New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

Abstract

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

The second-order problem for k-presymplectic Lagrangian field theories: application to the Einstein–Palatini model

“…The Einstein-Palatini model is a first order singular field theory. A multisymplectic formulation of the model has been developed in several works (see, for instance [7,8,22,36]). In particular, in [22] the constraints arising from the application of the constraint algorithm in the (pre)multisymplectic framework have been computed explicitly in coordinates.…”

“…A multisymplectic formulation of the model has been developed in several works (see, for instance [7,8,22,36]). In particular, in [22] the constraints arising from the application of the constraint algorithm in the (pre)multisymplectic framework have been computed explicitly in coordinates. They have a diverse origin and characteristics, which makes this system an interesting test for the theory developed in this article.…”

“…These equations appear in [22] (Proposition 3.7), and their solutions are obtained giving f ρσ = 0 and f α βγ = C β δ α γ + K α βγ , for functions C β and K α βγ such that K α αγ = 0 and K α βγ + K α γβ = 0. These last conditions can be rewritten in a more suitable way as follows: these functions K α βγ are a linear combination of the new functions…”

“…The resulting Lagrangian is affine and then singular; thus, as a previous step, it is very useful to make a preliminary analysis on the characteristics of the affine Lagrangians in general. In particular, the constraint algorithm for the Einstein-Palatini Lagrangian gives different kinds of constraints, and the results obtained here are discussed and compared with those obtained in the multisymplectic description of this model [22].…”

Griffiths variational multisymplectic formulation for Lovelock gravity

“…In particular, the multisymplectic techniques have been applied to describe the most standard models of General Relativity: the Einstein-Hilbert [25] and the Einstein-Palatini (or metricaffine) [6] models (see, for instance, [5,6,26,49]). In some of these applications, a unified formalism which joins the Lagrangian and Hamiltonian formalisms into a single one has been used.…”