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We prove in this paper the linear stability of the celebrated Schwarzschildfamily of black holes in general relativity: Solutions to the linearisation ofthe Einstein vacuum equations around a Schwarzschild metric arising fromregular initial data remain globally bounded on the black hole exterior and infact decay to a linearised Kerr metric. We express the equations in a suitabledouble null gauge. To obtain decay, one must in fact add a residual pure gaugesolution which we prove to be itself quantitatively controlled from initialdata. Our result a fortiori includes decay statements for general solutions ofthe Teukolsky equation (satisfied by gauge-invariant null-decomposed curvaturecomponents). These latter statements are in fact deduced in the course of theproof by exploiting associated quantities shown to satisfy the Regge--Wheelerequation, for which appropriate decay can be obtained easily by adaptingprevious work on the linear scalar wave equation. The bounds on the rate ofdecay to linearised Kerr are inverse polynomial, suggesting that dispersion issufficient to control the non-linearities of the Einstein equations in apotential future proof of nonlinear stability. This paper is self-contained andincludes a physical-space derivation of the equations of linearised gravityaround Schwarzschild from the full non-linear Einstein vacuum equationsexpressed in a double null gauge.

The linear stability of the Schwarzschild solution to gravitational perturbations