Stability of charged rotating black holes for linear scalar perturbations
University of Cambridge
Department of Pure Mathematics and Mathematical Statistics
Cambridge Centre for Analysis
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Civin, D. (2015). Stability of charged rotating black holes for linear scalar perturbations (doctoral thesis). https://doi.org/10.17863/CAM.16236
In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay (ILED) results are proved for solutions of the wave equation on the domain of outer communications. The main obstacles to the proof of these results are superradiance, trapping and their interaction. These difficulties are surmounted by localising solutions of the wave equation in phase space and applying the vector field method. Miraculously, as in the Kerr case, superradiance and trapping occur in disjoint regions of phase space and can be dealt with individually. Trapping is a high frequency obstruction to the proof whereas superradiance occurs at both high and low frequencies. The construction of energy currents for superradiant frequencies gives rise to an unfavourable boundary term. In the high frequency regime, this boundary term is controlled by exploiting the presence of a large parameter. For low superradiant frequencies, no such parameter is available. This difficulty is overcome by proving quantitative versions of mode stability type results. The mode stability result on the real axis is then applied to prove integrated local energy decay for solutions of the wave equation restricted to a bounded frequency regime. The (ILED) statement is necessarily degenerate due to the trapping effect. This implies that a nondegenerate (ILED) statement must lose differentiability. If one uses an (ILED) result that loses differentiability to prove (NEB), this loss is passed onto the (NEB) statement as well. Here, the geometry of the subextremal Kerr-Newman background is exploited to obtain the (NEB) statement directly from the degenerate (ILED) with no loss of differentiability.
Black hole stability, Einstein equations, General relativity, Geometric analysis, wave equation, superradiance, trapping, Kerr-Newman, Integrated local energy decay, Non degenerate energy bounds
I acknowledge essential financial support from the Engineering and Physical Sciences Research Council grant EP/H023348/1, the Cambridge Commonwealth Trust, the Cambridge Centre for Analysis and Sidney Sussex College.
This record's DOI: https://doi.org/10.17863/CAM.16236