In this section we describe the definition of asymptotic flatness according to Ashtekar [8].

Definition 2.4.2 A space-time (M , g) is asymptotically flat if there exists another larger space-time (M , g)˜ and a conformal mapping:

such that the following conditions are verified:

(1)Naming i0 spatial infinity, namely the locus in ψ(M ) where terminate all space-like curves, which is required to be a single point, we have:

M − ψ(M ) = J + i0 J − i0

(2) The boundary of M , named ∂M is decomposed as follows:

∂M = i0 J + J −

where by definition we have set:

J± = ∂J ± i0 − i0

(3)There exists a neighborhood V ∂ψ(M ) such that for every p V and every neighborhood Op of that point we can find a sub-neighborhood Up Op with the property that no causal curve intersects Up more than once.

(4)The conformal factor Ω can be extended to an overall function on the whole M

(5)The conformal factor Ω vanishes on J + and J − but its derivative μΩ does not on the same locus.

In order to appreciate all the points of the above definition it is convenient to look at Fig. 2.19 and compare with the case of Minkowski space. The starting point of the analysis is the obvious observation that any causal curve which departs from spatial infinity i0 ≡ (π, 0) cannot penetrate in the triangle representing Minkowski space and therefore lies in M − ψ(M ). If the causal curve is future-directed it goes up, while if it is past directed it goes down so that point (1) of Definition 2.4.2 is indeed verified. Let us next consider the boundary of the causal future and causal past of spatial infinity. They are given by the upper and lower side, respectively, of the triangle in Fig. 2.19, which intersect in i0. Hence point (2) of Definition 2.4.2 is also verified. Let us note that according to this definition J ± are just the Causal