Paris/Berkeley/Bonn/Zürich Analysis Seminar: The scalar wave equation on general asymptotically flat spacetimes: Stability and instability results
Seminar | November 29 | 9:10-10 a.m. | 238 Sutardja Dai Hall
Georgios Moschidis, Miller Institute and UC Berkeley
In this talk, we will examine how certain geometric conditions on general asymptotically flat spacetimes $(\mathcal M,g)$ are related to stability or instability properties of solutions to the scalar wave equation $\square _g\psi =0$ on $\mathcal M$. First, in the case when $(\mathcal M,g)$ possesses an event horizon with positive surface gravity and an ergoregion which is sufficiently small in terms of the near-horizon geometry, we will prove a logarithmic decay result for solutions to $\square _g\psi =0$, provided a uniform energy boundedness estimate holds on $(\mathcal M,g)$. This result, applicable also in the absence of a horizon and an ergoregion, generalises a result of Burq for the wave equation on the complement of an arbitrary compact obstacle in flat space. We will then proceed to enlarge our scope of asymptotically flat backgrounds by relaxing even further our assumptions on the properties of the ergoregion. In this case, we will present a rigorous proof of Friedman's ergosphere instability for scalar waves in the case when $(\mathcal M,g)$ possesses an ergoregion and no event horizon.