Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

Abstract

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 a ≪ M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal a < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, , pp. 132189, arXiv:1010]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

Linear Waves in the Interior of Extremal Black Holes I

“…Polynomial decay in time for solutions to (1.1) in the exterior was obtained for the full subextremal range |a| < M of Kerr spacetimes by Dafermos-RodnianskiShlapentokh-Rothman in [16]. This result is the culmination of many partial results by various different authors over the past decades.…”

“…This result is the culmination of many partial results by various different authors over the past decades. See [16,22] for comprehensive lists of references. The results of [16] have been generalised in [12] to include subextremal Kerr-Newman spacetimes.…”

Linear Waves in the Interior of Extremal Black Holes II

“…There is still no proof of pointwise and energy boundedness or decay for non-axisymmetric φ in the exterior region of extremal Kerr (cf. a complete picture of the boundedness and decay properties of the linear wave equation in the full subextremal range of Kerr-Newman) spacetimes has recently been obtained in [15,17]). …”

Superradiance on the Reissner–Nordstrøm metric

“…In 1983, S. Chandrasekhar's remarkable book [5] provided a complete and up-to-date account of the theory of black hole perturbations, with in particular calculations of scattering matrices (reflection and transmission coefficients) at fixed frequencies for Reissner-Nordstrøm and Kerr black holes. The last decade or so has seen a rather intensive analytical study of superradiance around rotating black holes (see for example [1,7,10,11]). It is now clear that in the slowly rotating case, the amount of energy extracted by scalar fields is finite and controlled by a fixed multiple of the initial energy of the field.…”