DOCSLIB.ORG

Tensor Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tensor Review Engineering Tensors A BEH430 review session by Thomas Gervais [email protected] References: • Long, RR, Mechanics of Solids and Fluids, Prentice-Hall, 1960, pp 1-32 • Deen, WD, Analysis of transport phenomena, Oxford, 1998, p. 551-563 • Goodbody, AM, Cartesian tensors: With applications to Mechanics, Fluid Mechanics and Elasticity, John Wiley & Sons, 1982 Friday November 16, 2001 16h30 -Muddy Charles Scalars, Vectors & Tensors Scalar: Quantity that is invariant in itself (does not depend on any referential) Also known as a zeroth order tensor. Ex: mass (non relativistic referentials), Temperature, Energy, Concentration Vector: Quantity that possess both a direction and a magnitude located somewhere in space. It is a first vorder tensor. v =a1eˆ1 + a2eˆ2 + a3eˆ3 = a'1 eˆ'1 +a'2 eˆ'2 +a'3 eˆ'3 Note: A vector possessesv two invariants with respect to coordinate space: Its magnitude, v , and its direction. Ex: Force, Electric field, Flux Higher order tensors 2nd order tensor: An ordered set of nine numbers, each of which possessing 2 directions. An helpful analogy would be to imagine a vector whose three components each would be a vector. A tensor of order n can be reprensented by 3n numbers in tridimensional space. It is a matrix of dimension n. A 2nd possesses 3 invariants which are the coefficients of its characteristic polynomial arising from det T - l I = 0 . The trace and the determinant of the tensor are two of its 3 invariants. In general, a tensor of order n will have n+1 invariants. Physical applications of tensors: Discipline Phenomenon Quantity Example Physics: E-M Light transmission in Refraction index n Optically active anisotropic media polarizers Dynamics Angular momentum Moment of Inertia I Gyroscope Fluid mechanics Hindered transport in Effective Ca++ diffusion in porous media permeability K muscles Solid mechanics Tensile properties of Young’s modulus E Duct Tape, anisotropic solids Muscles Dyadic Product (Tensor Product) This the general form of a tensor product: Scalar A (0th order tensor) v v v v A = A e + A e + A e Vector 1 1 2 2 3 3 (1st order tensor) v v v v v v v v A = A e + A e + A e A = A e + A e + A e 2nd order 1 11 1 21 v2 31 3 3 13 1 23 2 33 3 v v v tensor A2 = A12e1 + A22e2 + A32e3 etc. 3 3 • By dividing the Aij in three vector components, A = A eˆ eˆ we get a 3rd order tensor, and so on... å å ij i j i=1 j =1 Notation z z’ v v 3 3 A A = å Aieˆi = å A'i eˆ'i i=1 i=1 y The projection of the vector on the axis, x, y, z can each be decomposed in the basis (x’, y’, z’). The relation y’ between each of the components is then given by: x’ x 3 = Ax = a11A'x + a21A' y + a31A'z Ai å aij A' j j =1 Ay = a12 A'x + a22 A' y + a32 A'z Az = a13A'x + a23A' y + a33A'z Ai = aij A' j (Einstein’s notation) é a a a ù 3 3 11 12 13 ê ú a = a eˆ eˆ = a a a å å ij i j ê 21 22 23ú i=1 j =1 ëê a31 a32 a33ûú Dot and Cross products Definition of dot product: v v 3 3 3 3 3 3 3 A× B = å Aieˆi × å B jeˆ j = å å AiB j eˆi × eˆ j = å å AiB j d ij = å AiBi i=1 j =1 i=1 j =1 i=1 j =1 i=1 nd Where dij is the Kronecker delta, a 2 order tensor. Does it hold in tensor notation? 3 Let’s test it using a change of coordinate: Ai = å aij A' j j =1 v v 3 3 3 3 3 3 3 3 A × B = å å aki A'k eˆk × å å alj B'l eˆl = å å å å akialj A'k B'l d kl i=1 k =1 j= 1 l=1 i=1 j =1 k =1 l=1 If what we said about the conservation of the magnitude of a vector from v v 3 3 one cartesian referential to another, then A× B = å Ai Bi = å A' k B' k i=1 k =1 For this to be true, we need: d ij = akialjd kl (Einstein notation) Dot and Cross product (cont.) d ij = akialjd kl expanded yields: 2 2 2 a11 + a12 + a13 = 1 a21a31 + a22 a32 + a23a33 = 0 2 2 2 a21 + a22 + a23 = 1 a31a11 + a32 a12 + a33a13 = 0 2 2 2 a31 + a32 + a33 = 1 a11a21 + a12a22 + a13a23 = 0 These are the normalization and the orthogonality conditions that any orthonormal base respects. Definition of cross product: v v 3 3 3 3 3 3 3 A× B = å Aieˆi ´ å B jeˆ j = å å AiB j eˆi ´ eˆ j = å å å AiBj e ijkeˆk i=1 j =1 i=1 j =1 i=1 j=1 k =1 Where e ijk is called the Levi-Civita density and is the cross product equivalent of the Kroneker delta, d ij , for the dot product. It is a third order tensor. 0, i=j, j=k, or i=k This is the famous e ijk 1, ijk= 123, 231, 312 “right hand rule” -1, ijk= 132, 321, 213 Gradients and Divergence v v 3 ¶f 3 We know: Ñ f = å eˆj and A = å Aieˆi ¶ j =1 x j v iv=1 Q: What about the gradient of a vector Ñ A ? A: The generalizationv of a vector provided by tensor analysis implies: v v 3 ¶ A 3 3 ¶A i A second order tensor! Ñ A = å eˆ j = å å eˆieˆj j =1 ¶ x j i=1 j=1 ¶ x j • The gradient of a tensor increases its order by one Q: What about the divergence of a 2nd order tensor? A: Using our definition of the dot product: v 3 3 ¶ 3 Tij ¶Tii Ñ × T = å å d ijeˆieˆ j = å eˆi A vector! i=1 j=1 ¶ x j i=1 ¶xi • The divergence of a tensor decreases its order by one Practical problem: How to set up a tensor from physical reasoning? Consider the following problem: We want to stretch a piece of anisotropic tissue and find the components of the Young’s Moduli v v E2 F F E1 q Apply a given displacement and compute the force as a function of the angle (assuming constant strain and a Poisson coefficient of “0”) v 2 y' v 2 2 2 u = u eˆ å i i F = å Fieˆ'i = å å Eijuieˆj i=1 E2 x' i=1 i=1 j =1 E1 y q é F1 ù é E11 E12 ù é u1 ù x ê ú = ê ú ê ú ë F2 û ë E21 E22 û ëu2 û Practical problem: How to set up a tensor from physical reasoning? (cont.) We need to compute the 4 matrix elements: • F1 ® u1 : E11 = Ex cosq • F1 ® u2 : E12 = Ex sin q é F1 ù é Ex cosq Ex sinq ù é u1 ù = ê ú • F ® u : E = - E sin q ê ú - q q ê ú 2 1 21 y ë F2 û ë E y sin Ey cos û ëu2 û • F2 ® u2 : E22 = E y cosq We are measuring the magnitude of the force required to impose a unity displacement in the x direction. 2 v 1.8 é F ù é1ù x' 2 2 1.6 = [Eij ] Þ F = F + F ê ú ê ú x ' y ' 1.4 Fy' ë0û ë û 1.2 Effective modulus v 1 0 0.5 1 1.5 F angle (rad) = E = E 2 cos2 q + E 2 sin 2 q u eff x y Normalized effective modulus as a function of the angle for E2 double of E1 Same problem: other approach using tensors v v 2 y' 2 2 2 F = F eˆ' = E' u' eˆ' u = å uieˆi å i i å å ij j j i=1 i=1 j=1 i=1 E2 x' E1 é E1 0 ù y E'ij = ê ú q ë 0 E2 û x 2 u'i eˆ'i = aiju jeˆ j å v 2 2 j =1 F = å å E'ij aiju jeˆ j é cosq sinq ù i=1 j =1 aij = ê ú ë - sin q cosq û Q: Are expressions equivalent, i.e. E'ij aij = Eij ? E 0 cosq sinq E cosq E sin q é 1 ù é ù é 1 1 ù A: Yes! E'ij aij = ê ú ê ú = ê ú ë 0 E2 û ë - sin q cosq û ë - E2 sinq E2 cosq û Fluid mechanics: The stress and rate of strain tensors v 3 vz' A) Stress tensor y' S = å Sieˆi v n i=1 z S x' 3 v = × ˆ ˆ y Si å n (Siei )e j j=1 x Stress vector v B) Rate of strain tensor Sij = s ij = n × (Sieˆi )eˆieˆ j v v v v v v(r,t) v(r + d r,t) By Taylor expansion to the first order: v v v v v v v v v v v t v = v(r,t) + d v d v = d r × Ñ v(r,t) = (Ñ v(r,t)) × d r v v v Ñ v(r,t)is a tensor (gradient of a vector)! v v v v & 1 t v v e = [(Ñ v) + (Ñ v)] t & 2 (Ñv ) = e + x v v v v & 1 t x = [(Ñ v) - (Ñ v)] Vorticity tensor 2 Conclusion • 2nd order tensors establish relation between two sets of vectors.
Load more

Recommended publications
  • Package 'Einsum'
    Package ‘einsum’ May 15, 2021 Type Package Title Einstein Summation Version 0.1.0 Description The summation notation suggested by Einstein (1916) <doi:10.1002/andp.19163540702> is a concise mathematical notation that implicitly sums over repeated indices of n- dimensional arrays. Many ordinary matrix operations (e.g. transpose, matrix multiplication, scalar product, 'diag()', trace etc.) can be written using Einstein notation. The notation is particularly convenient for expressing operations on arrays with more than two dimensions because the respective operators ('tensor products') might not have a standardized name. License MIT + file LICENSE Encoding UTF-8 SystemRequirements C++11 Suggests testthat, covr RdMacros mathjaxr RoxygenNote 7.1.1 LinkingTo Rcpp Imports Rcpp, glue, mathjaxr R topics documented: einsum . .2 einsum_package . .3 Index 5 1 2 einsum einsum Einstein Summation Description Einstein summation is a convenient and concise notation for operations on n-dimensional arrays. Usage einsum(equation_string, ...) einsum_generator(equation_string, compile_function = TRUE) Arguments equation_string a string in Einstein notation where arrays are separated by ’,’ and the result is separated by ’->’. For example "ij,jk->ik" corresponds to a standard matrix multiplication. Whitespace inside the equation_string is ignored. Unlike the equivalent functions in Python, einsum() only supports the explicit mode. This means that the equation_string must contain ’->’. ... the arrays that are combined. All arguments are converted to arrays with
    [Show full text]
  • Notes on Manifolds
    Notes on Manifolds Justin H. Le Department of Electrical & Computer Engineering University of Nevada, Las Vegas [email protected] August 3, 2016 1 Multilinear maps A tensor T of order r can be expressed as the tensor product of r vectors: T = u1 ⊗ u2 ⊗ ::: ⊗ ur (1) We herein fix r = 3 whenever it eases exposition. Recall that a vector u 2 U can be expressed as the combination of the basis vectors of U. Transform these basis vectors with a matrix A, and if the resulting vector u0 is equivalent to uA, then the components of u are said to be covariant. If u0 = A−1u, i.e., the vector changes inversely with the change of basis, then the components of u are contravariant. By Einstein notation, we index the covariant components of a tensor in subscript and the contravariant components in superscript. Just as the components of a vector u can be indexed by an integer i (as in ui), tensor components can be indexed as Tijk. Additionally, as we can view a matrix to be a linear map M : U ! V from one finite-dimensional vector space to another, we can consider a tensor to be multilinear map T : V ∗r × V s ! R, where V s denotes the s-th-order Cartesian product of vector space V with itself and likewise for its algebraic dual space V ∗. In this sense, a tensor maps an ordered sequence of vectors to one of its (scalar) components. Just as a linear map satisfies M(a1u1 + a2u2) = a1M(u1) + a2M(u2), we call an r-th-order tensor multilinear if it satisfies T (u1; : : : ; a1v1 + a2v2; : : : ; ur) = a1T (u1; : : : ; v1; : : : ; ur) + a2T (u1; : : : ; v2; : : : ; ur); (2) for scalars a1 and a2.
    [Show full text]
  • The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics
    The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand.
    [Show full text]
  • Multilinear Algebra and Applications July 15, 2014
    Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3.
    [Show full text]
  • On Manifolds of Tensors of Fixed Tt-Rank
    ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest.
    [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]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • Abstract Tensor Systems As Monoidal Categories
    Abstract Tensor Systems as Monoidal Categories Aleks Kissinger Dedicated to Joachim Lambek on the occasion of his 90th birthday October 31, 2018 Abstract The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories. 1 Introduction This paper formalises the connection between monoidal categories and the ab- stract index notation developed by Penrose in the 1970s, which has been used by physicists directly, and category theorists implicitly, via the diagrammatic languages for traced symmetric monoidal and compact closed categories. This connection is given as a representation theorem for the free traced symmet- ric monoidal category as a syntactically-defined strict monoidal category whose morphisms are equivalence classes of certain kinds of terms called Einstein ex- pressions. Representation theorems of this kind form a rich history of coherence results for monoidal categories originating in the 1960s [17, 6]. Lambek’s con- arXiv:1308.3586v1 [math.CT] 16 Aug 2013 tribution [15, 16] plays an essential role in this history, providing some of the earliest examples of syntactically-constructed free categories and most of the key ingredients in Kelly and Mac Lane’s proof of the coherence theorem for closed monoidal categories [11].
    [Show full text]
  • Quantum Chromodynamics (QCD) QCD Is the Theory That Describes the Action of the Strong Force
    Quantum chromodynamics (QCD) QCD is the theory that describes the action of the strong force. QCD was constructed in analogy to quantum electrodynamics (QED), the quantum field theory of the electromagnetic force. In QED the electromagnetic interactions of charged particles are described through the emission and subsequent absorption of massless photons (force carriers of QED); such interactions are not possible between uncharged, electrically neutral particles. By analogy with QED, quantum chromodynamics predicts the existence of gluons, which transmit the strong force between particles of matter that carry color, a strong charge. The color charge was introduced in 1964 by Greenberg to resolve spin-statistics contradictions in hadron spectroscopy. In 1965 Nambu and Han introduced the octet of gluons. In 1972, Gell-Mann and Fritzsch, coined the term quantum chromodynamics as the gauge theory of the strong interaction. In particular, they employed the general field theory developed in the 1950s by Yang and Mills, in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.) First, QED Lagrangian… µ ! # 1 µν LQED = ψeiγ "∂µ +ieAµ $ψe − meψeψe − Fµν F 4 µν µ ν ν µ Einstein notation: • F =∂ A −∂ A EM field tensor when an index variable µ • A four potential of the photon field appears twice in a single term, it implies summation µ •γ Dirac 4x4 matrices of that term over all the values of the index
    [Show full text]
  • About Matrices, Tensors and Various Abbrevia- Tions
    About matrices, tensors and various abbrevia- tions The objects we deal with in this course are rather complicated. We therefore use a simplifying notation where at every step as much as possible of this complexity is hidden. Basically this means that objects that are made out of many elements or components. are written without indices as much as possible. We also have several different types of indices present. The ones that show up in this course are: • Dirac-Indices a; b; c • Lorentz indices, both upper and lower µ, ν; ρ • SU(2)L indices i; j; k • SU(3)c indices α; β; γ Towards the right I have written the type of symbols used in this note to denote a particular type of index. The course contains even more, three-vector indices or various others denoting sums over types of quarks and/or leptons. An object is called scalar or singlet if it has no index of a particular type of index, a vector if it has one, a matrix if it has two and a tensor if it has two or more. A vector can also be called a column or row matrix. Examples are b with elements bi: 0 1 b1 B C B b2 C b = (b1; b2; : : : ; bn) b = (b1 b2 ··· bn) b = B . C (1) B . C @ . A bn In the first case is a vector, the second a row matrix and the last a column vector. Objects with two indices are called a matrix or sometimes a tensor and de- noted by 0 1 c11 c12 ··· c1n B C B c21 c22 ··· c2n C c = (cij) = B .
    [Show full text]
  • 1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A
    1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n).
    [Show full text]
  • Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
    Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor.
    [Show full text]