[Paper Review] Optimal decay for solutions of the Teukolsky equation on the Kerr metric for the full subextremal range |a| < M

Pascal Millet|arXiv (Cornell University)|Feb 14, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes optimal pointwise decay rates for solutions of the Teukolsky equation on subextremal Kerr black holes for all half-integer spins, using spectral theory and microlocal analysis. It proves that compactly supported, Sobolev-regular initial data lead to decay matching Price’s law, while less decaying data yield decay of order $ t^{-1 - \alpha - s - |s| + \epsilon} $, with sharp asymptotics derived via resolvent analysis and contour deformation in the complex plane.

ABSTRACT

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. In the proof, we adopt the spectral point of view and make use of recent advances in microlocal analysis and non elliptic Fredholm theory which provide a robust framework to study linear operators on black hole type spacetimes.

Motivation & Objective

• To establish sharp large-time decay estimates for solutions of the Teukolsky equation on subextremal Kerr black holes across all half-integer spin values.
• To extend decay results beyond compactly supported smooth data to initial data with polynomial spatial decay $ r^{-1 - \alpha} $, $ \alpha \in (0,1) $.
• To rigorously derive the leading-order asymptotic behavior of solutions, confirming predictions from Price’s law in the context of linearized gravity and wave propagation on Kerr spacetimes.
• To develop and apply a robust spectral-theoretic framework based on non-elliptic Fredholm theory and semiclassical analysis for black hole spacetimes.

Proposed method

• Adopt a spectral approach by analyzing the resolvent of the Teukolsky operator $ \hat{T}_s(\sigma) $, treating it as a family of pseudodifferential operators on a compactified spacetime.
• Use microlocal techniques, including propagation of singularities and parametrix constructions, to analyze the behavior of the Hamiltonian flow associated with the principal symbol of $ \hat{T}_s(\sigma) $.
• Establish global Fredholm properties for $ \hat{T}_s(\sigma) $ by proving uniform estimates near the horizon, spatial infinity, and the trapped set, using weighted Sobolev spaces and semiclassical calculus.
• Apply a contour deformation argument in the complex $ \sigma $-plane to express the solution as a sum of a stationary term and a decaying remainder, leveraging the meromorphic structure of the resolvent.
• Use Mellin transforms with respect to the compactified radial variable $ \rho_I $ to analyze the behavior near spatial infinity and extract the leading-order decay term.
• Employ a partition of unity and cutoff techniques to localize the analysis and control the regularity and decay of the solution in different regions of spacetime.

Experimental results

Research questions

• RQ1 What is the optimal pointwise decay rate for solutions of the Teukolsky equation on subextremal Kerr black holes with compactly supported, Sobolev-regular initial data?
• RQ2 How does the decay rate depend on the spin $ s \in \frac{1}{2}\mathbb{Z} $ and the spatial decay of initial data, particularly when initial data decay like $ r^{-1 - \alpha} $?
• RQ3 Can the leading-order asymptotic behavior of solutions be rigorously derived using spectral and microlocal methods, confirming Price’s law in the full subextremal range $ |a| < M $?
• RQ4 What is the structure of the resolvent $ \hat{T}_s(\sigma)^{-1} $, and how does its meromorphic extension to the complex plane enable the derivation of time decay?

Key findings

• For compactly supported, $ H^s $-regular initial data with $ s $ sufficiently large, the solution decays pointwise like $ t^{-2 - 2|s|} $, matching the prediction of Price’s law for linearized gravity on Kerr spacetimes.
• For initial data decaying like $ r^{-1 - \alpha} $ with $ \alpha \in (0,1) $, the solution decays pointwise as $ t^{-1 - \alpha - s - |s| + \epsilon} $ for any $ \epsilon > 0 $, with the exponent sharp up to $ \epsilon $.
• The leading-order term in the large-time asymptotics is explicitly identified as a stationary mode associated with a simple pole of the resolvent at $ \sigma = -i $, arising from the Mellin transform analysis.
• The resolvent $ \hat{T}_s(\sigma)^{-1} $ is shown to be meromorphic on $ \mathbb{C} \setminus \{0\} $ with a single pole at $ \sigma = -i $, and its regularity is controlled via weighted Sobolev estimates.
• The Fredholm property of $ \hat{T}_s(\sigma) $ is established globally on the compactified spacetime, with uniform estimates near the horizon, spatial infinity, and the trapped set, using non-elliptic Fredholm theory.
• The solution operator is shown to be bounded and regular in weighted $ H^r_b $ spaces, and the contour deformation argument yields a precise decomposition of the solution into a stationary term and a decaying remainder with controlled regularity.