Curved space-times by crystallization of liquid fiber bundles

TL;DR

This paper develops a multisymplectic formulation of Einstein's gravity on a 10-dimensional manifold, leading to a new perspective where 4D spacetime emerges as a base of a fiber bundle with Einstein--Cartan solutions.

Contribution

It introduces a novel multisymplectic approach on principal bundles, resulting in a 10-dimensional theory that naturally yields Einstein--Cartan equations on a 4D base.

Findings

Derivation of Einstein--Cartan equations from a 10D multisymplectic framework

Fields represented as pairs of forms with Lie algebra coefficients

Spacetime emerges as a fiber bundle base in the theory

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 Poincar\'e 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.