Automorphisms and a Cartography of the Solution Space for Vacuum Bianchi Cosmologies: The Type III Case

TL;DR

This paper develops a symmetry-based algorithm to map the solution space of vacuum Bianchi Type III cosmologies, recovering known solutions and revealing a unified structure of the entire solution space.

Contribution

It introduces a novel symmetry-driven method that encapsulates the entire solution space into a single ODE and applies it to Type III, uncovering its structure and potential for unification across Bianchi types.

Findings

Recovered all known Type III solutions without extra assumptions

Enclosed the entire solution space into a single second-order ODE

Identified three disconnected regions within the solution space

Abstract

The theory of symmetries of systems of coupled, ordinary differential equations (ODE's) is used to develop a concise algorithm for cartographing the space of solutions to vacuum Bianchi Einstein's Field Equations (EFE). 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 eparameterization symmetry. Application of the method to Type III results in: a) the recovery of all known solutions without prior assumption of any extra symmetry, b) the enclosure of the entire unknown part of the solution space into a single, second order ODE in terms of one dependent variable and c) a partial solution to this ODE. It is also worth-mentioning the fact that the solution space is seen to be naturally partitioned into three distinct, disconnected pieces: one consisting of the known Siklos (pp-wave) solution, another occupied by the Type III member of the known Ellis-MacCallum family and the third described by the aforementioned ODE in which an one parameter subfamily of the known Kinnersley geometries resides. Lastly, preliminary results reported show that the unknown part of the solution space for other Bianchi Types is described by a strikingly similar ODE, pointing to a natural operational unification as far as the problem of solving the cosmological EFE's is concerned.