[Paper Review] Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\\ll M$

Mihalis Dafermos, Gustav Holzegel+1 more|arXiv (Cornell University)|Nov 21, 2017

Advanced Mathematical Physics Problems7 citations

TL;DR

This paper establishes boundedness and polynomial decay for solutions of the spin $\pm 2$ Teukolsky equation on slowly rotating Kerr spacetimes with $|a| \ll M$. By introducing generalized $P$ and $\underline{P}$ quantities inspired by Chandrasekhar's transformation theory, the authors extend the linear stability framework from Schwarzschild to Kerr, providing the first step toward proving full linear stability of the Kerr metric under gravitational perturbations.

ABSTRACT

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 case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

Motivation & Objective

• To extend the boundedness and decay results for the Teukolsky equation from the Schwarzschild case ($a=0$) to the slowly rotating Kerr case ($|a| \ll M$).
• To generalize the $P$ and $\underline{P}$ energy quantities used in the Schwarzschild stability analysis to the Kerr background.
• To establish foundational estimates for the linear stability of the Kerr metric under gravitational perturbations.
• To lay the groundwork for a full proof of linear stability in the general sub-extremal range $|a| < M$ in a follow-up paper.

Proposed method

• Introduce generalized $P^{[\pm 2]}$ and $\underline{P}^{[\pm 2]}$ quantities as higher-order energy norms adapted to the Kerr metric.
• Derive a generalized inhomogeneous Regge–Wheeler-type equation for the transformed Teukolsky variable $\Psi^{[\pm 2]}$.
• Use frequency-localized analysis in the separated form of the equation to control the dynamics in the region $r \in [A_1, A_2]$.
• Apply physical-space multiplier estimates using $T$, $\Phi$, $y$, redshift, and $r^p$ multipliers to control energy and integrated local energy decay.
• Construct a hierarchy of $r^p$-weighted energy estimates to derive polynomial decay rates for the Teukolsky variables.
• Leverage the structure of the Teukolsky equation and its separation via Teukolsky's ansatz to reduce the problem to radial ODE analysis.

Experimental results

Research questions

• RQ1 Can boundedness and polynomial decay be established for the spin $\pm 2$ Teukolsky equation on slowly rotating Kerr spacetimes with $|a| \ll M$?
• RQ2 How can the $P$ and $\underline{P}$ energy quantities from the Schwarzschild case be generalized to the Kerr background?
• RQ3 What is the role of Chandrasekhar-type transformations in extending the stability framework to rotating spacetimes?
• RQ4 How do frequency-localized estimates in the separated representation contribute to physical-space energy bounds?
• RQ5 What is the structure of the inhomogeneous term in the Regge–Wheeler-type equation derived from the Teukolsky equation?

Key findings

• The authors prove $L^2$-boundedness and polynomial decay in time for solutions of the spin $\pm 2$ Teukolsky equation on Kerr spacetimes with $|a| \ll M$.
• Generalized $P$ and $\underline{P}$ quantities are constructed and shown to control the energy and integrated local energy of the Teukolsky variables.
• The $r^p$-weighted energy hierarchy yields polynomial decay rates of order $t^{-p}$ for $p < 1$, with optimal decay for $p \to 1^-$.
• The redshift effect is quantified and used to control the energy near the horizon, ensuring uniform boundedness.
• The inhomogeneous term $\mathfrak{K}^{[\pm 2]}$ in the Regge–Wheeler-type equation is controlled via frequency-localized estimates.
• The analysis confirms the existence of a physical-space framework for energy control and decay, even in the absence of full symmetry.