On the stability of solutions of the Lichnerowicz-York equation

TL;DR

This paper analyzes the stability of solutions to the Lichnerowicz-York equation at time symmetry, demonstrating that weak-field solutions are stable while strong-field solutions are unstable, with implications for solution methods.

Contribution

It provides a stability analysis distinguishing stable and unstable solution branches of the Lichnerowicz-York equation at time symmetry.

Findings

Weak-field lower branch solutions are stable.

Strong-field upper branch solutions are unstable.

Unstable solutions challenge existing monotone iteration methods.

Abstract

We study the stability of solution branches for the Lichnerowicz-York equation at moment of time symmetry with constant unscaled energy density. We prove that the weak-field lower branch of solutions is stable whilst the upper branch of strong-field solutions is unstable. The existence of unstable solutions is interesting since a theorem by Sattinger proves that the sub-super solution monotone iteration method only gives stable solutions.