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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