Geometric and Extensor Algebras and the Differential Geometry of Arbitrary Manifolds

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U of M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field gamma defining a parallelism structure on U, which represents in a well defined way the action on U of the restriction there of some given connection del defined on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection operator fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor \slg defined on M and also introduce the concept a geometric structure (U,gamma,g) for U and study metric compatibility of covariant derivatives induced by the connection extensor gamma. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives operators in metric and geometrical structures, like ordinary and covariant Hodge coderivatives and some duality identities are exhibit.Comment: This paper is an improved version of material contained in math.DG/0501560, math.DG/0501561, math.DG/050200

Gravitation as a Plastic Distortion of the Lorentz Vacuum

In this paper we present a theory of the gravitational field where this field (a kind of square root of g) is represented by a (1,1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a real substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor in an appropriate way (see below) generating what may be interpreted as an effective Lorentzian metric extensor g and also it permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may have non null curvature and/or torsion and/or nonmetricity tensors. We thus have different possible effective geometries which may be associated to the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we developed with enough details the theory of multiform functions and multiform functionals that permitted us to successfully write a Lagrangian for h and to obtain its equations of motion, that results equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R^4. However, in our theory, differently from the case of General Relativity, trustful energy-momentum and angular momentum conservation laws exist. We express also the results of our theory in terms of the gravitational potential 1-form fields (living in Minkowski spacetime) in order to have results which may be easily expressed with the theory of differential forms. The Hamiltonian formalism for our theory (formulated in terms of the potentials) is also discussed. The paper contains also several important Appendices that complete the material in the main text.Comment: Misprints and typos have been corrected, Chapter 7 have been improved. Appendix E has been reformulated and Appendix F contains new remarks which resulted from a discussion with A. Lasenby. A somewhat modified version has been published in the Springer Series: Fundamental Theories of Physics vol. 168, 2010. http://www.ime.unicamp.br/~walrod/plastic2014.pd