Affine time space with Euclidean simultaneous slices and adapted charts

Let $V$ be a real vector space of dimension $4$ (Vector space over a field) and let $\tau :V\to \mathbb{R}$ be a surjective linear map (Linear map between vector spaces over the same field). Put

$K:=\ker \tau =\{v\in V:\tau (v)=0\},$

the kernel of $\tau$ (Kernel and image of a linear map). The kernel is a linear subspace (The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial), so the quotient vector space $V\mathrm{/}K$ is defined (The quotient vector space $V\mathrm{/}W$ and its canonical projection). The first isomorphism theorem gives the linear isomorphism $V\mathrm{/}K\to \mathrm{im}\tau =\mathbb{R}$, $v+K\mapsto \tau (v)$ (First isomorphism theorem for vector spaces: $V\mathrm{/}\ker T$ is isomorphic to $\mathrm{im}T$). Since $\mathbb{R}$ has the one-element standard basis, $\dim \mathbb{R}=1$ (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 $0$ with $F=\mathbb{R}$ and $n=1$); rank-nullity for the surjective map $\tau$ therefore gives $\dim K=4-1=3$ (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$).

Let $g$ be an inner product on $K$ in the sense of Real and complex inner product spaces, with the inner product linear in the first argument: a symmetric positive definite $\mathbb{R}$-bilinear form $g:K\times K\to \mathbb{R}$.

Let $A$ be a nonempty set equipped with an action of the additive group of $V$, written $(p,v)\mapsto p+v$, that is free and transitive: explicitly, $(p+v)+w=p+(v+w)$ and $p+0=p$ for all $p\in A$ and $v,w\in V$, and for all $p,q\in A$ there is exactly one $v\in V$ with $p+v=q$; write $q-p$ for that unique vector. The identities $p+(q-p)=q$, $(q-p)+(r-q)=r-p$ and $q-p=-(p-q)$ follow from the axioms together with uniqueness of the displacement. Such a set is an affine space modelled on $V$, and its elements are events. (This formulation restates the affine-space structure used here so that the item is self-contained; the general theory of affine spaces is supplied by the special-relativity collection for its own consumers.)

Fix an orientation of $K$ in the sense of Orientation of a finite-dimensional real vector space; this is a chosen orientation class, not a chosen spatial basis.

An adapted chart for the data $(V,\tau ,g,A)$ is a triple $(o,u,(e_1,e_2,e_3))$ consisting of

1. an origin $o\in A$;
2. a vector $u\in V$ with $\tau (u)=1$;
3. an ordered basis $(e_1,e_2,e_3)$ of $K$ that is $g$-orthonormal and positively oriented with respect to the fixed orientation of $K$ (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases, Orientation of a finite-dimensional real vector space).

An adapted chart exists for every such set of data and fixed orientation, by clause 3 of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices ↗.

For spatial coordinate vectors $x\in {\mathbb{R}}^3$, this item uses the one-based labels $x_i:=x(i-1)$ for $1\le i\le 3$; thus $(x_1,x_2,x_3)$ lists the values at $0,1,2$ of the function-space vector of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. The labels $e_1,e_2,e_3$ name the corresponding ordered spatial basis; they do not change the underlying index set of ${\mathbb{R}}^3$.

The adapted coordinates of a point $p\in A$ in this chart are the pair $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$ determined by

$p=o+tu+{\sum }_{i=1}^3{x}_i{e}_i,$

when such a pair exists (existence and uniqueness are Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices ↗, so that the notation is defined for every $p$). The number $t$ is the time coordinate and $x=(x_1,x_2,x_3)$ the spatial coordinate vector of $p$ in the chart. When $K$ is identified with ${\mathbb{R}}^3$ through the coordinates in $(e_1,e_2,e_3)$, the assignment $p\mapsto (t,x)$ is a bijection $A\to \mathbb{R}\times {\mathbb{R}}^3$.

Simultaneous slices. For $t\in \mathbb{R}$ set

$A_t:=\{p\in A:\tau (p-o)=t\}\mathrm{.}$

The family $\{A_t{\}}_{t\in \mathbb{R}}$ is the family of simultaneous slices of the chart. Each $A_t$ is an affine space modelled on $K$; the derivation is immediate from the affine identities and linearity of $\tau$ and is recorded in Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices ↗. The slice distance of two points $p,q$ in a common slice $A_t$ is

$d_g(p,q):=\sqrt{g(p-q,p-q)},$

the square root taken in $\mathbb{R}$; this is defined because $p-q\in K$ for points of a common slice. In adapted coordinates $d_g(p,q)=\mathrm{\parallel }x(p)-x(q){\mathrm{\parallel }}_2$, with $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

• The data $(V,\tau ,g,A)$ contain no inner product on $V$ and no identification of $V$ with ${\mathbb{R}}^4$: $g$ is defined only on $K$. Consequently a general displacement $v\in V$ with $\tau (v)\mathrm{\ne }0$ has no $g$-norm defined in this structure, and "the length of a temporal displacement" is not a quantity of the model. Only displacements inside a common slice carry a length, and those are exactly the elements of $K$.

• The time coordinate of an adapted chart is fixed by $\tau$ and the origin alone: if $p=o+tu+k$ with $k\in K$, then $\tau (p-o)=t$, so $t$ is independent of $u$ and of the spatial basis. The remaining freedom in the chart (choice of $o$, $u$, oriented orthonormal basis of $K$) is what produces the transformation law of Transformation between adapted charts and its kinematic consequences. The phrase "ten parameters" for the passage between two such charts refers to the dimensions of the transformation data $(b,a,v,R)$ and is not asserted here as a Lie-group dimension count.

• This is a mathematical model. No postulate that physical space and time are described by such data, that adapted charts are inertial frames, or that $g$ is measured by any apparatus is part of this definition; physical interpretations and their empirical qualifications are supplied by the physical items that consume it, with their own hypotheses.