Type D conformal initial data

TL;DR

This paper establishes necessary and sufficient conditions on initial data for the conformal equations to guarantee that the resulting spacetime development is a conformal extension of a vacuum type D solution, including Kerr.

Contribution

It provides a new set of criteria on initial data ensuring the development is a vacuum type D solution, with particular focus on Kerr spacetime.

Findings

Conditions for conformal data to yield type D solutions.

No obstructions to extending Petrov type in hyperboloidal data.

Characterization of initial data leading to Kerr solutions.

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.