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Observer fields and the strong energy condition. (English) Zbl 0861.53065

The authors investigate special observer fields in spacetimes satisfying the strong energy condition (= timelike convergence condition). About the first half of the paper is concerned with fundamentals and is essentially known. Nevertheless, the authors succeed in giving an exceptionally clear presentation. If \(T\) is irrotational and satisfies \(T\bullet \text{div}(T)\geq 0\) along some simply connected compact manifold \(L\) orthogonal to \(T\), then \(L\) is totally geodesic and \(T \bullet \text{div}(T)=0\). An observer field \(T\) is rigid if \(\nabla T\) is skew adjoint on \(T^\perp\).
Their main theorem states that a spacetime containing a rigid observer field \(T\) with the property that the sectional curvature of every plane containing \(T\) is positive is spacelike incomplete or splits isometrically.
Reviewer: M.Kriele (Berlin)

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C40 Gravitational energy and conservation laws; groups of motions
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