Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Energy-Morawetz estimates for Teukolsky equations in perturbations of Kerr

Siyuan Ma, Jérémie Szeftel

Abstract

In this paper, we prove energy and Morawetz estimates for solutions to Teukolsky equations in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. The Teukolsky equations are written in tensorial form using the non-integrable formalism in \cite{GKS22}, and we follow the approach in \cite{Ma} of relying on a Teukolsky wave/transport system. The estimates are proved by extending the ideas from our earlier result \cite{MaSz24} on the corresponding problem for the scalar wave, notably the use of $r$-foliation-adapted microlocal multipliers for the wave part, and by incorporating techniques from \cite{Ma} to control the linear coupling terms between the components of the Teukolsky wave/transport system. Additionally, in order to adapt the methodology of \cite{MaSz24} to tensorial waves, we introduce a well-suited regular scalarization procedure which is of independent interest. This result, alongside our companion paper \cite{MaSz24}, is an essential step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS22} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the Kerr stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.

Energy-Morawetz estimates for Teukolsky equations in perturbations of Kerr

Abstract

In this paper, we prove energy and Morawetz estimates for solutions to Teukolsky equations in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. The Teukolsky equations are written in tensorial form using the non-integrable formalism in \cite{GKS22}, and we follow the approach in \cite{Ma} of relying on a Teukolsky wave/transport system. The estimates are proved by extending the ideas from our earlier result \cite{MaSz24} on the corresponding problem for the scalar wave, notably the use of -foliation-adapted microlocal multipliers for the wave part, and by incorporating techniques from \cite{Ma} to control the linear coupling terms between the components of the Teukolsky wave/transport system. Additionally, in order to adapt the methodology of \cite{MaSz24} to tensorial waves, we introduce a well-suited regular scalarization procedure which is of independent interest. This result, alongside our companion paper \cite{MaSz24}, is an essential step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS22} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the Kerr stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.
Paper Structure (130 sections, 119 theorems, 1454 equations, 2 figures)

This paper contains 130 sections, 119 theorems, 1454 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kerr stability conjecture
  3. Teukolsky wave/transport system
  4. State of the art on energy-Morawetz estimates for Teukolsky equations
  5. Teukolsky equations in Kerr
  6. Teukolsky equations in perturbations of Kerr with $|a|\ll m$
  7. First version of the main result
  8. Strategy of the proof
  9. Regular scalarization of the tensorial Teukolsky wave/transport system
  10. Extension to a semi-global problem
  11. Main steps of the proof of Theorem \ref{['thm:main:Teu:rough']}
  12. EMF estimates for inhomogeneous scalarized tensorial wave equations
  13. EMF estimates conditional on the control of low frequencies
  14. EMF estimates for higher order derivatives
  15. Overview of the paper
  16. ...and 115 more sections

Key Result

Theorem 1.2

Let $({\mathcal{M}}, {\bf g})$ be a perturbationMore precisely, the spacetime $({\mathcal{M}},{\bf g})$ is assumed to satisfy the assumptions on a null pair, the metric perturbation, and a regular triplet of horizontal vectorfields made in Sections subsect:assumps:perturbednullframe, subsubsect:assu Here, $\mathbf{k}_s$ are integers measuring regularity, ${\bf E}^{(\mathbf{k}_s)}[{\cdot}](\tau)$,

Figures (2)

  • Figure 1: Penrose diagram of subextremal Kerr spacetimes.
  • Figure 2: Penrose diagram of $({\mathcal{M}}, {\bf g})$. $\Sigma(\tau_1)$ and $\Sigma(\tau_2)$ are two spacelike and asymptotically null level hypersurfaces of a function $\tau$, and ${\mathcal{A}}=\{r=r_+(1-\delta_{{\mathcal{H}}})\}$ is spacelike.

Theorems & Definitions (280)

  • Conjecture 1.1: Kerr stability conjecture
  • Theorem 1.2: Main theorem, rough version
  • Remark 1.3
  • Remark 1.4: Relevance to the Kerr stability conjecture in the full subextremal range
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 270 more