Spinorial Wave Equations and Stability of the Milne Spacetime

TL;DR

This paper develops a spinorial framework for conformal Einstein equations, constructs wave equations for these fields, and demonstrates the nonlinear stability of the Milne Universe under small perturbations.

Contribution

It introduces a spinorial approach to derive and analyze wave equations for conformal Einstein fields, establishing stability results for the Milne Universe.

Findings

Small perturbations lead to solutions resembling the Milne Universe

Constructed subsidiary wave equations ensure solutions satisfy Einstein equations

Unified spinorial method simplifies stability analysis

Abstract

The spinorial version of the conformal vacuum Einstein field equations are used to construct a system of quasilinear wave equations for the various conformal fields. As a part of the analysis we also show how to construct a subsidiary system of wave equations for the zero quantities associated to the various conformal field equations. This subsidiary system is used, in turn, to show that under suitable assumptions on the initial data a solution to the wave equations for the conformal fields implies a solution to the actual conformal Einstein field equations. The use of spinors allows for a more unified deduction of the required wave equations and the analysis of the subsidiary equations than similar approaches based on the metric conformal field equations. As an application of our construction we study the non-linear stability of the Milne Universe. It is shown that sufficiently small perturbations of initial hyperboloidal data for the Milne Universe gives rise to a solution to the Einstein field equations which exist towards the future and has an asymptotic structure similar to that of the Milne Universe.