We prove in this paper that the Schwarzschild family of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. In particular we improve on our recent work [33] by modifying the generalised wave gauge employed therein so as to establish asymptotic flatness of the associated linearised system. The result thus complements the seminal work [11] of Dafermos-Holzegel-Rodnianski in a similar vein as to how the work [40] of Lindblad-Rodnianski complemented that of Christodoulou-Klainerman [10] in establishing the nonlinear stability of the Minkowski space.

In this short note we shall demonstrate that given a smooth solution γ to the linearised Einstein equations on Schwarzschild which is supported on the l ≥ 2 spherical harmonics and expressed relative to a transverse and traceless gauge then one can construct from it a smooth solution to the sourced Maxwell equations expressed relative to a generalised Lorentz gauge. Here the Maxwell current is constructed from those gauge-invariant combinations of the components of γ which are determined by solutions to the Regge-Wheeler and Zerilli equations. The result thus provides an elegant link between the spin 1 and spin 2 equations on Schwarzschild.