  We show the stability of the geometric optics approximation in general
relativity by constructing a family $(g_\lambda)_{\lambda\in(0,1]}$ of
high-frequency metrics solutions to the Einstein vacuum equations in 3+1
dimensions without any symmetry assumptions. In the limit $\lambda\to 0$ this
family approaches a fixed background $g_0$ solution of the Einstein-null dust
system, illustrating the backreaction phenomenon. We introduce a precise second
order high-frequency ansatz and identify a generalised wave gauge as well as
polarization conditions at each order. The validity of our ansatz is ensured by
a weak polarized null condition satisfied by the quadratic non-linearity in the
Ricci tensor. The Einstein vacuum equations for $g_\lambda$ are recast as a
hierarchy of transport and wave equations, and their coupling induces a loss of
derivatives. In order to solve it, we take advantage of a null foliation
associated to $g_0$ as well as a Fourier cut-off adapted to our ansatz. The
construction of high-frequency initial data solving the constraint equations is
the content of our companion paper \cite{Touati2023a}.
