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Burnett's conjecture in generalized wave coordinates

Cécile Huneau, Jonathan Luk

Abstract

We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose $\{g_n\}_{n=1}^\infty$ is a sequence of Lorentzian metrics (in arbitrary dimensions $d \geq 3$) satisfying a generalized wave coordinate condition and such that $g_n\to g$ in a suitably weak and "high-frequency" manner, then the limit metric $g$ satisfies the Einstein--massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations.

Burnett's conjecture in generalized wave coordinates

Abstract

We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose is a sequence of Lorentzian metrics (in arbitrary dimensions ) satisfying a generalized wave coordinate condition and such that in a suitably weak and "high-frequency" manner, then the limit metric satisfies the Einstein--massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations.
Paper Structure (36 sections, 37 theorems, 175 equations)

This paper contains 36 sections, 37 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Ideas of the proof
  3. Ricci curvature of the limit
  4. Non-negativity of the measure $\mu$
  5. Propagation equation for $\mu$
  6. Related works
  7. Physics literature
  8. Related results on Burnett's conjecture
  9. Construction of high-frequency limits
  10. Outline of the paper
  11. Trilinear compensated compactness for null forms
  12. Notations
  13. The $\mathfrak Q_0^{(g)}$ null forms
  14. The $\mathfrak Q_{\mu\nu}$ null forms and the Ionescu--Pasauder estimate
  15. Einstein equations in wave coordinates
  16. ...and 21 more sections

Key Result

Theorem 1.5

Burnett's conjecture (Conjecture conj:Burnett) is true under Assumption ass:main. More precisely, under (1)--(4) in Assumption ass:main, define $\mu$ by where $\mu_{\alpha\beta\sigma\rho}$ is a Radon measureHere, the measure acts on continuous functions which are positively $2$-homogeneous in $\xi$; see Definition def:measure. on $S^* U$ definedThe well-definedness of $\mu_{\alpha\beta\sigma\rho}

Theorems & Definitions (81)

  • Conjecture 1.1: Burnett Burnett
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 71 more