The General Solution of Bianchi Type VII_h Vacuum Cosmology

2008/03/26 by Petros A. Terzis, T. Cristodoulakis · 1 citation
Physics and Astronomy · #gr-qc

paper · pdf · doi:10.1007/s10714-008-0678-5

published as Gen.Rel.Grav. 41 (2009) 469-495 · latex2e source file, 27 pages, 2 tables, no fiures

arxiv created 2008/03/26 · arxiv updated 2013/05/06

Abstract

The theory of symmetries of systems of coupled, ordinary differential equations (ODE) is used to develop a concise algorithm in order to obtain the entire space of solutions to vacuum Bianchi Einstein Field Equations (EFEs). The symmetries used are the well known automorphisms of the Lie algebra for the corresponding isometry group of each Bianchi Type, as well as the scaling and the time re-parametrization symmetry. The application of the method to Type VII_h results in (a) obtaining the general solution of Type VII₀ with the aid of the third Painlevé transcendental (b) obtaining the general solution of Type VII_h with the aid of the sixth Painlevé transcendental (c) the recovery of all known solutions (six in total) without a prior assumption of any extra symmetry (d) The discovery of a new solution (the line element given in closed form) with a G₃ isometry group acting on T₃, i.e. on time-like hyper-surfaces, along with the emergence of the line element describing the flat vacuum Type VII₀ Bianchi Cosmology.

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

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