Multisymplectic unified formalism for Einstein-Hilbert gravity

Abstract

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interesting when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the covariant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geometrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.

Unified formalism for Palatini gravity

“…The present work continues the exploration of geometrical formulations of the Lagrangian and Hamiltonian versions of General Relativity (GR from now on). More specifically, it deals with a unified formalism [BLEEdD · 07, ELM · 04, Vit10] for Palatini version of GR, pursuing the work initiated in [GRR17] for Einstein-Hilbert action. Also, a novel unified version of unimodular gravity is obtained as a by-product of the geometrical tools employed in the article.…”

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

“…Other recent discussions of gauge fields and gravity from the point of view of classical DW Hamiltonian theory and its geometrizations can be found in [30][31][32][33][34][35][36].…”

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

“…The multiplicity of the semiholonomic solutions of the Einstein-Hilbert model appears in the second derivative of the components of the metric (in the Hamiltonian formalism using the non-momentum coordinates). They are of the form (see [25])…”

“…In recent years, there is an increasing effort in understanding the covariant description of gravitational theories (General Relativity and other derived from it) using different kinds of geometric frameworks such as the multisymplectic or the polysymplectic manifolds. Thus, in [3,8,9,10,21,25,26,27,35,36,43,44,46] general aspects of the theory are studied in this way, meanwhile other papers are devoted to consider several particular problems. For instance, in [7,24,41,42] the reduction and projectability of higher-order theories (such as the Hilbert-Einstein model) is analized, in [47] the vielbein models of General Relativity are studied using the multisymplectic formulation and in [32,33,34] interesting contributions to the problem of the precanonical quantization of gravity are done.…”