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

Abstract

Abstract In this paper we continue our investigations of Rácz’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 Rácz’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 Rácz’s parabolic–hyperbolic formulation are stable, and, in particular, no modifications are necessary to obtain solutions which are asymptotically hyperboloidal.