On Killing vector fields and Newman–Penrose constants

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Abstract
Asymptotically flat spacetimes with one Killing vector field are studied. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e. series in pow- ers of 1/r and lnr), and solved order by order. The solution to the leading terms of these expansions yield the 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 condi- tions of the Killing equations are used to obtain constraints on the components of the Weyl tensor (�0, �1, �2) and on the shear (�). 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, the particular case of boost-rotation symmetric spacetimes is considered. The constraints on �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. The Newman-Penrose (NP) constants (20) are a set of five complex quantities, defined for asymp- totically flat spacetimes with a smooth null infinity, with the remarkable property of being abso- lutely conserved even in the presence of gravitational radiation. Recently (24), (26), it has been shown that in a more general setting —that of polyhomogeneous spacetimes— the NP constants are not conserved, but nevertheless, an adequate generalization of them (logarithmic Newman- Penrose constants) can be constructed which are indeed conserved. Polyhomogeneous spacetimes are spacetimes which are expanded asymptotically in terms of combinations of powers of 1/r and ln r. The introduction of this more general kind of expansion carries a drawback: null infinity (I ) is no longer smooth. For more details on these, and other aspects of polyhomogeneity, we

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