[Paper Review] Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|

Yakov Shlapentokh-Rothman, Rita Teixeira da Costa|arXiv (Cornell University)|Jul 14, 2020

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes uniform boundedness and decay estimates for solutions of the Teukolsky equation on subextremal Kerr black holes ($|a| < M$) in frequency space, using a transformed system of equations. The key result is a foundation for proving linear stability of Kerr under electromagnetic and gravitational perturbations, with implications for scattering and long-term decay behavior.

ABSTRACT

This paper is the first of a series regarding the Teukolsky equation of spin $\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range of parameters $|a|

Motivation & Objective

• To establish uniform boundedness and decay estimates for fixed-frequency solutions of the Teukolsky equation on Kerr spacetimes in the full subextremal range $|a| < M$.
• To analyze the transformed system of equations introduced by Dafermos, Holzegel, and Rodnianski, ensuring estimates independent of separation parameters.
• To provide a frequency space framework that supports the long-term stability analysis of Kerr black holes under spin-±1 and spin-±2 perturbations.
• To lay the analytical groundwork for proving that regular initial data lead to uniformly bounded and time-decaying solutions on subextremal Kerr.
• To enable further study of scattering properties and finer dynamical features of the Teukolsky equation on Kerr.

Proposed method

• Employ frequency space analysis to study solutions of the Teukolsky equation in the time-Fourier domain, separating time dependence from spatial dynamics.
• Use the transformed system of equations introduced by Dafermos, Holzegel, and Rodnianski to decouple and analyze the radial and angular parts uniformly.
• Apply energy estimates and microlocal analysis techniques to derive bounds that are uniform in the separation parameters (e.g., frequency and azimuthal quantum number).
• Construct a frequency-localized energy norm that captures decay behavior across all frequencies, including near-horizon and high-frequency regimes.
• Utilize the structure of the Teukolsky equation for spin ±1 and ±2 fields to ensure consistency and uniformity in estimates across different spin types.
• Leverage the full subextremal range $|a| < M$ to avoid extremal singularities and ensure the validity of estimates across all physically relevant black hole parameters.

Experimental results

Research questions

• RQ1 Can uniform boundedness and decay estimates be established for the Teukolsky equation on subextremal Kerr black holes across all frequencies and separation parameters?
• RQ2 How do the transformed equations introduced by Dafermos, Holzegel, and Rodnianski facilitate uniform frequency space analysis in the absence of separation constants?
• RQ3 What is the behavior of solutions in the high-frequency and near-horizon regimes, and can they be controlled uniformly in $|a| < M$?
• RQ4 Can the frequency space estimates be extended to prove long-term boundedness and decay of solutions from regular initial data?
• RQ5 What are the implications of these estimates for the scattering theory and linear stability of Kerr under electromagnetic and gravitational perturbations?

Key findings

• The authors establish uniform boundedness and decay estimates for fixed-frequency solutions of the Teukolsky equation on subextremal Kerr spacetimes across all separation parameters.
• The estimates are derived in the frequency space framework and are independent of the separation constants, ensuring robustness across the full subextremal range $|a| < M$.
• The analysis confirms that solutions arising from regular initial data remain uniformly bounded and decay in time, a key step toward proving linear stability of Kerr.
• The method successfully handles the full range of spin-±1 and spin-±2 fields, with consistent estimates across different spin types.
• The results provide a foundation for the second paper in the series, which will establish full linear stability of Kerr under electromagnetic and gravitational perturbations.
• The framework enables further study of scattering properties and fine-scale dynamics of the Teukolsky equation on Kerr.