Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients

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In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations.

• In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region.
• We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature.
• For the co-closed portion, we extend previous results by the first two authors, deriving Regge-Wheeler type equations for two gauge-invariant master quantities without the earlier paper's need of axisymmetry.