A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited

We present a new proof of linear stability of the Schwarzschild solution togravitational perturbations. Our approach employs the system of linearisedgravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the$|a|=0$ case. The proof fundamentally relies on the novel structure of thetransport equations in the system. Indeed, while exploiting the well-knowndecoupling of two gauge invariant linearised quantities into spin $\pm 2$Teukolsky equations, we make enhanced use of the red-shifted transportequations and their stabilising properties to control the gauge dependent partof the system. As a result, an initial-data gauge normalisation suffices toestablish both orbital and asymptotic stability for all the linearisedquantities in the system. The absence of future gauge normalisations is a novel element in the linearstability analysis of black hole spacetimes in geometric gauges governed bytransport equations. In particular, our approach simplifies the proof of\cite{DHR}, which requires a future normalised (double-null) gauge to establishasymptotic stability for the full system.