The General Solution of Bianchi Type III Vacuum Cosmology

TL;DR

This paper derives the complete general solution to vacuum Bianchi Type III cosmology equations using Painlevé transcendents, revealing known solutions and symmetries in Einstein's field equations.

Contribution

It provides the first complete integration of the key differential equation for Bianchi Type III vacuum cosmology, connecting it to Painlevé transcendents and symmetries.

Findings

Derived the general solution using Painlevé VI transcendent

Recovered known Kinnersley spacetime as a special case

Identified a hidden symmetry simplifying the solution process

Abstract

The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$. For particular representations of $P_{VI}$ we obtain the known Kinnersley two-parameter space-time and a solution of Euclidean signature. The imposition of the space-time generalization of a "hidden" symmetry of the generic Type III spatial slice, enables us to retrieve the two-parameter subfamily without considering the Painlev\'{e} transcendent.