Waldyr Alves Rodrigues

WHERE ARE ELKO SPINOR FIELDS IN LOUNESTO SPINOR FIELD CLASSIFICATION

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.

The Many Faces of Maxwell, Dirac and Einstein Equations

Preface.- Introduction.- Multivector and Extensor Calculus.- The Hidden Geometrical Nature of Spinors.- Some Differential Geometry.- Clifford Bundle Approach to the Differential Geometry of Branes.- Some Issues in Relativistic Spacetime Theories.- Clifford and Dirac-Hestenes Spinor Fields.- A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime.- Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes.- The DHE on a RCST and the Meaning of Active Local Lorentz Invariance.- On the Nature of the Gravitational Field.- On the Many Faces of Einstein Equations.- Maxwell, Dirac and Seiberg-Witten Equations.- Superparticles and Superfields.- Maxwell, Einstein, Dirac and Navier-Stokes Equations.- Magnetic Like Particles and Elko Spinor Fields.-Appendices A1-5.- Acronyms and Abbreviations.- List of Symbols.- Index.

Euclidean Clifford algebra

LetV be ann-dimensional real vector space. In this paper we introduce the concept ofeuclidean Clifford algebraCℓ (V, GE) for a given euclidean structure onV , i.e., a pair (V, GE) where GE is an euclidean metric forV (also called an euclidean scalar product). Our construction ofCℓ(V, GE) has been designed to produce a powerful computational tool. We start introducing the concept ofmultivectors overV. These objects are elements of a linear space over the real field, denoted by ΛV. We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of twocontraction operators on ΛV; and the concept of euclideaninterior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.

HIDDEN CONSEQUENCE OF ACTIVE LOCAL LORENTZ INVARIANCE

In this paper we investigate a hidden consequence of the hypothesis that Lagrangians and field equations must be invariant under active local Lorentz transformations. We show that this hypothesis implies in an equivalence between spacetime structures with several curvature and torsion possibilities.

Metric Clifford algebra

In this paper we introduce the concept of metric Clifford algebraCℓ(V; g) for ann-dimensional real vector spaceV endowed with a metric extensor g whose signature is (p; q), withp+q=n. The metric Clifford product onCℓ (V; g) appears as a well-defined deformation (induced by g) of an euclidean Clifford product onCℓ (V). Associated with the metric extensorg; there is a gauge metric extensorh which codifies all the geometric information just contained ing: The precise form of suchh is here determined. Moreover, we present and give a proof of the so-calledgolden formula, which is important in many applications that naturally appear in our studies of multivector functions, and differential geometry and theoretical physics.

Multivector functions of a real variable

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable.

GEOMETRIC ALGEBRAS AND EXTENSORS

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to metric geometric algebras Cl(V,G) and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in Cl(V,G) with easier ones in Cl(V,G_{E}) (e.g., a noticeable relation between the Hodge star operators associated to G and G_{E}). Several useful examples are worked in details fo the purpose of transmitting the tricks of the trade.

CLIFFORD VALUED DIFFERENTIAL FORMS, AND SOME ISSUES IN GRAVITATION, ELECTROMAGNETISM AND "UNIFIED" THEORIES

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the Lie algebra of Clifford bivectors is isomorphic to the Lie algebra of . In that way the pullback of the linear connection under a local trivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einsteins gravitational theory can be formulated in a way which is similar to a gauge theory. Such a theory is compared with other interesting mathematical formulations of Einsteins theory, and particularly with a supposedly unified field theory of gravitation and electromagnetism proposed by M. Sachs. We show that his identification of Maxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in the paper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the energy–momentum conservation. We show that many statements appearing in the literature are confusing or even wrong.

Finite energy superluminal solutions of Maxwell equations

Abstract We exhibit exact finite energy superluminal solutions of Maxwell equations in vacuum and discuss the physical meaning of these solutions.

The Maxwell and Navier-Stokes equations that follow from Einstein equation in a spacetime containing a Killing vector field

In this paper we are concerned to reveal that any spacetime structure 〈M, g, D, τg, ↑〉, which is a model of a gravitational field in General Relativity generated by an energy-momentum tensor T - and which contains at least one nontrivial Killing vector field A - is such that the 2-form field F = dA (where A = g(A,)) satisfies a Maxwell like equation - with a well determined current that contains a term of the superconducting type- which follows directly from Einstein equation. Moreover, we show that the resulting Maxwell like equations, under an additional condition imposed to the Killing vector field, may be written as a Navier-Stokes like equation as well. As a result, we have a set consisting of Einstein, Maxwell and Navier-Stokes equations, that follows sequentially from the first one under precise mathematical conditions and once some identifications about field variables are evinced, as explained in details throughout the text. We compare and emulate our results with others on the same subject appearing in the literature. In Appendix A we fix our notation and recall some necessary material concerning the theory of differential forms, Lie derivatives and the Clifford bundle formalism used in this paper. Moreover, we comment in Appendix B on some analogies (and main differences) between our results to the ones obtained long ago by Bergmann and Kommar which are reviewed and briefly criticized.In this paper we are concerned to reveal that any spacetime structure 〈M, g, D, τg, ↑〉, which is a model of a gravitational field in General Relativity generated by an energy-momentum tensor T - and which contains at least one nontrivial Killing vector field A - is such that the 2-form field F = dA (where A = g(A,)) satisfies a Maxwell like equation - with a well determined current that contains a term of the superconducting type- which follows directly from Einstein equation. Moreover, we show that the resulting Maxwell like equations, under an additional condition imposed to the Killing vector field, may be written as a Navier-Stokes like equation as well. As a result, we have a set consisting of Einstein, Maxwell and Navier-Stokes equations, that follows sequentially from the first one under precise mathematical conditions and once some identifications about field variables are evinced, as explained in details throughout the text. We compare and emulate our results with others on the same subject appea...