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

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 ${\square }_g\varphi =0$ 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 $\varphi$ are given by $r\varphi {|}_{{\mathcal{I}}^{+}}=C{u}^{-2}\log u+\mathcal{O}(u^{-2})$ along future null infinity, $\varphi {|}_{r=R>2M}=2C{\tau }^{-3}\log \tau +\mathcal{O}({\tau }^{-3})$ along hypersurfaces of constant $r$, and $\varphi {|}_{{\mathcal{H}}^{+}}=2C{v}^{-3}\log v+\mathcal{O}(v^{-3})$ along the event horizon. Moreover, the constant $C$ is given by $C=4M{I}_0^{(\mathrm{past})}[\varphi ]$, where $I_0^{(\mathrm{past})}[\varphi ]:={\lim }_{u\to -\infty }{r}^2{\partial }_u(r{\varphi }_{\ell =0})$ is the past Newman--Penrose constant of $\varphi$ on ${\mathcal{I}}^{-}$. Thus, the precise late-time asymptotics of $\varphi$ are completely determined by the early-time behaviour of the spherically symmetric part of $\varphi$ 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.

34 pages, 3 figures

References in corpus (3)

• The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background
• Price's law and precise late-time asymptotics for subextremal Reissner-Nordström black holes
• Price's law for spin fields on a Schwarzschild background

Cited by in corpus (5)

• Local supersymmetry and the square roots of Bondi-Metzner-Sachs supertranslations
• On the Relation Between Asymptotic Charges, the Failure of Peeling and Late-time Tails
• The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Scattering for wave equations with sources close to the lightcone and prescribed radiation fields