×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. Gödel, Rev. Mod. Phys. 21 pp 447– (1949) · Zbl 0041.56701 · doi:10.1103/RevModPhys.21.447
[2] I. Ozsváth, J. Math. Phys. 6 pp 590– (1965) · Zbl 0131.43203 · doi:10.1063/1.1704311
[3] D. L. Farnsworth, J. Math. Phys. 7 pp 1625– (1966) · doi:10.1063/1.1705075
[4] A. Banerjee, J. Phys. A 1 pp 188– (1968) · doi:10.1088/0305-4470/1/2/302
[5] M. J. Rebouças, Phys. Lett. 70A pp 161– (1979) · doi:10.1016/0375-9601(79)90193-2
[6] H. M. Raval, Curr. Sci. 36 pp 7– (1967)
[7] M. Novello, Phys. Rev. D 19 pp 2850– (1979) · doi:10.1103/PhysRevD.19.2850
[8] C. Hoenselaers, Gen. Relativ. Gravit. 1 pp 43– (1979) · Zbl 0407.76082 · doi:10.1007/BF00757022
[9] M. M. Som, Proc. R. Soc. London A304 pp 81– (1968) · doi:10.1098/rspa.1968.0073
[10] S. K. Chakraborty, Gen. Relativ. Gravit. 12 pp 925– (1980) · doi:10.1007/BF00757362
[11] A. K. Raychaudhuri, Phys. Rev. D 22 pp 802– (1980) · doi:10.1103/PhysRevD.22.802
[12] F. Bampi, Gen. Relativ. Gravit. 9 pp 393– (1978) · Zbl 0415.53052 · doi:10.1007/BF00759840
[13] S. C. Maitra, J. Math. Phys. 7 pp 1025– (1966) · doi:10.1063/1.1704993
[14] H. Stein, Philos. Sci. 37 pp 589– (1970) · doi:10.1086/288328
[15] I. D. Soares, in: Proceedings of the Third Brazilian Summer School of Cosmology and Gravitation, Rio de Janeiro, 1982 (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.