The black hole stability problem for linear scalar perturbations

We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_g{\psi} = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay has been shown in rapid developments over the last two years, following earlier progress in the Schwarzschild case a = 0. We then turn to the general subextremal range |a| < M, where we give here for the first time the essential elements of a proof of definitive decay bounds for solutions {\psi}. These developments give hope that the problem of the non-linear stability of the Kerr family of black holes might soon be addressed. This paper accompanies a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting, Paris, June 2009.Comment: 48 pages, 5 figures, to appear in Proceedings of the 12 Marcel Grossmann Meetin

Local Propagation of Impulsive Gravitational Waves

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data

Small-amplitude nonlinear waves on a black hole background

Let G(x) be a C^0 function such that |G(x)|\le K|x|^{p} for |x|\le c, for constants K,c>0. We consider spherically symmetric solutions of \Box_g\phi=G(\phi) where g is a Schwarzschild or more generally a Reissner-Nordstrom metric, and such that \phi and \nabla \phi are compactly supported on a complete Cauchy surface. It is proven that for p> 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, |\phi|\le C(\max\{v,1\})^{-1}, where v denotes an Eddington-Finkelstein advanced time coordinate.Comment: 24 pages, 8 figure