Lorentz Group - Conjugacy Classes

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 (e.g., 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

which has fixed points . Writing out the action and collecting terms, we find that our spinor map takes this to the (restricted) Lorentz transformation

This transformation represents a rotation about the z axis. The one-parameter subgroup it generates is obtained by simply taking 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 increases.

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

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

which also has fixed points . 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

This transformation represents a boost along the z axis. The one-parameter subgroup it generates is obtained by simply taking 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

which also has fixed points . Our homomorphism maps this to the Lorentz transformation

The one-parameter subgroup this generates is obtained by replacing with any real multiple of this complex constant. (If we let 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

which has the single fixed point 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)

This generates a one-parameter subgroup which is obtained by considering 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

Differentiating this transformation with respect to the group parameter and evaluate at, we read off the corresponding vector field (first order linear partial differential operator)

Apply this to an undetermined function . 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 on the right hand side now give three rational invariants describing how points (events) move under our parabolic transformation:

(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 with a hyperboloid . In particular, the reader may wish to sketch the case, 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 .

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 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, or by an appropriate orientation reversing diagonal matrix.