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Quantitative local recovery of Kerr-de Sitter parameters from high-frequency equatorial quasinormal modes

Ruiliang Li

Abstract

We study an inverse resonance problem for the scalar wave equation on the Kerr-de Sitter family. In a compact subextremal slow-rotation regime and at a fixed overtone index, high-frequency quasinormal modes admit semiclassical quantization and a real-analytic labeling by angular momentum indices. Using this structure, we first prove that a finite equatorial high-frequency package of quasinormal-mode frequencies determines the mass and rotation parameter $(M,a)$ (for fixed cosmological constant $Λ>0$), with a quantitative stability estimate. As a key geometric input we compute explicit second-order (in $a$) corrections to the equatorial photon-orbit invariants which control the leading real and imaginary parts of the quasinormal modes. Finally, allowing $Λ$ to vary in a compact interval, we show that adding one damping observable (the scaled imaginary part of a single equatorial mode) yields a three-parameter inverse theorem: a finite package of three independent real observables determines $(M,a,Λ)$ locally in the slow-rotation regime away from $a=0$.

Quantitative local recovery of Kerr-de Sitter parameters from high-frequency equatorial quasinormal modes

Abstract

We study an inverse resonance problem for the scalar wave equation on the Kerr-de Sitter family. In a compact subextremal slow-rotation regime and at a fixed overtone index, high-frequency quasinormal modes admit semiclassical quantization and a real-analytic labeling by angular momentum indices. Using this structure, we first prove that a finite equatorial high-frequency package of quasinormal-mode frequencies determines the mass and rotation parameter (for fixed cosmological constant ), with a quantitative stability estimate. As a key geometric input we compute explicit second-order (in ) corrections to the equatorial photon-orbit invariants which control the leading real and imaginary parts of the quasinormal modes. Finally, allowing to vary in a compact interval, we show that adding one damping observable (the scaled imaginary part of a single equatorial mode) yields a three-parameter inverse theorem: a finite package of three independent real observables determines locally in the slow-rotation regime away from .
Paper Structure (83 sections, 43 theorems, 325 equations)

This paper contains 83 sections, 43 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Geometric and analytic setup
  3. The Kerr--de Sitter exterior and the subextremal parameter region
  4. Boyer-- Lindquist form.
  5. Subextremality and horizons.
  6. A fixed reference manifold for varying parameters.
  7. Regular coordinates and the future extension across horizons
  8. Star coordinates.
  9. An extended spacetime.
  10. Killing fields.
  11. The wave operator and stationary reduction
  12. Wave operator.
  13. Stationary family.
  14. Azimuthal decomposition.
  15. Sobolev spaces, Fredholm setup, and the definition of quasinormal modes
  16. ...and 68 more sections

Key Result

Theorem 5

Fix $\Lambda>0$ and $(M,a)\in\mathcal{K}\Subset \mathcal{P}_\Lambda$. For each $\omega\in\mathbb{C}$ one can choose an outgoing variable order $\mathbf s_\omega$ such that is Fredholm of index $0$. Moreover, there exists $\omega_0\in\mathbb{R}$ such that for $\Im \omega>\omega_0$ the operator $P(\omega)$ is invertible, and the inverse family extends meromorphically to $\omega\in\mathbb{C}$ with

Theorems & Definitions (102)

  • Remark 1: Why we keep the full root structure
  • Remark 2: Varying the cosmological constant
  • Remark 3: Alternative Killing fields and the full subextremal range
  • Remark 4: Frequencies in $(t_*,\varphi_*)$ and Boyer--Lindquist $(t,\varphi)$
  • Theorem 5: Meromorphic resolvent and quasinormal modes
  • Definition 6: Quasinormal modes
  • Remark 7: Outgoing boundary conditions and regularity
  • Remark 8: Analytic dependence on parameters
  • Proposition 9: QNMs and common $\lambda$--poles
  • proof
  • ...and 92 more