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

TL;DR

This work analyzes the linear wave equation on Schwarzschild, showing that non-smooth null infinity near $i^0$ induces leading-order logarithmic corrections to Price's law tails near $i^+$. Building on the Angelopoulos–Aretakis–Gajic framework and Kerrburger insights, the authors relate early-time data to late-time logarithmic tails via modified Newman–Penrose constants $I_0^{\log}[\phi]$ and, when needed, time-inverted constants $I_0^{(1)}[\phi]$. The main result provides explicit asymptotics: $r\phi|_{\mathcal{I}^+}=C u^{-2}\log u+\mathcal{O}(u^{-2})$, $\phi|_{r=R}=2C\tau^{-3}\log\tau+\mathcal{O}(\tau^{-3})$, and $\phi|_{\mathcal{H}^+}=2C v^{-3}\log v+\mathcal{O}(v^{-3})$ with $C=4M I_0^{(\mathrm{past})}[\phi]$, illustrating how non-smoothness is physically measurable. The results generalize to other spherically symmetric settings and outline pathways to nonlinear extensions, highlighting the relevance of logarithmic tails for both theoretical and observational considerations in black-hole spacetimes.

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.

Figures (3)

• Figure 1: Schematic depiction of the results of the present paper. We can either impose data $r\phi|_{\Gamma}=C/t+\dots$, which will, by our previous results Kerrburger, lead to behaviour on $\mathcal{C}_{\mathrm{in}}$ as depicted, or we can directly impose data on $\mathcal{C}_{\mathrm{in}}$. In both cases, we obtain logarithmic late-time asymptotics near $i^+$ provided the data are extended to $\mathcal{H}^+$.
• Figure 2: Depiction of the spacelike-null foliation of the Schwarzschild manifold $\mathcal{M}_M$ by the hypersurfaces $\Sigma_\tau$.
• Figure 3: Depiction of $\mathcal{R}:=J^+(\Sigma_0)$ and its subsets $\mathcal{B}_\alpha$ (to the right of the blue curve), $\mathcal{R}\cap\{r\geq R\}\setminus \mathcal{B}_\alpha$, and $\mathcal{R}\cap\{r\leq R\}$. The blue curve, in turn, corresponds to $\gamma_\alpha$.

Theorems & Definitions (32)

• Remark 1.1: Higher $\ell$-modes
• Theorem 1.1
• Remark 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 4.1
• proof
• Proposition 4.2
• proof