Gravitation as Plastic Distortion of the Lorentz Vacuum

The Many Faces of Maxwell, Dirac and Einstein Equations

Oliveira

2016

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Section: Remark 157mentioning

confidence: 97%

Section: From Maxwell Equation To a Navier-stokes Equationmentioning

confidence: 99%

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Maxwell, Einstein, Dirac and Navier-Stokes Equations

The Many Faces of Maxwell, Dirac and Einstein Equations

Oliveira

2016

Self Cite

“…different Clifford products is an essential tool in the theory of the gravitational field as presented in [4], where this field is represented by the gauge metric extensor h (Sect. 2.8.1).…”

Section: Symmetric Automorphisms and Orthogonal Clifford Productsmentioning

confidence: 99%

Multivector and Extensor Calculus

The Many Faces of Maxwell, Dirac and Einstein Equations

Oliveira

2016

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This chapter is dedicated to a thoughtful exposition of the multiform and extensor calculus. Starting from the tensor algebra of a real n-dimensional vector space V we construct the exterior algebra V V of V. Equipping V with a metric tensor V g we introduce the Grassmann algebra and next the Clifford algebra C`.V; V g/ associated to the pair .V; V g/. The concept of Hodge dual of elements of V V (called nonhomogeneous multiforms) and of C`.V; V g/ (also called nonhomogeneous multiforms or Clifford numbers) is introduced, and the scalar product and operations of left and right contractions in these structures are defined. Several important formulas and "tricks of the trade" are presented. Next we introduce the concept of extensors which are multilinear maps from p subspaces of V V to q subspaces of V V and study their properties. Equipped with such concept we study some properties of symmetric automorphisms and the orthogonal Clifford algebras introducing the gauge metric extensor (an essential ingredient for theories presented in other chapters). Also, we define the concepts of strain, shear and dilation associated with endomorphisms. A preliminary exposition of the Minkoswski vector space is given and the Lorentz and Poincaré groups are introduced. In the remaining of the chapter we give an original presentation of the theory of multiform functions of multiform variables. For these objects we define the concepts of limit, continuity and differentiability. We study in details the concept of directional derivatives of multiform functions and solve several nontrivial exercises to clarify how to work with these notions, which in particular are crucial for the formulation of Chap. 8 which deals with a Clifford algebra Lagrangian formalism of field theory in Minkowski spacetime. Tensor AlgebraWe recall here some basic facts of tensor algebra. Let V be a vector space over the real field R of finite dimension, i.e., dimV D n, n 2 N. By V D V we denote the dual space of V. Recall that dim V D dim V. The elements of V are called vectors and the elements of V are called covectors or 1-forms.

“…21 Details about these possibilities are discussed in [10] where a theory of the gravitational field on a brane diffeomorphic to R 4 is discussed.…”

Section: A Maxwell Like Equation For a Brane World With A Killing Vecmentioning

confidence: 99%

Clifford Bundle Approach to the Differential Geometry of Branes

The Many Faces of Maxwell, Dirac and Einstein Equations

Wainer

2016

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We first recall using the Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on Cℓ(M, g) (the Clifford bundle of differential forms) the formulation of the intrinsic geometry of a differential manifold M equipped with a metric field g of signature (p, q) and an arbitrary metric compatible connection ∇ introducing the torsion (2 − 1)-extensor field τ , the curvature (2 − 2) extensor field R and (once fixing a gauge) the connection (1 − 2)-extensor ω and the Ricci operator ∂ ∧ ∂ (where ∂ is the Dirac operator acting on sections of Cℓ(M, g)) which plays an important role in this paper. Next, using the CBF we give a thoughtful presentation the Riemann or the Lorentzian geometry of an orientable submanifold M (dim M = m) living in a manifoldM (such thatM ≃ R n is equipped with a semi-Riemannian metricg with signature (p,q) andp +q = n and its Levi-Civita connectionD) and where there is defined a metric g = i * g , where i : M →M is the inclusion map. We prove several equivalent forms for the curvature operator R of M . Moreover we show a very important result, namely that the Ricci operator of M is the (negative) square of the shape operator S of M (object obtained by applying the restriction on M of the Dirac operator∂ of Cℓ(M ,g) to the projection operator P). Also we disclose the relationship between the (1 − 2)-extensor ω and the shape biform S (an object related to S). The results obtained are used to give a mathematical formulation to Clifford's theory of matter. It is hoped that our presentation will be useful for differential geometers and theoretical physicists interested, e.g., in string and brane theories and relativity theory by divulging, improving and expanding very important and so far unfortunately largely ignored results appearing in reference [13].