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The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background

Lionor M. A. Kehrberger

TL;DR

This paper advances the understanding of early-time, fixed-frequency wave behavior on Schwarzschild spacetimes by analyzing higher angular momentum modes (ℓ≥1) under a no-incoming-radiation condition. It develops a unified framework of approximate conservation laws and higher-order Newman–Penrose constants to derive explicit logarithmic corrections in the asymptotics of ∂_v(rφ_ℓ) near spatial infinity and i^0, for both timelike and null data. For the principal case ℓ=1 and then for general ℓ, the authors prove existence and precise decay of solutions, show logarithmic modifications to Price’s law, and reveal cancellations that affect whether logarithms appear at certain orders. The work also connects early-time behavior to late-time tails at i^+ and proposes conjectures about universal tails for compactly supported scattering data, thereby challenging the traditional smooth-null-infinity picture and informing ongoing analyses of gravitational radiation in curved backgrounds.

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions $φ_\ell$ to the wave equation $\Box_g φ_\ell=0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on $\mathcal I^-$ and polynomially decaying data, $rφ_\ell\sim t^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $\partial_v(rφ_\ell)$ along outgoing null hypersurfaces near spacelike infinity $i^0$ contains logarithmic terms at order $r^{-3-\ell}\log r$. In contrast, in case (II), we obtain that the asymptotic expansion of $\partial_v(rφ_\ell)$ near spacelike infinity $i^0$ contains logarithmic terms already at order $r^{-3}\log r$ (unless $\ell=1$). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity $i^+$ that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate logarithmic modifications to Price's law for each $\ell$-mode. On the other hand, the data of case (II) lead to much stronger deviations from Price's law. In particular, we conjecture that compactly supported scattering data on $\mathcal H^-$ and $\mathcal I^-$ lead to solutions that exhibit the same late-time asymptotics on $\mathcal I^+$ for each $\ell$: $rφ_\ell|_{\mathcal I^+}\sim u^{-2}$ as $u\to\infty$.

The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background

TL;DR

This paper advances the understanding of early-time, fixed-frequency wave behavior on Schwarzschild spacetimes by analyzing higher angular momentum modes (ℓ≥1) under a no-incoming-radiation condition. It develops a unified framework of approximate conservation laws and higher-order Newman–Penrose constants to derive explicit logarithmic corrections in the asymptotics of ∂_v(rφ_ℓ) near spatial infinity and i^0, for both timelike and null data. For the principal case ℓ=1 and then for general ℓ, the authors prove existence and precise decay of solutions, show logarithmic modifications to Price’s law, and reveal cancellations that affect whether logarithms appear at certain orders. The work also connects early-time behavior to late-time tails at i^+ and proposes conjectures about universal tails for compactly supported scattering data, thereby challenging the traditional smooth-null-infinity picture and informing ongoing analyses of gravitational radiation in curved backgrounds.

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions to the wave equation on a fixed Schwarzschild background () arising from the no incoming radiation condition on and polynomially decaying data, as , on either a timelike boundary of constant area radius (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of along outgoing null hypersurfaces near spacelike infinity contains logarithmic terms at order . In contrast, in case (II), we obtain that the asymptotic expansion of near spacelike infinity contains logarithmic terms already at order (unless ). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate logarithmic modifications to Price's law for each -mode. On the other hand, the data of case (II) lead to much stronger deviations from Price's law. In particular, we conjecture that compactly supported scattering data on and lead to solutions that exhibit the same late-time asymptotics on for each : as .

Paper Structure

This paper contains 101 sections, 46 theorems, 448 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. The spherically symmetric mode
  4. Higher l-modes
  5. Summary of the main results
  6. The case L=1
  7. The case of general L>0
  8. Future applications: Late-time asymptotics and the role of the modified Newman--Penrose constants
  9. The timelike case: A logarithmically modified Price's law for all L
  10. The null case: More severe deviations from Price's law
  11. Compactly supported scattering data on H- and I-
  12. Structure and guide to reading the paper
  13. Geometric preliminaries
  14. The Schwarzschild spacetime manifold
  15. The divergence theorem
  16. ...and 86 more sections

Key Result

Theorem 1.1

Given smooth boundary data satisfying eq:intro:ass1, there exists a unique smooth (finite-energy) solution to waveequation (restricted to the $(1,m)$-angular frequency) in the domain of dependence of $\Gamma\cup \mathcal{I}^-$ that restricts correctly to these data and satisfies eq:intro:nir. Moreov where $I_{\ell=1}^{\mathrm{past}}[\phi]$ is a constant which is non-vanishing as long as $C_\Gamma$

Figures (6)

  • Figure 1: Schematic depiction of the data setup considered in 1): We consider polynomially decaying data on a spherically symmetric timelike hypersurface $\Gamma$, and vanishing data on $\mathcal{I}^-$. The latter condition is to be thought of as the no incoming radiation condition.
  • Figure 2: Schematic depiction of the data setup considered in 2): We consider polynomially decaying data on a spherically symmetric ingoing null hypersurface $\mathcal{C}_{\mathrm{in}}$, and vanishing data on the part of $\mathcal{I}^-$ that lies to the future of $\mathcal{C}_{\mathrm {in}}$. The latter condition is to be thought of as the no incoming radiation condition.
  • Figure 3: Schematic depiction of the situation of § \ref{['intro:subsecNP1']}: Given smooth data for $r\phi_\ell$ on $\Gamma$ which decay like $1/t$ near $i^-$, the solution decays like $u^{-\ell-1}$ near $\mathcal{I}^-$ by Thm. \ref{['thm:intro:gtl']} and has finite logarithmically modified N--P constant on $\mathcal{I}^+$ by Thm. \ref{['thm:intro:gnl']}. The depicted late-time behaviour near $i^+$ should follow from the methods of AAG21 and should be independent of the data's extension towards $\mathcal{H}^+$.
  • Figure 4: Schematic depiction of the situation of § \ref{['intro:subsecNP2']}: Given data for $r\phi_\ell$ on $\mathcal{C}_{\mathrm{in} }$ which decay like $1/u^p$ near $\mathcal{I}^-$, the solution has finite $(\ell-p)$-modified N--P constant (see \ref{['intro:eq:y']}) on $\mathcal{I}^+$ by Thm. \ref{['thm:intro:moreg']}, provided that $p\leq \ell$. We also depicted the conjectured late-time behaviour near $i^+$.
  • Figure 5: Given compactly supported scattering data for $r\phi_\ell$ on $\mathcal{H}^-$ and $\mathcal{I}^-$, the solution generically decays like $1/u$ near $\mathcal{I}^-$ (away from the support of the data) unless $\ell=0$. It thus has finite $(\ell-1)$-modified N--P constant (see \ref{['intro:eq:y']}) on $\mathcal{I}^+$ by Thm. \ref{['thm:intro:moreg']}. We also depicted the conjectured late-time asymptotics for all $\ell\geq 0$.
  • ...and 1 more figures

Theorems & Definitions (135)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.4
  • Remark 1.5
  • ...and 125 more