Matching conditions at null infinity in the presence of logarithms: the role of advanced and retarded radiation

TL;DR

This paper investigates how dominant logarithmic terms in asymptotic expansions affect matching conditions at null infinity, highlighting the role of advanced and retarded radiation in electromagnetic and scalar fields.

Contribution

It introduces a new perspective on matching conditions at null infinity by explicitly considering logarithmic terms and their physical origin related to radiation.

Findings

Logarithmic terms are linked to advanced and retarded radiation.

Matching conditions stem from Coulombic behavior at spatial infinity.

Explicit analysis for scalar and electromagnetic fields.

Abstract

We provide a new perspective on the general matching conditions between the future of past null infinity and the past of future null infinity, emphasizing the impact of dominant logarithmic terms in the asymptotic expansion of the fields near null infinity. We explicitly consider the cases of a massless scalar field and of electromagnetism. Key in our derivation is the identification of the physical origin of these logarithms, which are associated with advanced and retarded radiation saturating the finite energy flux condition at null infinity (in a space of functions which is made precise). The matching conditions arise then from the requirement of Coulombic (i.e., $1/r$) behaviour at spatial infinity.