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General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Analysis of PDEs, math.AP,Mathematics, Mathematical Physics, math.MP,
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.
