Hidden symmetries and decay for the wave equation outside a Kerr black hole

Pieter Blue (Edinburgh)

Tuesday 3rd November, 2009 16:00-17:00 214


The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields. Outside the Schwarzschild black hole, a key intermediate step in proving decay involved proving a Morawetz estimate using a vector-field which pointed away from the photon sphere, where null geodesics orbit the black hole. Outside the Kerr black hole, the photon orbits have a more complicated structure. By using the hidden symmetry of Kerr, we can replace the Morawetz vector-field by a fifth-order operator which, in an appropriate sense, points away from the photon orbits. This allows us to prove the necessary Morawetz estimate. From this we can prove a decay estimate of almost $t^{-1}$ for fixed $r$ and the corresponding decay rates at the event horizon and null infinity. The major innovation in this result is that, by using the hidden symmetries with the energy-momentum, we can avoid taking Fourier tranforms in time. This is joint work with Lars Andersson.

Add to your calendar

Download event information as iCalendar file (only this event)