3+1 Decomposition of Einstein Equation

  • Éric Gourgoulhon
Part of the Lecture Notes in Physics book series (LNP, volume 846)


The fundamental equation for general relativity, the Einstein equation, is decomposed orthogonally with respect to a 3+1 foliation of spacetime. Then we introduce spatial coordinates on the hypersurfaces forming the foliation, thereby introducing the famous shift vector. This enables one to turn the 3+1 Einstein equation into a system of partial-differential equations. This system can be formulated as a Cauchy problem with constraints and we discuss briefly the known existence and uniqueness results regarding it. Finally we discuss the ADM Hamiltonian approach to general relativity, which is based on the 3+1 decomposition.


Cauchy Problem Einstein Equation Hamiltonian Formulation Spacelike Hypersurface Shift Vector 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Éric Gourgoulhon
    • 1
  1. 1.Lab. Univers et Théories (LUTH) UMR 8102 du CNRS, Observatoire de ParisUniversité Paris DiderotMeudonFrance

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