The shear-free condition and constant-mean-curvature hyperboloidal initial data. (3rd May 2016) - Allen, Paul T
Abstract: We consider the Einstein–Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conformal boundary at future null infinity; see (Andersson and Chruściel 1994 Commun. Math. Phys. 161 533–68 ). We work with initial data sets in a variety of regularity classes, primarily considering those data sets whose geometries are weakly asymptotically hyperbolic, as defined in (Allen et al 2015 arXiv:1506.03399 ). These metrics are C 1, 1 conformally compact, but not necessarily C 2 conformally compact. In order to ensure that the data sets we construct are indeed shear-free, we make use of the conformally covariant traceless Hessian introduced in (Allen et al 2015 arXiv:1506.03399 ). We furthermore construct a class of initial data sets with weakly asymptotically hyerbolic metrics that may be only C 0, 1 conformally compact; these data sets are insufficiently regular to make sense of the shear-free condition.