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Annales Henri Poincaré

, Volume 12, Issue 3, pp 419–482 | Cite as

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

  • Yvonne Choquet-Bruhat
  • Piotr T. ChruścielEmail author
  • José M. Martín-García
Article

Abstract

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.

Keywords

Cauchy Problem Einstein Equation Christoffel Symbol Null Hypersurface Null Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Yvonne Choquet-Bruhat
    • 1
  • Piotr T. Chruściel
    • 2
    Email author
  • José M. Martín-García
    • 3
  1. 1.Académie des SciencesParisFrance
  2. 2.Gravitational PhysicsUniversity of ViennaViennaAustria
  3. 3.Institut d’Astrophysique de Paris and Laboratoire Univers et Théories, UMR 8102, Observatoire de Paris-MeudonMeudonFrance

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