x 0x 1 x 2
x 0 = 0
(plane)|v|^2 < 0
(timelike)|v|^2 = 0
(light cone)|v|^2 > 0
(spacelike)Figure 40.1: Light cone structure of Minkowski spacetime.40.2 The Lorentz group and its Lie algebra
Recall that in 3 dimensions the group of linear transformations ofR^3 pre-
serving the standard inner product was the groupO(3) of 3 by 3 orthogonal
matrices. This group has two disconnected components:SO(3), the subgroup
of orientation preserving (determinant +1) transformations, and a component
of orientation reversing (determinant−1) transformations. In Minkowski space,
one has:
Definition(Lorentz group). The Lorentz groupO(3,1)is the group of linear
transformations preserving the Minkowski space inner product onR^4.
In terms of matrices, the condition for a 4 by 4 matrix Λ to be inO(3,1)
will be
ΛT
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Λ =
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
The Lorentz group has four components, with the component of the iden-
tity a subgroup calledSO(3,1) (which some callSO+(3,1)). The other three