Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

Abstract

Abstract. This paper gives a survey of wave dynamics in the Kerr spacetime geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin s = 0, 1 2 , 1, 2, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

“…Our results differ from the above in certain respects: the methods we use are based on constructing the Green's function and deriving the needed estimates on it. Previous works in this direction include mainly the series of papers [29], [30], [28] where the first pointwise decay result for Kerr black holes has been proved, see also [35] and [36] for Schwarzschild. In our approach, we freeze the angular momentum ℓ or, in other words, we project onto a spherical harmonic.…”

Global Analysis of Quasilinear Wave Equations on Asymptotically Kerr-de Sitter Spaces

“…On the non-linear side, Luk [33] established global existence for semilinear wave equations on Kerr space under a null condition. (There was also recent work by Marzuola, Metcalfe, Tataru and Tohaneanu [35] and Tohaneanu [44] on Strichartz estimates, which are applied to the study of semilinear wave equations with power non-linearities, and by Donninger, Schlag and Soffer [17] on L ∞ estimates on Schwarzschild black holes, following L ∞ estimates of Dafermos and Rodnianski [12,11], of Blue and Soffer [5] on non-rotating charged black holes giving L 6 estimates, and Finster, Kamran, Smoller and Yau [23,24] on Dirac waves on Kerr. )…”

On the linear stability of the extreme Kerr black hole under axially symmetric perturbations

“…In this calculation we have assumed that the background is stationary in this gauge (namely, we have assumed (28) and (29)). The difference between ε 2 andε 2 involves, of course, the second order perturbation σ 2 and ω 2 .…”