Linear Stability of Schwarzschild Spacetime

In this work, 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. Our method employs 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 both portions, we derive Regge-Wheeler type equations for decoupled, gauge-invariant quantities at the level of perturbed connection coefficients. A general framework for the analysis of Regge-Wheeler type equations is presented, identifying sufficient conditions for decay estimates. With the choice of an appropriate gauge in each of the two portions, such decay estimates on these decoupled quantities are used to establish decay of the linearized metric coefficients, completing the proof of linear stability. The initial value problem is formulated on Cauchy data sets, complementing the work of Dafermos-Holzegel-Rodnianski [6], where the linear stability of Schwarzschild is established for characteristic initial data sets.