The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes

Abstract

Starting from Post-Newtonian predictions for a system of $N$ infalling masses from the infinite past, we formulate and solve a scattering problem for the system of linearised gravity around Schwarzschild as introduced in [DHR19]. The scattering data are posed on a null hypersurface $\mathcal C$ emanating from a section of past null infinity $\mathcal I^-$, and on the part of $\mathcal I^-$ that lies to the future of this section: Along $\mathcal C$, we implement the Post-Newtonian theory-inspired hypothesis that the gauge-invariant components of the Weyl tensor $\alpha$ and $\underline{\alpha}$ (a.k.a. $\Psi_0$ and $\Psi_4$) decay like $r^{-3}$, $r^{-4}$, respectively, and we exclude incoming radiation from $\mathcal I^-$ by demanding the News function to vanish along $\mathcal I^-$. We also show that compactly supported gravitational perturbations along $\mathcal I^-$ induce very similar data, with $\alpha$, $\underline{\alpha}$ decaying like $r^{-3}$, $r^{-5}$ along $\mathcal C$. After constructing the unique solution to this scattering problem, we provide a complete analysis of the asymptotic behaviour of projections onto fixed spherical harmonic number $\ell$ near spacelike $i^0$ and future null infinity $\mathcal I^+$. Using our results, we also give constructive corrections to popular historical notions of asymptotic flatness such as Bondi coordinates or asymptotic simplicity. In particular, confirming earlier heuristics due to Damour and Christodoulou, we find that the peeling property is violated both near $\mathcal I^-$ and near $\mathcal I^+$, with e.g. $\alpha$ near $\mathcal I^+$ only decaying like $r^{-4}$ instead of $r^{-5}$. We also find that the resulting solution decays slower towards $i^0$ than often assumed, with $\alpha$ decaying like $r^{-3}$ towards $i^0$. The issue of summing up the fixed angular mode estimates in $\ell$ is dealt with in forthcoming work.