The null condition and asymptotic expansions for the Einstein equations

The global existence of solutions of non linear wave equations on the Minkowski spacetime of dimension 4 is linked with the non linearities satisfying the “null condition” discovered by Klainerman and Christodoulou. This null condition also improves local existence results, lowering the sufficient regularity of the Cauchy data. The non linear stability of the Minkowski spacetime among solutions of the vacuum Einstein equations relies on estimates for the Weyl tensor probably linked with some kind of null condition. It had been proved long ago that the significant part of an high frequency gravitational wave in General Relativity obeys a transport law along the rays which is linear, but inflicts a “back reaction” on the background metric. These two properties reflect the fact that the Einstein equations satisfy almost, but not quite, a “polarized” null condition. We will explicit this result, also for associated field equations, and explain its relation with the properties of high frequency waves.