DOCSLIB.ORG

Spinors in Curved Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spinors in Curved Space Spinors in Curved Space Erik Olsen December 4, 2008 1 Introduction The main problem that this paper will address is as follows: how can we in- clude a gravitational field g within a given Lagrangian density L? To solve this, we make use of the Principle of General Covariance. The first step is finding the special-relativistic equations of motion for the system where the gravita- tional field is neglected. The next step is to make several substitutions within these equations; all Lorentz tensors, derivatives and Minkowski tensor (η) terms will be replaced with objects that behave like tensors under general coordinate transformations, covariant derivatives and the metric tensor for curved space- time gµν respectively. These substitutions will make the equations generally covariant. It should be noted here that this method is only valid for space- time distances that are small in comparison to the space-time distances of the gravitational field. Also note that: [?] @ξα @ξβ g = η (1) µν @xµ @xν αβ Where ξ represents a coordinate system under the influence of gravity. The rank of the tensor within the Lagrangian density corresponds to the spin of the particle described. For example, a rank 0 tensor describes a spin-0 particle (π meson), a rank 1 tensor describes a spin-1 particle (photon) and a rank 2 tensor describes spin-2 particles. Notice however that these are all particles of integer spin; we are interested in particles with half-integer spin (electrons, quarks, etc). Such particles are described by spinors, not tensors [?]. Unfortunately, the substitution method described above does not work if we use spinors in our Lagrangian density instead of tensors. The reason for this can be seen by observing the group of linear invertible 4 X 4 matrices GL(4) and Lorentz transformations (which form a subgroup of GL(4)). The elements of GL(4) that are represented as tensors will act as tensors under a Lorentz transformation. However, if we look for representations of GL(4) that will act as spinors under an infinitesimal Lorentz transformation, we find that no such elements exist. To alleviate this problem, we shall use what are known as tetrads (or vier- beins). 2 Tetrads α The idea behind this method is to introduce normal local coordinates zX to each point in space-time X in addition to the generalized coordinates xµ one 1 would have to define their system. Using this, we can re-write the metric for curved space-time: @zα (x) @zβ (x) g (x) = X X η (2) µν @xµ @xν αβ Let: α @zX (x) α µ = Vµ (X) (3) @x x=X β @zX (x) β ν = Vν (X) (4) @x x=X α We define Vµ (X) to be a tetrad. Since we have two coordinate systems here (local inertial and general nonin- ertial), let's see what happens when we fix the local one and change the general from xµ to x0µ. This will cause the following change in the tetrad: α 0α Vµ ! Vµ (5) where @zα V 0α = X −! (6) µ @x0µ @zα @xν V 0α = X −! (7) µ @x0µ @xν @xν @zα V 0α = X (8) µ @x0µ @xν Thus we see [?]: @xν V 0α = V α (9) µ @x0µ ν From this we can conclude that a tetrad is not to be thought of as a tensor, but instead as four orthonormal vector fields. Still working with the coordinate systems, it is possible for us to take the Lorentz transform of the local system for every point in space-time [?]: α 0α α β zX −! zX = Λβ (X)zX (10) Plugging this into the definition of the tetrad: @ V α(X) = Λαzβ −! (11) µ @xµ β x @zβ V α(X) = Λα X (12) µ β @xµ Thus: α α β Vµ (X) ! Λβ Vµ (X) (13) α From this we see Vµ (X) transforms as a Lorentz contravariant vector. The usefulness of tetrads comes as follows: if we take a generally covariant µ vector Bµ and contract it into a tetrad Vα , we obtain the following [?]: µ Vα Bµ = Bα (14) 2 This particular object Bα has two useful properties: (1) Under a local Lorentz transformation it will behave as a vector. (2) Under a general coordinate transformation it will transform as four scalars. The reason tetrads are so useful is because now we can take any tensor field and contract it into a group of scalars, allowing us to use the substitution process discussed in the Introduction. This contraction method can even be used for a spinor field, which eliminates the problem of how spinors act under an infinitesimal Lorentz transformation. Covariant Derivatives It is not enough to focus only on the field; we must also look into the derivative of the field. This is because the derivative of a tensor field will itself be a tensor field in Minkowski space. To be useful within the Lagrangian density, the derivatives must be Lorentz vectors and coordinate scalars. We choose the following covariant derivative rα to act on a field where [?]: β rα ! Λα(x)D Λ(x) rβ (x) (15) where D(Λ) is the matrix representation of the infinitesimal Lorentz group. This expression is chosen since it is a Lorentz vector, coordinate scalar and transforms easily under a Lorentz transformation. To make the last property here possible, we define [?]: µ rα = Vα (@µ + Γµ) (16) where 1 Γ (x) = ΣαβV ν r V (x) (17) µ 2 α µ βν and Σαβ is the group generator for the Lorentz group. Also [?]: µ Vβν = gµν Vβ (18) We are now in a position to use the method described in the Introduction to generalize a Lagrangian density containing spinor fields into curved space. All it will involve is contracting all spinors using tetrads and replacing all partial derivatives with covariant ones. 1 3 Example: The Spin 2 Field The Lagrangian density for this particular case will be [?]: 1 L(x) = i( γα@ − @ γα ) − m (19) 2 α α Using what was developed in the previous section, we wish to generalize this to curved space-time. First, for the Dirac spinor field: 1 Σ = [γ ; γ ] (20) αβ 4 α β 3 where the γ terms are the Dirac matrices. Plugging this into our value of Γµ: 1 Γ (x) = [γ ; γ ]V ν (x)r g (x)V µ(x) (21) µ 8 α β α µ µν β Putting this into the definition rα and putting it into the Lagrangian density in place of @α, we get: n1 o L(x) = det V i γαV µr − V µ(r )γα − m (22) 2 α µ α µ Here we multiplied by det V to keep the Lagrangian density a scalar density. µ α Let γµ = Vα γ . Now we get [?]: n1 o L(x) = det V i γµr − (r )γµ − m (23) 2 µ µ We renamed the γ matrices because they are now the Dirac matrices in curved space. Taking their anti-commutator: fγµ; γν g = γµγν + γν γµ ! (24) µ α ν β ν β µ α Vα γ Vβ γ + Vβ γ Vα γ ! (25) µ ν α β Vα Vβ fγ ; γ g (26) By definition: α β fγ ; γ g = 2ηαβ (27) So: µ ν µ ν fγ ; γ g = 2Vα Vβ ηαβ (28) Thus we see: µ ν fγ ; γ g = 2gµν (29) This is the anti-commutator relation of the Dirac matrices generalized to curved space. 4 Conclusion With the ability to describe spinors in curved space, it becomes possible to see the effects of the gravitational action on systems of fermions. Of course, everything discussed here is still just an approximation, for we neglected to include quantum effects of gravity which are not completely understood at this time. References [1] G.B. Arfken and H.J Weber. Mathematical Methods for Physicists. Elsevier Adademic Press, sixth edition, 2005. [2] N.D. Birrell and P.C.W Davies. Quantum Fields in Curved Space. Cam- bridge University Press, 1982. [3] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley and Sons, 1972. 4.
Load more

Recommended publications
  • 1.2 Topological Tensor Calculus
    PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space.
    [Show full text]
  • Matrices and Tensors
    APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM..
    [Show full text]
  • General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
    General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ).
    [Show full text]
  • Tensor Calculus and Differential Geometry
    Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane .
    [Show full text]
  • The Language of Differential Forms
    Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects.
    [Show full text]
  • 3. Tensor Fields
    3.1 The tangent bundle. So far we have considered vectors and tensors at a point. We now wish to consider fields of vectors and tensors. The union of all tangent spaces is called the tangent bundle and denoted TM: TM = ∪x∈M TxM . (3.1.1) The tangent bundle can be given a natural manifold structure derived from the manifold structure of M. Let π : TM → M be the natural projection that associates a vector v ∈ TxM to the point x that it is attached to. Let (UA, ΦA) be an atlas on M. We construct an atlas on TM as follows. The domain of a −1 chart is π (UA) = ∪x∈UA TxM, i.e. it consists of all vectors attached to points that belong to UA. The local coordinates of a vector v are (x1,...,xn, v1,...,vn) where (x1,...,xn) are the coordinates of x and v1,...,vn are the components of the vector with respect to the coordinate basis (as in (2.2.7)). One can easily check that a smooth coordinate trasformation on M induces a smooth coordinate transformation on TM (the transformation of the vector components is given by (2.3.10), so if M is of class Cr, TM is of class Cr−1). In a similar way one defines the cotangent bundle ∗ ∗ T M = ∪x∈M Tx M , (3.1.2) the tensor bundles s s TMr = ∪x∈M TxMr (3.1.3) and the bundle of p-forms p p Λ M = ∪x∈M Λx . (3.1.4) 3.2 Vector and tensor fields.
    [Show full text]
  • Operator-Valued Tensors on Manifolds: a Framework for Field
    Operator-Valued Tensors on Manifolds: A Framework for Field Quantization 1 2 H. Feizabadi , N. Boroojerdian ∗ 1,2Department of pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran. Abstract In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commu- tative C⋆-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. Keywords: Operator-valued tensors, Operator-Valued Semi-Riemannian Met- rics, Levi-Civita Connection, Curvature, Hodge star operator MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. 0 Introduction arXiv:1501.05065v2 [math-ph] 27 Jan 2015 The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C∗-algebras, so C∗- algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C⋆-modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C⋆-algebras by studying Hilbert C⋆-modules over them. Hilbert C⋆-modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C⋆-algebra.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]
  • Tensors in 10 Minutes Or Less
    Tensors In 10 Minutes Or Less Sydney Timmerman Under the esteemed mentorship of Apurva Nakade December 5, 2018 JHU Math Directed Reading Program Theorem If there is a relationship between two tensor fields in one coordinate system, that relationship holds in certain other coordinate systems This means the laws of physics can be expressed as relationships between tensors! Why tensors are cool, kids This means the laws of physics can be expressed as relationships between tensors! Why tensors are cool, kids Theorem If there is a relationship between two tensor fields in one coordinate system, that relationship holds in certain other coordinate systems Why tensors are cool, kids Theorem If there is a relationship between two tensor fields in one coordinate system, that relationship holds in certain other coordinate systems This means the laws of physics can be expressed as relationships between tensors! Example Einstein's field equations govern general relativity, 8πG G = T µν c4 µν 0|{z} 0|{z} (2) tensor (2) tensor What tensors look like What tensors look like Example Einstein's field equations govern general relativity, 8πG G = T µν c4 µν 0|{z} 0|{z} (2) tensor (2) tensor Let V be an n-dimensional vector space over R Definition Given u; w 2 V and λ 2 R, a covector is a map α : V ! R satisfying α(u + λw) = α(u) + λα(w) Definition Covectors form the dual space V ∗ The Dual Space Definition Given u; w 2 V and λ 2 R, a covector is a map α : V ! R satisfying α(u + λw) = α(u) + λα(w) Definition Covectors form the dual space V ∗ The Dual Space Let V be
    [Show full text]
  • Lecture 12. Tensors
    Lecture 12. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. 12.1 Basic definitions We have already seen several examples of the idea we are about to introduce, namely linear (or multilinear) operators acting on vectors on M. For example, the metric is a bilinear operator which takes two vectors to give a real number, i.e. gx : TxM × TxM → R for each x is defined by u, v → gx(u, v). The difference between two connections ∇(1) and ∇(2) is a bilinear op- erator which takes two vectors and gives a vector, i.e. a bilinear operator Sx : TxM × TxM → TxM for each x ∈ M. Similarly, the torsion of a connec- tion has this form. Definition 12.1.1 A covector ω at x ∈ M is a linear map from TxM to R. The set of covectors at x forms an n-dimensional vector space, which we ∗ denote Tx M.Atensor of type (k, l)atx is a multilinear map which takes k vectors and l covectors and gives a real number × × × ∗ × × ∗ → R Tx : .TxM .../0 TxM1 .Tx M .../0 Tx M1 . k times l times Note that a covector is just a tensor of type (1, 0), and a vector is a tensor of type (0, 1), since a vector v acts linearly on a covector ω by v(ω):=ω(v). Multilinearity means that i1 ik j1 jl T c vi1 ,..., c vik , aj1 ω ,..., ajl ω i i j j 1 k 1 l . i1 ik j1 jl = c ...c aj1 ...ajl T (vi1 ,...,vik ,ω ,...,ω ) i1,...,ik,j1,...,jl 110 Lecture 12.
    [Show full text]
  • Solids at Rest
    Part II Solids at rest Copyright c 1998–2010 Benny Lautrup 96 Solids at rest In the following chapters we only consider matter at permanent rest without any macroscopic local or global motion. Microscopically, matter is never truly at rest because of random molec- ular motion (heat), and internal atomic motion. The essential chapters for a minimal curriculum are 6, 7, and 8. Readers that want to focus on fluid mechanics alone need only chapter 6. List of chapters 6. Stress: Friction is discussed as an example of shear stress. The stress tensor field is formally introduced. The equations of mechanical equilibrium in all matter are formu- lated. The symmetry of the stress tensor is discussed. 7. Strain: The displacement field is introduced and analyzed in the limit of infinitesimal strain gradients. The strain tensor is introduced and its geometrical meaning exposed. Work and energy are analyzed, and large deformation briefly discussed. 8. Elasticity: Hooke’s law is discussed and generalized to linear homogeneous materials. Young’s modulus and Poisson’s ratio are introduced. Fundamental examples of static uniform deformation are discussed. Expressions for the elastic energy are obtained. 9. Basic elastostatics: The most important elementary applications of linear elasticity are analyzed (gravitational settling, bending and twisting beams, spheres and tubes under pressure). 10. Slender rods: Slender elastic rods are analyzed and presented with characteristic ex- amples, such as cantilevers, bridges, and coiled springs. 11. Computational elastostatics: The basic theory behind numeric computation of linear elastostatic solutions is presented, and a typical example is analyzed and computed. Copyright c 1998–2010 Benny Lautrup 6 Stress Normal Force In fluids at rest pressure is the only contact force.
    [Show full text]
  • Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan
    Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics∗ Waldyr Alves Rodrigues Jr. Institute of Mathematics Statistics and Scientific Computation IMECC-UNICAMP CP 6065 13083760 Campinas SP Brazil e-mail: [email protected] or [email protected] May 28, 2018 Abstract In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry to some physical problems. A detailed account of the issues discussed in the paper appears in the table of contents. Contents 1 Introduction 3 arXiv:0712.3067v6 [math-ph] 4 Dec 2008 2 Classification of Metric Compatible Structures (M; g;D) 4 2.1 Levi-Civita and Riemann-Cartan Connections . 5 2.2 Spacetime Structures . 5 3 Absolute Differential and Covariant Derivatives 5 ∗published in: Ann. Fond. L. de Broglie 32 (special issue dedicate to torsion), 424-478 (2008). 1 ^ 4 Calculus on the Hodge Bundle ( T ∗M; ·; τg) 8 4.1 Exterior Product . 8 4.2 Scalar Product and Contractions . 9 4.3 Hodge Star Operator ? ........................ 9 4.4 Exterior derivative d and Hodge coderivative δ .
    [Show full text]