## Duke Mathematical Journal

### High-frequency backreaction for the Einstein equations under polarized $\mathbb{U}(1)$-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 $\mathbb{U}(1)$-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 $\mathbb{U}(1)$-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.

#### Article information

Source
Duke Math. J., Volume 167, Number 18 (2018), 3315-3402.

Dates
Revised: 15 May 2018
First available in Project Euclid: 16 November 2018

https://projecteuclid.org/euclid.dmj/1542337536

Digital Object Identifier
doi:10.1215/00127094-2018-0035

Mathematical Reviews number (MathSciNet)
MR3881199

Zentralblatt MATH identifier
07009768

Keywords
general relativity

#### Citation

Huneau, Cécile; Luk, Jonathan. High-frequency backreaction for the Einstein equations under polarized $\mathbb{U}(1)$ -symmetry. Duke Math. J. 167 (2018), no. 18, 3315--3402. doi:10.1215/00127094-2018-0035. https://projecteuclid.org/euclid.dmj/1542337536

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