Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity

TL;DR

This paper investigates the asymptotic charges of spin-1 and spin-2 fields at the critical sets of null infinity, establishing their relation at future and past null infinity using Friedrich's framework and initial data expansions.

Contribution

It provides a detailed analysis of the asymptotic charges at spatial infinity for spin-1 and spin-2 fields, revealing conditions for well-defined charges and their correspondence at null infinities.

Findings

Charges are determined by specific initial data satisfying regularity conditions.

A subset of initial data leads to well-defined asymptotic charges.

A natural correspondence between charges at future and past null infinity is established.

Abstract

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinity. To this end, we make use of Friedrich's framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. Expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, that satisfies certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinity.