Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2010-10-25
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
57 pages, 4 figures
Scientific paper
This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_g{\psi} = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general {\psi} or (II2) |a| < M, {\psi} axisymmetric. In either case, we prove a version of 'integrated local energy decay', specifically, that the 4-integral of an energy-type density (degenerating in a neighborhood of the Schwarzschild photon sphere and at infinity), integrated over the domain of dependence of a spacelike hypersurface {\Sigma} connecting the future event horizon with spacelike infinity or a sphere on null infinity, is bounded by a natural (non-degenerate) energy flux of {\psi} through {\Sigma}. (The case (II1) has in fact been treated previously in our Clay Lecture notes: Lectures on black holes and linear waves, arXiv:0811.0354.) In our forthcoming Part III, the restriction to axisymmetry for the general |a| < M case is removed. The complete proof is surveyed in our companion paper The black hole stability problem for linear scalar perturbations, which includes the essential details of our forthcoming Part III. Together with previous work (see our: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arxiv:0910.4957), this result leads, under suitable assumptions on initial data of {\psi}, to polynomial decay bounds for the energy flux of {\psi} through the foliation of the black hole exterior defined by the time translates of a spacelike hypersurface {\Sigma} terminating on null infinity, as well as to pointwise decay estimates, of a definitive form useful for nonlinear applications.
Dafermos Mihalis
Rodnianski Igor
No associations
LandOfFree
Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-94176