ABSTRACT:

The wave equation on the Kerr spacetime is interesting as a toy
model for the problem of stability of the Kerr black hole. In analyzing
waves on Kerr one encounters two important difficulties. The Kerr spacetime
has only two Killing fields, corresponding to stationarity and axial
symmetry. Further, the "photon sphere", i.e., the region filled with
rotating null geodesics, has codimension zero, which makes trapping a
serious problem.

In recent work with Pieter Blue, we have been able to circumvent these
difficulties and give a "physical space" approach to estimates for the wave
equation, by making use of the hidden symmetry of the Kerr spacetime,
discovered by Carter. By utilizing the fact that an operator related to the
Carter constant commutes with the Kerr wave operator, we are able to prove
energy bounds, trapping, and dispersive estimates for the wave equation on
Kerr.