Bondi-type systems near space-like infinity and the calculation of the NP-constants

TL;DR

This work builds a bridge between Bondi-type analyses at null infinity and a conformal, Cauchy-data framework near space-like infinity by using a regular finite initial-value problem and gauge relations between the F-gauge and the NP-gauge. The authors develop a systematic, third-order expansion of fields on the space-like infinity cylinder $I$ and show how initial data on a Cauchy surface $ ilde S$ determine near-null-infinity quantities, including the Newman–Penrose (NP) constants, for time-symmetric spacetimes. They derive explicit transformation formulae between gauges, solve transport equations on $I'$ and along ${\cal J}^+$, and demonstrate the absence of logarithmic singularities under asymptotic regularity, enabling NP-constants to be expressed purely in terms of initial data. The results provide a practical route to compute NP-constants from initial data, offer insights into the evolution toward null infinity, and have potential applications as consistency checks in numerical relativity and in understanding the asymptotic structure of gravitational fields.

Abstract

We relate Bondi systems near space-like infinity to another type of gauge conditions. While the former are based on null infinity, the latter are defined in terms of Einstein propagation, the conformal structure, and data on some Cauchy hypersurface. For a certain class of time symmetric space-times we study an expansion which allows us to determine the behavior of various fields arising in Bondi systems in the region of space-time where null infinity touches space-like infinity. The coefficients of these expansions can be read off from the initial data. We obtain in particular expressions for the constants discovered by Newman and Penrose (NP-constants) in terms of the initial data. For this purpose we calculate a certain expansion up to 3rd order.