Master Dissertation

Definition 2.2. Given some point x the forward cone of x is defined by
and the backwards cone of x is
V + (x) = {y|(y − x) 2 ≥ 0 and y 0 ≥ x 0 },
V − (x) = {y|(y − x) 2 ≥ 0 and y 0 ≤ x 0 }.
The n-dimensional generalizations are
Γ ± n (x) = {(x1, . . . , xn)|xj ∈ V ± (x) and ∀j = 1 = 1, . . . , n}.
That (y − x) 2 ≥ 0 just means that they have to be causally connected.
Later we will need a symbol to distinguish points in time.
Definition 2.3. Let A, B ⊂ M. If for all x ∈ A and y ∈ B we have
x 0 < y 0 , we write A < B.
2.2 The Poincaré Group
Definition 2.4. Let Λ ∈ L and a ∈ R 4 . By a Poincaré transformation
Π : R 4 → R 4 we mean Π(x) = Λx + a, and we write Π = (a, Λ).
The set P of all Poincaré transformations form a group under the
composition law
(a1, Λ1)(a2, Λ2) = (a1 + Λ1a2, Λ1Λ2).
We see that the Poincaré Group P is the semidirect product of L and the
group of space-time translations (R 4 , +), that is
P = R 4 ⊙ L
Further we define the Proper Poincaré Group as
P ↑
+ = R4 ⊙ L ↑
+
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