CHAPTER 2
Preliminaries
2.1. Spacetimes and c-boundaries
We will use typical background and terminology in Lorentzian Geometry as
in [4, 27, 29]. A spacetime will be a time-oriented connected smooth Lorentzian
manifold (V,g) (the time-orientation, and so, the choice of a future cone at each
tangent space, will be assumed implicitly) of signature (−,+,...,+). A tangent
vector v TpV , p V is called timelike (resp. lightlike; causal) if g(v,v) 0 (resp.
g(v,v) = 0, v = 0; v is either timelike or lightlike); null vectors include both, the
lightlike ones and the 0 vector. A causal vector is called future or past-directed if it
belongs to the future or past cone. Accordingly, a smooth curve γ : I V (I real
interval) is called timelike, lightlike, causal and future or past-directed if so is ˙(s) γ
for all s I.
Two events p,q V are chronologically related p q (resp. causally related
p q) if there exists some future-directed timelike (resp. either future-directed
causal or constant) curve from p to q. If p = q and p q but p q, then p is said
horismotically related to q. The chronological past (resp. future) of p,
I−(p)
(resp.
I+(p))
is defined as:
I−(p)
= {q V : q p} (resp.
I+(p)
= {q V : p q}).
In what follows, we will be especially interested in the chronological past
I−[γ]
=
∪s∈I
I−(γ(s))
(resp. future
I+[γ]
= ∪s∈I
I+(γ(s)))
of any future-directed (resp.
past-directed) timelike curve γ : I V .
Remark 2.1. Usually, in the literature a causal curve γ with a compact domain
I = [a,b] is allowed to be piecewise smooth, that is, it admits a partition a = t0
t1 · · · tn = b such that: (i) the restriction of γ to each subinterval [tj,tj+1]
is smooth and causal, and (ii) the left and right velocities of γ at each break tj
lie in the same cone. This is technically convenient, even though typically it does
not represent a true increment of generality (for example, in order to define the
relations , ≤), because any such curve admits a variation through smooth causal
ones with the same endpoints.
Here, we will use typically, say, future-directed timelike curves defined on a
half-open interval I = [a,b), which cannot be continuously extended to b. One can
assume also that such a curve ρ is piecewise smooth in the sense that there exists
a strictly increasing sequence {tj} b included in I such that conditions (i) and
(ii) above hold. In fact, on one hand, all the techniques to be used here will work
trivially in this general case. On the other, for any such a piecewise smooth curve
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