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 the 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.
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March 2018
Research Article|
March 22 2018
Multisymplectic unified formalism for Einstein-Hilbert gravity
Jordi Gaset;
Jordi Gaset
a)
Department of Mathematics, Universitat Politècnica de Catalunya–Barcelona Tech (UPC)
, Ed. C-3, Campus Norte, C/ Jordi Girona 1, 08034 Barcelona, Spain
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Narciso Román-Roy
Narciso Román-Roy
b)
Department of Mathematics, Universitat Politècnica de Catalunya–Barcelona Tech (UPC)
, Ed. C-3, Campus Norte, C/ Jordi Girona 1, 08034 Barcelona, Spain
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a)
Electronic mail: [email protected]
b)
Electronic mail: [email protected]
J. Math. Phys. 59, 032502 (2018)
Article history
Received:
August 01 2017
Accepted:
February 28 2018
Citation
Jordi Gaset, Narciso Román-Roy; Multisymplectic unified formalism for Einstein-Hilbert gravity. J. Math. Phys. 1 March 2018; 59 (3): 032502. https://doi.org/10.1063/1.4998526
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