Definitions

space inversion

Lorentz group

In physics (and mathematics), the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all (nongravitational) physical phenomena. The mathematical form of

are each invariant under Lorentz transformations. Therefore the Lorentz group can be said to express a fundamental symmetry of many of the known fundamental laws of nature.

Basic properties

The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski spacetime. The Lorentz transformations are precisely the isometries which 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(1,3), the matrix Lie group which preserves the quadratic form

(t,x,y,z) mapsto t^2-x^2-y^2-z^2
on R4. This quadratic form is interpreted in physics as the metric tensor of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime.

The Lorentz group is a 6-dimensional noncompact Lie group which is not connected, and whose connected components are not simply connected. The identity component (i.e. the component containing the identity element) of the Lorentz group is often called the restricted Lorentz group and is denoted SO+(1,3).

In pure mathematics, the restricted Lorentz group arises in another guise as the Möbius group, which is the symmetry group of conformal geometry on the Riemann sphere. This observation was taken by Roger Penrose as the starting point of twistor theory. It has a fascinating physical consequence for the appearance of the night sky as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars", which is discussed below.

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.

Note: the Lorentz group also preserves the quadratic form (t,x,y,z) mapsto x^2+y^2+z^2-t^2 and is therefore sometimes denoted O(3,1). A similar remark applies to its identity component and the subgroups introduced below.

Connected components

Because it is a Lie group, the Lorentz group O(1,3) is both a group and a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.

To see why, notice that a Lorentz transformation may or may not

Lorentz transformations which preserve the direction of time are called orthochronous. Those which preserve orientation are called proper, and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1,3). The subgroup of orthochronous transformations is often denoted O+(1,3).

The identity component of the Lorentz group is the set of all Lorentz transformations preserving both orientation and the direction of time. It is called the proper, orthochronous Lorentz group, or restricted Lorentz group, and it is denoted by SO+(1, 3). It is a normal subgroup of the Lorentz group which is also six dimensional.

Note: Some authors refer to SO(1,3) or even O(1,3) when they actually mean SO+(1, 3).

The quotient group O(1,3)/SO+(1,3) is isomorphic to the Klein four-group. Every element in O(1,3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group

{1, P, T, PT}
where P and T are the space inversion and time reversal operators:
P = diag(1, −1, −1, −1)
T = diag(−1, 1, 1, 1)
The four elements of this isomorphic copy of the Klein four-group label the four connected components of the Lorentz group.

The restricted Lorentz group

As stated above, the restricted Lorentz group is the identity component of the Lorentz group. This means that it consists of all Lorentz transformations which 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, 6 dimensions).

The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction). The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). 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, each generate a one-parameter subgroup. An arbitrary rotation is specified by 3 real parameters, as is an arbitrary boost. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real numbers (parameters) to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six dimensional. (We'll study this in more detail in a later section on the Lie algebra of the Lorentz group.) To specify an arbitrary Lorentz transformation requires a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite dimensional Lie groups.

Relation to the Möbius group

The restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group, which is, in turn, isomorphic to the projective special linear group PSL(2,C). It will be convenient to work at first with SL(2,C). This group consists of all two by two complex matrices with determinant one

P = left[begin{matrix} a & b c & d end{matrix} right], ; ad - bc = 1
We can write two by two Hermitian matrices in the form
X = left[begin{matrix} t+z & x-iy x+iy & t-z end{matrix} right]
This trick has the pleasant feature that
det , X = t^2 - x^2 - y^2 - z^2
Therefore, we have identified the space of Hermitian matrices (which is four dimensional, as real vector space) 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. But now SL(2,C) acts on the space of Hermitian matrices via
X rightarrow P X P^*
where P^* is the Hermitian transpose of P, and this action preserves the determinant. Therefore, SL(2,C) acts on Minkowski spacetime by (linear) isometries. We have now constructed a homomorphism of Lie groups from SL(2,C) onto SO+(1,3), which we will call the spinor map. The kernel of the spinor map is the two element subgroup ±I. Therefore, the quotient group PSL(2,C) is isomorphic to SO+(1,3).

Appearance of the night sky

This isomorphism has a very interesting physical interpretation. We can identify the complex number

xi = u+iv
with a null vector in Minkowski space
left[begin{matrix} u^2+v^2+1 2u -2v u^2+v^2-1 end{matrix} right]
or the Hermitian matrix
N = 2left[begin{matrix} u^2+v^2 & u+iv u-iv & 1 end{matrix} right]

The set of real scalar multiples of this null vector, which we can call a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event which we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars.

But by stereographic projection, we can identify xi with a point on the Riemann sphere. Putting it all together, we have identified the points of the celestial sphere with certain Hermitian matrices, and also with lines of sight. This implies that the Möbius transformations of the Riemann sphere precisely represent the way that Lorentz transformations change the appearance of the celestial sphere.

For our purposes here, we can pretend that the "fixed stars" live in Minkowski spacetime. Then, the Earth is moving at a nonrelativistic velocity with respect to a typical astronomical object which might be visible at night. But, an observer who is moving at relativistic velocity with respect to the Earth would see the appearance of the night sky (as modeled by points on the celestial sphere) transformed by a Möbius transformation.

Conjugacy classes

Because the restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group PSL(2,C), its conjugacy classes also fall into four classes:

  • elliptic transformations
  • hyperbolic transformations
  • loxodromic transformations
  • parabolic transformations

(To be utterly pedantic, the identity element is in a fifth class, all by itself.)

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.

We will discuss a particularly simple example of each type, and in particular, the effect on the appearance of the night sky of the one-parameter subgroup which it generates. At the end of the section we will briefly indicate how we can understand the effect of general Lorentz transformations on the appearance of the night sky in terms of these examples.

A typical elliptic element of SL(2,C) is

P_1 = left[begin{matrix} exp(i theta/2) & 0 0 & exp(-i theta/2) end{matrix} right]
which has fixed points xi = 0, infty. Writing out the action X rightarrow P_1 X {P_1}^* and collecting terms, we find that our spinor map takes this to the (restricted) Lorentz transformation
Q_1 = left[begin{matrix} 1 & 0 & 0 & 0
                                   0 & cos(theta) & -sin(theta) & 0 
                                   0 & sin(theta) &  cos(theta) & 0 
0 & 0 & 0 & 1 end{matrix} right] This transformation represents a rotation about the z axis. The one-parameter subgroup it generates is obtained by simply taking theta 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 two fixed points, the North and South pole. They move all other points around latitude circles. In other words, this group yields a continuous counterclockwise rotation about the z axis as theta increases.

Notice the angle doubling; this phenomenon is a characteristic feature of spinorial double coverings.

A typical hyperbolic element of SL(2,C) is

P_2 = left[begin{matrix} exp(beta/2) & 0 0 & exp(-beta/2) end{matrix} right]
which also has fixed points xi = 0, infty. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin. Our homomorphism maps this to the Lorentz transformation
Q_2 = left[begin{matrix} cosh(beta) & 0 & 0 & sinh(beta)
                                   0            & 1 & 0 & 0 
                                   0            & 0 & 1 & 0 
sinh(beta) & 0 & 0 & cosh(beta) end{matrix} right] This transformation represents a boost along the z axis. The one-parameter subgroup it generates is obtained by simply taking beta 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 along longitudes away from the South pole and toward the North pole.

A typical loxodromic element of SL(2,C) is

P_3 = P_2 P_1 = P_1 P_2
= left[begin{matrix} exp left((beta+itheta)/2 right) & 0
                                   0                                    & exp left(-(beta+itheta)/2 right)
end{matrix} right] which also has fixed points xi = 0, infty. Our homomorphism maps this to the Lorentz transformation
Q_3 = Q_2 Q_1 = Q_1 Q_2
The one-parameter subgroup this generates is obtained by replacing beta + i theta with any real multiple of this complex constant. (If we let beta,theta vary independently, we obtain a two-dimensional abelian subgroup, consisting of simultaneous rotations about the z axis and boosts along the z axis; in contrast, the one-dimensional subgroup we are discussing here consists of those elements of this two-dimensional subgroup such that the rapidity of the boost and angle of the rotation have a fixed ratio.) The corresponding continuous transformations of the celestial sphere (always 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 called loxodromes. Each loxodrome spirals infinitely often around each pole.

A typical parabolic element of SL(2,C) is

P_4 = left[begin{matrix} 1 & alpha 0 & 1 end{matrix} right]
which has the single fixed point xi=infty on the Riemann sphere. Under stereographic projection, it appears as ordinary translation along the real axis. Our homomorphism maps this to the matrix (representing a Lorentz transformation)
Q_4 = left[begin{matrix} 1+alpha^2/2 & alpha & 0 & -alpha^2/2
                             alpha       & 1      & 0 & -alpha      
                             0            & 0      & 1 & 0            
alpha^2/2 & alpha & 0 & 1-alpha^2/2 end{matrix} right] This generates a one-parameter subgroup which is obtained by considering alpha to be a real variable rather than a constant. The corresponding continuous transformations of the celestial sphere move points along a family of circles which are all tangent at the North pole to a certain great circle. All points other than the North pole itself move along these circles. (Except, of course, for the identity transformation.)

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 (elliptic, hyperbolic, loxodromic, parabolic), we will show how to determine the effect of our example of a parabolic Lorentz transformation on Minkowski spacetime, leaving the other examples as exercises for the reader. From the matrix given above we can read off the transformation

left[begin{matrix} t x y z end{matrix} right] rightarrow left[begin{matrix} t x y z end{matrix} right] + alpha ; left[begin{matrix} x t-z 0 x end{matrix} right] + frac{alpha^2}{2} ; left[begin{matrix} t-z 0 0 t-z end{matrix} right] Differentiating this transformation with respect to the group parameter alpha and evaluate at alpha=0, we read off the corresponding vector field (first order linear partial differential operator)
x , left(partial_t + partial_z right) + (t-z) , partial_x
Apply this to an undetermined function f(t,x,y,z). The solution of the resulting first order linear partial differential equation can be expressed in the form
f(t,x,y,z) = F(y, , t-z , , t^2-x^2-z^2)
where F is an arbitrary smooth function. The arguments on the right hand side now give three rational invariants describing how points (events) move under our parabolic transformation:
y = c_1, ; t-z = c_2, ; t^2-x^2-z^2 = c_3
(The reader can verify that these quantities standing on the left hand sides are invariant under our transformation.) Choosing real values for the constants standing on the right hand sides gives three conditions, and thus defines a curve in Minkowski spacetime. This curve is one of the flowlines of our transformation. We see from the form of the rational invariants that these flowlines (or orbits) have a very simple description: suppressing the inessential coordinate y, we see that each orbit is the intersection of a null plane t = z+c_2 with a hyperboloid t^2-x^2-z^2 = c_3. In particular, the reader may wish to sketch the case c_3 = 0, in which the hyperboloid degenerates to a light cone; then orbits are parabolas lying in null planes just mentioned.

Parabolic transformations lead to the gauge symmetry of massless particles with helicity |h|geq 1.

Notice that a particular null line lying in the light cone is left invariant; this corresponds to the unique (double) fixed point on the Riemann sphere which was 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 alpha increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.

The Möbius transformations are precisely the conformal transformations of the Riemann sphere (or celestial sphere). It follows that by conjugating with an arbitrary element of SL(2,C), we can obtain from the above examples arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the flow lines of the corresponding one-parameter subgroups is to transform the pattern seen in our examples by some conformal transformation. Thus, an arbitrary elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points will still flow along circular arcs from one fixed point toward the other. Similarly for the other cases.

Finally, arbitrary Lorentz transformations can be obtained from the restricted ones by multiplying by a matrix which reflects across t=0, or by an appropriate orientation reversing diagonal matrix.

The Lie algebra of the Lorentz group

As with any Lie group, the best way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group is SO^+(1,3), its Lie algebra is reducible and can be decomposed to two copies of the Lie algebra of SL(2,R), as will be shown explicitly below (this is the Minkowski space analog of the SO(4) rightarrow SU(2)timesSU(2) decomposition in a Euclidean space). In particle physics, a state that is invariant under one of these copies of SL(2,R) is said to have chirality, and is either left-handed or right-handed, according to which copy of SL(2,R) it is invariant under.

The Lorentz group is a subgroup of the diffeomorphism group of R4 and therefore its Lie algebra can identified with vector fields on R4. In particular, the vectors which 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 immediately write down an obvious set of six generators:

  • vector fields on R4 generating three rotations

-y partial_x + x partial_y, ; -z partial_y + y partial_z, ; -x partial_z + z partial_x

  • vector fields on R4 generating three boosts

x partial_t + t partial_x, ; y partial_t + t partial_y, ; z partial_t + t partial_z

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

-y partial_x + x partial_y
The corresponding initial value problem is
frac{partial x}{partial lambda} = -y, ; frac{partial y}{partial lambda} = x, ; x(0) = x_0, ; y(0) = y_0
The solution can be written
x(lambda) = x_0 cos(lambda) - y_0 sin(lambda), ; y(lambda) = x_0 sin(lambda) + y_0 cos(lambda)
or
left[begin{matrix} t x y z end{matrix} right]
= left[begin{matrix} 1 & 0 & 0 & 0
                       0 & cos(lambda) & -sin(lambda) & 0 
                       0 & sin(lambda) & cos(lambda)  & 0 
0 & 0 & 0 & 1 end{matrix} right] left[begin{matrix} t_0 x_0 y_0 z_0 end{matrix} right] where we easily recognize the one-parameter matrix group of rotations about the z axis. Differentiating with respect to the group parameter and setting lambda=0 in the result, we recover the matrix
left[begin{matrix} 0 & 0 & 0 & 0 0 & 0 & -1 & 0 0 & 1 & 0 & 0 0 & 0 & 0 & 0 end{matrix} right]
which corresponds to the vector field we started with. This shows 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 which correspond to our six generators arise from exponentiating respectively frac{theta}{2} (for the three boosts) or frac{i theta}{2} (for the three rotations) times the three Pauli matrices

sigma_1 = left[begin{matrix} 0 & 1 1 & 0 end{matrix} right], ; ;
sigma_2 = left[begin{matrix} 0 & -i i & 0 end{matrix} right], ; ; sigma_3 = left[begin{matrix} 1 & 0 0 & -1 end{matrix} right]

For our purposes, another generating set is more convenient. We list the six generators in the following table, in which

  • the first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a real 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 partial_z direction),
  • three elliptics (rotations about x,y,z axes respectively).

Vector field on R2 One-parameter subgroup of SL(2,C),
representing Möbius transformations
One-parameter subgroup of SO+(1,3),
representing Lorentz transformations
Vector field on R4
Parabolic
partial_u,!

left[begin{matrix} 1 & alpha 0 & 1 end{matrix} right]

left[begin{matrix} 1+alpha^2/2 & alpha & 0 & -alpha^2/2
                               alpha        & 1      & 0 & -alpha     
                               0             & 0      & 1 & 0           
alpha^2/2 & alpha & 0 & 1-alpha^2/2 end{matrix} right]

X_1 = ,!
x (partial_t + partial_z) + (t-z) partial_x ,!
partial_v,!

left[begin{matrix} 1 & i alpha 0 & 1 end{matrix} right]

left[begin{matrix} 1+alpha^2/2 & 0 & alpha & -alpha^2/2
                               0       & 1 & 0      & 0      
                               alpha            & 0 & 1      & -alpha            
alpha^2/2 & 0 & alpha & 1-alpha^2/2 end{matrix} right]

X_2 = ,!
y (partial_t + partial_z) + (t-z) partial_y ,!
Hyperbolic
frac{1}{2} left(u partial_u + v partial_v right)

left[begin{matrix} exp left(frac{beta}{2}right) & 0 0 & exp left(-frac{beta}{2}right) end{matrix} right]

left[begin{matrix} cosh(beta) & 0 & 0 & sinh(beta)
                               0            & 1 & 0 & 0            
                               0            & 0 & 1 & 0            
sinh(beta) & 0 & 0 & cosh(beta) end{matrix} right]

X_3 = ,!
z partial_t + t partial_z ,!
Elliptic
frac{1}{2} left(-v partial_u + u partial_v right)

left[begin{matrix} exp left(frac{i theta}{2} right) & 0 0 & exp left(frac{-i theta}{2} right) end{matrix} right]

left[begin{matrix} 1 & 0 & 0 & 0
                               0 & cos(theta) & -sin(theta) & 0 
                               0 & sin(theta) &  cos(theta) & 0 
0 & 0 & 0 & 1 end{matrix} right]

X_4 = ,!
-y partial_x + x partial_y ,!
frac{v^2-u^2-1}{2} partial_u - u v , partial_v

left[begin{matrix} cos left(frac{theta}{2} right) & -sin left(frac{theta}{2} right) sin left(frac{theta}{2} right) & cos left(frac{theta}{2} right) end{matrix} right]

left[begin{matrix} 1 & 0 & 0 & 0
                               0 & cos(theta)  & 0 & sin(theta) 
                               0 & 0             & 1 & 0            
0 & -sin(theta) & 0 & cos(theta) end{matrix} right]

X_5 = ,!
-x partial_z + z partial_x ,!
u v , partial_u + frac{1-u^2+v^2}{2} partial_v

left[begin{matrix} cos left(frac{theta}{2} right) & i sin left(frac{theta}{2} right) i sin left(frac{theta}{2} right) & cos left(frac{theta}{2} right) end{matrix} right]

left[begin{matrix} 1 & 0 & 0 & 0
                               0 & 1 & 0            & 0 
                               0 & 0 & cos(theta) & -sin(theta) 
0 & 0 & sin(theta) & cos(theta) end{matrix} right]

X_6 = ,!
-z partial_y + y partial_z ,!

Let's verify one line in this table. Start with

sigma_2 = left[begin{matrix} 0 & i -i & 0 end{matrix} right]
Exponentiate:
exp left(frac{ i theta}{2} , sigma_2 right) =
left[begin{matrix} cos(theta/2) & -sin(theta/2) sin(theta/2) & cos(theta/2) end{matrix} right] This element of SL(2,C) represents the one-parameter subgroup of (elliptic) Möbius transformations:
xi rightarrow frac{ cos(theta/2) , xi - sin(theta/2) }{ sin(theta/2) , xi + cos(theta/2) }
Next,
frac{dxi}{dtheta} |_{theta=0} = -frac{1+xi^2}{2}
The corresponding vector field on C (thought of as the image of S2 under stereographic projection) is
-frac{1+xi^2}{2} , partial_xi
Writing xi = u + i v, this becomes the vector field on R2
-frac{1+u^2-v^2}{2} , partial_u - u v , partial_v
Returning to our element of SL(2,C), writing out the action X rightarrow P X P^* and collecting terms, we find that the image under the spinor map is the element of SO+(1,3)
left[begin{matrix} 1 & 0 & 0 & 0
                               0 & cos(theta)  & 0 & sin(theta) 
                               0 & 0             & 1 & 0            
0 & -sin(theta) & 0 & cos(theta) end{matrix} right] Differentiating with respect the theta at theta=0, we find that the corresponding vector field on R4 is
z partial_x - x partial_z ,!
This is evidently the generator of counterclockwise rotation about the y 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. (See the book by Hall cited below for the details.) 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:

  • X_1 generates a one-parameter subalgebra of parabolics SO(0,1),
  • X_3 generates a one-parameter subalgebra of boosts SO(1,1),
  • X_4 generates a one-parameter of rotations SO(2),
  • X_3 + a X_4 (for any a neq 0) generates a one-parameter subalgebra of loxodromic transformations.

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

  • X_1, X_2 generate an abelian subalgebra consisting entirely of parabolics,
  • X_1, X_3 generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group A(1),
  • 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 are:

  • X_1,X_2,X_3 generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties,
  • X_1,X_2,X_4 generate a Bianchi VII_0 subalgebra, isomorphic to the Lie algebra of E(2), the euclidean group,
  • X_2,X_2,X_3 + a X_4, where a neq 0, generate a Bianchi VII_a subalgebra,
  • X_1,X_3,X_5 generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2,R), the group of isometries of the hyperbolic plane,
  • X_4,X_5,X_6 generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group.

(Here, the Bianchi types refer to the classification of three dimensional Lie algebras by the Italian mathematician Luigi Bianchi.) The four dimensional subalgebras are all conjugate to

  • X_1,X_2,X_3,X_4 generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean similitudes.

The subalgebras form a lattice (see the figure), 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. We briefly describe some of them here:

  • the group Sim(2) is the stabilizer of a null line, i.e. of a point on the Riemann sphere, so the homogeneous space SO+(1,3)/Sim(2) is the Kleinian geometry which represents conformal geometry on the sphere S2,
  • the (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO+(1,3)/SE(2) is the momentum space of a massless particle; geometrically, this Kleinian geometry represents the degenerate geometry of the light cone in Minkowski spacetime,
  • the rotation group SO(3) is the stabilizer of a timelike vector, so the homogeneous space SO+(1,3)/SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H3.

Covering groups

In a previous section we constructed a homomorphism SL(2,C) rightarrow SO+(1,3), which we called the spinor map. Since SL(2,C) is simply connected, it is the covering group of the restricted Lorentz group SO+(1,3). By restriction we obtain a homomorphism SU(2) rightarrow SO(3). Here, the special unitary group SU(2), 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(3). 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.

Warning: in applications to quantum mechanics the special linear group SL(2, C) is sometimes called the Lorentz group.

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

Spin+(1,3)=SL(2,C) rightarrow SO+(1,3)
Spin(3)=SU(2) rightarrow SO(3)
we have the double coverings
Pin(1,3) rightarrow O(1,3)
Spin(1,3) rightarrow SO(1,3)
Spin+(1,2) = SU(1,1) rightarrow SO(1,2)

These spinorial double coverings are all closely related to Clifford algebras.

Topology

The left and right groups in the double covering

SU(2) rightarrow SO(3)
are deformation retracts of the left and right groups, respectively, in the double covering
SL(2,C) rightarrow SO+(1,3)
But the homogeneous space SO+(1,3)/SO(3) is homeomorphic to hyperbolic 3-space H3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H3. Since the latter is homeomorphic to R3, while SO(3) 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.

General dimensions

The concept of the Lorentz group has a natural generalization to any spacetime dimension. Mathematically, the Lorentz group of n+1 dimensional Minkowski space is the group O(n,1) (or O(1,n)) of linear transformations of Rn+1 which preserve the quadratic form

(x_1,x_2,ldots ,x_n,x_{n+1})mapsto x_1^2+x_2^2+cdots +x_n^2-x_{n+1}^2.
Many of the properties of the Lorentz group in four dimensions (n=3) generalize straightforwardly to arbitrary n. For instance, the Lorentz group O(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 component SO+(n,1) is an SO(n)-bundle over hyperbolic n-space Hn.

The low dimensional cases n=1 and n=2 are often useful as "toy models" for the physical case n=3, while higher dimensional Lorentz groups are used in physical theories such as string theory which posit the existence of hidden dimensions. The Lorentz group O(n,1) is also the isometry group of n-dimensional de Sitter space dSn, which may be realized as the homogeneous space O(n,1)/O(n-1,1). In particular O(4,1) is the isometry group of the de Sitter universe dS4, a cosmological model.

See also

References

  • Artin, Emil (1957). Geometric Algebra. New York: Wiley. ISBN 0-471-60839-4. See Chapter III for the orthogonal groups O(p,q).
  • Carmeli, Moshe (1977). Group Theory and General Relativity, Representations of the Lorentz Group and Their Applications to the Gravitational Field. McGraw-Hill, New York. ISBN 0-07-009986-3. A canonical reference; see chapters 1-6 for representations of the Lorentz group.
  • Frankel, Theodore (2004). The Geometry of Physics (2nd Ed.). Cambridge: Cambridge University Press. ISBN 0-521-53927-7. An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics.
  • See Lecture 11 for the irreducible representations of SL(2,C).
  • Hall, G. S. (2004). Symmetries and Curvature Structure in General Relativity. Singapore: World Scientific. ISBN 981-02-1051-5. See Chapter 6 for the subalgebras of the Lie algebra of the Lorentz group.
  • Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79540-0. See also the See Section 1.3 for a beautifully illustrated discussion of covering spaces. See Section 3D for the topology of rotation groups.
  • Naber, Gregory (1992). The Geometry of Minkowski Spacetime. New York: Springer-Verlag. ISBN 0-486-43235-1 (Dover reprint edition). An excellent reference on Minkowski spacetime and the Lorentz group.
  • Needham, Tristam (1997). Visual Complex Analysis. Oxford: Oxford University Press. ISBN 0-19-853446-9. See Chapter 3 for a superbly illustrated discussion of Möbius transformations.

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