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.

44 pp. The paper has been substantially revised: A new Section (V) about symmetries and conservation laws has been added. Some changes of notation in Section II have been done in order to do it more clear and readable. Some notational errors and misprints have been corrected along the paper. The bibliography is updated and some new references have been added

References in corpus (9)

• Hamiltonian approach to GR - Part 1: covariant theory of classical gravity
• Manifest Covariant Hamiltonian Theory of General Relativity
• On precanonical quantization of gravity
• Noether's Theorem in Multisymplectic Geometry
• Unified formalism for Palatini gravity
• Order reduction, projectability and constraints of second-order field theories and higher-order mechanics
• On A Covariant Hamiltonian Description of Palatini's Gravity on Manifolds with Boundary
• Energy-momentum tensors in classical field theories - a modern perspective
• 10-plectic formulation of gravity and Cartan connections

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• Covariant Momentum Map Thermodynamics for Parametrized Field Theories
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• Symmetries and gauge symmetries in multisymplectic first and second-order Lagrangian field theories: electromagnetic and gravitational fields
• Griffiths Variational Multisymplectic Formulation for Lovelock Gravity
• Skinner-Rusk formalism for k-contact systems
• On Hamiltonian formulations of the Dirac system
• Multisymplectic formulation of Lagrangian models in gravitation
• Multisymplectic Formalism for Cubic Horndeski Theories