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