In this paper, we provide an elementary, unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background: the celebrated blue-shift at the Cauchy horizon (familiar from the strong cosmic censorship conjecture) and the time-reversed red-shift at the event horizon (relevant in classical scattering theory). Our first theorem concerns the latter and constructs solutions to the wave equation on Kerr spacetimes such that the radiation field along the future event horizon vanishes and the radiation field along future null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the future event horizon. Our second theorem constructs solutions to the wave equation on rotating Kerr spacetimes such that the radiation field along the past event horizon (extended into the black hole) vanishes and the radiation field along past null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the Cauchy horizon. The results make essential use of the scattering theory developed in Dafermos, Rodnianski and Shlapentokh-Rothman (A scattering theory for the wave equation on Kerr black hole exteriors (2014). arXiv:1412.8379) and exploit directly the time-translation invariance of the scattering map and the non-triviality of the transmission map.
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 ( g =0 ) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 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 on ( H ^ + ) . Several new analytical features of degenerate horizons are also presented. We describe the evolution of a scalar test field on the interior of a Reissner-Nordstroem black hole. For a wide variety of initial field configurations the energy density in the scalar field is shown to develop singularities in a neighborhood of the geometry's Cauchy horizon, suggesting that for a stellar collapse curvature singularities will develop prior to encountering the Cauchy horizon. The extension to the interior of stationary perturbations due to exterior sources is shown not to disrupt the Cauchy horizon. Abstract We study the Cauchy problem for the wave equation □ g ψ = 0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ 0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ 0 . We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T , and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N . Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H + . The stability of the inner Reissner-Nordstroem geometry is studied with test massless integer-spin fields. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate most of the results. Monochromatic waves which are of small amplitude and ingoing near the outer horizon develop infinite energy densities near the inner Cauchy horizon (as measured by a freely falling observer). Previous work has shown that certain derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon (r = r sub - ) if energy densities are to remain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past null infinity (I sup - ), cannot be localized near r sub + , the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the black hole. Price, Sibgatullin, and Alekseev have shown that such moments are radiated away as ''tails'' which travel outward and are rescattered inward yielding a wave field with a time dependence t sup -p , p > 0. This decay in time is sufficiently slow that themore » tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with the time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. have further shown that even perturbations which are localized as they cross the outer horizon produce singularities at the inner horizon. It is shown that this singularity arises when the incoming radiation is first scattered just inside the outer horizon« less  We consider solutions to the linear wave equation in the interior region of extremal Reissner–Nordstrom black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and, moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner–Nordstrom black holes, where the local energy was shown to generically blow up at the Cauchy horizon. The behaviour, on the Cauchy horizon, of a flux of gravitational and or electromagnetic radiation crossing the event horizon of a Reissner-Nordstrom black-hole is investigated as a problem in the theory of one-dimensional potential-scattering. It is shown that the flux of radiation received by an observer crossing the Cauchy horizon, along a radial time-like geodesic, diverges for all physically reasonable perturbations crossing the event horizon, even including those with compact support. This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation ( g =0 ) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon ( H ^ + ). We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon ( H ^ + ). A hierarchy of conservations laws on degenerate horizons is also derived. This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I II: the cases jaj M or axisymmetry, arXiv:1010.5132], providing the complete proof of denitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal jaj < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most dicult mul 