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

Oct, 2011
52 pages
Published in:
  • Commun.Pure Appl.Math. 66 (2013) 1751-1802
  • Published: 2013
e-Print:

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Abstract: (arXiv)
This paper investigates the decay properties of solutions to the massive linear wave equation gψ+αl2ψ=0\Box_g \psi + \frac{{\alpha}}{l^2} \psi =0 for gg the metric of a Kerr-AdS spacetime satisfying al<r+2|a|l<r_+^2 and α<9/4\alpha<9/4 satisfying the Breitenlohner Freedman bound. We prove that the non-degenerate energy of ψ\psi with respect to an appropriate foliation of spacelike slices decays like (logt)2(\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\psi at null-infinity.
Note:
  • 47 pages, 1 figure; typos corrected, minor improvement of the results
  • black hole: anti-de Sitter
  • space-time: Kerr
  • scalar particle: decay rate