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Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

Yannis Angelopoulos, Stefanos Aretakis, Dejan Gajic

TL;DR

The work tackles second-order late-time behavior of scalar radiation on 4D, spherically symmetric, asymptotically flat spacetimes, including Schwarzschild and sub-extremal Reissner–Nordström. It develops purely physical-space methods to derive logarithmic corrections to leading-order decay along null infinity and expresses the second-order coefficients in terms of initial data via the Newman–Penrose constants and their time-integrals. The results cover two regimes: nonzero $I_0[\psi]$, where the leading radiation behaves like $\sim 2 I_0[\psi]/u$ with a logarithmic $\log u/u^{2}$ correction, and zero $I_0[\psi]$, where explicit logarithmic corrections appear in terms of $I_0^{(1)}[\psi]$, including $r\psi|_{\mathcal{I}^+}$. The findings sharpen our understanding of late-time tails, with implications for black hole stability, cosmic censorship, and gravitational-wave propagation in curved spacetimes.

Abstract

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.

Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

TL;DR

The work tackles second-order late-time behavior of scalar radiation on 4D, spherically symmetric, asymptotically flat spacetimes, including Schwarzschild and sub-extremal Reissner–Nordström. It develops purely physical-space methods to derive logarithmic corrections to leading-order decay along null infinity and expresses the second-order coefficients in terms of initial data via the Newman–Penrose constants and their time-integrals. The results cover two regimes: nonzero , where the leading radiation behaves like with a logarithmic correction, and zero , where explicit logarithmic corrections appear in terms of , including . The findings sharpen our understanding of late-time tails, with implications for black hole stability, cosmic censorship, and gravitational-wave propagation in curved spacetimes.

Abstract

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.

Paper Structure

This paper contains 23 sections, 10 theorems, 147 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Introduction and background
  3. The main results
  4. Discussion on the full asymptotic expansion
  5. Preliminaries
  6. The geometric assumptions
  7. The Lorentzian manifolds $(\mathcal{M},g)$
  8. The spacelike-null foliation $\Sigma_{\tau}$
  9. Assumptions for the wave equation
  10. The Newman--Penrose constant $I_{0}[\psi]$ at null infinity
  11. The time-integral $\psi^{(1)}$ and the time-inverted Newman--Penrose constant $I_{0}^{(1)}$
  12. Leading-order asymptotics
  13. Higher-order estimates in the region $\{ r \geq R \}$
  14. Estimates for $\partial_v(r\psi)$ in the case $I_0 [\psi ] \neq 0$
  15. Estimate for $r\psi$ in the case $I_0 [\psi ] \neq 0$
  16. ...and 8 more sections

Key Result

Theorem 1.1

(Second-order asymptotics for $r\psi|_{\mathcal{I}^+}$ with $I_{0}[\psi]=0$) For all spherically symmetric solutions $\psi$ to the wave equation we on the spacetimes $(\mathcal{M},g)$ given in Section thegeometricassumptions with compactly supported initial data, there exists a constant $C>0$ that d The constant $I_0^{(1)}[\psi]$ is explicitly given in terms of the initial data of $\psi$ on $\Sigm

Figures (4)

  • Figure 1: Penrose diagram of $\mathcal{M}$ in the case $r_{\text{min}}>0$. The hypersurfaces $\Sigma_{\tau}$ intersect the future event horizon $\mathcal{H}^{+}$.
  • Figure 2: The Penrose diagram of $\mathcal{M}$ in the case $r_{\text{min}}=0$.
  • Figure 3: The region $\{r\geq R\}\cap\{u\geq 0\}$ and the curve $\gamma$.
  • Figure 4: The shaded region depicts the integration region appearing on the very left-hand side of \ref{['eq:spacetimeint']}.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Proposition 3.1
  • ...and 10 more