  We derive an energy conservation law for the system of gravitational
perturbations on the Schwarzschild spacetime expressed in a double null gauge.
The resulting identity involves only first derivatives of the metric
perturbation. Exploiting the gauge invariance up to boundary terms of the
fluxes that appear, we are able to establish positivity of the flux on any
outgoing null hypersurface to the future of the initial data. This allows us to
bound the total energy flux through any such hypersurface, including the event
horizon, in terms of initial data. We similarly bound the total energy radiated
to null infinity. Our estimates provide a direct approach to a weak form of
stability, thereby complementing the proof of the full linear stability of the
Schwarzschild solution recently obtained in [M. Dafermos, G. Holzegel and I.
Rodnianski \emph{The linear stability of the Schwarzschild solution to
gravitational perturbations}, arXiv:1601.06467].
