)}^{\mathrm{\prime }}{V}^c$. At any parameter value replace the coordinate functions of $y$ and $V$ locally by affine functions with the same values and first derivatives. This gives a smooth curve and smooth field with the same values and first derivatives; the displayed formula agrees there with [F10]. In another chart the component formula and its transformation depend only on those values and first derivatives and the smooth chart transition, so the intrinsic smooth derivative in [F10] proves coordinate independence also for the $C^1$ data. The ordinary product rule and the compatibility identity [F9] give $(g(V,W){)}^{\mathrm{\prime }}=g(D_{s}V,W)+g(V,D_{s}W)$ for $C^1$ fields, since differentiating $g_{ab}(y)V^a{W}^b$ produces exactly the two connection terms. Derivatives at included endpoints are one-sided. Apply this definition also along each parameter curve of $H$; $H\in C^2$ makes its tangent fields $C^1$, so all derivatives below exist and are continuous.

Write $T_{ϵ}={\mathrm{\partial }}_{s}H$ and $J={\mathrm{\partial }}_{ϵ}H{\mathrm{\mid }}_0$. Differentiating ${\mathrm{\ell }}_{ϵ}=\sqrt{-g(T_{ϵ},T_{ϵ})}$ and using metric compatibility gives ${\mathrm{\partial }}_{ϵ}\mathrm{\ell }{\mathrm{\mid }}_0=-g(D_{ϵ}T,T)\mathrm{/}\mathrm{\ell }$. In a chart, the difference of the two covariant derivatives is ${\mathrm{\Gamma }}^a{}_{bc}(J^b{T}^c-T^b{J}^c)$; it vanishes because [F9] gives ${\mathrm{\Gamma }}^a{}_{bc}={\mathrm{\Gamma }}^a{}_{cb}$ and coordinate mixed partials commute by [F6]. Thus $D_{ϵ}T=D_{s}J$. By [F6], mixed derivatives commute; [F2] permits differentiation through the compact integral, so $\delta S=mc{\int }_a^{b}g(D_{s}J,T\mathrm{/}\mathrm{\ell })ds$.

Metric compatibility gives $\frac{d}{ds}g(J,T\mathrm{/}\mathrm{\ell })=g(D_{s}J,T\mathrm{/}\mathrm{\ell })+g(J,D_s(T\mathrm{/}\mathrm{\ell }))$. Integrating and applying [F3], the boundary term vanishes because $J(a)=J(b)=0$, yielding the displayed first-variation formula.

If $D_s(T\mathrm{/}\mathrm{\ell })=0$, step 3.1 makes every fixed-endpoint first variation vanish. Conversely, if this continuous vector field is nonzero at an interior point, [F5] gives a local chart component field $Z$ with $g(Z,D_s(T\mathrm{/}\mathrm{\ell }))$ strictly positive on a smaller interval. Multiply $Z$ by the nonnegative bump [F4], equal to $1$ near that point. Choose the support inside one chart interval and define $H(ϵ,s)={\chi }^{-1}(\chi (x(s))+ϵJ^{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}(s))$ there for a chart $\chi$, with $H=x$ off the support. The compact support makes this a $C^2$ fixed-endpoint variation; timelikeness persists for small $\mathrm{\mid }ϵ\mathrm{\mid }$ by openness and compactness. The pairing is bounded below by some $\delta >0$ on an interval where the bump equals $1$, so [F8] makes the integral in step 3.1 strictly positive and its prefactor $-mc<0$ makes the variation nonzero, a contradiction. Thus stationarity forces $D_s(T\mathrm{/}\mathrm{\ell })=0$ everywhere; a nonzero endpoint value would by continuity give a nonzero interior value.

Since $U=cT\mathrm{/}\mathrm{\ell }$ and $d\tau \mathrm{/}ds=\mathrm{\ell }\mathrm{/}c$, the chain rule gives $D_s(T\mathrm{/}\mathrm{\ell })=(\mathrm{\ell }\mathrm{/}{c}^2)D_{\tau }U$, so the two equations are equivalent. For an orientation-preserving $C^1$ diffeomorphism $s=\varphi (r)$, $T_r(r)={\varphi }^{\mathrm{\prime }}(r)T(\varphi (r))$ and ${\mathrm{\ell }}_r(r)={\varphi }^{\mathrm{\prime }}(r)\mathrm{\ell }(\varphi (r))$; [F7] therefore gives ${\int }_{a^{\mathrm{\prime }}}^{b^{\mathrm{\prime }}}{\mathrm{\ell }}_r(r)dr={\int }_a^b\mathrm{\ell }(s)ds$. If $\varphi$ is $C^2$, composition preserves the stated $C^2$ curve and variation classes, and $T_r\mathrm{/}{\mathrm{\ell }}_r=(T\mathrm{/}\mathrm{\ell })\circ \varphi$ gives $D_r(T_r\mathrm{/}{\mathrm{\ell }}_r)(r)={\varphi }^{\mathrm{\prime }}(r)D_s(T\mathrm{/}\mathrm{\ell })(\varphi (r))\mathrm{.}$ Thus the stationarity equation, and hence stationarity under the mapped fixed-endpoint variations, is covariant for class-preserving reparametrizations.