Topological censorship from the initial data point of view

Details Topological censorship from the initial data point of view Topological censorship from the initial data point of view

2012-04-01

arxiv.org/abs/1204.0278v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

16 pages

Scientific paper

We introduce a natural generalization of marginally outer trapped surfaces (MOTSs) in an initial data set, called immersed MOTSs, and prove that for 3-dimensional asymptotically flat initial data sets (V,h,K), either V is diffeomorphic to R^3 or V contains an immersed MOTS. We also establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed MOTS generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Thus the former result may be seen as a purely initial data version of the Gannon-Lee singularity theorem. It can also be viewed as a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that any asymptotically flat Riemannian 3-manifold that is not diffeomorphic to R^3 contains an embedded stable minimal surface. As the Gannon-Lee singularity theorem may be viewed as a precursor to the space-time principle of topological censorship, we go further to obtain an initial data version of topological censorship. We establish, under a natural physical assumption, the topological simplicity of 3-dimensional asymptotically flat initial data sets with MOTS boundaries. We also obtain a generalization of these results to higher dimensions.

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