Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

Abstract

Abstract. We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant t. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localised energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in t. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.

Superradiance on the Reissner–Nordstrøm metric

“…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. Recent results [2] show that the phenomenon is similar for Maxwell fields. In a different spirit, conditions on the stress-energy tensor under which fields or matter outside a Kerr black hole can extract rotational energy are derived in a recent paper by J.-P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, R. Narayan [15] ; these extend the conditions for energy extraction in the Penrose process.…”

Lectures on Linear Stability of Rotating Black Holes

“…More detailed estimates of metric perturbations in Schwarzschild were obtained in [9,34]. For the Kerr black hole, linear stability under perturbations of general spin has been an open problem for many years, which was solved in the dynamical setting in [30] (for related results obtained with different methods see [35,3,2,10] and the references in these papers). A key ingredient to our proof is the so-called mode stability result obtained by Whiting [45], who proved that the Teukolsky equation does not admit solutions which decay both at spatial infinity and at the event horizon and increase exponentially in time.…”

Quasilinear waves and trapping: Kerr-de Sitter space

“…(See also [16] for more background and additional references.) For instance, polynomial decay on Kerr space was shown recently by Tataru and Tohaneanu [43,42] and Dafermos, Rodnianski and Shlapentokh-Rothman [15,14,17], while electromagnetic waves were studied by Andersson and Blue [1] (see also Bachelot [2] in the Schwarzschild case), after pioneering work of Kay and Wald in [33] and [47] in the Schwarzschild setting. Normal hyperbolicity of the trapping, corresponding to null-geodesics that do not escape through the event horizons, in Kerr space was realized and proved by Wunsch and Zworski [48]; later Dyatlov extended and refined the result [22,23].…”