Stability and Instability of Extreme Reissner-Nordstrom Black Hole Spacetimes for Linear Scalar Perturbations I

We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation square(g)psi = 0 on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Sigma(0) crossing the future event horizon H+. We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon H+. The fundamental new aspect of this problem is the degeneracy of the redshift. Several new analytical features of degenerate horizons are also presented.