Because the restricted Lorentz group is isomorphic to the Möbius group , its
conjugacy classes also fall into five classes: •
Elliptic transformations •
Hyperbolic transformations •
Loxodromic transformations •
Parabolic transformations • The trivial
identity transformation 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 (e.g., on the appearance of the night sky). The Möbius transformations are the
conformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of obtains the following examples of 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 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 is : P_1 = \begin{bmatrix} \exp\left(\frac{i}{2}\theta\right) & 0 \\ 0 & \exp\left(-\frac{i}{2}\theta\right) \end{bmatrix} and has fixed points = 0, ∞. Writing the action as and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation : Q_1 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & \sin(\theta) & 0 \\ 0 & -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \exp\left(\theta\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right) ~. 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 (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 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 is : P_2 = \begin{bmatrix} \exp\left(\frac{\eta}{2}\right) & 0 \\ 0 & \exp\left(-\frac{\eta}{2}\right) \end{bmatrix} 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 : Q_2 = \begin{bmatrix} \cosh(\eta) & 0 & 0 & \sinh(\eta) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh(\eta) & 0 & 0 & \cosh(\eta) \end{bmatrix} = \exp \left( \eta \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}\right)~. 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 (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.
Loxodromic A loxodromic element of is : P_3 = P_2 P_1 = P_1 P_2 = \begin{bmatrix} \exp \left(\frac{1}{2}(\eta + i\theta)\right) & 0 \\ 0 & \exp \left(-\frac{1}{2}(\eta + i\theta)\right) \end{bmatrix} and has fixed points = 0, ∞. The spinor map converts this to the Lorentz transformation : Q_3 = Q_2 Q_1 = Q_1 Q_2 = \begin{bmatrix} \cosh(\eta) & 0 & 0 & \sinh(\eta) \\ 0 & \cos(\theta) & \sin(\theta) & 0 \\ 0 & -\sin(\theta) & \cos(\theta) & 0 \\ \sinh(\eta) & 0 & 0 & \cosh(\eta) \end{bmatrix} = \exp\begin{bmatrix} 0 & 0 & 0 & \eta \\ 0 & 0 & \theta & 0 \\ 0 & -\theta & 0 & 0 \\ \eta & 0 & 0 & 0 \end{bmatrix} ~. The one-parameter subgroup this generates is obtained by replacing with any real multiple of this complex constant. (If , vary independently, then a
two-dimensional abelian subgroup is obtained, consisting of simultaneous rotations about the axis and boosts along the -axis; in contrast, the
one-dimensional subgroup discussed 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 (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.
Parabolic A parabolic element of is : P_4 = \begin{bmatrix} 1 & \alpha \\ 0 & 1 \end{bmatrix} 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 (representing a Lorentz transformation) : \begin{align} Q_4 &= \begin{bmatrix} 1 + \frac{1}{2}\vert\alpha\vert^2 & \operatorname{Re}(\alpha) & -\operatorname{Im}(\alpha) & -\frac{1}{2}\vert\alpha\vert^2 \\ \operatorname{Re}(\alpha) & 1 & 0 & -\operatorname{Re}(\alpha) \\ -\operatorname{Im}(\alpha) & 0 & 1 & \operatorname{Im}(\alpha) \\ \frac{1}{2}\vert\alpha\vert^2 & \operatorname{Re}(\alpha) & -\operatorname{Im}(\alpha) & 1 - \frac{1}{2}\vert\alpha\vert^2 \end{bmatrix} \\[6pt] &= \exp \begin{bmatrix} 0 & \operatorname{Re}(\alpha) & -\operatorname{Im}(\alpha) & 0 \\ \operatorname{Re}(\alpha) & 0 & 0 & -\operatorname{Re}(\alpha) \\ -\operatorname{Im}(\alpha) & 0 & 0 & \operatorname{Im}(\alpha) \\ 0 & \operatorname{Re}(\alpha) & -\operatorname{Im}(\alpha) & 0 \end{bmatrix} ~. \end{align} 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 (except for the identity transformation) 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 (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 : \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix} \rightarrow \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix} + \operatorname{Re}(\alpha) \; \begin{bmatrix} x \\ t - z \\ 0 \\ x \end{bmatrix} - \operatorname{Im}(\alpha) \; \begin{bmatrix} y \\ 0 \\ z - t \\ y \end{bmatrix} + \frac{\vert\alpha\vert^2}{2} \; \begin{bmatrix} t - z \\ 0 \\ 0 \\ t - z \end{bmatrix}. Now,
without loss of generality, pick . Differentiating this transformation with respect to the now real group parameter and evaluating at produces the corresponding vector field (first order linear partial differential operator), : x\, \left( \partial_t + \partial_z \right) + (t - z)\, \partial_x. 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 : f(t, x, y, z) = F\left(y,\, t - z,\, t^2 - x^2 - z^2\right), where is an
arbitrary smooth function. The arguments of give three
rational invariants describing how points (events) move under this parabolic transformation, as they themselves do not move, : 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 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 (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 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 (such as
photons) with
helicity || ≥ 1. In the above explicit example, a massless particle moving in the direction, so with 4-momentum , is not affected at all by the -boost and -rotation combination 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 of have disappeared. , in the special case discussed. (The other similar generator, as well as it and comprise altogether the little group of the light-like vector, isomorphic to .)
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 (a
light-like vector) in Minkowski space : \begin{bmatrix} u^2 + v^2 + 1 \\ 2u \\ -2v \\ u^2 + v^2 - 1 \end{bmatrix} or, in the Weyl representation (the spinor map), the Hermitian matrix : N = 2\begin{bmatrix} u^2 + v^2 & u + iv \\ u - iv & 1 \end{bmatrix}. 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 (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the
celestial sphere (equivalently, lines of sight) are identified with certain Hermitian matrices.
Projective geometry and different views of the 2-sphere This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on the
projective celestial sphere. This is the space of non-zero null vectors with t>0 under the given quotient for projective spaces: (t,x,y,z)\sim (t',x',y',z') if (t',x',y',z') = (\lambda t, \lambda x, \lambda y, \lambda z) for \lambda > 0. This is referred to as the celestial sphere as this allows us to rescale the time coordinate t to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere. From the Möbius side, acts on complex projective space , 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 group . Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If \xi is a projective vector, it can be tensored with its Hermitian conjugate to produce a 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 the
Bloch sphere. == Lie algebra ==