Editing Lorentz group (section)

== Basic properties ==
The Lorentz group is a [[subgroup]] of the [[Poincaré group]]—the group of all [[Isometry#Generalizations|isometries]] of [[Minkowski spacetime]]. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the [[Group action (mathematics)#Fixed points and stabilizer subgroups|isotropy subgroup]] with respect to the origin of the [[isometry group]] of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the '''homogeneous Lorentz group''' while the Poincaré group is sometimes called the ''inhomogeneous Lorentz group''. Lorentz transformations are examples of [[Linear map|linear transformation]]s; general isometries of Minkowski spacetime are [[affine transformation]]s.

=== Physics definition ===
Assume two [[inertial reference frames]] {{math|(''t'', ''x'', ''y'', ''z'')}} and {{math|(''t''′, ''x''′, ''y''′, ''z''′)}}, and two points {{math|''P''{{sub|1}}}}, {{math|''P''{{sub|2}}}}, the Lorentz group is the set of all the transformations between the two reference frames that preserve the [[speed of light]] propagating between the two points:
: <math>c^2(\Delta t')^2 - (\Delta x')^2 - (\Delta y')^2 - (\Delta z')^2 = c^2(\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2</math>
In matrix form these are all the linear transformations {{math|Λ}} such that:
: <math>\Lambda^\textsf{T} \eta \Lambda = \eta\qquad\eta = \operatorname{diag}(1,-1,-1,-1)</math>
These are then called [[Lorentz transformations]].

=== Mathematical definition ===
Mathematically, the Lorentz group may be described as the [[indefinite orthogonal group]] {{math|O(1, 3)}}, the [[matrix Lie group]] that preserves the [[quadratic form]]
: <math>(t, x, y, z) \mapsto t^2 - x^2 - y^2 - z^2</math>

on {{math|'''R'''{{sup|4}}}} (the vector space equipped with this quadratic form is sometimes written {{math|'''R'''{{sup|1,3}}}}). This quadratic form is, when put on matrix form (see ''[[Classical group#O(p, q) and O(n) – the orthogonal groups|Classical orthogonal group]]''), interpreted in physics as the [[metric tensor]] of Minkowski spacetime.

=== Note on Notation ===

Both {{math|O(1, 3)}} and {{math|O(3, 1)}} are in common use for the Lorentz group.  The first refers to matrices which preserve a metric of signature with one + and three -'s, and the second refers to a metric of signature with one - and three +'s.  Because the overall sign of the metric is irrelevant in the defining equation, the resulting groups of matrices are identical.  There appears to be a modern push from some sectors to adopt (1,3) notation versus (3,1), but the latter still finds plenty of use in current practice, and a great deal of the historical literature employed it.   Everything described in this article applies to O(3,1) notation as well, [[mutatis mutandis]].  These considerations extend to related definitions as well (ex. {{math|SO{{sup|+}}(1, 3)}} vs {{math|SO{{sup|+}}(3, 1)}}.

=== Mathematical properties ===
The Lorentz group is a six-[[dimension]]al [[Compact space|noncompact]] [[Non-abelian group|non-abelian]] [[real Lie group]] that is not [[Connected space|connected]]. The four [[Connected component (topology)|connected component]]s are not [[simply connected]].<ref>{{harvnb|Weinberg|2002}}</ref> The [[identity component]] (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the '''restricted Lorentz group''', and is denoted {{math|SO{{sup|+}}(1, 3)}}. The restricted Lorentz group consists of those Lorentz transformations that preserve both the [[Orientation (mathematics)|orientation]] of space and the direction of time. Its [[fundamental group]] has order 2, and its universal cover, the [[spin group#Indefinite signature|indefinite spin group]] {{math|Spin(1, 3)}}, is isomorphic to both the [[special linear group]] {{math|SL(2, '''C''')}} and to the [[symplectic group]] {{math|Sp(2, '''C''')}}. These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably [[spinor]]s. Thus, in [[relativistic quantum mechanics]] and in [[quantum field theory]], it is very common to call {{math|SL(2, '''C''')}} the Lorentz group, with the understanding that {{math|SO{{sup|+}}(1, 3)}} is a specific representation (the vector representation) of&nbsp;it.

A recurrent representation of the action of the Lorentz group on Minkowski space uses [[biquaternion]]s, which form a [[composition algebra]]. The isometry property of Lorentz transformations holds according to the composition property {{tmath|1= {{!}} p q {{!}} = {{!}} p {{!}} \times {{!}} q {{!}} }}.

Another property of the Lorentz group is ''conformality'' or preservation of angles. Lorentz boosts act by [[hyperbolic rotation]] of a spacetime plane, and such "rotations" preserve [[hyperbolic angle]], the measure of [[rapidity]] used in relativity. Therefore, the Lorentz group is a subgroup of the [[conformal group#Conformal group of spacetime|conformal group of spacetime]].

Note that this article refers to {{math|O(1, 3)}} as the "Lorentz group", {{math|SO(1, 3)}} as the "proper Lorentz group", and {{math|SO{{sup|+}}(1, 3)}} as the "restricted Lorentz group".  Many authors (especially in physics) use the name "Lorentz group" for {{math|SO(1, 3)}} (or sometimes even {{math|SO{{sup|+}}(1, 3)}}) rather than {{math|O(1, 3)}}.  When reading such authors it is important to keep clear exactly which they are referring to.

=== Connected components ===
[[File:World line.svg|300px|right|thumb|Light cone in 2D space plus a time dimension.]]
Because it is a [[Lie group]], the Lorentz group {{math|O(1, 3)}} is a group and also has a topological description as a [[smooth manifold]]. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.

The four connected components can be categorized by two transformation properties its elements have:
* Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing [[timelike vector]] would be inverted to a past-pointing vector
* Some elements have orientation reversed by '''improper Lorentz transformations''', for example, certain [[vierbein]] (tetrads)

Lorentz transformations that preserve the direction of time are called '''{{Visible anchor|orthochronous}}'''. The subgroup of orthochronous transformations is often denoted {{math|O{{sup|+}}(1, 3)}}. Those that preserve orientation are called '''proper''', and as linear transformations they have determinant {{math|+1}}.  (The improper Lorentz transformations have determinant {{math|−1}}.)  The subgroup of proper Lorentz transformations is denoted {{math|SO(1, 3)}}.

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the '''proper, orthochronous Lorentz group''' or '''restricted Lorentz group''', and is denoted by {{math|SO{{sup|+}}(1, 3)}}.{{efn|Note that some authors refer to {{math|SO(1, 3)}} or even {{math|O(1, 3)}} when they mean {{math|SO{{sup|+}}(1, 3)}}.}}

The set of the four connected components can be given a group structure as the [[quotient group]] {{math|O(1, 3) / SO{{sup|+}}(1, 3)}}, which is isomorphic to the [[Klein four-group]]. Every element in {{math|O(1, 3)}} can be written as the [[semidirect product]] of a proper, orthochronous transformation and an element of the [[discrete group]]
: {{math|{{mset|1, ''P'', ''T'', ''PT''}}}}

where ''P'' and ''T'' are the [[P-symmetry|parity]] and [[T-symmetry|time reversal]] operators:
: {{math|1=''P'' = diag(1, −1, −1, −1)}}
: {{math|1=''T'' = diag(−1, 1, 1, 1)}}.

Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components.  This pattern is typical of finite-dimensional Lie groups.