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.
This paper establishes a mathematical proof of the blue-shift instability at the sub-extremal Kerr Cauchy horizon for the linearised vacuum Einstein equations. More precisely, we exhibit conditions on the $s=+2$ Teukolsky field, consisting of suitable integrated upper and lower bounds on the decay along the event horizon, that ensure that the Teukolsky field, with respect to a frame that is regular at the Cauchy horizon, becomes singular. The conditions are in particular satisfied by solutions of the Teukolsky equation arising from generic and compactly supported initial data by the recent wor…
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, le…
This second part of the series treats spin $\pm2$ components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin $\pm 2$ components is proved by treating this system. This is a first step in a joint work (Andersson et al. in Stability fo…
We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters $|a|<M$. As in the Schwarzschild …
We derive the large time asymptotics of initially regular and localized solutions of the Teukolsky equation on the exterior of a subextremal Kerr black hole for any half integer spin. More precisely, we obtain the leading order term (predicted by Price's law) in the large time regime assuming that the initial data have compact support and have enough (but finite) Sobolev regularity. For initial data with less spatial decay (typically decaying like r^{--1--$α$} with $α$ $\in$ (0, 1)), we prove that the solution has a pointwise decay of order t^{--1--$α$--s--|s|+} on spatially compact regions. I…
We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the $|a|=0$ case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin $\pm 2$ Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent pa…
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for g…
We relate the recent derivation of the linearised Einstein equation on an AdS background from holographic entanglement entropy arguments to the Sparling construction: we derive the differential form whose exterior derivative gives the Einstein equation from the Sparling formalism. We develop the study of perturbations within the context of the Sparling formalism and find that the Sparling form vanishes for linearised perturbations on flat space.
This first part of the series treats the Maxwell equations in the exterior of a slowly rotating Kerr black hole. By performing a first-order differential operator on each extreme Newman-Penrose (N-P) scalar in a Kinnersley tetrad, the resulting equation and the Teukolsky master equation for the extreme N-P component are both in the form of an inhomogeneous \textquotedblleft{spin-weighted Fackerell-Ipser equation\textquotedblright} (SWFIE) and constitute a weakly coupled system. We first prove energy estimate and integrated local energy decay (Morawetz) estimate for this type of inhomogeneous S…
A reformulation of the Schwarzschild solution of the linearised Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further a…
In a recent seminal paper \cite{D-H-R} of Dafermos, Holzegel and Rodnianski the linear stability of the Schwarzschild family of black hole solutions to the Einstein vacuum equations was established by imposing a double null gauge. In this paper we shall prove that the Schwarzschild family is linearly stable as solutions to the Einstein vacuum equations by imposing instead a generalised wave gauge: all sufficiently regular solutions to the system of equations that result from linearising the Einstein vacuum equations, as expressed in a generalised wave gauge, about a fixed Schwarzschild solutio…
We study the connection between many-body quantum chaos and energy dynamics for the holographic theory dual to the Kerr-AdS black hole. In particular, we determine a partial differential equation governing the angular profile of gravitational shock waves that are relevant for the computation of out-of-time ordered correlation functions (OTOCs). Further we show that this shock wave profile is directly related to the behaviour of energy fluctuations in the boundary theory. In particular, we demonstrate using the Teukolsky formalism that at complex frequency $ω_* = i 2 πT$ there exists an extra i…
EMRI Dephasing from a Torsion-Inspired Near-Zone Kerr Deformation: Motivated by Spin-Polarized Dark Matter
Faculty of Physics, Lomonosov Moscow State University（莫斯科国立大学物理系）；International School, I.M. Sechenov First Moscow State Medical University（伊万·谢切诺夫第一莫斯科国立医科大学国际学院）；Steklov Mathematical Institute, Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
