Covariant acceleration along a $C^2$ curve

Let $M$ be a smooth manifold with an affine connection $\mathrm{\nabla }$, let $I\subseteq \mathbb{R}$ be any interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $z:I\to M$ be continuous. We say that $z$ is $C^2$ on $I$ in this definition when the following chartwise conditions hold. At each interior parameter value ${\tau }_0\in \mathrm{int}I$, there are a chart $x:U\to x(U)\subseteq {\mathbb{R}}^n$ around $z({\tau }_0)$ and an open interval $J\subseteq \mathrm{int}I$ containing ${\tau }_0$, with $z(J)\subseteq U$, such that $x\circ z$ is twice continuously differentiable on $J$. At each included endpoint ${\tau }_0\in I\setminus \mathrm{int}I$, there are such a chart and a relative neighborhood $J=I\cap ({\tau }_0-\delta ,{\tau }_0+\delta )$, with $\delta >0$ and $z(J)\subseteq U$, for which the first and second derivatives on $J\cap \mathrm{int}I$ extend continuously to ${\tau }_0$; their endpoint values are the corresponding one-sided derivatives. Here $n=\dim M$, and differentiability of $x\circ z$ is componentwise as a map of one real variable. Smooth chart transitions and $C^2$ composition make this condition independent of the chosen chart ($C^r$ and smooth maps between smooth manifolds, $C^k$ Euclidean maps are closed under componentwise algebra and composition); at an included endpoint, the first- and second-derivative chain-rule expressions have the corresponding continuous one-sided limits. On a time-oriented Lorentzian four-manifold, this is exactly the curve regularity specified in clause (a) of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold; that definition's causal conditions are not part of the present regularity convention.

The parameter $\tau$ is an arbitrary real parameter here; no causal or nonzero-velocity condition is imposed. Put $U(\tau ):=z^{\mathrm{\prime }}(\tau )$, using the one-sided derivative at included endpoints. A vector field along $z$ assigns $V(\tau )\in T_{z(\tau )}M$ to each $\tau \in I$; it is continuous when its component functions are continuous in every chart along $z$. In a chart $x=(x^1,\dots ,x^n)$ covering a segment of $z$, write $U^a(\tau ):=\frac{d}{d\tau }(x^a\circ z)(\tau )$ and let ${\mathrm{\Gamma }}^a{}_{bc}$ be the Christoffel symbols of $\mathrm{\nabla }$. The components $U^a$ transform by the tangent-map Jacobian on chart overlaps, by the chain rule, so $U$ is a vector field along $z$. All indices range from $1$ to $n$, and repeated upper/lower indices are summed. Define the covariant acceleration of $z$ on this segment by the component formula

$({\mathrm{\nabla }}_{U}U{)}^a(\tau ):=\frac{d{U}^a}{d\tau }(\tau )+{\mathrm{\Gamma }}^a{}_{bc}(z(\tau ))U^b(\tau )U^c(\tau )\mathrm{.}$

The local $C^2$ regularity makes $U^a$ continuously differentiable, including one-sided endpoint derivatives, so these components are continuous on the relative parameter interval. The name and notation extend the pullback covariant derivative of Covariant derivative along a curve from smooth curves; the coordinate expression is well-defined on overlaps by The covariant acceleration formula is well-defined along a $C^2$ curve ↗.