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Mathematics, Analysis of PDEs, math.AP,General Relativity and Quantum Cosmology, gr-qc,
Abstract:
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.
