The global structure of simple space-times
According to the author a space-time satisfying the chronology condition is called simple if it is equipped with future and past null infinities \({\mathcal I}^+\) and \({\mathcal I}^-\) in the usual way such that all endless null geodesics originate from \({\mathcal I}^-\) and end at \({\mathcal I}^+\). This situation is more general than that of an asymptotically simple and empty space-time [see the book of \textit{S. W. Hawking} and \textit{G. F. R. Ellis}, The large scale structure of space-time (Cambridge University Press. XI, 1973, Zbl 0265.53054)] where \({\mathcal I}^+\) and \({\mathcal I}^-\) are required to satisfy the condition of strong causality. The objective of this paper is to identify the principal causal and topological properties of simple space-times. Main goal is to generalize the topological classifications, obtained by R. Penrose, R. Geroch, S. W. Hawking and G. F. R. Ellis under additional constraints on the structure of the null infinities, to the case of simple space-times. The topological structure of such a space-time is characterized by the author as follows: \({\mathcal I}^+\) is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal region \({\mathcal I}^+_ 0\) of \({\mathcal I}^+\) is diffeomorphic to \(S^ 2\times {\mathbb{R}}\), and every compact connected spacelike 2-surface in \({\mathcal I}^+\) is contained in \({\mathcal I}^+_ 0\) and is a strong deformation retract of both \({\mathcal I}^+_ 0\) and \({\mathcal I}^+\). Moreover, the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to \({\mathbb{R}}^ 3\). In addition to these main results the paper contains a lot of other useful information on the geometrical structure of simple space-times.
scientific article; zbMATH DE number 4119159