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Homogeneity of Riemannian space-times of Gödel type. (English) Zbl 0959.83513

Summary: The conditions for space-time homogeneity of a Riemannian manifold with a Gödel-type metric are examined. The Raychaudhuri-Thakurta necessary conditions for space-time homogeneity are shown also to be sufficient and to lead to five linearly independent Killing vectors. These vector fields are exhibited for the most general case and their algebra is examined. The irreducible set of isometrically independent space-time-homogeneous Gödel-type metrics is shown to be given, in cylindrical coordinates, by \[ ds^2=[dt+(4\Omega/m^2)\sinh^2(mr/2)d\varphi]^2- (1/m^2)\sinh^2(mr)d\varphi^2-dr^2-dz^2, \] where \(\Omega\) is the vorticity and \(-\infty\leq m^2\leq+\infty\), \(m^2=2\Omega^2\) corresponding to the Godel metric. Sources of Einstein’s equations leading to these metrics as solutions are examined, and it is shown that the inclusion of a scalar field extends the previously known region of solutions \(-\infty\leq m^2\leq 2\Omega^2\) to \(-\infty\leq m^2\leq 4\Omega^2\). The problem of ambiguity of physical sources of the same metric and that of violation of causality in Gödel-type space-time-homogeneous universes are examined. In the case \(m^2=4\Omega^2\), we obtain the first exact Gödel-type solution of Einstein’s equations describing a completely causal space-time-homogeneous rotating universe.

MSC:

83C40 Gravitational energy and conservation laws; groups of motions
53C80 Applications of global differential geometry to the sciences
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
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