Lie algebra automorphisms as Lie point symmetries and the solution space for Bianchi Type I, II, IV, V vacuum geometries

Abstract

Lie group symmetry analysis for systems of coupled, nonlinear ordinary differential equations is performed in order to obtain the entire solution space to Einstein's field equations for vacuum Bianchi spacetime geometries. The symmetries used are the automorphisms of the Lie algebra of the corresponding three-dimensional isometry group acting on the hyper-surfaces of simultaneity for each Bianchi Type, as well as the scaling and the time reparametrization symmetry. A detailed application of the method is presented for Bianchi Type IV. The result is the acquisition of the general solution of Type IV in terms of sixth Painleve transcendent PVI, along with the known pp-wave solution. For Bianchi Types I, II, V the known entire solution space is attained and very briefly listed, along with two new Type V solutions of Euclidean and neutral signature and a Type I pp-wave metric.

LaTeX source file, 37 pages, no figures. Accepted to CQG. arXiv admin note: substantial text overlap with arXiv:0803.3710, arXiv:gr-qc/0410123

References in corpus (3)

• The General Solution of Bianchi Type III Vacuum Cosmology
• The General Solution of Bianchi Type $VI{I}_h$ Vacuum Cosmology
• Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

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• Faithful representations of Lie algebras and Homogeneous Spaces
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