The General Solution of Bianchi Type $VII_h$ Vacuum Cosmology

TL;DR

This paper develops a symmetry-based algorithm to find the complete set of solutions for vacuum Bianchi Type VII_h cosmologies, utilizing Painlevé transcendents and recovering known solutions while discovering a new one.

Contribution

It introduces a concise symmetry-based method to derive all solutions of vacuum Bianchi Type VII_h Einstein equations, including new solutions and connections to Painlevé transcendents.

Findings

Derived the general solution of Type VII_0 using the third Painlevé transcendental.

Obtained the general solution of Type VII_h using the sixth Painlevé transcendental.

Discovered a new solution with a G_3 isometry group acting on T_3.

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 VII_0 with the aid of the third Painlev\'{e} transcendental (b) obtaining the general solution of Type $VII_h$ with the aid of the sixth Painlev\'{e} 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 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 VII_0 Bianchi Cosmology.