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Space-time distributions. (English) Zbl 0934.83004

Summary: The space-time foliation \(\Sigma\) compatible with the gravitational field \(g\) on a 4-manifold \(M\) determines a fibration \(\pi\) of \(M, \pi : M\to \mathcal N\) is surjective submersion over the 1-dimensional leaves space \(\mathcal N\). \(M\) is then written as a disjoint union of the leaves of \(\Sigma\) , which are 3-dimensional spacelike surfaces on \(M\). The decomposition, \(TM =\Sigma\otimes T^0 M\), also implies that we can define a lift of the curves on \(\mathcal N\) to curves (non-spacelike) on \(M\). The stable causality condition holds on \(M\) means that \(\Sigma\) is a causal space-time distribution, generated by an exact timelike 1-form \(\omega^0 = dt\) where t is some real function on \(M\). In this case \(M\) is written as a disjoint union of a family of spacelike 3-surfaces of constant \(t\), which cover \(D^+(S)\) of a initial 3-surface \(S\) of \(M\).

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53C12 Foliations (differential geometric aspects)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
57R30 Foliations in differential topology; geometric theory
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