Gravity Seminar - Peter Zimmerman Seminar

Abstract: We first introduce a formalism to treat higher order metric perturbations of the Kerr spacetime in a Teukolsky framework, showing that solutions to the linearized Einstein equation with nonvanishing stress tensor can be decomposed into a pure gauge part plus a zero mode, a perturbation arising from a certain scalar potential, as well as a so-called "corrector tensor." As higher order perturbations are subject to a linearized Einstein equation with a stress tensor obtained from the lower order perturbations, our method also applies iteratively to higher order metric perturbations. As such it can be used to analyze nonlinear quasinormal mode coupling of perturbations in the near-extremal Kerr spacetime, where weakly turbulent behaviour has been conjectured to occur. To study this mode coupling, we introduce a novel bilinear form under which quasinormal modes exhibit orthogonality. Under this bilinear form the QNMs have properties (such as orthogonality) similar to normal modes. Using the bilinear form as a projector allows the computation of mode coupling coefficients. In the near-horizon region of nearly extremal Kerr, these coefficients are expected to take the form of "Clebsch-Gordan coefficients" of the relevant near-horizon symmetry group.