Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed

(Component form of $D_t$.) On a chart domain around a parameter value, let $e_a:={\mathrm{\partial }}_a\circ \gamma$ be the pulled-back coordinate frame and write ${\gamma }^{\mathrm{\prime }}={\sum }_b{\dot{\gamma }}^b{e}_b$, $V={\sum }_a{V}^a{e}_a$, with coefficient functions $V^a$ on the parameter interval. By [L1], $D_t{e}_c=(\mathrm{\nabla }{\mathrm{\partial }}_c{)}_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t))={\sum }_{a,b}{\mathrm{\Gamma }}^a{}_{bc}{\dot{\gamma }}^b{e}_a$. The pullback product rule therefore gives $D_{t}V={\sum }_a({\mathrm{\partial }}_t{V}^a+{\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}{\dot{\gamma }}^b{V}^c)e_a$. This uses ambient extensions only for the coordinate frame, not for the arbitrary coefficient functions along the curve, and hence also applies to repeated points and constant base curves. The same formula and pairing calculation hold for $C^1$ fields along a $C^1$ curve, with the derivatives interpreted at included endpoints one-sided.

(Metric pairing.) With the notation of step 1.1, $g(V,W)={\sum }_{a,b}{g}_{ab}{V}^a{W}^b$, so $\frac{d}{dt}g(V,W)={\sum }_{a,b}({\mathrm{\partial }}_c{g}_{ab}{\dot{\gamma }}^c{V}^a{W}^b+g_{ab}{\mathrm{\partial }}_t{V}^a{W}^b+g_{ab}{V}^a{\mathrm{\partial }}_t{W}^b)$. Substituting the compatibility identity of [L2] for ${\mathrm{\partial }}_c{g}_{ab}$ and the component formula of step 1.1 for $D_{t}V$ and $D_{t}W$, the terms in which $\mathrm{\Gamma }$ multiplies $V$ or $W$ match, using symmetry $g_{ab}=g_{ba}$ and the relabelling of the summed indices $b,c,d$; the remaining terms are exactly the two derivatives ${\mathrm{\partial }}_t{V}^a{W}^b{g}_{ab}$ and $V^a{\mathrm{\partial }}_t{W}^b{g}_{ab}$. Hence the displayed pairing identity holds; if $D_{t}V=D_{t}W=0$, the derivative of the smooth function $g(V,W)$ vanishes on each chart, hence on $I$; [L7] gives constancy.

(Parallel transport.) On a singleton, [L4] and the transport definition give the unique prescribed vector and identity map, which preserves the metric. For $a<b$, let $\gamma :[a,b]\to M$ be piecewise smooth and $v\in T_{\gamma (a)}M$. The existence and uniqueness of a continuous piecewise parallel section through $v$ is [L4]; the map $P_{\gamma }$ is therefore well defined, and it is linear because the difference of two parallel sections with initial values $v,w$ is a parallel section (linearity of $D_t$ from step 1.1) with initial value $v-w$. It is bijective with inverse given by the same construction for the reversed curve. For $v,w$ apply claim 1 to their parallel sections $V,W$ on each smooth piece: $g(V,W)$ has zero derivative there and is continuous across corners, hence constant on $[a,b]$ by [L7], so $g(P_{\gamma }v,P_{\gamma }w)=g(v,w)$.

(Constant speed and causal character.) Let $\gamma$ be a geodesic, so $D_s{\gamma }^{\mathrm{\prime }}=0$ by [L3]. Applying the pairing identity of step 2.1 with $V=W={\gamma }^{\mathrm{\prime }}$ gives $\frac{d}{ds}g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=2g(D_s{\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=0$ on every chart of the domain; hence $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})$ is a constant by [L7]. By [L4], the tangent ${\gamma }^{\mathrm{\prime }}$ is the parallel section with its value prescribed at any one parameter, and parallel transport is bijective. Thus if ${\gamma }^{\mathrm{\prime }}$ vanishes once it vanishes everywhere, so $\gamma$ is constant; otherwise it is nowhere zero. In the latter case its constant squared norm fixes its timelike, null or spacelike character on all of $I$. If it is causal, [L6] makes its continuous pairing with $N$ nowhere zero, so its time orientation is also constant, including the null case.

(Timelike geodesics and proper time.) Let $\gamma$ be a future-directed timelike geodesic with $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=-{\lambda }^2$, $\lambda >0$. Its proper-time density is $\sqrt{{\lambda }^2}\mathrm{/}c=\lambda \mathrm{/}c$, constant, so for $t_0<t_1$ the increment over $[t_0,t_1]$ is $\lambda (t_1-t_0)\mathrm{/}c$ by [L5]; hence proper time is an affine function of the parameter and an affine parameter of $\gamma$. The four-velocity is $U=d\gamma \mathrm{/}d\tau =(c\mathrm{/}\lambda ){\gamma }^{\mathrm{\prime }}$; since $D_t{\gamma }^{\mathrm{\prime }}=0$ and $d\tau \mathrm{/}dt=\lambda \mathrm{/}c$ is constant, $D_{\tau }U=(c\mathrm{/}\lambda )D_t((c\mathrm{/}\lambda ){\gamma }^{\mathrm{\prime }})=0$. A nonconstant null geodesic has nonzero null tangent by step 3.2, so the integrand in [L5] is zero and the curve is not a future timelike worldline; it admits no proper-time parametrization.

(Affine reparametrizations.) Let $\gamma :I\to M$ have nowhere-zero tangent and be a geodesic; let $h:J\to I$ be a $C^2$ diffeomorphism onto its open image with $h^{\mathrm{\prime }}\mathrm{\ne }0$, and put $\tilde{\gamma }=\gamma \circ h$. In a chart and on each parameter interval where $h$ and $\gamma$ are taken on a fixed chart, $\tilde{\gamma }{}^{\mathrm{\prime }}=h^{\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ h)$, so by step 1.1 $D_s\tilde{\gamma }{}^{\mathrm{\prime }}=h^{\mathrm{\prime }\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ h)+(h^{\mathrm{\prime }}{)}^2(D_t{\gamma }^{\mathrm{\prime }})\circ h=h^{\mathrm{\prime }\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ h),$ where the last equality is the geodesic equation for $\gamma$ and the two terms arise exactly as in the product rule $D_s(h^{\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ h))=h^{\mathrm{\prime }\mathrm{\prime }}({\gamma }^{\mathrm{\prime }}\circ h)+h^{\mathrm{\prime }}{D}_s({\gamma }^{\mathrm{\prime }}\circ h)$ with $D_s({\gamma }^{\mathrm{\prime }}\circ h)=h^{\mathrm{\prime }}(D_t{\gamma }^{\mathrm{\prime }})\circ h$. Since ${\gamma }^{\mathrm{\prime }}\circ h$ is nowhere zero, $\tilde{\gamma }$ is a geodesic if and only if $h^{\mathrm{\prime }\mathrm{\prime }}\equiv 0$ on $J$, that is $h(s)=as+b$ with constants $a=h^{\mathrm{\prime }}\mathrm{\ne }0$ and $b$. If $\gamma$ is constant, then $\tilde{\gamma }$ is constant and geodesic for every $h$ by [L3]. Conversely, every $C^2$ diffeomorphism with $h^{\mathrm{\prime }\mathrm{\prime }}=0$ has this affine form, and for a nonconstant geodesic starting from a fixed affine parameter $t$, every other affine parameter $s$ is related by $s=at+b$ with $a\mathrm{\ne }0$ by applying the criterion to $h=s\circ t^{-1}$.