Lorentz Group - Basic Properties

Basic Properties

The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski space-time. 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

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 six-dimensional noncompact non-abelian real Lie group which is not connected. All four of its connected components are not simply connected. The identity component (i.e. the component containing the identity element) of the Lorentz group is itself a group and is often called the restricted Lorentz group and is denoted SO+(1,3). The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time. The restricted Lorentz group has often been presented through a facility of biquaternion algebra.

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.