High-frequency backreaction for the Einstein equations under polarized U(1)-symmetry

Abstract

Known examples in plane symmetry or Gowdy symmetry show that, given a $1$-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb{U}\left(1\right)$-symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized $\mathbb{U}\left(1\right)$-symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.