Hidden symmetries and decay for the wave equation outside a Kerr black hole
|
Tue, 05/05/2009 12:00 |
Pieter Blue (Edinburgh) |
Relativity Seminar  |
L3 |
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  for fixed  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. |
|||