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Curved Space-Times by Crystallization of Liquid Fiber Bundles

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. 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. 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. 3.

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

  4. 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. 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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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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Keywords

  • General relativity
  • Einstein–Cartan equations
  • Cartan connections