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Relativistic gravitational fields with close Newtonian analogs. (English) Zbl 0977.53515

Harvey, Alex (ed.), On Einstein’s path: essays in honor of Engelbert Schücking. A symposium, New York Univ., New York, NY, USA, December 12-13, 1996. New York, NY: Springer. 329-337 (1999).
Summary: Given a Newtonian velocity field \({\mathbf v}({\mathbf x}, t)\), one considers the manifold \(\mathbb{R}^4\) with the Lorentz metric \(g=(d{\mathbf x}- {\mathbf v} d t)^2-dt^2\). The Riemann tensor is computed and used to characterize flat space-times with \(g\) of this form. Among nonflat solutions of Einstein’s equations for such a \(g\) there are some cosmological models, the Schwarzschild and Kasner metrics and their generalizations to include matter fields and the cosmological constant. If \(|{\mathbf v}|=1\), then the vector field \(\partial/ \partial t\) is null and has vanishing divergence; it is geodetic and shear-free if and only if \(\partial {\mathbf v}/\partial t\) is parallel to \({\mathbf v}\).
For the entire collection see [Zbl 0913.00041].

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53C80 Applications of global differential geometry to the sciences
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