On Killing vector fields and Newman-Penrose constants

TL;DR

This paper analyzes asymptotically flat spacetimes with a Killing vector field, solving the Killing equations asymptotically, and explores their implications for Newman-Penrose constants and spacetime symmetries.

Contribution

It provides an asymptotic solution to the Killing equations using polyhomogeneous expansions and discusses the classification of Killing fields and their impact on Newman-Penrose constants.

Findings

Asymptotic form of Killing vectors derived

Constraints on Weyl tensor components established

Behavior of Newman-Penrose constants analyzed

Abstract

Asymptotically flat spacetimes with one Killing vector field are considered. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e. series in powers of 1/r an ln r), and solved order by order. The solution to the leading terms of these expansions yield the asymptotic form of the Killing vector field. The possible classes of Killing fields are discussed by analysing their orbits on null infinity. The integrability conditions of the Killing equations are used to obtain constraints on the components of the Weyl tensor (\Psi_0, \Psi_1, \Psi_2) and on the shear (\sigma). The behaviour of the solutions to the constraint equations is studied. It is shown that for Killing fields that are non-supertranslational the characteristics of the constraint equations are the orbits of the restriction of the Killing field to null infinity. As an application, boost-rotation symmetric spacetimes are considered. The constraints on \Psi_0 are used to study the behaviour of the coefficients that give rise to the Newman-Penrose constants, if the spacetime is non-polyhomogeneous, or the logarithmic Newman-Penrose constants if the spacetime is polyhomogeneous.