Geometric Optics Approximation for the Einstein Vacuum Equations.

We show the stability of the geometric optics approximation in general relativity by constructing a family (g λ) λ ∈ (0 , 1 ] of high-frequency metrics solutions to the Einstein vacuum equations in 3 + 1 dimensions without any symmetry assumptions. In the limit λ → 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 λ 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 (Touati in Commun Math Phys, 2023). [ABSTRACT FROM AUTHOR]