Inphysics andmathematics, theLorentz group is thegroup of allLorentz transformations ofMinkowski spacetime, theclassical andquantum setting for all (non-gravitational)physical phenomena. The Lorentz group is named for theDutch physicistHendrik Lorentz.

For example, the following laws, equations, and theories respect Lorentz symmetry:

• Thekinematical laws ofspecial relativity
• Maxwell's field equations in the theory ofelectromagnetism
• TheDirac equation in the theory of theelectron
• TheStandard Model of particle physics

The Lorentz group expresses the fundamentalsymmetry of space and time of all known fundamentallaws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.

The Lorentz group is asubgroup of thePoincaré group—the group of allisometries ofMinkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is theisotropy subgroup with respect to the origin of theisometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called thehomogeneous Lorentz group while the Poincaré group is sometimes called theinhomogeneous Lorentz group. Lorentz transformations are examples oflinear transformations; general isometries of Minkowski spacetime areaffine transformations.

Assume twoinertial reference frames(t,x,y,z) and(t′,x′,y′,z′), and two pointsP₁,P₂, the Lorentz group is the set of all the transformations between the two reference frames that preserve thespeed of light propagating between the two points:

${\displaystyle c^2(\mathrm{\Delta }{t}^{\prime }{)}^2-(\mathrm{\Delta }{x}^{\prime }{)}^2-(\mathrm{\Delta }{y}^{\prime }{)}^2-(\mathrm{\Delta }{z}^{\prime }{)}^2=c^2(\mathrm{\Delta }t{)}^2-(\mathrm{\Delta }x{)}^2-(\mathrm{\Delta }y{)}^2-(\mathrm{\Delta }z{)}^2}$ 

In matrix form these are all the linear transformationsΛ such that:

${\displaystyle {\mathrm{\Lambda }}^{\mathsf{\text{T}}}\eta \mathrm{\Lambda }=\eta \eta =\mathrm{diag}(1,-1,-1,-1)}$ 

These are then calledLorentz transformations.

Mathematically, the Lorentz group may be described as theindefinite orthogonal groupO(1, 3), thematrix Lie group that preserves thequadratic form

${\displaystyle (t,x,y,z)\mapsto t^2-x^2-y^2-z^2}$ 

onR⁴ (the vector space equipped with this quadratic form is sometimes writtenR^1,3). This quadratic form is, when put on matrix form (seeClassical orthogonal group), interpreted in physics as themetric tensor of Minkowski spacetime.

BothO(1, 3) andO(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.SO⁺(1, 3) vsSO⁺(3, 1).

The Lorentz group is a six-dimensionalnoncompactnon-abelianreal Lie group that is notconnected. The fourconnected components are notsimply connected.^[1] Theidentity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called therestricted Lorentz group, and is denotedSO⁺(1, 3). The restricted Lorentz group consists of those Lorentz transformations that preserve both theorientation of space and the direction of time. Itsfundamental group has order 2, and its universal cover, theindefinite spin groupSpin(1, 3), is isomorphic to both thespecial linear groupSL(2,C) and to thesymplectic groupSp(2,C). These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notablyspinors. Thus, inrelativistic quantum mechanics and inquantum field theory, it is very common to callSL(2,C) the Lorentz group, with the understanding thatSO⁺(1, 3) is a specific representation (the vector representation) of it.

A recurrent representation of the action of the Lorentz group on Minkowski space usesbiquaternions, which form acomposition algebra. The isometry property of Lorentz transformations holds according to the composition property⁠${\displaystyle |pq|=|p|\times |q|}$ ⁠.

Another property of the Lorentz group isconformality or preservation of angles. Lorentz boosts act byhyperbolic rotation of a spacetime plane, and such "rotations" preservehyperbolic angle, the measure ofrapidity used in relativity. Therefore, the Lorentz group is a subgroup of theconformal group of spacetime.

Note that this article refers toO(1, 3) as the "Lorentz group",SO(1, 3) as the "proper Lorentz group", andSO⁺(1, 3) as the "restricted Lorentz group". Many authors (especially in physics) use the name "Lorentz group" forSO(1, 3) (or sometimes evenSO⁺(1, 3)) rather thanO(1, 3). When reading such authors it is important to keep clear exactly which they are referring to.

Because it is aLie group, the Lorentz groupO(1, 3) is a group and also has a topological description as asmooth 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-pointingtimelike vector would be inverted to a past-pointing vector
• Some elements have orientation reversed byimproper Lorentz transformations, for example, certainvierbein (tetrads)

Lorentz transformations that preserve the direction of time are calledorthochronous. The subgroup of orthochronous transformations is often denotedO⁺(1, 3). Those that preserve orientation are calledproper, and as linear transformations they have determinant+1. (The improper Lorentz transformations have determinant−1.) The subgroup of proper Lorentz transformations is denotedSO(1, 3).

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called theproper, orthochronous Lorentz group orrestricted Lorentz group, and is denoted bySO⁺(1, 3).^[a]

The set of the four connected components can be given a group structure as thequotient groupO(1, 3) / SO⁺(1, 3), which is isomorphic to theKlein four-group. Every element inO(1, 3) can be written as thesemidirect product of a proper, orthochronous transformation and an element of thediscrete group

{1,P,T,PT}

whereP andT are theparity andtime reversal operators:

P = diag(1, −1, −1, −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.

The restricted Lorentz groupSO⁺(1, 3) is theidentity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by acontinuous curve lying in the group. The restricted Lorentz group is a connectednormal subgroup of the full Lorentz group with the same dimension, in this case with dimension six.

The restricted Lorentz group is generated by ordinaryspatial rotations andLorentz boosts (which are rotations in a hyperbolic space that includes a time-like direction^[2]). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by3 real parameters) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional. (See also theLie algebra of the Lorentz group.)

The set of all rotations forms aLie subgroup isomorphic to the ordinaryrotation groupSO(3). The set of all boosts, however, doesnot form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates toThomas rotation.) A boost in some direction, or a rotation about some axis, generates aone-parameter subgroup.

If a groupG acts on a spaceV, then a surfaceS ⊂V is asurface of transitivity ifS is invariant underG (i.e.,∀g ∈G, ∀s ∈S:gs ∈S) and for any two pointss₁,s₂ ∈S there is ag ∈G such thatgs₁ =s₂. By definition of the Lorentz group, it preserves the quadratic form

${\displaystyle Q(x)=x_0^2-x_1^2-x_2^2-x_3^2.}$ 

The surfaces of transitivity of the orthochronous Lorentz groupO⁺(1, 3),Q(x) = const. acting on flatspacetimeR^1,3 are the following:^[3]

• Q(x) > 0, x₀ > 0 is the upper branch of ahyperboloid of two sheets. Points on this sheet are separated from the origin by a futuretime-like vector.
• Q(x) > 0, x₀ < 0 is the lower branch of this hyperboloid. Points on this sheet are the pasttime-like vectors.
• Q(x) = 0, x₀ > 0 is the upper branch of thelight cone, the future light cone.
• Q(x) = 0, x₀ < 0 is the lower branch of the light cone, the past light cone.
• Q(x) < 0 is a hyperboloid of one sheet. Points on this sheet arespace-like separated from the origin.
• The originx₀ =x₁ =x₂ =x₃ = 0.

These surfaces are3-dimensional, so the images are not faithful, but they are faithful for the corresponding facts aboutO⁺(1, 2). For the full Lorentz group, the surfaces of transitivity are only four since the transformationT takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.

An equivalent way to formulate the above surfaces of transitivity is as asymmetric space in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient spaceSO⁺(1, 3) / SO(3), due to theorbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensionalhyperbolic space.

These observations constitute a good starting point for finding allinfinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method ofinduced representations.^[4] One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are calledlittle groups by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as(m, 0, 0, 0). For eachm ≠ 0, the vector pierces exactly one sheet. In this case the little group isSO(3), therotation group, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles (as far as is known). Standard vectors on the one-sheeted hyperbolas would correspond totachyons. Particles on the light cone arephotons, and more hypothetically,gravitons. The "particle" corresponding to the origin is the vacuum.

Several other groups are either homomorphic or isomorphic to the restricted Lorentz groupSO⁺(1, 3). These homomorphisms play a key role in explaining various phenomena in physics.

• Thespecial linear groupSL(2,C) is adouble covering of the restricted Lorentz group. This relationship is widely used to express theLorentz invariance of theDirac equation and the covariance of spinors. In other words, the (restricted) Lorentz group is isomorphic toSL(2,C) / Z₂
• Thesymplectic groupSp(2,C) is isomorphic toSL(2,C); it is used to constructWeyl spinors, as well as to explain how spinors can have a mass.
• Thespin groupSpin(1, 3) is isomorphic toSL(2,C); it is used to explain spin and spinors in terms of theClifford algebra, thus making it clear how to generalize the Lorentz group to general settings inRiemannian geometry, including theories ofsupergravity andstring theory.
• The restricted Lorentz group isisomorphic to theprojective special linear groupPSL(2,C) which is, in turn, isomorphic to theMöbius group, thesymmetry group ofconformal geometry on theRiemann sphere. This relationship is central to the classification of the subgroups of the Lorentz group according to an earlier classification scheme developed for the Möbius group.

TheWeyl representation orspinor map is a pair ofsurjectivehomomorphisms fromSL(2,C) toSO⁺(1, 3). They form a matched pair under parity transformations, corresponding to left and rightchiral spinors.

One may define an action ofSL(2,C) on Minkowski spacetime by writing a point of spacetime as a two-by-twoHermitian matrix in the form

 

in terms ofPauli matrices.

This presentation, the Weyl presentation, satisfies

${\displaystyle det\overline{X}=(ct{)}^2-x^2-y^2-z^2.}$ 

Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as areal vector space) with Minkowski spacetime, in such a way that thedeterminant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An elementS ∈ SL(2,C) acts on the space of Hermitian matrices via

${\displaystyle \overline{X}\mapsto S\overline{X}{S}^{†},}$ 

where${\displaystyle S^{†}}$  is theHermitian transpose ofS. This action preserves the determinant and soSL(2,C) acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is

${\displaystyle X=ct11-\overrightarrow{x}\cdot \overrightarrow{\sigma }}$ 

which transforms as

${\displaystyle X\mapsto {\left(S^{-1}\right)}^{†}X{S}^{-1}}$ 

That this is the correct transformation follows by noting that

${\displaystyle \overline{X}X=\left(c^2{t}^2-\overrightarrow{x}\cdot \overrightarrow{x}\right)11=\left(c^2{t}^2-x^2-y^2-z^2\right)11}$ 

remains invariant under the above pair of transformations.

These maps aresurjective, andkernel of either map is the two element subgroup±I. By thefirst isomorphism theorem, the quotient groupPSL(2,C) = SL(2,C) / {±I} is isomorphic toSO⁺(1, 3).

The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism ofSL(2,C). These two distinct coverings corresponds to the two distinctchiral actions of the Lorentz group onspinors. The non-overlined form corresponds to right-handed spinors transforming as⁠${\displaystyle {\psi }_R\mapsto S{\psi }_R}$ ⁠, while the overline form corresponds to left-handed spinors transforming as⁠${\displaystyle {\psi }_L\mapsto {\left(S^{†}\right)}^{-1}{\psi }_L}$ ⁠.^[b]

It is important to observe that this pair of coverings doesnot survive quantization; when quantized, this leads to the peculiar phenomenon of thechiral anomaly. The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of theAtiyah–Singer index theorem.

In physics, it is conventional to denote a Lorentz transformationΛ ∈ SO⁺(1, 3) as⁠${\displaystyle {{\mathrm{\Lambda }}^{\mu }}_{\nu }}$ ⁠, thus showing the matrix with spacetime indexesμ,ν = 0, 1, 2, 3. Afour-vector can be created from the Pauli matrices in two different ways: as${\displaystyle {\sigma }^{\mu }=(I,\overrightarrow{\sigma })}$  and as⁠${\displaystyle {\overline{\sigma }}^{\mu }=\left(I,-\overrightarrow{\sigma }\right)}$ ⁠. The two forms are related by aparity transformation. Note that⁠${\displaystyle {\overline{\sigma }}_{\mu }={\sigma }^{\mu }}$ ⁠.

Given a Lorentz transformation⁠${\displaystyle x^{\mu }\mapsto x^{\mathrm{\prime }\mu }={{\mathrm{\Lambda }}^{\mu }}_{\nu }{x}^{\nu }}$ ⁠, the double-covering of the orthochronous Lorentz group byS ∈ SL(2,C) given above can be written as

${\displaystyle x^{\mathrm{\prime }\mu }{\overline{\sigma }}_{\mu }={\overline{\sigma }}_{\mu }{{\mathrm{\Lambda }}^{\mu }}_{\nu }{x}^{\nu }=S{x}^{\nu }{\overline{\sigma }}_{\nu }{S}^{†}}$ 

Dropping the${\displaystyle x^{\mu }}$  this takes the form

${\displaystyle {\overline{\sigma }}_{\mu }{{\mathrm{\Lambda }}^{\mu }}_{\nu }=S{\overline{\sigma }}_{\nu }{S}^{†}}$ 

The parity conjugate form is

${\displaystyle {\sigma }_{\mu }{{\mathrm{\Lambda }}^{\mu }}_{\nu }={\left(S^{-1}\right)}^{†}{\sigma }_{\nu }{S}^{-1}}$ 

That the above is the correct form for indexed notation is not immediately obvious, partly because, when working in indexed notation, it is quite easy to accidentally confuse a Lorentz transform with its inverse, or its transpose. This confusion arises due to the identity${\displaystyle \eta {\mathrm{\Lambda }}^{\mathsf{\text{T}}}\eta ={\mathrm{\Lambda }}^{-1}}$  being difficult to recognize when written in indexed form. Lorentz transforms arenot tensors under Lorentz transformations! Thus a direct proof of this identity is useful, for establishing its correctness. It can be demonstrated by starting with the identity

${\displaystyle \omega {\sigma }^k{\omega }^{-1}=-{\left({\sigma }^k\right)}^{\mathsf{\text{T}}}=-{\left({\sigma }^k\right)}^{\ast }}$ 

where${\displaystyle k=1,2,3}$  so that the above are just the usual Pauli matrices, and${\displaystyle (\cdot {)}^{\mathsf{\text{T}}}}$  is the matrix transpose, and${\displaystyle (\cdot {)}^{\ast }}$  is complex conjugation. The matrix${\displaystyle \omega }$  is

 

Written as the four-vector, the relationship is

${\displaystyle {\sigma }_{\mu }^{\mathsf{\text{T}}}={\sigma }_{\mu }^{\ast }=\omega {\overline{\sigma }}_{\mu }{\omega }^{-1}}$ 

This transforms as

 

Taking one more transpose, one gets

${\displaystyle {\sigma }_{\mu }{{\mathrm{\Lambda }}^{\mu }}_{\nu }={\left(S^{-1}\right)}^{†}{\sigma }_{\nu }{S}^{-1}}$ 

The symplectic groupSp(2,C) is isomorphic toSL(2,C). This isomorphism is constructed so as to preserve asymplectic bilinear form onC², that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as

${\displaystyle \mathrm{Sp}(2,\mathbf{C})=\left\{S\in \mathrm{GL}(2,\mathbf{C}):S^{\mathsf{\text{T}}}\omega S=\omega \right\}}$ 

where

 

Other common notations are${\displaystyle \omega =ϵ}$  for this element; sometimesJ is used, but this invites confusion with the idea ofalmost complex structures, which are not the same, as they transform differently.

Given a pair of Weyl spinors (two-component spinors)

${\displaystyle u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right],v=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]}$ 

the invariant bilinear form is conventionally written as

${\displaystyle ⟨u,v⟩=-⟨v,u⟩=u_1{v}_2-u_2{v}_1=u^{\mathsf{\text{T}}}\omega v}$ 

This form is invariant under the Lorentz group, so that forS ∈ SL(2,C) one has

${\displaystyle ⟨Su,Sv⟩=⟨u,v⟩}$ 

This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariantmass term inLagrangians. There are several notable properties to be called out that are important to physics. One is that${\displaystyle {\omega }^2=-1}$  and so${\displaystyle {\omega }^{-1}={\omega }^{\mathsf{\text{T}}}={\omega }^{†}=-\omega }$ 

The defining relation can be written as

${\displaystyle \omega S^{\mathsf{\text{T}}}{\omega }^{-1}=S^{-1}}$ 

which closely resembles the defining relation for the Lorentz group

${\displaystyle \eta {\mathrm{\Lambda }}^{\mathsf{\text{T}}}{\eta }^{-1}={\mathrm{\Lambda }}^{-1}}$ 

where${\displaystyle \eta =\mathrm{diag}(+1,-1,-1,-1)}$  is themetric tensor forMinkowski space and of course,${\displaystyle \mathrm{\Lambda }\in \mathrm{SO}(1,3)}$  as before.

SinceSL(2,C) is simply connected, it is theuniversal covering group of the restricted Lorentz groupSO⁺(1, 3). By restriction, there is a homomorphismSU(2) → SO(3). Here, thespecial unitary group SU(2), which is isomorphic to the group of unitnormquaternions, is also simply connected, so it is the covering group of the rotation groupSO(3). Each of thesecovering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group aredoubly connected. This means that thefundamental group of the each group isisomorphic to the two-elementcyclic groupZ₂.

Twofold coverings are characteristic ofspin groups. Indeed, in addition to the double coverings

Spin⁺(1, 3) = SL(2,C) → SO⁺(1, 3)

Spin(3) = SU(2) → SO(3)

we have the double coverings

Pin(1, 3) → O(1, 3)

Spin(1, 3) → SO(1, 3)

Spin⁺(1, 2) = SU(1, 1) → SO(1, 2)

These spinorialdouble coverings are constructed fromClifford algebras.

The left and right groups in the double covering

SU(2) → SO(3)

aredeformation retracts of the left and right groups, respectively, in the double covering

SL(2,C) → SO⁺(1, 3).

But the homogeneous spaceSO⁺(1, 3) / SO(3) ishomeomorphic tohyperbolic 3-spaceH³, so we have exhibited the restricted Lorentz group as aprincipal fiber bundle with fibersSO(3) and baseH³. Since the latter is homeomorphic toR³, whileSO(3) is homeomorphic to three-dimensionalreal projective spaceRP³, we see that the restricted Lorentz group islocally homeomorphic to the product ofRP³ withR³. Since the base space is contractible, this can be extended to a global homeomorphism.^{[clarification needed]}

Because the restricted Lorentz groupSO⁺(1, 3) is isomorphic to the Möbius groupPSL(2,C), itsconjugacy classes also fall into five classes:

• Elliptic transformations
• Hyperbolic transformations
• Loxodromic transformations
• Parabolic transformations
• The trivialidentity transformation

In the article onMöbius transformations, it is explained how this classification arises by considering thefixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here tonulleigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime.

An example of each type is given in the subsections below, along with the effect of theone-parameter subgroup it generates (e.g., on the appearance of the night sky).

The Möbius transformations are theconformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element ofSL(2,C) obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on theflow lines of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar.

An elliptic element ofSL(2,C) is

 

and has fixed pointsξ = 0, ∞. Writing the action asX ↦P₁X P₁^† and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation

 

This transformation then represents a rotation about thez axis, exp(iθJ_z). The one-parameter subgroup it generates is obtained by takingθ to be a real variable, the rotation angle, instead of a constant.

The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about thez axis asθ increases. Theangle doubling evident in the spinor map is a characteristic feature ofspinorial double coverings.

A hyperbolic element ofSL(2,C) is

 

and has fixed pointsξ = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.

The spinor map converts this to the Lorentz transformation

 

This transformation represents a boost along thez axis withrapidityη. The one-parameter subgroup it generates is obtained by takingη to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points alonglongitudes away from the South pole and toward the North pole.

A loxodromic element ofSL(2,C) is

 

and has fixed pointsξ = 0, ∞. The spinor map converts this to the Lorentz transformation

 

The one-parameter subgroup this generates is obtained by replacingη + iθ with any real multiple of this complex constant. (Ifη,θ vary independently, then atwo-dimensionalabelian subgroup is obtained, consisting of simultaneous rotations about thez axis and boosts along thez-axis; in contrast, theone-dimensional subgroup discussed here consists of those elements of this two-dimensional subgroup such that therapidity of the boost andangle of the rotation have afixed ratio.)

The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves calledloxodromes. Each loxodrome spirals infinitely often around each pole.

A parabolic element ofSL(2,C) is

 

and has the single fixed pointξ = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinarytranslation along thereal axis.

The spinor map converts this to the matrix (representing a Lorentz transformation)

 

This generates a two-parameter abelian subgroup, which is obtained by consideringα a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles that are all tangent at the North pole to a certaingreat circle. All points other than the North pole itself move along these circles.

Parabolic Lorentz transformations are often callednull rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.

The matrix given above yields the transformation

${\displaystyle \left[\begin{array}{c}t\\ x\\ y\\ z\end{array}\right]\to \left[\begin{array}{c}t\\ x\\ y\\ z\end{array}\right]+\mathrm{Re}(\alpha )\left[\begin{array}{c}x\\ t-z\\ 0\\ x\end{array}\right]-\mathrm{Im}(\alpha )\left[\begin{array}{c}y\\ 0\\ z-t\\ y\end{array}\right]+\frac{|\alpha {|}^2}{2}\left[\begin{array}{c}t-z\\ 0\\ 0\\ t-z\end{array}\right].}$ 

Now,without loss of generality, pickIm(α) = 0. Differentiating this transformation with respect to the now real group parameterα and evaluating atα = 0 produces the corresponding vector field (first order linear partial differential operator),

${\displaystyle x\left({\mathrm{\partial }}_t+{\mathrm{\partial }}_z\right)+(t-z){\mathrm{\partial }}_x.}$ 

Apply this to a functionf(t,x,y,z), and demand that it stays invariant; i.e., it is annihilated by this transformation. The solution of the resulting first order linearpartial differential equation can be expressed in the form

${\displaystyle f(t,x,y,z)=F\left(y,t-z,t^2-x^2-z^2\right),}$ 

whereF is anarbitrary smooth function. The arguments ofF give threerational invariants describing how points (events) move under this parabolic transformation, as they themselves do not move,

${\displaystyle y=c_1,t-z=c_2,t^2-x^2-z^2=c_3.}$ 

Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation.

The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinatey, each orbit is the intersection of anull plane,t =z + c₂, with ahyperboloid,t² − x² − z² =c₃. The casec₃ = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.

A particular null line lying on the light cone is leftinvariant; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above. The other null lines through the origin are "swung around the cone" by the transformation. Following the motion of one such null line asα increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.

A choiceRe(α) = 0 instead, produces similar orbits, now with the roles ofx andy interchanged.

Parabolic transformations lead to the gauge symmetry of massless particles (such asphotons) withhelicity |h| ≥ 1. In the above explicit example, a massless particle moving in thez direction, so with 4-momentumP = (p, 0, 0,p), is not affected at all by thex-boost andy-rotation combinationK_x − J_y defined below, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector,P itself is now invariant; i.e., all traces or effects ofα have disappeared.c₁ =c₂ =c₃ = 0, in the special case discussed. (The other similar generator,K_y +J_x as well as it andJ_z comprise altogether the little group of the light-like vector, isomorphic toE(2).)

This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars".

Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated withξ =u +iv, a complex number that corresponds to the point on theRiemann sphere, and can be identified with a null vector (alight-like vector) in Minkowski space

${\displaystyle \left[\begin{array}{c}{u}^2+v^2+1\\ 2u\\ -2v\\ u^2+v^2-1\end{array}\right]}$ 

or, in the Weyl representation (the spinor map), the Hermitian matrix

 

The set of real scalar multiples of this null vector, called anull line through the origin, represents aline of sight from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of thecelestial sphere (equivalently, lines of sight) are identified with certain Hermitian matrices.

This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on theprojective celestial sphere. This is the space of non-zero null vectors with${\displaystyle t>0}$  under the given quotient for projective spaces:${\displaystyle (t,x,y,z)\sim (t^{\prime },x^{\prime },y^{\prime },z^{\prime })}$  if${\displaystyle (t^{\prime },x^{\prime },y^{\prime },z^{\prime })=(\lambda t,\lambda x,\lambda y,\lambda z)}$  for${\displaystyle \lambda >0}$ . This is referred to as the celestial sphere as this allows us to rescale the time coordinate${\displaystyle t}$  to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere.

From the Möbius side,SL(2,C) acts on complex projective spaceCP¹, which can be shown to be diffeomorphic to the 2-sphere – this is sometimes referred to as the Riemann sphere. The quotient on projective space leads to a quotient on the groupSL(2,C).

Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If${\displaystyle \xi }$  is aCP¹ projective vector, it can be tensored with its Hermitian conjugate to produce a${\displaystyle 2\times 2}$  Hermitian matrix. From elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in the Riemann sphere into a matrix is known as theBloch sphere.

As with any Lie group, a useful way to study many aspects of the Lorentz group is via itsLie algebra. Since the Lorentz groupSO(1, 3) is amatrix Lie group, its corresponding Lie algebra${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)}$  is a matrix Lie algebra, which may be computed as^[5]

${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)=\left\{4\times 4\mathbf{R}\text{-valued matrices}X\mid e^{tX}\in \mathrm{S}\mathrm{O}(1,3)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\right\}}$ .

If${\displaystyle \eta }$  is the diagonal matrix with diagonal entries(1, −1, −1, −1), then the Lie algebra${\displaystyle \mathfrak{o}(1,3)}$  consists of${\displaystyle 4\times 4}$  matrices${\displaystyle X}$  such that^[6]

${\displaystyle \eta X\eta =-X^{\mathsf{\text{T}}}}$ .

Explicitly,${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)}$  consists of${\displaystyle 4\times 4}$  matrices of the form

 ,

where${\displaystyle a,b,c,d,e,f}$  are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)}$  consisting of elements in which${\displaystyle a}$ ,${\displaystyle b}$ , and${\displaystyle c}$  equal to zero is isomorphic to${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ .

The full Lorentz groupO(1, 3), the proper Lorentz groupSO(1, 3) and the proper orthochronous Lorentz groupSO⁺(1, 3) (the component connected to the identity) all have the same Lie algebra, which is typically denoted⁠${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)}$ ⁠.

Since the identity component of the Lorentz group is isomorphic to a finite quotient ofSL(2,C) (see the section above on the connection of the Lorentz group to the Möbius group), the Lie algebra of the Lorentz group is isomorphic to the Lie algebra⁠${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbf{C})}$ ⁠. As a complex Lie algebra${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbf{C})}$  is three dimensional, but is six dimensional when viewed as a real Lie algebra.

The standard basis matrices can be indexed as${\displaystyle M^{\mu \nu }}$  where${\displaystyle \mu ,\nu }$  take values in{0, 1, 2, 3}. These arise from taking only one of${\displaystyle a,b,\cdots ,f}$  to be one, and others zero, in turn. The components can be written as

${\displaystyle {(M^{\mu \nu }{)}^{\rho }}_{\sigma }={\eta }^{\nu \sigma }{{\delta }^{\rho }}_{\mu }-{\eta }^{\mu \sigma }{{\delta }^{\rho }}_{\nu }}$ .

The commutation relations are

${\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }{\eta }^{\nu \rho }-M^{\nu \sigma }{\eta }^{\mu \rho }+M^{\nu \rho }{\eta }^{\mu \sigma }-M^{\mu \rho }{\eta }^{\nu \sigma }.}$ 

There are different possible choices of convention in use. In physics, it is common to include a factor of${\displaystyle i}$  with the basis elements, which gives a factor of${\displaystyle i}$  in the commutation relations.

Then${\displaystyle M^{0i}}$  generate boosts and${\displaystyle M^{ij}}$  generate rotations.

The structure constants for the Lorentz algebra can be read off from the commutation relations. Any set of basis elements which satisfy these relations form a representation of the Lorentz algebra.

The Lorentz group can be thought of as a subgroup of thediffeomorphism group ofR⁴ and therefore its Lie algebra can be identified with vector fields onR⁴. In particular, the vectors that generate isometries on a space are itsKilling vectors, which provides a convenient alternative to theleft-invariant vector field for calculating the Lie algebra. We can write down a set of sixgenerators:

• Vector fields onR⁴ generating three rotationsiJ,

  ${\displaystyle -y{\mathrm{\partial }}_x+x{\mathrm{\partial }}_y\equiv i{J}_z,-z{\mathrm{\partial }}_y+y{\mathrm{\partial }}_z\equiv i{J}_x,-x{\mathrm{\partial }}_z+z{\mathrm{\partial }}_x\equiv i{J}_y;}$ 

• Vector fields onR⁴ generating three boostsiK,

  ${\displaystyle x{\mathrm{\partial }}_t+t{\mathrm{\partial }}_x\equiv i{K}_x,y{\mathrm{\partial }}_t+t{\mathrm{\partial }}_y\equiv i{K}_y,z{\mathrm{\partial }}_t+t{\mathrm{\partial }}_z\equiv i{K}_z.}$ 

The factor ofi appears to ensure that the generators of rotations are Hermitian.

It may be helpful to briefly recall here how to obtain a one-parameter group from avector field, written in the form of a first orderlinearpartial differential operator such as

${\displaystyle \mathcal{L}=-y{\mathrm{\partial }}_x+x{\mathrm{\partial }}_y.}$ 

The corresponding initial value problem (consider${\displaystyle r=(x,y)}$  a function of a scalar${\displaystyle \lambda }$  and solve${\displaystyle {\mathrm{\partial }}_{\lambda }r=\mathcal{L}r}$  with some initial conditions) is

${\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }\lambda }=-y,\frac{\mathrm{\partial }y}{\mathrm{\partial }\lambda }=x,x(0)=x_0,y(0)=y_0.}$ 

The solution can be written

${\displaystyle x(\lambda )=x_0\cos (\lambda )-y_0\sin (\lambda ),y(\lambda )=x_0\sin (\lambda )+y_0\cos (\lambda )}$ 

or

 

where we easily recognize the one-parameter matrix group of rotationsexp(iλJ_z) about the z-axis.

Differentiating with respect to the group parameterλ and setting itλ = 0 in that result, we recover the standard matrix,

 

which corresponds to the vector field we started with. This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra. Theexponential map plays this special role not only for the Lorentz group but for Lie groups in general.

Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectivelyη/2 (for the three boosts) oriθ/2 (for the three rotations) times the threePauli matrices

 

Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which

• The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as areal vector field on the Euclidean plane.
• The second column gives the corresponding one-parameter subgroup of Möbius transformations.
• The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup).
• The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.

Notice that the generators consist of

• Two parabolics (null rotations)
• One hyperbolic (boost in the${\displaystyle {\mathrm{\partial }}_z}$  direction)
• Three elliptics (rotations about thex,y,z axes, respectively)

Start with

 

Exponentiate:

 

This element ofSL(2,C) represents the one-parameter subgroup of (elliptic) Möbius transformations:

${\displaystyle \xi \mapsto {\xi }^{\prime }=\frac{\cos \left(\frac{\theta }{2}\right)\xi -\sin \left(\frac{\theta }{2}\right)}{\sin \left(\frac{\theta }{2}\right)\xi +\cos \left(\frac{\theta }{2}\right)}.}$ 

Next,

${\displaystyle {\frac{d{\xi }^{\prime }}{d\theta }|}_{\theta =0}=-\frac{1+{\xi }^2}{2}.}$ 

The corresponding vector field onC (thought of as the image ofS² under stereographic projection) is

${\displaystyle -\frac{1+{\xi }^2}{2}{\mathrm{\partial }}_{\xi }.}$ 

Writing${\displaystyle \xi =u+iv}$ , this becomes the vector field onR²

${\displaystyle -\frac{1+u^2-v^2}{2}{\mathrm{\partial }}_u-uv{\mathrm{\partial }}_v.}$ 

Returning to our element ofSL(2,C), writing out the action${\displaystyle X\mapsto PX{P}^{†}}$  and collecting terms, we find that the image under the spinor map is the element ofSO⁺(1, 3)

 

Differentiating with respect toθ atθ = 0, yields the corresponding vector field onR^1,3,

${\displaystyle z{\mathrm{\partial }}_x-x{\mathrm{\partial }}_z.}$ 

This is evidently the generator of counterclockwise rotation about they-axis.

The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which theclosed subgroups of the restricted Lorentz group can be listed, up to conjugacy. (See the book by Hall cited below for the details.) These can be readily expressed in terms of the generators${\displaystyle X_n}$  given in the table above.

The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:

• ${\displaystyle X_1}$  generates a one-parameter subalgebra of parabolicsSO(0, 1),
• ${\displaystyle X_3}$  generates a one-parameter subalgebra of boostsSO(1, 1),
• ${\displaystyle X_4}$  generates a one-parameter of rotationsSO(2),
• ${\displaystyle X_3+a{X}_4}$  (for any${\displaystyle a\ne 0}$ ) generates a one-parameter subalgebra of loxodromic transformations.

(Strictly speaking the last corresponds to infinitely many classes, since distinct${\displaystyle a}$  give different classes.)The two-dimensional subalgebras are:

• ${\displaystyle X_1,X_2}$  generate an abelian subalgebra consisting entirely of parabolics,
• ${\displaystyle X_1,X_3}$  generate a nonabelian subalgebra isomorphic to the Lie algebra of theaffine groupAff(1),
• ${\displaystyle X_3,X_4}$  generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.

The three-dimensional subalgebras use theBianchi classification scheme:

• ${\displaystyle X_1,X_2,X_3}$  generate aBianchi V subalgebra, isomorphic to the Lie algebra ofHom(2), the group ofeuclidean homotheties,
• ${\displaystyle X_1,X_2,X_4}$  generate aBianchi VII₀ subalgebra, isomorphic to the Lie algebra ofE(2), theeuclidean group,
• ${\displaystyle X_1,X_2,X_3+a{X}_4}$ , where${\displaystyle a\ne 0}$ , generate aBianchi VII_a subalgebra,
• ${\displaystyle X_1,X_3,X_5}$  generate aBianchi VIII subalgebra, isomorphic to the Lie algebra ofSL(2,R), the group of isometries of thehyperbolic plane,
• ${\displaystyle X_4,X_5,X_6}$  generate aBianchi IX subalgebra, isomorphic to the Lie algebra ofSO(3), the rotation group.

TheBianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematicianLuigi Bianchi.

The four-dimensional subalgebras are all conjugate to

• ${\displaystyle X_1,X_2,X_3,X_4}$  generate a subalgebra isomorphic to the Lie algebra ofSim(2), the group of Euclideansimilitudes.

The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation aclosed subgroup of the restricted Lie group. From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group.

As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, orhomogeneous spaces, have considerable mathematical interest. A few, brief descriptions:

• The groupSim(2) is the stabilizer of anull line; i.e., of a point on the Riemann sphere—so the homogeneous spaceSO⁺(1, 3) / Sim(2) is theKleinian geometry that representsconformal geometry on the sphereS².
• The (identity component of the) Euclidean groupSE(2) is the stabilizer of anull vector, so the homogeneous spaceSO⁺(1, 3) / SE(2) is themomentum space of a massless particle; geometrically, this Kleinian geometry represents thedegenerate geometry of the light cone in Minkowski spacetime.
• The rotation groupSO(3) is the stabilizer of atimelike vector, so the homogeneous spaceSO⁺(1, 3) / SO(3) is themomentum space of a massive particle; geometrically, this space is none other than three-dimensionalhyperbolic spaceH³.

The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of (n + 1)-dimensional Minkowski space is theindefinite orthogonal groupO(n, 1) of linear transformations ofRⁿ⁺¹ that preserves the quadratic form

${\displaystyle (x_1,x_2,\dots ,x_n,x_{n+1})\mapsto x_1^2+x_2^2+\cdots +x_n^2-x_{n+1}^2.}$ 

The groupO(1,n) preserves the quadratic form

${\displaystyle (x_1,x_2,\dots ,x_n,x_{n+1})\mapsto x_1^2-x_2^2-\cdots -x_{n+1}^2}$ 

O(1,n) is isomorphic toO(n, 1), and both presentations of the Lorentz group are in use in the theoretical physics community. The former is more common in literature related to gravity, while the latter is more common in particle physics literature.

A common notation for the vector spaceRⁿ⁺¹, equipped with this choice of quadratic form, isR^1,n.

Many of the properties of the Lorentz group in four dimensions (wheren = 3) generalize straightforwardly to arbitraryn. For instance, the Lorentz groupO(n, 1) has four connected components, and it acts by conformal transformations on the celestial(n − 1)-sphere in(n + 1)-dimensional Minkowski space. The identity componentSO⁺(n, 1) is anSO(n)-bundle over hyperbolicn-spaceHⁿ.

The low-dimensional casesn = 1 andn = 2 are often useful as "toy models" for the physical casen = 3, while higher-dimensional Lorentz groups are used in physical theories such asstring theory that posit the existence of hidden dimensions. The Lorentz groupO(n, 1) is also the isometry group ofn-dimensionalde Sitter spacedS_n, which may be realized as the homogeneous spaceO(n, 1) / O(n − 1, 1). In particularO(4, 1) is the isometry group of thede Sitter universedS₄, a cosmological model.