Yakov Shlapentokh-Rothman

Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes

For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to

Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes

{\frac{\left|am\right|}{2Mr_+}}

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

am2Mr+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation.

Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime

We construct one-parameter families of solutions to the Einstein–Klein–Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein–Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein–Klein–Gordon equations.

A scattering theory for the wave equation on Kerr black hole exteriors

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.

Stationary axisymmetric black holes with matter

YS acknowledges support from the NSF Postdoctoral Research Fellowship under award no. 1502569. MD acknowledges support through NSF grant DMS-1709270 and EPSRC grant EP/K00865X/1.