Decay of Solutions of the Wave Equation in the Kerr Geometry

By F. Finster, Niky Kamran, J. Smoller, Shing-Tung Yau. 2005

139 citations recorded by the source corpus.

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L∞loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/033e8269847de14561009d1156df676f4baa95ae

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Citations in the corpus

Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “Finally, we recall that decay for the wave equation has previously been obtained [21] from an explicit representation of solutions [20] using the complete separability of the Kerr wave equation.”
Price's Law on Nonstationary Space-Times (2011, unknown): “Switching to Kerr, the first decay results there were obtained by Finster-KamranSmoller-Yau [28].”
A new physical-space approach to decay for the wave equation with applications to black hole spacetimes (2009, unknown): “We also mention previous work concerning non-quantitative statements for azimuthal modes [23, 24].”
A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds (2008, background): “The statement (116) is contained in [ 27 ]f org exactly Kerr, with r restricted however to r> 2M (i.e.”
Local decay of waves on asymptotically flat stationary space-times (2009, methodology): “As far as the Kerr space-time is concerned, partial results were first obtained by Finster-Smoller [19], and Finster-Kamran-Smoller-Yau [18].”
Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole (2010, background): “The separation of variables techniques and the related symmetries have been used in many papers, including [2], [9], [11], [15, 16], [19, 20], [31], [37]; however, these mostly consider the case of zero cosmological constant, where other difficulties occur at zero energy and a global meromorphic con”
Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric (2007, unknown): “See also the paper of Finster–Kamran–Smoller–Yau for the Kerr space-time [13].”
The red‐shift effect and radiation decay on black hole spacetimes (2005, background): “For the more general Kerr family, even uniform boundedness remains an open problem (see however [ 20 ]).”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, result): “(There was also recent work by Marzuola, Metcalf, Tataru and Tohaneanu [36] on Strichartz estimates, and by Donninger, Schlag and Soffer [18] on L∞ estimates on Schwarzschild black holes, following L∞ estimates of Dafermos and Rodnianski [12, 10], of Blue and Soffer [4] on non-rotating charged black”