Linear and affine Lorentz maps, and the Lorentz group

Let $(V,\eta )$ be a $4$-dimensional Lorentz vector space (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space) equipped with a time orientation $C^{+}$ (Time orientation, future and past timelike/null/causal vectors, and the closed cones), and let $(A,V,\eta ,C^{+})$ be the corresponding oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)).

(a) Linear Lorentz maps. A linear Lorentz map is a linear map $L:V\to V$ (Linear map between vector spaces over the same field) such that

$\eta (Lv,Lw)=\eta (v,w)\text{for all }v,w\in V\mathrm{.}$

The set of all linear Lorentz maps is denoted $O(V,\eta )$.

(b) Matrix criterion. Let $(e_0,e_1,e_2,e_3)$ be an ordered $\eta$-orthonormal basis of $V$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)), so that ${\eta }_0:=\mathrm{diag}(-1,1,1,1)$ is the matrix of $\eta$ in that basis, and let $M$ be the matrix of $L$ in that basis (Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases). Then $L$ is a linear Lorentz map if and only if

$M^T{\eta }_{0}M={\eta }_0,$

where the product is matrix multiplication (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes). Every linear Lorentz map is a linear isomorphism of $V$ (Invertible linear maps, linear isomorphisms, and inverse linear maps), and the criterion itself does not depend on the $\eta$-orthonormal basis used. The three assertions of this clause are proved in Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups ↗.

(c) Proper and orthochronous maps; the Lorentz group. A linear Lorentz map $L$ is proper if $\det L=+1$ (The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space) and orthochronous if $L(C^{+})=C^{+}$, a condition on the image of the set $C^{+}$. The set of proper orthochronous linear Lorentz maps is denoted $S{O}^{+}(V,\eta )$. The set $O(V,\eta )$ is called the (full) Lorentz group of $(V,\eta )$ and $S{O}^{+}(V,\eta )$ the proper orthochronous Lorentz group; that these sets are groups under composition is proved in Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups ↗, which also shows that $L$ is orthochronous if and only if $L(C^{+})$ is contained in $C^{+}$.

(d) Affine Lorentz maps. An affine Lorentz map is a map $F:A\to A$ for which there exist $p_0\in A$ and a linear Lorentz map $L$ with

$F(p_0+v)=F(p_0)+Lv\text{for all }v\in V\mathrm{.}$

Equivalently $F(p)=F(p_0)+L(p-p_0)$ for all $p\in A$, and $L$ is then unique: it is recovered as $Lv=F(p+v)-F(p)$ for every $p\in A$, so it is independent of the base event. $F$ is called proper (respectively orthochronous) when its linear part $L$ is. If $x$ is the chart of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model), then

$x(F(p))=Mx(p)+b\text{for all }p\in A,$

where $M$ is the matrix of $L$ in the frame's ordered basis and $b:=x(F(o))-Mx(o)\in {\mathbb{R}}^4$; the two sides are column vectors in ${\mathbb{R}}^4$.

(e) Convention. In this framework, unless stated otherwise, the term Lorentz transformation means a proper orthochronous affine Lorentz map. Improper or non-orthochronous Lorentz maps are always named explicitly.

The composition of two affine Lorentz maps is affine Lorentz with linear part the composition of the linear parts, and the identity and the inverse of an affine Lorentz map are again such maps; these facts are part of Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups ↗.