Foundations of Physics

, Volume 47, Issue 1, pp 1–41 | Cite as

Curved Space-Times by Crystallization of Liquid Fiber Bundles

  • Frédéric Hélein
  • Dimitri Vey


Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \(((\alpha ,\omega ),\varpi )\), where \((\alpha ,\omega )\) is a 1-form with coefficients in the Lie algebra of the Poincaré group and \(\varpi \) is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations.


General relativity Einstein–Cartan equations Cartan connections 



we thank Friedrich W. Hehl for indicating us the nice paper of Lurçat [37], to the Referee for drawing our attention to the very interesting work of Toller [38, 39] and to Igor Kanatchikov for comments on a first version of this paper.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.IMJ-PRG, UMR CNRS 7586Université Paris 7–Paris Diderot, UFR de MathématiquesParis Cedex 13France
  2. 2.Nomad Institute for Quantum GravityParisFrance

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