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

TL;DR

The paper addresses how dominant logarithmic terms modify the standard matching conditions between past and future null infinity for massless scalar and electromagnetic fields in four-dimensional Minkowski space. It provides a physically transparent mechanism: advanced radiation saturating the finite incoming-energy flux generates $rac{ ext{log }r}{r}$ terms at $\\\mathscr{I}^{+}$, while the first retarded wave yields corresponding contributions near $\\\mathscr{I}^{-}$, with the Coulombic $1/r$ behavior at spatial infinity enforcing precise matching. The results reproduce and complement prior spatial-infinity analyses (e.g., Fuentealba 2024, 2025) and generalize to all multipoles, yielding mixed $P$ and $Q$-branch matching conditions that are sensitive to the presence of leading logarithms. The work offers a physically intuitive derivation of these conditions and discusses potential extensions to gravity and polyhomogeneous asymptotics.

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.

Figures (1)

• Figure 1: The double cover of the Penrose diagram for Minkowski space depicting, in blue, the trace of the last advanced modes at future null infinity $\mathscr{I}^+$ and, in red, the first retarded modes at past null infinity $\mathscr{I}^-$. The advanced branch corresponds to solutions whose logarithmic falloff $\log(r)/r$ at $\mathscr{I}^+$ originates from finite-energy data entering through $\mathscr{I}^-$, while the retarded branch encodes the matching conditions at early times. Together, they illustrate how logarithmic terms at null infinity naturally emerge from the interplay between advanced and retarded solutions saturating the finite-energy condition.