The non-linear perturbation of a black hole by gravitational waves. III. Newman–Penrose constants
Jörg Frauendiener, Alex Goodenbour, Chris Stevens
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math> . The integral expressions for the NP constants as given originally are expressed in a general gauge to enable their computation from numerically generated data. Canonical methods of fixing inherent degrees of freedom in their definitions are discussed. The NP expressions are then computed for a variety of different ingoing wave profiles in axisymmetry, and then with no symmetry assumptions in a general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> setting for which all five complex quantities are non-zero. We discuss the consequence of the (numerical) constancy of the NP expressions for the (non)-smoothness of null-infinity.