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Curved Space-Times by Crystallization of Liquid Fiber Bundles

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Abstract

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.

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Notes

  1. An alternative approach would consist in building a suitable reduction of the geometry of connections on a \(\mathfrak {G}\)-principal bundle as for instance in [2, 3].

  2. The \(\left( 110 + \frac{110!}{100!10!}\right) \)-dimensional universal Lepage–Dedecker manifold \(\Lambda ^{10}T^*(\mathfrak {p}\otimes T^*\mathcal {P})\) is far too big.

  3. Beware that sign conventions below are different from [25].

  4. For instance in the degenerate case where \(\textsf {K}= 0\), if \((\alpha ,\omega )\) is a solution of the HVDW equations, then K is locally the pull-back by the fibration map of the pseudo-Riemannian metric on the quotient space of leaves found in Lemma 5.1.

  5. This holds if, e.g., one assumes that \(\varpi {_A}^{cd}-\kappa ^{cd}_A\), \(\varpi {_A}^{ck}\) and \(\varpi {_A}^{jk}\) have compact support in \(\mathcal {P}\) or decay at infinity.

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Acknowledgments

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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Authors and Affiliations

  1. IMJ-PRG, UMR CNRS 7586, Université Paris 7–Paris Diderot, UFR de Mathématiques, Case 7012, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Frédéric Hélein

  2. Nomad Institute for Quantum Gravity, Paris, France

    Dimitri Vey

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  1. Frédéric Hélein
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Correspondence to Frédéric Hélein.

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Hélein, F., Vey, D. Curved Space-Times by Crystallization of Liquid Fiber Bundles. Found Phys 47, 1–41 (2017). https://doi.org/10.1007/s10701-016-0039-2

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