C^k tensor fields on affine Minkowski space and their inertial components

Let $(A,V,\eta ,C^{+})$ be the affine Minkowski space of Oriented time-oriented affine Minkowski space (the spacetime primitives), with its inertial smooth structure (The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4, Smooth manifolds and their smooth charts). Let $\mathrm{\Omega }\subseteq A$ be open, let $r,s\in \mathbb{N}$, and let $k\in \mathbb{N}\cup \{\mathrm{\infty }\}$. Empty $\mathrm{\Omega }$ is allowed; it has the unique empty field of each type.

Write $T^{r,s}(V)$ for the real vector space, under pointwise operations, of multilinear maps $(V^{\ast }{)}^r\times V^s\to \mathbb{R}$ (A type $(r,s)$ tensor on a finite-dimensional vector space, Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). Identify $T_{p}A$ with $V$ by The tangent space of the affine Minkowski model at any event is canonically V. For an inertial frame $(o;e_0,e_1,e_2,e_3)$ with chart $x$, let $(e^{\ast a})$ be its dual basis, defined by $e^{\ast a}(e_b)={\delta }_b^a$ (The dual family $(b^{\ast }{)}_{b\in B}$ associated to a Hamel basis $B$, defined by $b^{\ast }(c)={\delta }_{bc}$). The components of a map $T:\mathrm{\Omega }\to T^{r,s}(V)$ are

$T^{a_1\dots a_r}{}_{b_1\dots b_s}(p):=T(p)(e^{\ast a_1},\dots ,e^{\ast a_r},e_{b_1},\dots ,e_{b_s})\mathrm{.}$

A ${\mathcal{C}}^k$ type-$(r,s)$ tensor field on $\mathrm{\Omega }$ is such a map for which each component, regarded as a function of $u=x(p)$ on the open set $x(\mathrm{\Omega })\subseteq {\mathbb{R}}^4$, is of class $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms; for $k=\mathrm{\infty }$ all finite orders are required.

Finite-dimensional construction and intrinsic regularity. Put $N=4^{r+s}\ge 1$, with the component array ordered lexicographically. Every array of $N$ real numbers defines exactly one tensor: expand each input as ${\alpha }_i={\sum }_a{\alpha }_i(e_a)e^{\ast a}$ and $v_j={\sum }_b{e}^{\ast b}(v_j)e_b$, then sum the array entries multiplied by ${\prod }_i{\alpha }_i(e_{a_i}){\prod }_j{e}^{\ast b_j}(v_j)$. This is multilinear and recovers the array by evaluation on the indicated basis inputs; multilinearity forces the same formula for any tensor. Thus the component map is a linear isomorphism $T^{r,s}(V)\to {\mathbb{R}}^N$. Pulling back the Euclidean norm gives a norm on this space (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms), and every norm makes it Banach by Every finite-dimensional normed space is Banach. Any two norms are equivalent by transporting them to ${\mathbb{R}}^N$ and applying For $n\ge 1$ all norms on ${\mathbb{R}}^n$ are equivalent.

A change of inertial frame expresses each new basis and dual-basis vector as a constant finite linear combination of the old ones; multilinearity therefore makes the new component array an invertible constant linear transformation of the old array. The coordinate precomposition is an invertible affine map by The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4. Finite linear combinations and affine precomposition preserve $C^k$: each first derivative follows from The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, and successive derivatives repeat that rule with constant affine coefficients. Applying the inverse transformations proves the converse. Hence the definition is independent of frame. Continuous partials give total derivatives by If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, and induction on order gives the equivalent finite-dimensional Banach-space $C^k$ definition of C k map between Banach spaces for $T\circ x^{-1}$. The norm equivalence inequalities preserve the derivative remainder limits and continuity of each iterated derivative, so this formulation is also independent of norm. No definition of a Banach-space map with manifold domain is presumed.

A type-$(0,0)$ field is a scalar field, a $C^k$ real-valued function. A type-$(1,0)$ field is a vector field, with the vector $v$ identified with $\alpha \mapsto \alpha (v)$: conversely a linear functional $Q$ on $V^{\ast }$ has $v={\sum }_{a}Q(e^{\ast a})e_a$ and $Q(\alpha )=\alpha (v)$. A type-$(0,1)$ field is a covector field, taking values in $V^{\ast }$.