Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with inertial chart $x:A\to {\mathbb{R}}^4$ (Inertial frames and inertial charts of the affine Minkowski model), let $c>0$ be the framework's conversion constant, and let $\delta$ be the Euclidean spatial form of that frame on $n^{\perp }$, $n:=e_0$, with norm $\mathrm{\mid }\cdot \mathrm{\mid }$ and positive spatial orientation (The Euclidean spatial structure and cross product of an inertial frame). Fix a nonempty open interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and a nonempty open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, and put

$U:=\{p\in A:\text{ there are }t\in I,y\in \mathrm{\Omega }\text{ with }{x}^0(p)=ct,(x^1,x^2,x^3)(p)=y\}\mathrm{.}$

Then:

(a) [Chart restriction] $x(U)=\{(ct,y):t\in I,y\in \mathrm{\Omega }\}$ and $\mathrm{\Phi }:U\to I\times \mathrm{\Omega }$, $\mathrm{\Phi }(p):=(x^0(p)\mathrm{/}c,(x^1,x^2,x^3)(p))$, is a bijection, because the inertial chart $x:A\to {\mathbb{R}}^4$ is a bijection and $c>0$; its components are the laboratory time coordinate $t(p):=x^0(p)\mathrm{/}c$ and the laboratory position $r(p):=(x^1(p),x^2(p),x^3(p))$. In the chart coordinates the pair $(t(p),r(p))$ is the reading of the laboratory decomposition of $p$. Writing $r(p)=(r^1(p),r^2(p),r^3(p))\in {\mathbb{R}}^3$, the standard basis of ${\mathbb{R}}^3$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$) labels the three coordinate directions, and the identification ${\mathbb{R}}^3\cong n^{\perp }$ is the linear map ${\sum }_{i=1}^3{r}^i{e}_i$ into the three-dimensional real vector space $n^{\perp }$ (Vector space over a field, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

(b) [Slices and spatial structure] For each $t\in I$ the slice $U_t:=\{p\in U:t(p)=t\}$ is carried bijectively onto $\mathrm{\Omega }$ by $r$; the constant vector $n=e_0$ is future unit timelike, and $(e_1,e_2,e_3)$ is the $\delta$-orthonormal positively oriented basis of $n^{\perp }$ in which the coordinates of $r$ are expressed.

(c) [Laboratory fields] For a nonnegative integer $m$, a laboratory field of rank $m$ is a map $F:\mathrm{\Omega }\times I\to {\mathbb{R}}^m$; it is identified with the map $U\to {\mathbb{R}}^m$, $p\mapsto F(r(p),t(p))$, and with the family $(F(\cdot ,t){)}_{t\in I}$ of maps $\mathrm{\Omega }\to {\mathbb{R}}^m$, its values at fixed time. Such a field is of class $C^k$, for $m\ge 1$, when the map $(y,t)\mapsto F(y,t)$ is $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. For $m=0$, set ${\mathbb{R}}^0:=\{()\}$; the unique field into this singleton is declared $C^{\mathrm{\infty }}$ (and $C^k$ for every finite $k$), and each of its partial derivatives is the same singleton-valued zero map. For $m\ge 1$ and $k\ge 1$, ${\mathrm{\partial }}_{y_i}F$ (at fixed $t$) and ${\mathrm{\partial }}_{t}F$ (at fixed $y$) are the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. No regularity of a field is asserted beyond the class stated for it.

(d) [Conventions] The product decomposition is a coordinate identification inside the single inertial frame $(o;e_0,e_1,e_2,e_3)$; it is not a Galilean transformation, it does not compare two inertial frames, and it does not assert absolute simultaneity. The objects $I,\mathrm{\Omega },t,r,\delta$ and the identification ${\mathbb{R}}^3\cong n^{\perp }$ belong to the named frame; changes of inertial description are the proper orthochronous maps adopted in Adopted inertial covariance of the model equations. The frame is an origin together with an ordered basis; it is not a congruence of observers, and no moving observer field is defined here.