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.
Conclusions and Future Roadmap
Conclusions:
- The article presents a scattering problem for a system of linearized gravity around Schwarzschild.
- The scattering data are posed on a null hypersurface emanating from past null infinity.
- The gauge-invariant components of the Weyl tensor decay along the null hypersurface.
- Gravitational perturbations along past null infinity induce similar data.
- A unique solution to the scattering problem is constructed.
- An analysis of the asymptotic behavior of projections onto fixed spherical harmonic number is provided.
- Corrections to popular historical notions of asymptotic flatness are given.
- The peeling property is found to be violated near both past and future null infinity.
- The resulting solution decays slower towards spacelike infinity than previously assumed.
Future Roadmap:
- Further work is needed to address the issue of summing up the fixed angular mode estimates in ell.
- Explore and develop the implications of the constructed solution to other areas of study.
- Investigate the impact of violating the peeling property near past and future null infinity on gravitational phenomena.
- Study the consequences of slower decay towards spacelike infinity on the understanding of black hole dynamics.
- Continuously compare and refine the corrections to asymptotic flatness notions, such as Bondi coordinates or asymptotic simplicity.
Overall, the article provides valuable insights into the behavior of a system of linearized gravity around Schwarzschild and its implications for the understanding of asymptotic flatness and gravitational perturbations. It opens up avenues for further research and invites exploration of the consequences of violating the peeling property and slower decay towards spacelike infinity.