Overconnections and the energy-tensors of gauge and gravitational fields

Canarutto, Daniel

doi:10.1016/j.geomphys.2016.03.027

Cited by 5 publications

(18 citation statements)

References 34 publications

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“…In the standard, torsion-free formulation of General Relativity, the stress-energy tensor in the right-hand side of the Einstein equation is divergence-free on-shell (namely when the field equations are taken into account). This well-known result [4,48] is a consequence of the naturality of the Lagrangian, and holds in particular for gauge theories provided that the stress-energy tensor contains the contributions of the matter field and the gauge field [19]. This property is interpreted as local energy-conservation.…”

Section: Divergencesmentioning

confidence: 87%

“…Here the * stands for the 'Hodge isomorphism' of exterior forms, 4 namely * F ab j i = g ac g bd |θ| F i cdj , * ∇φ a αi = g ab |θ| ∇ b φ αi , and d[A] and d[Γ ⊗ A] are the exterior covariant differentials with respect to the connections indicated between brackets. A generalized version of the so-called 'replacement principle' states that these differ from the usual 'covariant divergences' by torsion terms [19]. In fact we have the identities…”

Section: Field Equationsmentioning

confidence: 97%

“…3.10. A precise geometric construction and a discussion of the meaning of this object can be found in previous work [50,19].…”

Section: Gauge Field Theories In Tetrad-affine Gravitymentioning

confidence: 99%

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On the Notions of Energy Tensors in Tetrad-Affine Gravity

Canarutto¹

2018

Gravit. Cosmol.

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We are concerned with the precise modalities by which mathematical constructions related to energy-tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor-the canonical tensor and the stress-energy tensor-exactly coincide with no need for tweaking. Moreover we show how both notions of energy-tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and a spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy-conservation in the presence of torsion.1 Indeed, a straightforward physical interpretation of the Ricci tensor in terms of energy is problematic as, for example, Schwarzschild spacetime has non-zero gravitational energy while the Ricci tensor vanishes.2 The notion of covariant differential of vector-valued forms, which has been variously present in the literature for several years, is strictly related to the Frölicher-Nijenhuis bracket [38,5,39,40,41]. In this paper we will just write down the needed coordinate expressions.

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Section: Divergencesmentioning

confidence: 87%

Section: Field Equationsmentioning

confidence: 97%

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On the Notions of Energy Tensors in Tetrad-Affine Gravity

Canarutto¹

2018

Gravit. Cosmol.

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“…which in turn can be regarded [2] as the covariant derivative of κ with respect to a certain "overconnection", i.e. a connection (associated with κ itself) of the bundle of linear connections of E. Let x a , y i be linear fibered coordinates on E. We denote the induced fiber coordinates on ⊗ r T * M and ⊗ r T * M ⊗ E by the shorthands z a 1 ...ar ≡ ∂x a 1 ⊗ ··· ⊗ ∂x ar , z i a 1 ...ar ≡ z a 1 ...ar ⊗ y i , and use the same symbols for their restrictions to ∧ r T * M and Ω r E. Moreover we set…”

Section: Frölicher-nijenhuis Bracket and Covariant Differentialmentioning

confidence: 99%

Covariant-differential formulation of Lagrangian field theory

Canarutto

2018

Int. J. Geom. Methods Mod. Phys.

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Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable "covariant prolongation bundle"; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into this formalism. The examples of bosonic fields and spin one-half fields, interacting with non-Abelian gauge fields, are worked out. The "metric-affine" description of the gravitational field is naturally included, too.2010 MSC: 53B05, 53B50, 70Sxx.

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“…Evaluated through the field, the canonical energy-tensor is a section U : M → TM ⊗ ∧ 3 T * M . In non-trivial bundles U can be introduced as a geometrically welldefined tensor field with the intervention of a suitable connection; possibly, the latter can be the gauge field itself [11]. In the case of the theory under consideration we find…”

Section: Lagrangianmentioning

confidence: 99%

A first-order Lagrangian theory of fields with arbitrary spin

Canarutto¹

2018

Int. J. Geom. Methods Mod. Phys.

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The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic 'building block'. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann-Wigner equations and the 2(2j + 1)-dimensional representation of the angular-momentum algebra needed for the Joos-Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric 'main sector'-carrying an irreducible representation of the angularmomentum algebra-and an induced sequence of 'ghost sectors'. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime one gets plane-wave solutions that are characterised by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.MSC 2010: 81R20, 81R25.

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Overconnections and the energy-tensors of gauge and gravitational fields

Cited by 5 publications

References 34 publications

On the Notions of Energy Tensors in Tetrad-Affine Gravity

On the Notions of Energy Tensors in Tetrad-Affine Gravity

Covariant-differential formulation of Lagrangian field theory

A first-order Lagrangian theory of fields with arbitrary spin

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