Asymptotically hyperboloidal initial data sets from a parabolic-hyperbolic formulation of the Einstein vacuum constraints

TL;DR

This paper investigates the stability of Rácz's parabolic-hyperbolic formulation of Einstein vacuum constraints in asymptotically hyperboloidal settings, demonstrating through numerical and analytical methods that the solutions are stable without modifications.

Contribution

It extends previous work on asymptotically flat cases by showing stability of solutions in the hyperboloidal setting without needing modifications.

Findings

Solutions are stable in the hyperboloidal setting.

No modifications are needed for asymptotically hyperboloidal solutions.

Numerical and analytical evidence supports stability.

Abstract

In this paper, we continue our investigations of R\'acz's parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our previous studies of the asymptotically flat setting provided strong evidence for unstable asymptotics which we were able to resolve by introducing a certain modification of R\'acz's parabolic-hyperbolic formulation. The primary focus of the present paper here is the asymptotically hyperboloidal setting. We provide evidence through a mixture of numerical and analytical methods that the asymptotics of the solutions of R\'acz's parabolic-hyperbolic formulation is stable, and, in particular, no modifications are necessary to obtain solutions that are asymptotically hyperboloidal.