Lorentz group


In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
For example, the following laws, equations, and theories respect Lorentz symmetry:
The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics.

Basic properties

The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup 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 transformations; general isometries of Minkowski spacetime are affine transformations.
Mathematically, the Lorentz group may be described as the generalized orthogonal group O, the matrix Lie group that preserves the quadratic form
on R4. This quadratic form is, when put on matrix form, interpreted in physics as the metric tensor of Minkowski spacetime.
The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected. The identity component of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted SO+. The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time, and it can be described using biquaternions. Its fundamental group has order 2, and its universal cover, the indefinite spin group Spin, turns out to be isomorphic to the special linear group SL.
The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the point symmetry group of a certain ordinary differential equation. This fact also has physical significance.

Connected components

Because it is a Lie group, the Lorentz group O is both a group and admits 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:
Lorentz transformations that preserve the direction of time are called . The subgroup of orthochronous transformations is often denoted O+. Those that preserve orientation are called proper, and as linear transformations they have determinant +1. The subgroup of proper Lorentz transformations is denoted SO.
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 SO+. or even O when they actually mean SO+
The set of the four connected components can be given a group structure as the quotient group O/SO+, which is isomorphic to the Klein four-group. Every element in O can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group
where P and T are the space inversion and time reversal operators:
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.

Restricted Lorentz group

The restricted Lorentz group is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group. The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six.
The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts. Since every proper, orthochronos Lorentz transformation can be written as a product of a rotation and a boost, 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.
The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO. The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. A boost in some direction, or a rotation about some axis, generates a one-parameter subgroup.

Surfaces of transitivity

If a group acts on a space, then a surface is a surface of transitivity if is invariant under, i.e.,, and for any two points there is a such that. By definition of the Lorentz group, it preserves the quadratic form
The surfaces of transitivity of the orthochronous Lorentz group, of spacetime are the following:
These surfaces are, so the images are not faithful, but they are faithful for the corresponding facts about. For the full Lorentz group, the surfaces of transitivity are only four since the transformation takes an upper branch of a hyperboloid to a lower one and vice versa.
These observations constitute a good starting point for finding all infinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method of induced representations. One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called little 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. For each, the vector pierces exactly one sheet. In this case the little group is, the rotation 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. Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum.

Relation to the Möbius group

The restricted Lorentz group SO+ is isomorphic to the projective special linear group PSL which is, in turn, isomorphic to the Möbius group, the symmetry group of conformal geometry on the Riemann sphere.
This may be shown by constructing a surjective homomorphism of Lie groups from SL to SO+, dubbed
the spinor map. It proceeds as follows.
One may define an action of SL on Minkowski spacetime by writing a point of spacetime as a two-by-two Hermitian matrix in the form
in terms of Pauli matrices.
This presentation satisfies
Therefore, one has identified the space of Hermitian matrices with Minkowski spacetime, in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime.
SL acts on the space of Hermitian matrices via
where * is the Hermitian transpose of, and this action preserves the determinant.
Therefore, SL acts on Minkowski spacetime by isometries. This defines a map from SL to the Lorentz group SO+, and the map is evidently a homomorphism. This is the spinor map.
The kernel of the spinor map is the two element subgroup ±I, and it happens that the map is surjective. By the first isomorphism theorem, the quotient group PSL = SL / is isomorphic to SO+.

Appearance of the night sky

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 with, a complex number that corresponds to the point on the Riemann sphere, and can be identified with a null vector in Minkowski space
or, in the Weyl representation, the Hermitian matrix
The set of real scalar multiples of this null vector, called a null line through the origin, represents a line of sight from an observer at a particular place and time to various distant objects, such as stars. Then the points of the celestial sphere are identified with certain Hermitian matrices.

Conjugacy classes

Because the restricted Lorentz group SO+ is isomorphic to the Möbius group PSL, its conjugacy classes also fall into five classes:
In the article on Möbius transformations, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to null eigenspaces 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 the one-parameter subgroup it generates.
The Möbius transformations are the conformal transformations of the Riemann sphere. Then conjugating with an arbitrary element of SL obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic Lorentz transformations, respectively. The effect on the flow 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.

Elliptic

An elliptic element of SL is
and has fixed points = 0, ∞. Writing the action as and collecting terms, the spinor map converts this to the Lorentz transformation
This transformation then represents a rotation about the axis, exp. The one-parameter subgroup it generates is obtained by taking to be a real variable, the rotation angle, instead of a constant.
The corresponding continuous transformations of the celestial sphere 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 counterclockwise rotation about the axis as increases. The angle doubling evident in the spinor map is a characteristic feature of spinorial double coverings.

Hyperbolic

A hyperbolic element of SL is
and has fixed points = 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 the axis with rapidity. The one-parameter subgroup it generates is obtained by taking to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere all share the same fixed points, and they move all other points along longitudes away from the South pole and toward the North pole.

Loxodromic

A loxodromic element of SL is
and has fixed points = 0, ∞. The spinor map converts this to the Lorentz transformation
The one-parameter subgroup this generates is obtained by replacing β+iθ with any real multiple of this complex constant.
The corresponding continuous transformations of the celestial sphere all share the same two fixed points. They move all other points away from the South pole and toward the North pole, along a family of curves called loxodromes. Each loxodrome spirals infinitely often around each pole.

Parabolic

A parabolic element of SL is
and has the single fixed point = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary translation along the real axis.
The spinor map converts this to the matrix
This generates a two-parameter abelian subgroup, which is obtained by considering a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere move points along a family of circles that are all tangent at the North pole to a certain great circle. All points other than the North pole itself move along these circles.
Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations, 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
Now, without loss of generality, pick. Differentiating this transformation with respect to the now real group parameter and evaluating at α=0 produces the corresponding vector field,
Apply this to a function, and demand that it stays invariant, i.e., it is annihilated by this transformation. The solution of the resulting first order linear partial differential equation can be expressed in the form
where is an arbitrary smooth function. The arguments of give three rational invariants describing how points move under this parabolic transformation, as they themselves do not move,
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 have a simple description: suppressing the inessential coordinate, each orbit is the intersection of a null plane,, with a hyperboloid, . The case 3 = 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 left invariant; this corresponds to the unique 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 as increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.
A choice instead, produces similar orbits, now with the roles of and interchanged.
Parabolic transformations lead to the gauge symmetry of massless particles with helicity || ≥ 1. In the above explicit example, a massless particle moving in the direction, so with 4-momentum P=, is not affected at all by the -boost and -rotation combination displayed above, 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 of have disappeared. 1 = 2 = 3 = 0, in the special case discussed.

Lie algebra

As with any Lie group, a useful way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group SO is a matrix Lie group, its Lie algebra so is an algebra of matrices, which may be computed as
If is the diagonal matrix with diagonal entries, then the Lie algebra o consists of matrices such that
Explicitly, so consists of matrices of the form
where are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of so consisting of elements in which,, and equal zero is isomorphic to so.
Note that the full Lorentz group O, the proper Lorentz group SO and the proper orthochronous Lorentz group all have the same Lie algebra, which is typically denoted so.
Since the identity component of the Lorentz group is isomorphic to a finite quotient of SL, the [|Lie algebra of the Lorentz group] is isomorphic to the Lie algebra sl. Note that sl is three dimensional when viewed as a complex Lie algebra, but six dimensional when viewed as a real Lie algebra.
Meanwhile, the Lorentz group can also be thought of as a subgroup of the diffeomorphism group of R4 and therefore its Lie algebra can be identified with vector fields on R4. In particular, the vectors that generate isometries on a space are its Killing vectors, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can write down a set of six generators:
It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order linear partial differential operator such as
The corresponding initial value problem is
The solution can be written
or
where we easily recognize the one-parameter matrix group of rotations exp about the z axis.
Differentiating with respect to the group parameter and setting it λ=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.
Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively β/2 or iθ/2 times the three Pauli matrices
For our purposes, another generating set is more convenient. The following table lists the six generators, in which
Notice that the generators consist of
Let's verify one line in this table. Start with
Exponentiate:
This element of SL represents the one-parameter subgroup of Möbius transformations:
Next,
The corresponding vector field on C is
Writing, this becomes the vector field on R2
Returning to our element of SL, writing out the action and collecting terms, we find that the image under the spinor map is the element of SO+
Differentiating with respect to at =0, yields the corresponding vector field on R4,
This is evidently the generator of counterclockwise rotation about the axis.

Subgroups of the Lorentz group

The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which we can list the closed subgroups of the restricted Lorentz group, up to conjugacy. We can readily express the result in terms of the generating set given in the table above.
The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
The two-dimensional subalgebras are:
The three-dimensional subalgebras are:
The four-dimensional subalgebras are all conjugate to
The subalgebras form a lattice, and each subalgebra generates by exponentiation a closed 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, or homogeneous spaces, have considerable mathematical interest. A few, brief descriptions:
In [|a previous section], we constructed a homomorphism, which we called the spinor map. Since is simply connected, it is the universal covering group of the restricted Lorentz group. By restriction we obtain a homomorphism. Here, the special unitary group SU, which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO. Each of these covering 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 are doubly connected. This means that the fundamental group of the each group is isomorphic to the two-element cyclic group Z2.
Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings
we have the double coverings
These spinorial double coverings are all closely related to Clifford algebras.

Topology

The left and right groups in the double covering
are deformation retracts of the left and right groups, respectively, in the double covering
But the homogeneous space SO+/SO is homeomorphic to hyperbolic 3-space H3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO and base H3. Since the latter is homeomorphic to R3, while SO is homeomorphic to three-dimensional real projective space RP3, we see that the restricted Lorentz group is locally homeomorphic to the product of RP3 with R3. Since the base space is contractible, this can be extended to a global homeomorphism.

Generalization to higher dimensions

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 the group O of linear transformations of Rn+1 that preserves the quadratic form
Many of the properties of the Lorentz group in four dimensions generalize straightforwardly to arbitrary n. For instance, the Lorentz group O has four connected components, and it acts by conformal transformations on the celestial -sphere in n+1-dimensional Minkowski space. The identity component SO+ is an SO-bundle over hyperbolic n-space Hn.
The low-dimensional cases and are often useful as "toy models" for the physical case, while higher-dimensional Lorentz groups are used in physical theories such as string theory that posit the existence of hidden dimensions. The Lorentz group O is also the isometry group of n-dimensional de Sitter space dSn, which may be realized as the homogeneous space O/O. In particular O is the isometry group of the de Sitter universe dS4, a cosmological model.