The non-linear perturbation of a black hole by gravitational waves. III. Newman–Penrose constants

Abstract

In this paper we continue our study of the non-linear response of a Schwarzschild black hole to an ingoing gravitational wave by computing the Newman-Penrose (NP) constants. The NP constants are five complex, supertranslation-invariant quantities defined on null infinity $\mathscr{I}^+$ and although put forward in the 60's, they have never been computed in a non-stationary setting. We accomplish this through a numerical implementation of Friedrich's generalized conformal field equations whose semi-global evolution yields direct access to $\mathscr{I}^+$. Generalizations of the NP constants' integral expressions are made to allow their computation in a more general gauge that better suits the output of a numerical evolution. Canonical methods of fixing inherent degrees of freedom in their definitions are discussed. The NP constants are then computed for a variety of different ingoing wave profiles in axisymmetry, and then with no symmetry assumptions in 3+1 for which all five are non-zero.