Gauge theory of a group of diffeomorphisms. III. The fiber bundle description

Lord, Eric A.; Goswami, Partha S.

doi:10.1063/1.528183

Cited by 24 publications

(40 citation statements)

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“…The introduction of torsion related to spin gives rise to a strong link between gravitation and particle physics, because it extends the holonomy group to the translations. An enlightening discussion of gauge translations can be found for example in [53][54]. In particular, the introduction of [54] clarifies from the very beginning the main geometric role played by the translations in the gauge group : they change a principal fibre bundle having no special relationship between the points on the fibres and the base manifold into the bundle of linear frames of the base manifold.…”

Section: Singularities For Theories With Torsionmentioning

confidence: 99%

Mathematical Structures of Space-Time

Esposito

1992

Fortschr. Phys.

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Abstract. At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables.Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves 1 Mathematical Structures of Space-Time whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christoffel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity.Our work should be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic curvature tensor and by the variation formulae. * Fortschritte der Physik, 40, 1-30 (1992). 2Mathematical Structures of Space-Time

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Section: Singularities For Theories With Torsionmentioning

confidence: 99%

Mathematical Structures of Space-Time

Esposito

1992

Fortschr. Phys.

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“…Rewriting G A Θ = D B A ΘG B , differentiating with respect to g λ and taking the limit g = (id) G , we arrive at the commutation relations [27]:…”

Section: Appendix a Maurer-cartan 1-formsmentioning

confidence: 99%

“…The tetrads fields arise in their formulation as a result of the reduction of the structure group of the tangent bundle from the general linear to Lorentz group. In 1987, Lord and Goswami [26,27] developed the NLR in the fiber bundle formalism based on the bundle structure G (G/H, H) as suggested by Ne'eman and Regge [28]. In this approach the quotient space G/H is identified with physical spacetime.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Realization of the Local Conform-Affine Symmetry Group for Gravity in the Composite Fiber Bundle Formalism

Ali

Capozziello

2007

Int. J. Geom. Methods Mod. Phys.

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A gauge theory of gravity based on a nonlinear realization (NLR) of the local Conform-Affine (CA) group of symmetry transformations is presented. The coframe fields and gauge connections of the theory are obtained. The tetrads and Lorentz group metric are used to induce a spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxillary dynamical fields associated with the dilation, special conformal and deformational (shear) degrees of freedom inherent in the bundle manifold. The bundle curvature of the theory is determined. Boundary topological invariants are constructed. They serve as a prototype (source free) gravitational Lagrangian. The Bianchi identities, covariant field equations and gauge currents are obtained.

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“…(6.6) of [19]) one recognizes the fundamental equation for nonlinear realizations [26] [6] [7]. The nonlinear gauge transformations of fields induced by (13) are deduced in Section VIII of [19].…”

Section: B Nonlinear Realizations In Composite Bundlesmentioning

confidence: 99%

“…Conceived as an alternative to the standard general relativistic metric approach to gravity, gauge theories of spacetime groups describe gravitational forces in close analogy to the remaining interactions [ [7] [8] [9]. The Lorentz group and the GL(4 , R) group are usual candidates proposed by different authors [2] [3] [10] [11] to play the role of local symmetries.…”

Section: Introductionmentioning

confidence: 99%