Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry

TL;DR

This paper proves integrated local energy decay for solutions to the scalar wave equation on Kerr black hole backgrounds in two specific cases, advancing understanding of wave behavior in these spacetimes.

Contribution

It establishes integrated local energy decay for the wave equation on Kerr backgrounds with small or axisymmetric angular momentum, extending previous results and setting the stage for full generality.

Findings

Bounded energy flux of scalar waves on Kerr spacetimes

Decay estimates for wave solutions in specified Kerr cases

Framework for future nonlinear stability analysis

Abstract

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.