On the Relation Between Asymptotic Charges, the Failure of Peeling and Late-time Tails

TL;DR

<3-5 sentence high-level summary> The paper develops a framework based on $f(r)$-modified Newman–Penrose charges to connect asymptotics of massless fields in different regions of black hole spacetimes, deriving a dictionary between data near $\mathcal I^-$ and late-time behavior at $i^+$ (and vice versa). It shows that physically motivated scattering data, following a quadrupole-like model for infalling masses, generically violates the peeling property and yields slower late-time tails (notably $r\Psi^{[4]}|_{\mathcal I^+} \sim u^{-3}$ for gravitational perturbations, with scalar tails involving logarithmic factors). The approach provides explicit steps (Steps 0–3) to extract late-time behavior from initial data and extends to higher angular modes and to gravitational perturbations, offering a principled link between initial scattering data and observable radiation tails. The results challenge conventional peeling assumptions and point to observable signatures in late-time gravitational radiation and potential extensions to Kerr and nonlinear regimes.

Abstract

The last few years have seen considerable mathematical progress concerning the asymptotic structure of gravitational radiation in dynamical, astrophysical spacetimes. In this paper, we distil some of the key ideas from recent works and assemble them in a new way in order to make them more accessible to the wider general relativity community. We also announce some new physical findings in this process. First, we introduce the conserved $f(r)$-modified Newman--Penrose charges on asymptotically flat spacetimes, and we show that these charges provide a dictionary that relates asymptotics of massless, general spin fields in different regions: Asymptotic behaviour near $i^+$ ("late-time tails") can be read off from asymptotic behaviour towards $\mathcal I^+$, and, similarly, asymptotic behaviour towards $\mathcal I^+$ can be read off from asymptotic behaviour near $i^-$ or $\mathcal I^-$. Using this dictionary, we then explain how: (I) the quadrupole approximation for a system of $N$ infalling masses from $i^-$ causes the "peeling property towards $\mathcal I^+$" to be violated, and (II) this failure of peeling results in deviations from the usual predictions for tails in the late-time behaviour of gravitational radiation: Instead of the Price's law rate $rΨ^{[4]}|_{\mathcal I^+}\sim u^{-6}$ as $u\to\infty$, we predict that $rΨ^{[4]}|_{\mathcal I^+}\sim u^{-3}$, with the coefficient of this latter decay rate being a multiple of the monopole and quadrupole moments of the matter distribution in the infinite past.

Figures (8)

• Figure 1: Depiction of the problems A) and B) described in the paragraph above.
• Figure 2: Summary of the main results of the paper, focussing on the spherically symmetric ($\ell=0$) part of the scalar field ($s=0$).
• Figure 3: A Penrose diagrammatic depiction of $\mathcal{M}_M$ with the asymptotically hyperboloidal foliation by $\Sigma_{\tau}$, constant-$t$ level sets and $u$- and $v$-null hypersurfaces.
• Figure 4: Step 1: integrating $\partial_u\partial_v\phi_0$ in $u$.
• Figure 5: Step 2: integrating $\partial_v\phi_{0}$ in $v$ from $\gamma$.

Theorems & Definitions (1)

• Theorem 5.1