The case against smooth null infinity II: A logarithmically modified Price’s Law
Leonhard M. A. Kehrberger
Abstract
In this paper, we expand on results from our previous paper "The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples" [1] by showing that the failure of "peeling" (and, thus, of smooth null infinity) in a neighbourhood of $i^0$ derived therein translates into logarithmic corrections at leading order to the well-known Price's law asymptotics near $i^+$. This suggests that the non-smoothness of $\mathcal{I}^+$ is physically measurable. More precisely, we consider the linear wave equation $\Box_g ϕ=0$ on a fixed Schwarzschild background ($M>0$), and we show the following: If one imposes conformally smooth initial data on an ingoing null hypersurface (extending to $\mathcal{H}^+$ and terminating at $\mathcal{I}^-$) and vanishing data on $\mathcal{I}^-$ (this is the no incoming radiation condition), then the precise leading-order asymptotics of the solution $ϕ$ are given by $rϕ|_{\mathcal{I}^+}=C u^{-2}\log u+\mathcal{O}(u^{-2})$ along future null infinity, $ϕ|_{r=R>2M}=2Cτ^{-3}\logτ+\mathcal{O}(τ^{-3})$ along hypersurfaces of constant $r$, and $ϕ|_{\mathcal{H}^+}=2Cv^{-3}\log v+\mathcal{O}(v^{-3})$ along the event horizon. Moreover, the constant $C$ is given by $C=4M I_0^{(\mathrm{past})}[ϕ]$, where $I_0^{(\mathrm{past})}[ϕ]:=\lim_{u\to -\infty} r^2\partial_u(rϕ_{\ell=0})$ is the past Newman--Penrose constant of $ϕ$ on $\mathcal{I}^-$. Thus, the precise late-time asymptotics of $ϕ$ are completely determined by the early-time behaviour of the spherically symmetric part of $ϕ$ near $\mathcal{I}^-$. Similar results are obtained for polynomially decaying timelike boundary data. The paper uses methods developed by Angelopoulos--Aretakis--Gajic and is essentially self-contained.