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On almost Ehlers-Geren-Sachs theorems

Ho Lee, Ernesto Nungesser, John Stalker

Abstract

We show assuming small data that massless solutions to the reflection symmetric Einstein-Vlasov system with Bianchi VII$_0$ symmetry which are not locally rotational symmetric, can be arbitrarily close to and will remain close to isotropy as regards {to} the shear. However in general the shear will not tend to zero and the Hubble normalised Weyl curvature will blow up. This generalises the work \cite{NHW,WHU}, which considered a non-tilted radiation fluid to the massless Vlasov case. This represents another example of the fact that almost Ehlers-Geren-Sachs theorems do not hold in general and that collisionless matter behaves differently than a perfect fluid.

On almost Ehlers-Geren-Sachs theorems

Abstract

We show assuming small data that massless solutions to the reflection symmetric Einstein-Vlasov system with Bianchi VII symmetry which are not locally rotational symmetric, can be arbitrarily close to and will remain close to isotropy as regards {to} the shear. However in general the shear will not tend to zero and the Hubble normalised Weyl curvature will blow up. This generalises the work \cite{NHW,WHU}, which considered a non-tilted radiation fluid to the massless Vlasov case. This represents another example of the fact that almost Ehlers-Geren-Sachs theorems do not hold in general and that collisionless matter behaves differently than a perfect fluid.

Paper Structure

This paper contains 16 sections, 3 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. The massless Einstein-Vlasov system
  3. The massless Einstein-Vlasov system with Bianchi symmetry
  4. Vlasov equation with Bianchi symmetry
  5. The equations for Bianchi VII$_0$ with reflection symmetry for massless Vlasov particles close to the isotropic case
  6. WHU variables
  7. Future asymptotic behaviour
  8. Estimate of $M$
  9. Suppressing the oscillations
  10. The truncated system
  11. Closing the bootstrap argument
  12. Bounds for the matter terms
  13. Estimate of the quotient of $w_-$ and $X$
  14. Estimate of $Y$
  15. The Weyl parameter and the shear at late times
  16. ...and 1 more sections

Key Result

Theorem 1

Consider any $C^\infty$-solution of the massless Einstein-Vlasov system with reflection and Bianchi VII$_0$ symmetry, which is not LRS, given by the equations vlasovstructure, sc7, M--psi, w+--w- and xi1--xi5 with initial data satisfying the constraint CE1 and the conditions $X(\tau_0)\neq 0$ and $w then the following estimates hold:

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 1
  • proof
  • Corollary
  • proof