Curved space-times by crystallization of liquid fiber bundles

Espaces-temps courbes par cristallisation de fibrés liquides

Frédéric Hélein (1) , Dimitri Vey (1)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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 ((α,ω),ϖ)((α,ω),ϖ) , where (α,ω)(α,ω) is a 1-form with coefficients in the Lie algebra of the Poincaré group and ϖϖ 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.

Mots clés: Cartan geometry; Relativity and gravitation; Fiber bundle

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]; Mathématiques [math]

Frédéric Hélein, Dimitri Vey. Curved space-times by crystallization of liquid fiber bundles. Foundations of Physics, 2017, 47 (1), pp.1-41. ⟨10.1007/s10701-016-0039-2⟩. ⟨hal-01205784v2⟩