Vacuum type D initial data

TL;DR

This paper provides a systematic characterization of vacuum type D initial data sets in general relativity, involving only the fundamental forms and differential equations, with applications to Kerr black hole stability.

Contribution

It introduces a new set of differential conditions involving vacuum constraints that fully characterize regular vacuum type D initial data sets.

Findings

Characterization of vacuum type D initial data via differential equations

Conditions applicable to data development containing Kerr solutions

Framework useful for studying Kerr black hole stability

Abstract

A vacuum type D initial data set is a vacuum initial data set of the Einstein field equations whose data development contains a region where the space-time is of Petrov type D. In this paper we give a {\em systematic} characterisation of a vacuum type D initial data set. By systematic we mean that the only quantities involved are those appearing in the vacuum constraints, namely the first fundamental form (Riemannian metric) and the second fundamental form. Our characterisation is a set of conditions consisting of the vacuum constraints and some additional differential equations for the first and second fundamental forms. These conditions can be regarded as a system of partial differential equations on a Riemannian manifold and the solutions of the system contain all possible {\em regular} vacuum type D initial data sets. As an application we particularise our conditions for the case of vacuum data whose data development is a subset of the Kerr solution. This has applications in the formulation of the non-linear stability problem of the Kerr black hole.