  We prove in this paper the linear stability of the celebrated Schwarzschild
family of black holes in general relativity: Solutions to the linearisation of
the Einstein vacuum equations around a Schwarzschild metric arising from
regular initial data remain globally bounded on the black hole exterior and in
fact decay to a linearised Kerr metric. We express the equations in a suitable
double null gauge. To obtain decay, one must in fact add a residual pure gauge
solution which we prove to be itself quantitatively controlled from initial
data. Our result a fortiori includes decay statements for general solutions of
the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature
components). These latter statements are in fact deduced in the course of the
proof by exploiting associated quantities shown to satisfy the Regge--Wheeler
equation, for which appropriate decay can be obtained easily by adapting
previous work on the linear scalar wave equation. The bounds on the rate of
decay to linearised Kerr are inverse polynomial, suggesting that dispersion is
sufficient to control the non-linearities of the Einstein equations in a
potential future proof of nonlinear stability. This paper is self-contained and
includes a physical-space derivation of the equations of linearised gravity
around Schwarzschild from the full non-linear Einstein vacuum equations
expressed in a double null gauge.
