Brill Waves with Slow Fall-Off Towards Spatial Infinity
Lydia Bieri, David Garfinkle, James Wheeler
TL;DR
This work extends Brill wave initial data to slower fall-off toward spatial infinity and establishes a rigorous existence/uniqueness framework for the associated conformal factor via a Schrödinger-type equation $P U = f$ with $P = \Delta - \phi$. It proves that, for broad classes of $q$ in weighted Sobolev spaces, there exists a unique $\Psi$ (with $\Psi = 1 + U$) whose decay is controlled by $q$ up to a constant mass term, and shows that type (A) data must have an isotropic mass. Complementing the analysis, the authors develop a robust axisymmetric numerical scheme to compute $\Psi$ and generate explicit slow-fall-off Brill data, including an example lacking antipodal symmetry at infinity, with detailed studies of the curvature decay. The results broaden the space of physically admissible Brill data, offer practical tools for constructing initial data for evolutions, and provide new insights into asymptotic symmetry properties relevant to the Strominger antipodal conjecture. Overall, the paper combines rigorous PDE techniques with computational methods to explore slow fall-off Brill waves and their geometric and physical implications.
Abstract
We compute families of solutions to the Einstein vacuum equations of the type of Brill waves, but with slow fall-off towards spatial infinity. We prove existence and uniqueness of solutions for physical data and numerically construct some representative solutions. We numerically construct an explicit example with slow-off which does not exhibit antipodal symmetry at spatial infinity.