The Euclidean spatial structure and cross product of an inertial frame

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space and let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame (Inertial frames and inertial charts of the affine Minkowski model). Put

$S:=\mathrm{span}(e_1,e_2,e_3)=\{w\in V:\eta (e_0,w)=0\},$

the equality holding because a vector $w={\sum }_a{w}^a{e}_a$ satisfies $\eta (e_0,w)=-w^0$ by $\eta$-orthonormality, so $\eta (e_0,w)=0$ exactly when $w$ is a linear combination of $e_1,e_2,e_3$. In particular $S$ is the linear subspace of $V$ spanned by the three spatial basis vectors of the frame, and $V=S\oplus \mathrm{span}(e_0)$.

Let $\delta$ be the restriction of $\eta$ to $S$, i.e. $\delta (w,z):=\eta (w,z)$ for $w,z\in S$. Writing $w={\sum }_{i=1}^3{w}^i{e}_i$ and $z={\sum }_{i=1}^3{z}^i{e}_i$ and using $\eta (e_i,e_j)={\delta }_{ij}$ for $i,j\ge 1$, one has

$\delta (w,z)=w^1{z}^1+w^2{z}^2+w^3{z}^3,\delta (w,w)=(w^1{)}^2+(w^2{)}^2+(w^3{)}^2\mathrm{.}$

Hence $\delta$ is a symmetric bilinear form on $S$ that is positive definite: $\delta (w,w)\ge 0$, with equality exactly for $w=0$ by uniqueness of coordinates in the basis $(e_1,e_2,e_3)$ of $S$ (A finite list $v:n\to V$ is an ordered basis if and only if every $x\in V$ equals ${\sum }_{i<n}{\lambda }_i{v}_i$ for exactly one $\lambda :n\to F$; those scalars are the coordinates of $x$ in that ordered basis). Thus $\delta$ is an inner product on the real vector space $S$ (Real and complex inner product spaces, with the inner product linear in the first argument); the induced function

$\mid w\mid :=\sqrt{\delta (w,w)}=\parallel (w^1,w^2,w^3){\parallel }_2$

is the Euclidean norm of the coordinate vector of $w$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), and it is a norm on $S$ (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality). The ordered basis $(e_1,e_2,e_3)$ of $S$ is $\delta$-orthonormal, and it is declared positively oriented in $S$ by the convention that an ordered triple $(u_1,u_2,u_3)$ of elements of $S$ is positively oriented when the ordered basis $(e_0,u_1,u_2,u_3)$ of $V$ is positively oriented for the fixed orientation of $V$. The pair $(S,\delta )$, together with this orientation and the frame's basis, is the Euclidean spatial structure of the inertial frame $(o;e_0,e_1,e_2,e_3)$.

Vector product. For $w={\sum }_{i=1}^3{w}^i{e}_i$ and $z={\sum }_{i=1}^3{z}^i{e}_i$ in $S$, define the vector product $w\times z$ to be the unique element of $S$ whose coordinates are

$(w\times z{)}^1=w^2{z}^3-w^3{z}^2,(w\times z{)}^2=w^3{z}^1-w^1{z}^3,(w\times z{)}^3=w^1{z}^2-w^2{z}^1\mathrm{.}$

This is well defined by uniqueness of coordinates in the basis $(e_1,e_2,e_3)$. It is bilinear in $(w,z)$, alternating ($z\times w=-(w\times z)$ and $w\times w=0$), and it satisfies, for all $w,z,y\in S$,

$\delta (w\times z,y)=\det \left(\begin{array}{ccc}{w}^1& z^1& y^1\\ w^2& z^2& y^2\\ w^3& z^3& y^3\end{array}\right),\mid w\times z{\mid }^2=\mid w{\mid }^2\mid z{\mid }^2-\delta (w,z{)}^2\mathrm{.}$

The first identity is the expansion of the determinant along its columns, both sides equalling ${\sum }_i(w\times z{)}^i{y}^i$; the second is Lagrange's identity, obtained by expanding $\mid w\times z{\mid }^2+\delta (w,z{)}^2$ and comparing termwise with $\mid w{\mid }^2\mid z{\mid }^2$ using $\mid w{\mid }^2=\delta (w,w)$ (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix).

Frame dependence. The objects $S$, $\delta$, $\mid \cdot \mid$ and $\times$ belong to the spatial three-space of the named inertial frame: replacing the spatial basis by another $\delta$-orthonormal positively oriented basis of $S$ transforms coordinates by an orientation-preserving orthogonal matrix, under which $\delta$, the norm and the displayed identities are unchanged, while a general change of inertial frame with a different time axis $e_0$ changes $S$ itself and hence changes the domains of $\delta$ and $\times$. The Lorentz form, the fixed orientation of $V$, and the time orientation $C^{+}$ are data of the model independent of the named inertial frame (Oriented time-oriented affine Minkowski space (the spacetime primitives)). The Euclidean spatial structure instead requires the choice of a time axis $e_0$; it is not an additional frame-independent structure on the whole of $V$.