The Case Against Smooth Null Infinity II: A Logarithmically Modified Price's Law

TL;DR

This paper demonstrates that Price's law asymptotics near future null infinity are modified by logarithmic terms due to non-smoothness at spatial infinity, with the leading behavior determined by initial data and the Newman-Penrose constant.

Contribution

It establishes the presence of logarithmic corrections to Price's law in Schwarzschild spacetime, linking late-time decay to initial data and the Newman-Penrose constant.

Findings

Logarithmic corrections appear in Price's law asymptotics.

Late-time behavior is determined by initial data near past null infinity.

Results are applicable to polynomially decaying boundary data.

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 $\Box_g \phi=0$ 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 $\phi$ are given by $r\phi|_{\mathcal{I}^+}=C u^{-2}\log u+\mathcal{O}(u^{-2})$ along future null infinity, $\phi|_{r=R>2M}=2C\tau^{-3}\log\tau+\mathcal{O}(\tau^{-3})$ along hypersurfaces of constant $r$, and $\phi|_{\mathcal{H}^+}=2Cv^{-3}\log v+\mathcal{O}(v^{-3})$ along the event horizon. Moreover, the constant $C$ is given by $C=4M I_0^{(\mathrm{past})}[\phi]$, where $I_0^{(\mathrm{past})}[\phi]:=\lim_{u\to -\infty} r^2\partial_u(r\phi_{\ell=0})$ is the past Newman--Penrose constant of $\phi$ on $\mathcal{I}^-$. Thus, the precise late-time asymptotics of $\phi$ are completely determined by the early-time behaviour of the spherically symmetric part of $\phi$ 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.