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Abstract

In this paper we prove integrated energy and pointwise decay estimates for
solutions of the vacuum linearized Einstein equation on the Kerr black hole
exterior. The estimates are valid for the full, subextreme range of Kerr black
holes, provided integrated energy estimates for the Teukolsky Master Equation
holds. For slowly rotating Kerr backgrounds, such estimates are known to hold,
due to the work of one of the authors arXiv:1708.07385. The results in this
paper thus provide the first stability results for linearized gravity on the
Kerr background, in the slowly rotating case, and reduce the linearized
stability problem for the full subextreme range to proving integrated energy
estimates for the Teukolsky equation. This constitutes an essential step
towards a proof of the black hole stability conjecture, i.e. the statement that
the Kerr family is dynamically stable, one of the central open problems in
general relativity.
The proof relies on three key steps. First, there are energy decay estimates
for the Teukolsky equation, proved by applying weighted multiplier estimates to
a system of spin-weighted wave equations derived from the Teukolsky equation,
and making use of the pigeonhole principle for the resulting hierarchy of
weighted energy estimates. Second, working in the outgoing radiation gauge, the
linearized Einstein equations are written as a system of transport equations,
driven by one of the Teukolsky scalars. Third, expansions for the relevant
curvature, connection, and metric components can be made near null infinity. An
analysis of the dynamics on future null infinity, together with the Teukolsky
Starobinsky Identity plays an important role in the argument.