Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants

TL;DR

This paper explores the asymptotic behavior of zero-rest-mass fields and gravity, focusing on polyhomogeneity and its relation to Newman-Penrose constants, revealing how logarithmic terms naturally arise and can be interpreted physically.

Contribution

It introduces a detailed analysis of polyhomogeneous expansions for zero-rest-mass fields, including their relation to NP constants and the development of a regular auxiliary field for gravitational asymptotics.

Findings

Logarithmic terms appear naturally in asymptotic expansions without the Peeling theorem.

A new auxiliary field for gravity is constructed, capturing asymptotic information.

Logarithmic NP constants are derived from the auxiliary field at future infinity.

Abstract

A discussion of polyhomogeneity (asymptotic expansions in terms of $1/r$ and $\ln r$) for zero-rest-mass fields and gravity and its relation with the Newman-Penrose (NP) constants is given. It is shown that for spin-$s$ zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms in the asymptotic expansion appear naturally if the field does not obey the ``Peeling theorem''. The terms that give rise to the slower fall-off admit a natural interpretation in terms of advanced field. The connection between such fields and the NP constants is also discussed. The case when the background spacetime is curved and polyhomogeneous (in general) is considered. The free fields have to be polyhomogeneous, but the logarithmic terms due to the connection appear at higher powers of $1/r$. In the case of gravity, it is shown that it is possible to define a new auxiliary field, regular at null infinity, and containing some relevant information on the asymptotic behaviour of the spacetime. This auxiliary zero-rest-mass field ``evaluated at future infinity ($i^+$)'' yields the logarithmic NP constants.