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’s 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 V I I h results in (a) obtaining the general solution of Type V I I 0 with the aid of the third Painlevé transcendental P I I I ; (b) obtaining the general solution of Type V I I h with the aid of the sixth Painlevé transcendental P V I ; (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 G 3 isometry group acting on T 3, i.e., on time-like hyper-surfaces, along with the emergence of the line element describing the flat vacuum Type V I I 0 Bianchi Cosmology.
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Terzis, P.A., Christodoulakis, T. The general solution of Bianchi type V I I h vacuum cosmology. Gen Relativ Gravit 41, 469–495 (2009). https://doi.org/10.1007/s10714-008-0678-5
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DOI: https://doi.org/10.1007/s10714-008-0678-5