Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity
Ma, S.
Abstract
This second part of the series treats spin $\\pm2$ components (or extreme components) of the linearized gravitational perturbations (linearized gravity) in the exterior of a slowly rotating Kerr black hole, following the hierarchy introduced in our first part [15] on the Maxwell field. This hierarchy lies in the fact that for each of these two components defined in Kinnersley tetrad, the resulting equations by performing some first-order differential operator on it once and twice, together with the Teukolsky master equation, are in the form of an "inhomogeneous spin-weighted wave equation" (ISWWE) with different potentials and constitute a linear spin-weighted wave system. We then prove energy and integrated local energy decay (Morawetz) estimates for this type of ISWWE, and utilize them to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the regular extreme Newman-Penrose components defined in the regular Hawking-Hartle tetrad.