The shear-free condition and constant-mean-curvature hyperboloidal initial data

Abstract

Abstract. We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constantmean-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 [4]. 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 [1]. 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 [1]. 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.

Smoothly Compactifiable Shear-Free Hyperboloidal Data is Dense in the Physical Topology

“…Indeed, it was the approximation result here that motivated several of the results in [1], where continuity of the conformal method for construction of solutions to the constraint equations is established in a topology strong enough to detect the shear-free condition.…”

“…Here we present a discussion of the details needed in order to make precise our approximation result. As we make use of several results from [11], [2], and [1], we maintain conventions similar to the conventions in those works.…”

“…Asymptotically hyperbolic data sets may be viewed as intersecting future null infinity in the asymptotically flat spacetime containing a future development of the data set; we refer the reader to [1], and the references therein, for a more detailed discussion of asymptotically hyperbolic initial data sets and asymptotic flatness. The formula for the change of scalar curvature under conformal deformation, together with (1.1a), implies…”

“…Thus we refer to initial data satisfying the shear-free condition as hyperboloidal, distinguished among those solutions to the constraint equations having asymptotically hyperbolic geometry. Our recent work [1], joint with James Isenberg and John M. Lee, contains a systematic study of CMC hyperboloidal initial data, including a parametrization of all such data in the "weakly asymptotically hyperbolic" setting (see also [2]).…”

“…This is not, however, the end of the story. Even if one restricts attention to shearfree data, the initial data constructed in [4] and [1] may not be sufficiently regular at the conformal boundary to obtain a spacetime development admitting conformal compactification. For example, the existing evolution theorems of Helmut Friedrich [8], [9], etc., all require more regularity of the conformal compactification.…”