Lorentz Group - Connected Components

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

• reverse the direction of time (or more precisely, transform a future-pointing timelike vector into a past-pointing one),
• reverse the orientation of a vierbein (tetrad).

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.