Type D conformal initial data

Abstract

For a vacuum initial data set of the Einstein field equations it is possible to carry out a conformal rescaling or conformal compactification of the data giving rise to an initial data set for the Friedrich vacuum conformal equations. When will the data development with respect to the conformal equations of this set be a conformal extension of a type D solution? In this work we provide a set of necessary and sufficient conditions on a set of initial data for the conformal equations that guarantees that the data development of the conformal equations has a subset that is conformal to a vacuum type D solution of the Einstein's equations. In particular we find the conditions under which this vacuum solution corresponds to the Kerr solution. Using our results we are able to show that there are no obstructions to extend the Petrov type of the physical spacetime to the unphysical spacetime if the conformal data are hyperboloidal.

arXiv comment22 pages, 2 figures. New section added and reference list enlarged. To appear in General Relativity and Gravitation

References in corpus (5)

• xPerm: fast index canonicalization for tensor computer algebra
• A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies
• On ``hyperboloidal'' Cauchy data for vacuum Einstein equations and obstructions to smoothness of ``null infinity''
• The extended Conformal Einstein field equations with matter: the Einstein-Maxwell field
• Vacuum type D initial data