The Einstein constraints: uniqueness and non-uniqueness in the conformal thin sandwich approach

TL;DR

This paper investigates the non-uniqueness of solutions in Einstein's constraint equations, showing that the Hamiltonian constraint alone can admit multiple solutions, which explains the behavior observed in the conformal thin-sandwich approach.

Contribution

It demonstrates that the Hamiltonian constraint can have multiple solutions and links this non-uniqueness to specific terms in the equations, extending understanding of Einstein constraint decompositions.

Findings

Hamiltonian constraint admits two solution branches.

Non-uniqueness is related to the sign of a key term.

Analytic solutions constructed for a spherical star model.

Abstract

We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended conformal thin-sandwich decomposition. We show that the Hamiltonian constraint alone, when expressed in a certain way, admits two branches of solutions with properties very similar to those found by Pfeiffer and York. We construct these two branches analytically for a constant-density star in spherical symmetry, but argue that this behavior is more general. In the case of the Hamiltonian constraint this non-uniqueness is well known to be related to the sign of one particular term, and we argue that the extended conformal thin-sandwich equations contain a similar term that causes the breakdown of uniqueness.