Non-uniqueness in conformal formulations of the Einstein constraints

TL;DR

This paper demonstrates the existence of multiple solutions in conformal formulations of Einstein's constraints using non-linear analysis, highlighting potential non-uniqueness issues in these methods.

Contribution

It introduces a parabolic branching analysis of solutions in conformal Einstein constraint equations, extending previous numerical findings.

Findings

Existence of solution branches with unscaled sources

Kernel solutions lead to parabolic solution curves

Implications for constrained evolution methods

Abstract

Standard methods in non-linear analysis are used to show that there exists a parabolic branching of solutions of the Lichnerowicz-York equation with an unscaled source. We also apply these methods to the extended conformal thin sandwich formulation and show that if the linearised system develops a kernel solution for sufficiently large initial data then we obtain parabolic solution curves for the conformal factor, lapse and shift identical to those found numerically by Pfeiffer and York. The implications of these results for constrained evolutions are discussed.