Editing Lorentz transformation (section)

===Homogeneous Lorentz group===

Writing the coordinates in column vectors and the [[Minkowski metric]] {{mvar|η}} as a square matrix
<math display="block"> X' = \begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} \,, \quad \eta = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \,, \quad X = \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}  </math>
the spacetime interval takes the form (superscript {{math|T}} denotes [[transpose]])
<math display="block"> X \cdot X = X^\mathrm{T} \eta X = {X'}^\mathrm{T} \eta {X'} </math>
and is [[Invariant (physics)|invariant]] under a Lorentz transformation
<math display="block">X' = \Lambda X  </math>
where {{math|Λ}} is a square matrix which can depend on parameters.

The [[set (mathematics)|set]] of all Lorentz transformations <math>\Lambda  </math> in this article is denoted <math>\mathcal{L}</math>. This set together with matrix multiplication forms a [[group (mathematics)|group]], in this context known as the ''[[Lorentz group]]''. Also, the above expression {{math|''X''·''X''}} is a [[quadratic form]] of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the [[indefinite orthogonal group]] O(3,1), a [[Lie group]]. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are [[matrix Lie group]]s. In this context the operation of composition amounts to [[matrix multiplication]].

From the invariance of the spacetime interval it follows
<math display="block">\eta = \Lambda^\mathrm{T} \eta \Lambda </math>
and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the [[determinant]] of the equation using the product rule<ref group=nb>For two square matrices {{mvar|A}} and {{mvar|B}}, {{math|1=det(''AB'') = det(''A'')det(''B'')}}</ref> gives immediately
<math display="block">\left[\det (\Lambda)\right]^2 = 1 \quad \Rightarrow \quad \det(\Lambda) = \pm 1 </math>

Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
<math display="block">\eta = \begin{bmatrix}-1 & 0 \\ 0 & \mathbf{I}\end{bmatrix} \,, \quad \Lambda=\begin{bmatrix}\Gamma & -\mathbf{a}^\mathrm{T}\\-\mathbf{b} & \mathbf{M}\end{bmatrix} \,, </math>
carrying out the block matrix multiplications obtains general conditions on {{math|Γ, '''a''', '''b''', '''M'''}} to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results
<math display="block">\Gamma^2 = 1 + \mathbf{b}^\mathrm{T}\mathbf{b}</math>
is useful; {{math|'''b'''{{sup|T}}'''b''' ≥ 0}} always so it follows that
<math display="block"> \Gamma^2 \geq 1 \quad \Rightarrow \quad \Gamma \leq - 1 \,,\quad \Gamma \geq  1 </math>

The negative inequality may be unexpected, because {{math|Γ}} multiplies the time coordinate and this has an effect on [[Time translation symmetry|time symmetry]]. If the positive equality holds, then {{math|Γ}} is the Lorentz factor.

The determinant and inequality provide four ways to classify '''L'''orentz '''T'''ransformations (''herein '''LT'''s for brevity''). Any particular LT has only one determinant sign ''and'' only one inequality. There are four sets which include every possible pair given by the [[Intersection (set theory)|intersection]]s ("n"-shaped symbol meaning "and") of these classifying sets.

{| class="wikitable"
|-
! Intersection, ∩
! '''Antichronous''' (or non-orthochronous) LTs

:<math> \mathcal{L}^\downarrow = \{ \Lambda  \, : \, \Gamma \leq -1 \} </math>
! '''Orthochronous''' LTs

:<math> \mathcal{L}^\uparrow = \{ \Lambda  \, : \, \Gamma \geq 1 \} </math>
|-
! '''Proper''' LTs

:<math> \mathcal{L}_{+} = \{ \Lambda  \, : \, \det(\Lambda) = +1 \} </math>
| '''Proper antichronous''' LTs
:<math>\mathcal{L}_+^\downarrow = \mathcal{L}_+ \cap \mathcal{L}^\downarrow </math>
|'''Proper orthochronous''' LTs
:<math>\mathcal{L}_+^\uparrow = \mathcal{L}_+ \cap \mathcal{L}^\uparrow </math>
|-
! '''Improper''' LTs

:<math> \mathcal{L}_{-} = \{ \Lambda  \, : \, \det(\Lambda) = -1 \} </math>
|'''Improper antichronous''' LTs
:<math>\mathcal{L}_{-}^\downarrow = \mathcal{L}_{-} \cap \mathcal{L}^\downarrow </math>
|'''Improper orthochronous''' LTs
:<math>\mathcal{L}_{-}^\uparrow = \mathcal{L}_{-} \cap \mathcal{L}^\uparrow </math>
|-
|}

where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.

The full Lorentz group splits into the [[Union (set theory)|union]] ("u"-shaped symbol meaning "or") of four [[disjoint set]]s
<math display="block"> \mathcal{L} = \mathcal{L}_{+}^\uparrow \cup \mathcal{L}_{-}^\uparrow \cup \mathcal{L}_{+}^\downarrow \cup \mathcal{L}_{-}^\downarrow </math>

A [[subgroup]] of a group must be [[Closure (mathematics)|closed]] under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations {{math|Λ}} and {{mvar|L}} from a particular subgroup, the composite Lorentz transformations {{math|Λ''L''}} and {{math|''L''Λ}} must be in the same subgroup as {{math|Λ}} and {{mvar|L}}. This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets <math>\mathcal{L}_+^\uparrow </math>, <math>\mathcal{L}_+</math>, <math>\mathcal{L}^\uparrow</math>, and <math>\mathcal{L}_0 = \mathcal{L}_+^\uparrow \cup \mathcal{L}_{-}^\downarrow</math> all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. <math>\mathcal{L}_+^\downarrow </math>, <math>\mathcal{L}_{-}^\downarrow </math>, <math>\mathcal{L}_{-}^\uparrow </math>) do not form subgroups.