Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature

Marica Minucci, Juan Antonio Valiente Kroon

Abstract

In this article, it is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of de Sitter-like spacetimes with spatial sections of negative scalar curvature. This class of spacetimes admits a smooth conformal extension with a space-like conformal boundary. Central to the analysis is the use of conformal Gaussian systems to obtain a hyperbolic reduction of the conformal Einstein field equations for which standard Cauchy stability results for symmetric hyperbolic systems can be employed. The use of conformal methods allows us to rephrase the question of the global existence of solutions to the Einstein field equations into considerations of finite existence time for the conformal evolution system.

A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature

Abstract

In this article, it is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of de Sitter-like spacetimes with spatial sections of negative scalar curvature. This class of spacetimes admits a smooth conformal extension with a space-like conformal boundary. Central to the analysis is the use of conformal Gaussian systems to obtain a hyperbolic reduction of the conformal Einstein field equations for which standard Cauchy stability results for symmetric hyperbolic systems can be employed. The use of conformal methods allows us to rephrase the question of the global existence of solutions to the Einstein field equations into considerations of finite existence time for the conformal evolution system.

Paper Structure

This paper contains 41 sections, 10 theorems, 226 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The extended conformal Einstein field equations
  3. Weyl connections
  4. A frame formalism
  5. The frame version of the extended conformal Einstein field equations
  6. The conformal constraint equations
  7. Properties of the background solution
  8. A class of conformal geodesics
  9. Conformal geodesics
  10. Reparametrisation as conformal geodesic
  11. The canonical factor associated to the congruence of conformal geodesics
  12. The background spacetime as a solution to the conformal Einstein field equations
  13. The frame
  14. The connection coefficients
  15. Conformal fields
  16. ...and 26 more sections

Key Result

Theorem 1

Given smooth initial data $({\bm h},{\bm K})$ for the Einstein field equations on $\mathcal{S}$ which is suitably close (as measured by a suitable Sobolev norm) to the data implied by the metric BackgroundMetricIntro, there exists a smooth metric $\tilde{{\bm g}}$ defined over $[0,\infty)\times\math

Figures (1)

  • Figure 1: Penrose diagram of the background solution. The conformal representation discussed in the main text has compact sections of negative scalar curvature. The vertical lines $\Gamma_1$ and $\Gamma_2$ correspond to axes of symmetry. The solution has a singularity in the past and a spacelike future conformal boundary. Hence, in our discussion we only consider future evolution of the initial hypersurface $\mathcal{S}_\star$.

Theorems & Definitions (41)

  • Theorem
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • ...and 31 more