ABSTRACT
The issue of asymptotic structure of space-times is tied with the question: what is the appropriate space-time which corresponds to an ‘isolated body’ or a source of gravitation confined to a compact region? This means that we expect there are regions in the manifold where the matter stress tensor vanishes and hence the metric satisfies the vacuum Einstein equation. Allowing for the possibility of a cosmological constant, there are precisely three matterfree space-times which are simplest in the sense that they have maximum possible symmetry. These are: the Minkowski space-time (Λ = 0), De Sitter (Λ > 0) and the anti-De Sitter (Λ < 0). The space-times exterior to the matter sources are expected to be approaching these special solutions as one goes ‘sufficiently far away’ from the sources. Thus we need to understand what the ‘sufficiently far away’ (or infinity) from the ‘origin’ (a point interior to a compact region) means for these special solutions. This is non-trivial because the Lorentzian signature implies there are different ways to approach an infinity. For instance, in a Euclidean space, we can ‘go to large r’ in different directions. For Lorentzian space-times, there is a further possibility of ‘going to large r’ with different speeds along different directions. In particular the speeds are delineated by the speed of light. These asymptotic approaches can be understood in terms of taking (say) affine parameters to their asymptotic values along time-like/light-like/space-like geodesics. The solutions of basic wave equation also shows possibilities of different asymptotic behaviours. Secondly, without any preferred coordinate system, specification of asymptotic fall-off behaviours of metric is at best ambiguous. It would conceivably be easier if we could bring the ‘infinity’ - region of infinite coordinate values - to region of finite coordinate values. Let us see how this could work. We will focus on Λ = 0 case first and comment on the other cases at the end.
