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Maximal hypersurfaces in stationary asymptotically flat spacetimes. (English) Zbl 0803.53041

The authors prove the existence of a maximal hypersurface foliation in asymptotically flat, strongly causal spacetimes which are stationary near infinity and satisfy the strong energy condition (\(\text{Ric}(v,v) \geq 0\) for all timelike vectors \(v\)). This is an improvement on earlier work by R. Bartnik [Commun. Math. Phys. 94, 155-175 (1984; Zbl 0548.53054)] by R. Bartnik, P. T. Chruściel and N. Ó. Murchadha [ibid. 130, No. 1, 95-109 (1990; Zbl 0703.53053)] who had to assume stationarity everywhere. Notice that this global stationarity assumption is violated even for the Schwarzschild solution.
The exact statements of their theories is technical. Their main assumption is: (a) there exists a slice which does not contain any white or black hole and which is compact ‘away from infinity’, i.e., it does not intersect any singularity; or (b) spacetime contains a black and a white hole and there exists a slice which has a compact boundary \(S\) such that \(S\) is the intersection of the horizons of the white and the black hole. Moreover, the slice is compact ‘away from infinity’.
They give many examples and application and discuss their theorems critically. The heart of their proof is a causal argument which ensures that the domains of dependence of compact sets are compact.
Reviewer: M.Kriele (Berlin)

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C75 Space-time singularities, cosmic censorship, etc.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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