arXiv subjects

Hamed Masaood

Publications and source records attributed to Hamed Masaood.

4 recordsLinked to original sources

arXiv2026

On the image of the scattering map for horizon-regular solutions of the linear scalar wave equation on the Schwarzschild black hole exterior

We construct Hilbert-space isomorphisms identifying the spaces of radiation fields on the event horizon and null infinity that are induced by the forward scattering map on the Schwarzschild exterior for fixed spherical harmonic mode solutions to the linear scalar wave equation arising from Cauchy data sets at $t=0$ which are regular at the event horizon, in the sense that the local energy at the event horizon is finite. We show that fixed spherical harmonic mode solutions with no radiation on the event horizon must decay exponentially in time, at a rate that is determined by the surface gravity. On the other hand, we construct examples of fixed spherical harmonic mode solutions with no radiation on null infinity and which have polynomial decay along the event horizon. Finally, we construct examples of polynomially decaying solutions to the linear scalar wave equation which are regular at the event horizon, have unbounded support in spherical harmonic modes, and induce no radiation on the event horizon.

gr-qc↗

arXiv2024

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

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 $α$ and $\underlineα$ (a.k.a. $Ψ_0$ and $Ψ_4$) 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 $α$, $\underlineα$ 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. $α$ 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 $α$ 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.

gr-qc↗

arXiv2022

A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole II: The Full System

We construct a scattering theory for the linearised Einstein equations on a Schwarzschild background in a double null gauge. We build on the results of Part I \cite{Mas20}, where we used the energy conservation enjoyed by the Regge--Wheeler equation associated with the stationarity of the Schwarzschild background to construct a scattering theory for the Teukolsky equations of spin $\pm2$. We now extend the scattering theory of Part I to the full system of linearised Einstein equations by treating it as a system of transport equations which is sourced by solutions to the Teukolsky equations, leading to Hilbert space-isomorphisms between spaces of finite energy initial data and corresponding spaces of scattering states under suitably chosen gauge conditions on initial and scattering data. As a corollary, we show that for a solution which is Bondi-normalised at both past and future null infinity, past and future linear memories are related by an antipodal map.

gr-qc↗

arXiv2020

A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations

We construct a scattering theory for the spin $\pm2$ Teukolsky equations on the exterior of the Schwarzschild spacetime, as a first step towards developing a scattering theory for the linearised Einstein equations in double null gauge. This is done by exploiting a physical-space version of the Chandrasekhar transformation used by Dafermos, Holzegel and Rodnianski in [arxiv:1601.06467] to prove the linear stability of the Schwarzschild solution. We also address the Teukolsky-Starobinsky correspondence and construct an isomorphism between scattering data for the $+2$ and $-2$ Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin $+2$ and $-2$ Teukolsky equations and connected via the Teukolsky-Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.

gr-qc↗