On Killing vector fields and Newman–Penrose constants

Abstract

Asymptotically flat space–times with one Killing vector field are studied. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e., series in powers of 1/r and ln r), and solved order by order. The solution to the leading terms of these expansions yields the asymptotic form of the Killing vector field. The possible classes of Killing fields are discussed by analyzing their orbits on null infinity. The integrability conditions of the Killing equations are used to obtain constraints on the components of the Weyl tensor (Ψ0,Ψ1,Ψ2) and on the shear (σ). The behavior 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, the particular case of boost-rotation symmetric space–times is considered. The constraints on Ψ0 are used to study the behavior of the coefficients that give rise to the Newman–Penrose constants, if the space–time is non-polyhomogeneous, or the logarithmic Newman–Penrose constants, if the space–time is polyhomogeneous.