Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds

Stefanos Aretakis

TL;DR

<3-5 sentence high-level summary>

Abstract

We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of S. We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for solutions to the wave equation. Our estimates hold up to and including the event horizon. We remark that these results do not yield decay for the derivatives transversal to the horizon, and this is suggestive that these derivatives may satisfy instability properties analogous to those shown in our previous work on extreme Reissner-Nordstrom backgrounds.

Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds

TL;DR

<3-5 sentence high-level summary>

Abstract

We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of S. We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for solutions to the wave equation. Our estimates hold up to and including the event horizon. We remark that these results do not yield decay for the derivatives transversal to the horizon, and this is suggestive that these derivatives may satisfy instability properties analogous to those shown in our previous work on extreme Reissner-Nordstrom backgrounds.

Paper Structure

This paper contains 69 sections, 31 theorems, 224 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Schwarzschild and Kerr for $|a|<M$
  4. Extreme Reissner-Nordström
  5. Overview of Results and Techniques
  6. Conservation of Degenerate Energy
  7. Integrated Local Energy Decay
  8. Uniform Boundedness of Energy
  9. Energy and Pointwise Decay
  10. The Non-Axisymmetric Case
  11. The Uniqueness Problem and Extremality
  12. Geometry of Extreme Kerr
  13. The Metric
  14. The Differential Structure
  15. The Penrose Diagram
  16. ...and 54 more sections

Key Result

Theorem 1

(Integrated Local Energy Decay) Let $\delta>0$ and $r_{e}>M$. There exists a constant $C_{\delta}$ which depends only on $M,r_{e}$ and $\delta$ such that for all axisymmetric solutions $\psi$ to the wave equation we have

Figures (17)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 12 more figures

Theorems & Definitions (53)

  • Theorem 1
  • Remark 3.1
  • Theorem 2
  • Remark 3.2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 4.3.1
  • Proposition 4.4.1
  • Proposition 4.4.2
  • ...and 43 more