Characterization of (asymptotically) Kerr–de Sitter-like spacetimes at null infinity**Preprint UWThPh-2016-5.

Walter Simon,Tim-Torben Paetz,Marc Mars,José M M Senovilla

doi:10.1088/0264-9381/33/15/155001

Abstract

We investigate solutions to Einstein's vacuum field equations with positive cosmological constant Λ which admit a smooth past null infinity à la Penrose and a Killing vector field whose associated Mars–Simon tensor (MST) vanishes. The main purpose of this work is to provide a characterization of these spacetimes in terms of their Cauchy data on . Along the way, we also study spacetimes for which the MST does not vanish. In that case there is an ambiguity in its definition which is captured by a scalar function Q. We analyze properties of the MST for different choices of Q. In doing so, we are led to a definition of ‘asymptotically Kerr–de Sitter-like spacetimes’, which we also characterize in terms of their asymptotic data on .

Highlights

We investigate solutions (M, g) to Einsteins vacuum field equations with positive cosmological constant Λ which admit a smooth past null infinity I- à la Penrose and a Killing vector field whose associated Mars–Simon tensor (MST) vanishes
We are led to a definition of ‘asymptotically Kerr–de Sitter-like spacetimes’, which we characterize in terms of their asymptotic data on I
Combining theorem 3.1 and remark 3.2 it follows that a L > 0-vacuum spacetime admitting a Killing vector field (KVF) X with vanishing associated MST for some Q and for which 2 = 0 somewhere cannot admit a smooth I, unless the spacetime is locally isometric to de Sitter

Summary

Introduction

Kerr–de Sitter metric or one of the related metrics classified in [21] (M, g) is called ‘asymptotically Kerr–de Sitter-like’ at a connected component I- of I if it admits a KVF X which induces a CKVF Y. on I-, which satisfies ∣Y∣2 > 0, such satisfied, or, equivalently, such that the that the rescaled condit~io(nevs) (i) MST mnsr is and (ii) in theorem regular at I-. As will be shown later (cf corollary 4.17), KdS-like space-times have a vanishing MST, whence, as shown, the condition ∣Y∣2 > 0 follows automatically. We derive an analog to the Bianchi equation rCmns r = 0 for the MST (menvs) r.

The conformally rescaled spacetime

Cauchy data at I À

The function Q

Properties of the MST on I

The rescaled MST on I

The functions c and k and their constancy

The functions cb ðY Þ and kb ðY Þ

Constancy of cb ðY Þ

The Ernst potential on I

Alternative definition of the function Q

Xs mn mnsr 4

Main result We end up with the following main result