General Relativity as a Constrained Gauge Theory

Vignolo, Stefano; Cianci, Roberto; Bruno, Danilo

doi:10.1142/s0219887806001818

Cited by 10 publications

(17 citation statements)

References 6 publications

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“…The set ker 4 π 1 Ω L EP consists of multivector fields of the form (5) whose coefficients satisfy the connection and metric equations (8) and (9) respectivelly. But the equations (9) are not compatible.…”

Section: 21mentioning

confidence: 99%

“…Proposition 3. On the submanifold S T , the general solutions to the equations (8) and (9) are, respectively,…”

Section: 21mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Gaset

Román‐Roy

2019

Journal of Geometric Mechanics

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We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

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Section: 21mentioning

confidence: 99%

“…Proposition 3. On the submanifold S T , the general solutions to the equations (8) and (9) are, respectively,…”

Section: 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gaset

Román‐Roy

2019

Journal of Geometric Mechanics

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“…J -bundles have been recently used to provide new geometric formulations of gauge theories and GR [1][2][3][29][30][31][32].…”

Section: The Geometric Frameworkmentioning

confidence: 99%

f(R) GRAVITY WITH TORSION: A GEOMETRIC APPROACH WITHIN THE $\mathcal{J}$-BUNDLES FRAMEWORK

Capozziello

Cianci

Stornaiolo

et al. 2008

Int. J. Geom. Methods Mod. Phys.

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We discuss the f (R)-theories of gravity with torsion in the framework of J -bundles. Such an approach is particularly useful since the components of the torsion and curvature tensors can be chosen as fiber J -coordinates on the bundles and then the symmetries and the conservation laws of the theory can be easily achieved. Field equations of f (R)-gravity are studied in empty space and in presence of various forms of matter as Dirac fields, Yang-Mills fields and spin perfect fluid. Such fields enlarge the jet bundles framework and characterize the dynamics. Finally we give some cosmological applications and discuss the relations between f (R)-gravity and scalar-tensor theories.

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“…The geometrisation of the theories of gravitation (General Relativity) and, in particular the multisymplectic framework, allows us to do a covariant description of these theories, considering and understanding several inherent characteristics of it, and it has been studied by different authors. For instance, relevant references devoted to develop geometrically general aspects of the theory are [2,5,6,7,15,22,26,40], the reduction of the order of the theory and the projectability of the Poincaré-Cartan form associated with the Hilbert-Einstein action is explained in [4,38,39], meanwhile in [32,33] different aspects of the theory are studied using Lepage-Cartan forms, and in [45,46] a multisymplectic analysis of the vielbein formalism of General Relativity is done. Finally, some general features of the gravitational theory following the polysymplectic version of the multisymplectic formalism are described in [21,41], including the problem of its precanonical quantization [27,28,29].…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic unified formalism for Einstein-Hilbert gravity

Gaset

Román‐Roy

2018

Journal of Mathematical Physics

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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.

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General Relativity as a Constrained Gauge Theory

Abstract: The formulation of General Relativity presented in [1] and the Hamiltonian formulation of Gauge theories described in [2] are made to interact. The resulting scheme allows to see General Relativity as a constrained Gauge theory.

Cited by 10 publications

References 6 publications

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

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

f(R) GRAVITY WITH TORSION: A GEOMETRIC APPROACH WITHIN THE $\mathcal{J}$-BUNDLES FRAMEWORK

Multisymplectic unified formalism for Einstein-Hilbert gravity

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