Solving Einstein's Equation Numerically on Manifolds with Non-Orientable Spatial Slices

TL;DR

This paper develops numerical solutions to Einstein's equations on manifolds with non-orientable spatial slices, demonstrating models with various curvatures and inhomogeneities to evaluate the methods' effectiveness.

Contribution

It introduces numerical techniques for solving Einstein's equations on non-orientable manifolds and explores their capabilities and limitations with diverse cosmological models.

Findings

Solutions include models with positive, negative, and zero scalar curvature.

Some solutions are locally indistinguishable from homogeneous cosmologies.

The study tests and evaluates the numerical methods and code used.

Abstract

This paper presents solutions to Einstein's equation -- and the numerical methods used to construct them -- that describe simple cosmological models on manifolds with compact non-orientable spatial slices. These solutions have been constructed on a selection of manifolds having positive, negative, and vanishing spatial scalar curvatures. One example is shown to be indistinguishable locally from a homogeneous Friedman cosmological model, others are constructed with significant inhomogeneities. Together these examples are used to explore the strengths and the limitations of the numerical methods used in this study, and to test the code used to implement them.