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

TL;DR

This paper constructs and parametrizes constant-mean-curvature hyperboloidal initial data satisfying the shear-free condition for Einstein-Maxwell-fluid equations, using the conformal method and considering various regularity classes of asymptotically hyperbolic metrics.

Contribution

It introduces a method to generate shear-free hyperboloidal initial data with different regularity levels, extending previous work to less regular geometries.

Findings

Constructed shear-free initial data with $C^{1,1}$ regularity.

Extended construction to $C^{0,1}$ regularity where shear-free condition is not well-defined.

Provided parametrization of initial data sets satisfying the shear-free condition.

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. 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 [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 [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.