Multisymplectic unified formalism for Einstein-Hilbert gravity
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 interest- ing when...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2018 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/117223 |
| Online Access: | https://hdl.handle.net/2117/117223 https://dx.doi.org/10.1063/1.4998526 |
| Access Level: | Open access |
| Keyword: | Symplectic geometry Fiber spaces (Mathematics) Field theory (Physics) General relativity (Physics) 2nd-order classical field theories Higher-order jet bundles Multisymplectic forms Hilbert- Einstein action Einstein equations. Geometria simplèctica Espais fibrats (Matemàtica) Camps, Teoria dels (Física) Relativitat general (Física) Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry Classificació AMS::55 Algebraic topology::55R Fiber spaces and bundles Classificació AMS::70 Mechanics of particles and systems::70S Classical field theories Classificació AMS::83 Relativity and gravitational theory::83C General relativity Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica aplicada a les ciències |
| Summary: | 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 interest- ing 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 co- variant 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 multimomen- tum (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 geo- metrical 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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