Some Global Results for Asymptotically Simple Spacetimes

The aim of this paper is to present some rigidity results for asymptotically simple spacetimes. By a spacetime we mean a smooth connected time oriented Lorentzian manifold (M, g) of dimension ≥ 3, having signature (−+ · · ·+). We use standard notation for causal theoretic notions, e.g., for A ⊂ M , I(A,M), the timelike future of A in M (resp., I(A,M), the timelike past of A in M), is the set of all points that can be reached from A by a future directed (resp. past directed) timelike curve in M . For other basic definitions and results in spacetime geometry and causal theory we refer the reader to the standard references [12, 17]. Penrose’s treatment of infinity [16, 18] in asymptotically flat spacetimes (and spacetimes with other asymptotic structures, as well) is based on his notion of asymptotic simplicity. A 4-dimensional, chronological spacetime (M, g) is asymptotically simple provided there exists a smooth spacetime-with-boundary (M̃, g̃) such that,