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Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes

Gustav Holzegel, Jacques Smulevici

Abstract

This paper investigates the decay properties of solutions to the massive linear wave equation $\Box_g ψ+ \frac{α}{l^2} ψ=0$ for $g$ the metric of a Kerr-AdS spacetime satisfying $|a|l<r_+^2$ and $α<9/4$ satisfying the Breitenlohner Freedman bound. We prove that the non-degenerate energy of $ψ$ with respect to an appropriate foliation of spacelike slices decays like $(\log t^\star)^{-2}$. Our estimates are expected to be sharp from heuristic and numerical arguments in the physics literature suggesting that general solutions will only decay logarithmically. The underlying reason for the slow decay rate can be traced back to a stable trapping phenomenon for asymptotically anti de Sitter black holes which is in turn a consequence of the reflecting boundary conditions for $ψ$ at null-infinity.

Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes

Abstract

This paper investigates the decay properties of solutions to the massive linear wave equation for the metric of a Kerr-AdS spacetime satisfying and satisfying the Breitenlohner Freedman bound. We prove that the non-degenerate energy of with respect to an appropriate foliation of spacelike slices decays like . Our estimates are expected to be sharp from heuristic and numerical arguments in the physics literature suggesting that general solutions will only decay logarithmically. The underlying reason for the slow decay rate can be traced back to a stable trapping phenomenon for asymptotically anti de Sitter black holes which is in turn a consequence of the reflecting boundary conditions for at null-infinity.

Paper Structure

This paper contains 48 sections, 28 theorems, 201 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Logarithmic Decay
  3. Absence of time-periodic solutions
  4. Quasi-normal modes
  5. The non-linear stability problem
  6. Outline
  7. Outline
  8. The Kerr-AdS family of spacetimes
  9. The fixed manifold with boundary $\mathcal{R}$
  10. The parameter space
  11. Kerr-AdS-star coordinates
  12. The tortoise coordinate $r^*$
  13. Boyer-Lindquist coordinates
  14. The Kerr-AdS metric for fixed $(a,M,l)$
  15. Coordinate Vectorfields and Isometries
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $\left(\mathcal{M},g \right)$ be a Kerr-AdS spacetime with parameters $\left(M,a,l\right)$ for which both the Hawking-Reall condition $r_+^2 > |a| l$ and the regularity condition $|a|<l$ hold. Let $\psi$ be a $CH_{AdS}^2$ solution of (mwe) with $-\infty < \alpha< \frac{9}{4}$ on this background. Then, for the foliation of spacelike slices $\Sigma_{t^\star}$ (intersecting the event horizon) def

Figures (1)

  • Figure 1: The spacelike hypersurfaces $\Sigma_t$ and $\Sigma_{t^\star}$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Remark 1.1
  • Conjecture 1.1
  • Theorem 4.1: Well-posedness in the energy class, Holzegelwp
  • Theorem 4.2
  • Lemma 5.1: Spectral properties of $P_{(\alpha)}+\xi^2$
  • proof
  • ...and 44 more