Unified formalism for Palatini gravity

Abstract

The present article is devoted to the construction of a unified formalism for Palatini and unimodular gravity. The basic idea is to employ a relationship between unified formalism for a Griffiths variational problem and its classical Lepage-equivalent variational problem. As a way to understand from an intuitive viewpoint the Griffiths variational problem approach considered here, we may say the variations of the Palatini Lagrangian are performed in such a way that the so called metricity condition, i.e. (part of) the condition ensuring that the connection is the Levi-Civita connection for the metric specified by the vielbein, is preserved. From the same perspective, the classical Lepage-equivalent problem is a geometrical implementation of the Lagrange multipliers trick, so that the metricity condition is incorporated directly into the Palatini Lagrangian. The main geometrical tools involved in these constructions are canonical forms living on the first jet of the frame bundle for the spacetime manifold Å . These forms play an essential rôle in providing a global version of the Palatini Lagrangian and expressing the metricity condition in an invariant form; with their help, it was possible to formulate an unified formalism for Palatini gravity in a geometrical fashion. Moreover, we were also able to find the associated equations of motion in invariant terms and, by using previous results from the literature, to prove their involutivity. As a bonus, we showed how this construction can be used to provide a unified formalism for the so called unimodular gravity by employing a reduction of the structure group of the principal bundle ÄÅ to the special linear group ËĴѵ Ñ dim Å .

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

“…We have recovered the gauge symmetries discussed in [11], showing that there are no more. As it is known [5,11], it is possible to recover the Einstein-Hilbert model by a gauge fixing in the Einstein-Palatini model, which consists in imposing the trace of the torsion to vanish. This particular gauge fixing transforms the torsion and the pre-metricity constraints, which are a consequence of the constraint algorithm, to the torsionless and the metricity conditions respectively (Proposition 4).…”

“…The multisymplectic and polysymplectic techniques have been also applied to treat different aspects of one of the most classical approaches in General Relativity: the Einstein-Palatini or Metric-Affine model [4,5,31,37,38]. In particular, in [5] an exhaustive study of the multisymplectic description of the model has been done, using a unified formalism which joins both the Lagrangian and Hamiltonian formalisms into a single one. This unified framework had been previously stated to do a covariant multisymplectic formulation of the Hilbert-Einstein model in General Relativity [25].…”

Routh Reduction of Palatini Gravity in Vacuum

“…We choose to focus on variational problems of Griffiths type because there exists a description of Palatini gravity in terms of this kind of variational problems [5]. The present section is devoted to give a brief account of the geometrical ingredients involved in this construction.…”

“…It is time to discuss the restrictions we must impose on the sections of τ 1 in order to have a characterization of a gravity field in this description. Our aim is to describe a metric and a connection on the spacetime, and the restrictions to be considered will establish the relationship between them; this approach has been extensively discussed in the references [3,5].…”

“…The relevance of the unified formalism in dealing with the variational problems posed by Definition 4.1 is guaranteed by the following result [5].…”