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

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 $r{\Psi }^{[4]}{|}_{{\mathcal{I}}^{+}}\sim u^{-6}$ as $u\to \infty$, we predict that $r{\Psi }^{[4]}{|}_{{\mathcal{I}}^{+}}\sim u^{-3}$, 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.

24 pages, 8 figures. v2: Typo in (2.9) fixed. Prediction for the decay rate of Psi_4 changed from u^{-4} to u^{-3}, further details given in section 6

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• The Case Against Smooth Null Infinity IV: Linearised Gravity Around Schwarzschild -- An Overview
• Late-time asymptotics for geometric wave equations with inverse-square potentials
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces