The flat covariant derivative on affine Minkowski space and fields along curves

Let $(A,V,\eta ,C^{+})$ be the affine Minkowski space with tangent spaces canonically identified with $V$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), The tangent space of the affine Minkowski model at any event is canonically V), and let $\mathrm{\Omega }\subseteq A$ be a nonempty open subset. All tensor fields below are $C^k$ fields in the sense of C^k tensor fields on affine Minkowski space and their inertial components, with $k=\mathrm{\infty }$ unless a clause says otherwise.

(a) Vector fields. For smooth vector fields $X,Y$ on $\mathrm{\Omega }$ define the covariant derivative of $Y$ in the direction $X$ at $p\in \mathrm{\Omega }$ by

$\left({\mathrm{\nabla }}_{X}Y\right)(p):=\frac{d}{dt}{\mid }_{t=0}Y(p+tX(p)),$

the derivative of the $V$-valued composite $t\mapsto Y(p+tX(p))$, which is defined for $t$ in a neighbourhood of $0$ because $\mathrm{\Omega }$ is open (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). In the inertial chart $x$ of any inertial frame one has $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$; the value does not depend on the inertial chart: for another inertial chart $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+b$ one has ${\mathrm{\partial }}_a^{\mathrm{\prime }}=({\mathrm{\Lambda }}^{-1}{)}_a^b{\mathrm{\partial }}_b$ and $Y^{\mathrm{\prime }a}={\mathrm{\Lambda }}_b^a{Y}^b$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps, Inertial frames and inertial charts of the affine Minkowski model), and $\mathrm{\Lambda }$ is constant, so $({\mathrm{\Lambda }}^{-1}{)}_c^a{X}^{\mathrm{\prime }b}{\mathrm{\partial }}_b^{\mathrm{\prime }}{Y}^{\mathrm{\prime }c}=X^b{\mathrm{\partial }}_b{Y}^a$. The field ${\mathrm{\nabla }}_{X}Y$ is again a smooth vector field on $\mathrm{\Omega }$.

(b) Covector and general tensor fields. For a smooth covector field $\alpha$ set $({\mathrm{\nabla }}_X\alpha )(Y):=X(\alpha (Y))-\alpha ({\mathrm{\nabla }}_{X}Y)$ for every smooth vector field $Y$, and for a type $(r,s)$ tensor field $T$ define ${\mathrm{\nabla }}_{X}T$ by the derivation rule on the evaluation of $T$: for $T:(V^{\ast }{)}^r\times V^s\to \mathbb{R}$,

$({\mathrm{\nabla }}_{X}T)({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)=X[T({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)]-{\sum }_{i=1}^{r}T({\alpha }_1,\dots ,{\mathrm{\nabla }}_X{\alpha }_i,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)-{\sum }_{j=1}^{s}T({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,{\mathrm{\nabla }}_X{Y}_j,\dots ,Y_s),$

where the vector and covector arguments are smooth fields on $\mathrm{\Omega }$. In inertial coordinates every component of ${\mathrm{\nabla }}_{X}T$ is $X^b{\mathrm{\partial }}_b$ applied to that component, with no connection coefficients.

(c) Properties. The operator $\mathrm{\nabla }$ is $\mathbb{R}$-bilinear, is $C^{\mathrm{\infty }}(\mathrm{\Omega })$-linear in $X$, and satisfies the Leibniz rules ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$ and the analogous rules for tensor fields; it is metric compatible, $\mathrm{\nabla }\eta =0$, because ${\mathrm{\partial }}_a{\eta }_{bc}=0$ in every inertial chart; it is torsion-free, ${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]$, by the coordinate formula for the Lie bracket (The Lie bracket of smooth vector fields, Coordinate formula for the Lie bracket); and it has zero curvature, ${\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z=0$, by equality of mixed partial derivatives of the components (Young's theorem: total differentiability of the first partials forces equality of mixed partials).

(d) Fields along a curve. If $\gamma :I\to A$ is a $C^1$ curve and $W:I\to V$ is a $C^1$ field along it (Vector field and section along a smooth curve), the covariant derivative of $W$ along $\gamma$ is the ordinary $V$-valued derivative $DW\mathrm{/}ds:=W^{\mathrm{\prime }}$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral), because the canonical identification $T_{p}A=V$ is independent of $p$; consequently parallel transport along any curve is the identity and a field is parallel exactly when it is constant. This derivative is the one used for the acceleration of congruences, for Fermi–Walker transport and for the kinematic decomposition below; the model needs no Christoffel symbols and no parallel-transport ODE in this formulation.

(e) First-order $C^1$ version. For $C^1$ vector fields $X,Y$ and $C^1$ covector or tensor fields on $\mathrm{\Omega }$ the same directional and evaluation definitions are well-defined and have continuous coefficients: in an inertial chart $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$, and each tensor component is differentiated in the same way. The component recovery and the subtraction of all argument derivatives remain valid; $C^1$ scalar multiplication satisfies the same first-order Leibniz rules; metric compatibility remains $\mathrm{\nabla }\eta =0$; and the first-order torsion identity uses the continuous coordinate bracket $[X,Y{]}^a=X^b{\mathrm{\partial }}_b{Y}^a-Y^b{\mathrm{\partial }}_b{X}^a$. No $C^1$ closure of this bracket and no second covariant derivative of arbitrary $C^1$ data are asserted. For $C^2$ vector fields the second derivatives needed for the zero-curvature identity of (c) exist, and that identity follows by equality of the continuous mixed second partials. All smooth-field properties above and the $C^1$ along-curve derivative in (d) are retained.

• Well-definedness of (a). The composite $t\mapsto Y(p+tX(p))$ is the composition of the smooth curve $t\mapsto p+tX(p)$ (which stays in $\mathrm{\Omega }$ for small $\mathrm{\mid }t\mathrm{\mid }$ because $\mathrm{\Omega }$ is open and $X(p)\in V$ is a fixed vector) with the smooth map $Y:\mathrm{\Omega }\to V$; its derivative at $0$ exists in the finite-dimensional real vector space $V$ by The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, and the coordinate expression $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$ is the chain rule of The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$. Chart independence uses the transformation laws of Transformation of tensor, covector, derivative and current components under affine Lorentz maps together with the constancy of the linear part $\mathrm{\Lambda }$ of an inertial transition (Inertial frames and inertial charts of the affine Minkowski model): the second derivatives of the transition vanish, so no inhomogeneous term appears. Smoothness of ${\mathrm{\nabla }}_{X}Y$ follows because $X^b{\mathrm{\partial }}_b{Y}^a$ is a smooth function of $x(p)$ in an inertial chart, and $C^k$ tensor fields are exactly those with $C^k$ inertial components (C^k tensor fields on affine Minkowski space and their inertial components).

• Well-definedness of (b). In an inertial chart use the constant coordinate frame and its constant dual frame; their covariant derivatives vanish, so evaluating the derivation rule on those constant bases leaves only $X$ acting on each component of $T$. For variable arguments ${\alpha }_1,\dots ,Y_s$ the ordinary product rule expressed by Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$ produces exactly the subtracted terms displayed in (b), and all their derivatives cancel; this proves the stated coordinate formula and shows that the evaluation rule is well-defined, once the covector case $({\mathrm{\nabla }}_X\alpha )(Y)=X(\alpha (Y))-\alpha ({\mathrm{\nabla }}_{X}Y)$ is fixed by the requirement that ${\mathrm{\nabla }}_X$ commute with the pairing. No connection coefficients appear because $\eta$ is constant and the inertial transition matrices are constant.

• Well-definedness of (c). Torsion-freeness is the coordinate identity $X^b{\mathrm{\partial }}_b{Y}^a-Y^b{\mathrm{\partial }}_b{X}^a=[X,Y{]}^a$ (Coordinate formula for the Lie bracket); metric compatibility is the constancy of the component matrix ${\eta }_{ab}$ of the metric in inertial charts; and the curvature identity follows from the equality of mixed second partial derivatives (Young's theorem: total differentiability of the first partials forces equality of mixed partials) after expanding the two iterated derivatives of the components. The Leibniz and linearity properties are immediate from the definitions.

• Well-definedness of (d) and (e). In (d) the canonical isomorphism $T_{p}A\to V$ is independent of $p$ (The tangent space of the affine Minkowski model at any event is canonically V), so a field along a curve is simply a $V$-valued map and its covariant derivative is its ordinary derivative; parallel transport is the identity because the equation $W^{\mathrm{\prime }}=0$ has exactly the constant solutions on an interval: each real component of $W$ is constant by A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant, and a constant vector-valued map has derivative zero. In (e) a $C^1$ field $Y$ has a continuous differential, and the derivative of $t\mapsto Y(p+tX(p))$ at $0$ is $d{Y}_p[X(p)]$, which is a continuous function of $p$ when $Y$ and $X$ are $C^1$; only first derivatives occur in the Leibniz, metric-compatibility and torsion assertions, and the second covariant derivatives in the curvature identity require the stated $C^2$ hypotheses. No choice principle is used anywhere; all objects are finite-dimensional or explicitly parametrized.