Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces

TL;DR

This paper investigates the asymptotic behavior of spin-0 fields in n-dimensional Minkowski spaces using conformal geometry, revealing the existence and regularity conditions of conserved charges at infinity.

Contribution

It introduces a method to compute asymptotic charges for spin-0 fields near infinity and characterizes their regularity in various dimensions.

Findings

In even dimensions, infinitely many regular asymptotic charges exist.

In odd dimensions, no non-trivial regular asymptotic charges are found.

Conditions for initial data to produce regular solutions are derived.

Abstract

We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin-$0$ fields (solutions to the wave equation) on $n$-dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin-$0$ charges. It is shown that if one considers the most general initial data within the class considered in this paper, the expansion is poly-homogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain regular solutions and well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In even dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for odd higher dimensions there exists no non-trivial asymptotic charges that remain regular at the critical sets.