Master Dissertation

Clearly the composition of Lorentz transformations is itself a Lorentz
transformation. And since
1 = − det g = − det(Λ T gΛ) = − det Λ det g det Λ = (det Λ) 2 ,
we must have det Λ = ±1. Thus all Lorentz transformations are invertible
and the set of all Lorentz transformations form group called the Lorentz
Group L, also denoted O(3, 1).
Further we denote the group of Lorentz transformations Λ with det Λ = 1
by L+ = SO(1, 3). From equation (2.3) we get 10 independent quadratic
equations for the components of Λ. The first is
Which implies that either
When Λ 0 0
(Λ 0 0) 2 − (Λ 1 0) 2 − (Λ 2 0) 2 − (Λ 3 0) 2 = 1. (2.4)
Λ 0 0 ≥ 1 or Λ 0 0 ≤ −1. (2.5)
≥ 1 time is not reversed and we call the subgroup defined by
the Proper Lorentz Group.
L ↑
+ := {Λ ∈ L+|Λ 0 0 ≥ 1},
The Lorentz group is normally split into 4 disjoint classes. Namely, L ↑
+
and the 3 following
Name det Λ Λ 0 0
L ↑
− −1 ≥ +1
L ↓
− −1 ≤ −1
L ↓
+ +1 ≤ −1
Further we mention 3 important discrete Lorentz transformations, namely:
I, the identity, P = g, space inversion or the parity transform, and
T = −g, time inversion.
There are 3 types of vectors in M. A vector x is said to be time-like if
x 2 > 0, space-like if x 2 < 0 and light-like if x 2 = 0. Since x 2 is constant
under Lorentz transformation the categories stay the same under the
transformation.
Information cannot propagate faster than the speed of light. Equivalently,
two events at x µ and y ν cannot influence each other if they are separated
by a space-like distance,
(x − y) 2 < 0.
They have no causal relation. Causality plays a key role in this thesis.
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