Antonio Manuel Moya - Academia.edu (original) (raw)

Papers by Antonio Manuel Moya

Clifford Algebras and their Applications in Mathematical Physics, 2000

Page 397. Covariant Derivatives on Minkowski Manifolds Virginia V. Fernandez, Antonio M. Moya, an... more Page 397. Covariant Derivatives on Minkowski Manifolds Virginia V. Fernandez, Antonio M. Moya, and Waldyr A. Rodrigues, Jr. ABSTRACT We present a general theory of covariant derivative operators (linear connections) on ...

Desenvolvemos o formalismo Lagrangiano para os chamados campos relativisticos utilizando o calcul... more Desenvolvemos o formalismo Lagrangiano para os chamados campos relativisticos utilizando o calculo do espaco-tempo, i.e., um calculo multivetorial baseado na algebra do espaco-tempo. Derivamos rigorosamente a equacao de campo, associada a Lagrangiana para um campo multivetorial (rotor ou spinor), a partir do principio de minima acao. Derivamos as formulas gerais para os extensores canonicos da energia-momento e do momento angular, e obtemos duas formas equivalentes para os correspondentes teoremas de conservacao, com campos multivetoriais (rotores) e campos spinoriais tratados de um modo completamente unificado. Demonstramos que aparte antisimetrica do extensor de energia-momento e de grande importância no tratamento correto do momento angular, ela esta relacionada a fonte do spin Abstract

Clifford Algebras and their Applications in Mathematical Physics, 2000

Advances in Applied Clifford Algebras, 2001

In this paper we develop with considerable details a theory of multivector functions of a p-vecto... more In this paper we develop with considerable details a theory of multivector functions of a p-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., the A-directional derivative (where A is a p-vector) and the generalized concepts of curl, divergence and gradient. The derivation rules for different types of products of multivector functions and for compositon of multivector functions are proved.

Advances in Applied Clifford Algebras, 2001

This paper is an introduction to the theory of multivector functions of a real variable. The noti... more 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.

This paper, the third in a series of eight introduces some of the basic concepts of the theory of... more This paper, the third in a series of eight introduces some of the basic concepts of the theory of extensors needed for our formulation of the differential geometry of smooth manifolds . Key notions such as the extension and generalization operators of a given linear operator (a (1,1)-extensor) acting on a real vector space V are introduced and studied in

Advances in Applied Clifford Algebras, 2001

In this paper we give a comparison between the formulation of the concept of metric for a real ve... more In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of tensors and extensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do not have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations.

We introduce the key concepts of duality mappings and metric extensor. The fundamental identities... more We introduce the key concepts of duality mappings and metric extensor. The fundamental identities involving the duality mappings are presented, and we disclose the logical equivalence between the so-called metric tensor and the metric extensor. By making use of the duality mappings and the metric extensor, we construct the so-called metric products, i.e., scalar product and contracted products of both multivectors and multiforms. The so-known identities involving the metric products are obtained. We find the fundamental formulas involving the metric extensor and, specially, we try its surprising inversion formula. This proposal unveils, once and for all, an unsuspected meaning of the metric products.

Eprint Arxiv Math 0501561, 2005

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in the... more This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in the previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated metric compatible connection extensor field, the associated Christoffel operators and a notable decomposition of those objects are given The paper introduces also the concept of a geometrical structure for a manifold M as a triple (M, g, γ), where γ is a connection extensor field defining a parallelism structure for M. Next, the theory of metric compatible covariant derivatives is given and a relationship between the connection extensor fields and covariant derivatives of two deformed (metric compatible) geometrical structures (M, g, γ) and (M, η, γ ′) is determined.

In this paper we introduce the concept of euclidean Clifford algebra C (V, G E) for a given eucli... more In this paper we introduce the concept of euclidean Clifford algebra C (V, G E) for a given euclidean structure on V , i.e., a pair (V, G E) where G E is an euclidean metric for V (also called an euclidean scalar product). Our construction of C (V, GE) has been designed to produce a powerful computational tool. We start introducing the concept of multivectors over V. These objects are elements of a linear space over the real field, denoted by V. We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two contraction operators on V, and the concept of euclidean interior 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.

In this paper we introduce the concept of metric Clifford algebra Cℓ(V, g) for a n-dimensional re... more In this paper we introduce the concept of metric Clifford algebra Cℓ(V, g) for a n-dimensional real vector space V endowed with a metric extensor g whose signature is (p, q), with p+q = n. The metric Clifford product on Cℓ(V, g) appears as a well-defined deformation (induced by g) of an euclidean Clifford product on Cℓ(V). Associated with the metric extensor g, there is a gauge metric extensor h which codifies all the geometric information just contained in g. The precise form of such h is here determined. Moreover, we present and give a proof of the so-called golden formula, which is important in many applications that naturally appear in ours studies of multivector functions, and differential geometry and theoretical physics.

Eprint Arxiv Math 0502003, Jan 31, 2005

Here (the last paper in a series of four) we end our presentation of the basics of a systematical... more Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U , i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi-Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivat... more A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived.

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivat... more A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

Using the theory of extensors developed in a previous paper we present a theory of the parallelis... more Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the concept of deformed parallelism structures and relative parallelism structures which play

In this paper we study in details the properties of the duality product of multivectors and multi... more In this paper we study in details the properties of the duality product of multivectors and multiforms (used in the definition of the hyperbolic Clifford algebra of multivefors) and introduce the theory of the k multivector and l multiform variables multivector (or multiform) extensors over V studying their properties with considerable detail.

International Journal of Geometric Methods in Modern Physics, 2007

Here (the last paper in a series of four) we end our presentation of the basics of a systematical... more Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

International Journal of Geometric Methods in Modern Physics, 2007

We give in this paper which is the third in a series of four a theory of covariant derivatives of... more 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 ⊂ 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 γ defining a parallelism structure on U ⊂ M, which represents in a well-defined way the action on U of the restriction there of some given connection ∇ 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 g define...

Clifford Algebras and their Applications in Mathematical Physics, 2000

Advances in Applied Clifford Algebras, 2001

Eprint Arxiv Math 0501561, 2005

Eprint Arxiv Math 0502003, Jan 31, 2005

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivat... more A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

International Journal of Geometric Methods in Modern Physics, 2007

Here (the last paper in a series of four) we end our presentation of the basics of a systematical... more Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

International Journal of Geometric Methods in Modern Physics, 2007