25.07.2024, 16:30
– Raum 0.14
Forschungsseminar Differentialgeometrie
Dirac eigenvalues, hyperspherical radius and applications
Christian Bär (UP)
We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant $t$. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localised energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in $t$. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.