Tensor Calculus and Applications: Simplified Tools and Techniques [1 ed.] 0367138069, 9780367138066

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Tensor Calculus and Applications: Simplified Tools and Techniques [1 ed.]
 0367138069, 9780367138066

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
About the Book
Author
Part I: Formalism of Tensor Calculus
1. Prerequisites for Tensors
1.1 Ideas of Coordinate Systems
1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity)
1.3 Quadratic Forms, Properties, and Classifications
1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials
Exercises
2. Concept of Tensors
2.1 Some Useful Definitions
2.2 Transformation of Coordinates
2.3 Second and Higher Order Tensors
2.4 Operations on Tensors
2.5 Symmetric and Antisymmetric (or Skew-Symmetric) Tensors
2.6 Quotient Law
Exercises
3. Riemannian Metric and Fundamental Tensors
3.1 Riemannian Metric
3.2 Cartesian Coordinate System and Orthogonal Coordinate System
3.3 Euclidean Space of n Dimensions, Euclidean Co-Ordinates, and Euclidean Geometry
3.4 The Metric Functions g[sub(ij)] Are Second-Order Covariant Symmetric Tensors
3.5 The Function g[sub(ij)] Is a Contravariant Second-Order Symmetric Tensor
3.6 Scalar Product and Magnitude of Vectors
3.7 Angle Between Two Vectors and Orthogonal Condition
Exercises
4. Christoffel Three-Index Symbols (Brackets) and Covariant Differentiation
4.1 Christoffel Symbols (or Brackets) of the First and Second Kinds
4.2 Two Standard Applicable Results of Christoffel Symbols
4.3 Evolutionary Basis of Christoffel Symbols (Brackets)
4.4 Use of Symmetry Condition for the Ultimate Result
4.5 Coordinate Transformations of Christoffel Symbols
4.5.1 Transformation of the First Kind ┌[sub(ij, k)]
4.5.2 Transformation of the Second Kind ┌[sup(ij)][sub(k)]
4.6 Covariant Derivative of Covariant Tensor of Rank One
4.7 Covariant Derivative of Contravariant Tensor of Rank One
4.8 Covariant Derivative of Covariant Tensor of Rank Two
4.9 Covariant Derivative of Contravariant Tensor of Rank Two
4.10 Covariant Derivative of Mixed Tensor of Rank Two
4.10.1 Generalization
4.11 Covariant Derivatives of g[sub(ij)] g[sup(ij)] and also g[sub(i)][sub(j)]
4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors
4.13 Gradient of an Invariant Function
4.14 Curl of a Vector
4.15 Divergence of a Vector
4.16 Laplacian of a Scalar Invariant
4.17 Intrinsic Derivative or Derived Vector of v
4.18 Definition: Parallel Displacement of Vectors
4.18.1 When Magnitude Is Constant
4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude
Exercises
5. Properties of Curves in V[sub(n)] and Geodesics
5.1 The First Curvature of a Curve
5.2 Geodesics
5.3 Derivation of Differential Equations of Geodesics
5.4 Aliter: Differential Equations of Geodesics as Stationary Length
5.5 Geodesic Is an Autoparallel Curve
5.6 Integral Curve of Geodesic Equations
5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates
5.7.1 Another Form of Condition for Geodesic Coordinates
5.8 If a Curve Is a Geodesic of a Space (V[sub(m)]), It Is also a Geodesic of Any Space V[sub(n)] in Which It Lies (V[sub(n)] a Subspace)
Exercises
6. Riemann Symbols (Curvature Tensors)
6.1 Introduction
6.2 Riemannian Tensors (Curvature Tensors)
6.3 Derivation of the Transformation Law of Riemannian Tensor R[sup(α)][sub(abc)]
6.4 Properties of the Curvature Tensor RR[sup(α)][sub(ijk)]
6.5 Covariant Curvature Tensor
6.6 Properties of the Curvature Tensor R[sub(hijk)] of the First Kind
6.7 Bianchi Identity
6.8 Einstein Tensor Is Divergence Free
6.9 Isometric Surfaces
6.10 Three-Dimensional Orthogonal Cartesian Coordinate Metric and Two-Dimensional Curvilinear Coordinate Surface Metric Imbedded in It
6.11 Gaussian Curvature of the Surface S immersed in E[sub(3)]
Exercises
Part II: Application of Tensors
7. Application of Tensors in General Theory of Relativity
7.1 Introduction
7.2 Curvature of a Riemannian Space
7.3 Flat Space and Condition for Flat Space
7.4 Covariant Differential of a Vector
7.5 Motion of Free Particle in a Curvilinear Co-Ordinate System for Curved Space
7.6 Necessity of Ricci Tensor in Einstein’s Gravitational Field Equation
8. Tensors in Continuum Mechanics
8.1 Continuum Concept
8.2 Mathematical Tools Required for Continuum Mechanics
8.3 Stress at a Point and the Stress Tensor
8.4 Deformation and Displacement Gradients
8.5 Deformation Tensors and Finite Strain Tensors
8.6 Linear Rotation Tensor and Rotation Vector in Relation to Relative Displacement
9. Tensors in Geology
9.1 Introduction
9.2 Equation for the Determination of Shearing Stresses on Any Plane Surface
9.3 General Transformation and Maximum and Minimum Longitudinal Strains
9.4 Determination of the Two Principal Strains in a Plane
10. Tensors in Fluid Dynamics
10.1 Introduction
10.2 Equations of Motion for Newtonian Fluid
10.3 Navier–Stokes Equations for the Motion of Viscous Fluids
Appendix
Remarks
Bibliography
Index

Citation preview

Tensor Calculus and ­Applications

Mathematics and Its Applications: Modelling, Engineering, and Social Sciences Series Editor: Hemen Dutta Discrete Mathematical Structures: A Succinct Foundation Beri Venkatachalapathy Senthil Kumar and Hemen Dutta

Concise Introduction to Logic and Set Theory Iqbal H. Jebril and Hemen Dutta

Tensor Calculus and Applications: Simplified Tools and Techniques Bhaben Chandra Kalita For more information on this series, please visit: www.crcpress.com/ Mathematics-and-its-applications/book-series/MES

Tensor Calculus and ­Applications

Simplified Tools and Techniques

Bhaben Chandra Kalita

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-0-367-13806-6 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.­ copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Kalita, Bharat Chandra, 1937- author. Title: Tensor calculus and applications : simplified tools and techniques / authored by Bhaben Chandra Kalita. Description: Boca Raton : Taylor & Francis, [2019] | Includes bibliographical references. Identifiers: LCCN 2018052527 | ISBN 9780367138066 (hardback :acid-free paper) | ISBN 9780429028670 (e-book) Subjects: LCSH: Calculus of tensors. | Geometry, Differential. Classification: LCC QA433 .K345 2019 | DDC 515/.63—dc23 LC record available at https://lccn.loc.gov/2018052527 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Preface.......................................................................................................................ix About the Book........................................................................................................xi Author.................................................................................................................... xiii

Part I  Formalism of Tensor Calculus 1. Prerequisites for Tensors...............................................................................3 1.1 Ideas of Coordinate Systems................................................................3 1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity)��������������������������������������������������3 1.3 Quadratic Forms, Properties, and Classifications.............................7 1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials���������������������������������������������������������9 Exercises........................................................................................................... 11 2. Concept of Tensors........................................................................................ 13 2.1 Some Useful Definitions..................................................................... 13 2.2 Transformation of Coordinates.......................................................... 14 2.3 Second and Higher Order Tensors.................................................... 17 2.4 Operations on Tensors......................................................................... 18 2.5 Symmetric and Antisymmetric (or Skew-Symmetric) Tensors..... 20 2.6 Quotient Law........................................................................................ 25 Exercises........................................................................................................... 26 3. Riemannian Metric and Fundamental Tensors...................................... 29 3.1 Riemannian Metric.............................................................................. 29 3.2 Cartesian Coordinate System and Orthogonal Coordinate System��������������������������������������������������������������������������������������������������� 29 3.3 Euclidean Space of n Dimensions, Euclidean Co-Ordinates, and Euclidean Geometry�������������������������������������������������������������������30 3.4 The Metric Functions gij Are Second-Order Covariant Symmetric Tensors������������������������������������������������������������������������������30 3.5 The Function gij Is a Contravariant Second-Order Symmetric Tensor������������������������������������������������������������������������������� 33 3.6 Scalar Product and Magnitude of Vectors........................................ 38 3.7 Angle Between Two Vectors and Orthogonal Condition............... 38 Exercises........................................................................................................... 39

v

vi

Contents

4. Christoffel Three-Index Symbols (Brackets) and Covariant Differentiation............................................................................................... 41 4.1 Christoffel Symbols (or Brackets) of the First and Second Kinds����������������������������������������������������������������������������������� 41 4.2 Two Standard Applicable Results of Christoffel Symbols.............42 4.3 Evolutionary Basis of Christoffel Symbols (Brackets).....................43 4.4 Use of Symmetry Condition for the Ultimate Result...................... 49 4.5 Coordinate Transformations of Christoffel Symbols...................... 50 4.5.1 Transformation of the First Kind ij , k .................................. 50 i 4.5.2 T  ransformation of the Second Kind jk............................... 51 4.6 Covariant Derivative of Covariant Tensor of Rank One................ 55 4.7 Covariant Derivative of Contravariant Tensor of Rank One......... 56 4.8 Covariant Derivative of Covariant Tensor of Rank Two................ 57 4.9 Covariant Derivative of Contravariant Tensor of Rank Two......... 59 4.10 Covariant Derivative of Mixed Tensor of Rank Two...................... 60 4.10.1 Generalization......................................................................... 61 4.11 Covariant Derivatives of g ij ′ g ij and also g ij........................................ 62 4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors������������������������������������������������������������������������������63 4.13 Gradient of an Invariant Function..................................................... 66 4.14 Curl of a Vector..................................................................................... 67 4.15 Divergence of a Vector......................................................................... 68 4.16 Laplacian of a Scalar Invariant........................................................... 69  4.17 Intrinsic Derivative or Derived Vector of v...................................... 71 4.18 Definition: Parallel Displacement of Vectors................................... 72 4.18.1 When Magnitude Is Constant............................................... 72 4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude������������������������������������������������������������������������������� 73 Exercises........................................................................................................... 76 5. Properties of Curves in Vn and Geodesics................................................77 5.1 The First Curvature of a Curve..........................................................77 5.2 Geodesics............................................................................................... 78 5.3 Derivation of Differential Equations of Geodesics......................... 78 5.4 Aliter: Differential Equations of Geodesics as Stationary Length���������������������������������������������������������������������������������������������������80 5.5 Geodesic Is an Autoparallel Curve.................................................... 82 5.6 Integral Curve of Geodesic Equations..............................................85 5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates������������������������������������������ 86 5.7.1 Another Form of Condition for Geodesic Coordinates....... 88 5.8 If a Curve Is a Geodesic of a Space (Vm), It Is also a Geodesic of Any Space Vn in Which It Lies (Vn a Subspace)�������������������������� 89 Exercises........................................................................................................... 91

Contents

vii

6. Riemann Symbols (Curvature Tensors).................................................... 93 6.1 Introduction.......................................................................................... 93 6.2 Riemannian Tensors (Curvature Tensors)........................................ 93 6.3 Derivation of the Transformation Law of Riemannian α Tensor Rabc ��������������������������������������������������������������������������������������������� 95 α 6.4 Properties of the Curvature Tensor Rijk ............................................. 97 6.5 Covariant Curvature Tensor............................................................... 99 6.6 Properties of the Curvature Tensor Rhijk of the First Kind........... 100 6.7 Bianchi Identity.................................................................................. 101 6.8 Einstein Tensor Is Divergence Free................................................. 102 6.9 Isometric Surfaces.............................................................................. 103 6.10 Three-Dimensional Orthogonal Cartesian Coordinate Metric and Two-Dimensional Curvilinear Coordinate Surface Metric Imbedded in It�������������������������������������������������������� 103 6.11 Gaussian Curvature of the Surface S immersed in E3.................. 104 Exercises......................................................................................................... 108

Part II  Application of Tensors 7. Application of Tensors in General Theory of Relativity.................... 113 7.1 Introduction........................................................................................ 113 7.2 Curvature of a Riemannian Space................................................... 114 7.3 Flat Space and Condition for Flat Space......................................... 118 7.4 Covariant Differential of a Vector................................................... 118 7.5 Motion of Free Particle in a Curvilinear Co-Ordinate System for Curved Space����������������������������������������������������������������� 119 7.6 Necessity of Ricci Tensor in Einstein’s Gravitational Field Equation��������������������������������������������������������������������������������������������� 120 8. Tensors in Continuum Mechanics........................................................... 123 8.1 Continuum Concept.......................................................................... 123 8.2 Mathematical Tools Required for Continuum Mechanics........... 123 8.3 Stress at a Point and the Stress Tensor............................................ 125 8.4 Deformation and Displacement Gradients.................................... 126 8.5 Deformation Tensors and Finite Strain Tensors............................ 127 8.6 Linear Rotation Tensor and Rotation Vector in Relation to Relative Displacement���������������������������������������������������������������������� 130 9. Tensors in Geology..................................................................................... 133 9.1 Introduction........................................................................................ 133 9.2 Equation for the Determination of Shearing Stresses on Any Plane Surface����������������������������������������������������������������������� 135

viii

Contents

9.3 9.4

 eneral Transformation and Maximum and Minimum G Longitudinal Strains������������������������������������������������������������������������� 137 Determination of the Two Principal Strains in a Plane................ 140

10. Tensors in Fluid Dynamics....................................................................... 143 10.1 Introduction........................................................................................ 143 10.2 Equations of Motion for Newtonian Fluid..................................... 143 10.3 Navier–Stokes Equations for the Motion of Viscous Fluids........ 145 Appendix.............................................................................................................. 149 Remarks................................................................................................................ 153 Bibliography......................................................................................................... 155 Index...................................................................................................................... 157

Preface There is a great demand from students for a book on tensors with simple and conceivable presentations. The theoretical development of the subject “tensor calculus” is critical to be understood by students because of its complex nature of uses of the subscripts and superscripts. The simplification in the working process with repeated/nonrepeated indices of a mixed tensor makes it rather more complex for readers if some special clues are not mentioned. Moreover, the fields of application, namely, non-isotropic media and the exact situation of deformation of bodies, cannot be identified easily in the true sense. Through concrete citation of applicable media and physical bodies, the subject can be made conceivable. For example, in elastic media such as motion of viscous fluids or dirty water, the application of tensors is inevitable. Research on viscous media use of Navier–Stokes equation governed by tensors is one of the primary prerequisites. The investigation causing deformation with elasticity in physics needs application of tensors. However, a clear concept to make use of the different classes of tensors in such fields is of paramount importance. Only then correct results of investigation can be unearthed. Eventually, calculus of tensors can be considered as the most appropriate tool to know the physical field theories. Hence, applied mathematicians, physicists, engineering scientists, and geologists cannot excel without the knowledge of tensors. Emphasis is given primarily on the subject, and only to motivate the readers, some physical fields are described for real interest. During the past 37 years of teaching this subject at the MSc level, I could clearly read the minds of students on why they found tensor calculus difficult to understand. For the skillful teaching arts adopted in the class lectures, hundreds of students requested that I write a book on tensors for the benefit of students. Students’ feedback and suggestions from many colleagues inspired me to undertake this venture of writing this book. With the teaching arts based on some individual special techniques of changing the indices of tensors, I attempted to place a book on tensors in the hands of students. The complexity that arises out of the use of shorthand notations was removed so that readers can easily understand them. If the students ­capitalize the techniques provided in the book, in addition to its elegant ­presentation and simple language, my belief will become a reality. I will be highly obliged if this book renders at least some ­service to students and researchers so that they can understand its ­tremendous ­importance in research gates such as relativity, physics, continuum ­mechanics, and geology. Overall, this book is designed to cater to the needs of students of mathematics, physics, engineering, and geology from all universities. Further, I will acknowledge the readers with sincere gratitude if they point out mistakes in ix

x

Preface

the formative stage of the book on tensors, which is very difficult to publish without any printing ­mistakes. Dr. Bhaben Chandra Kalita Professor Emeritus Department of Mathematics Gauhati University Guwahati-781014 India

About the Book The book Tensor Calculus and Applications is not elementary in nature; rather, it is physically motivated in the sense of application. Theoretically, the subject “tensor calculus” is critical for students to understand the ­complex nature of using subscripts and superscripts. Besides the lack of knowledge about the fields of application in non-isotropic media and of identifying deformation situations of bodies poses rather more difficulty to earn the concepts. The elegant nature of description of the theory with specific style of changing suffixes and prefixes and reasons to recover meaningful results of the subsequent fields can only make the subject easier. With this objective in mind, the author was inspired to write the book for the benefit of readers. In the opinion of the author, the old books written by L. P. Eisenhart and C. E. Weatherburn could not serve this purpose though they are of ­fundamental nature from a theoretical standpoint but not available in the market and ­conceivable at the same time. The experience derived from teaching the ­subject for more than 37 years to the postgraduate students and the psychology gathered from the feedback of students are the intense feeling of the author to write a book on tensors using special techniques. The techniques adopted in the book with directions will definitely encourage the students to read the book to develop concepts and make use of them in appropriate geometrical fields and space. For example, the curvature of space (a geometrical entity) is the manifestation of gravity, and hence, ­tensor calculus becomes the fundamental tool as discovered by Einstein for ­general theory of relativity. The book is designed to discuss the fundamental ingredients such as Riemannian tensors, which are essential to enter into the threshold of research in general theory of relativity. Tensor being an intrinsic concept independent of any referential systems, different from Newtonian mechanics, is the essence of invariance for physical laws. Besides, in nonisotropic media such as viscous fluids, elastic media, deformation of bodies similar to structural geology, uses of tensors are essential ingredients. To amplify the uses of tensors in these fields, some relevant ideas are included in the book. In this context, the preface is written to manifest its suitability and necessary background why the author has written the book. The book consists of 10 chapters. Chapter 1 is devoted to giving some ­prerequisites of the subjects. Chapter 2 deals with the fundamental concepts of quadratic forms and their properties. Chapter 3 discusses the essential concept of generating space of any dimensions and corresponding geometry like the Riemannian metric inherent in fundamental tensors. Chapter 4 is devoted to developing the subject with the use of shorthand symbols called Christoffel symbols and the important tensorial operation covariant differentiations theoretically. Chapter 5 includes the geometrical concept xi

xii

About the Book

“geodesics” primarily required for dynamical scenario. Chapter 6 discusses the curvature tensors or Riemannian tensors, the fundamental ingredients of general theory of relativity along with properties. Chapters 7–10 narrate the applications of tensors in general theory of relativity, continuum mechanics, geology, and fluid dynamics, respectively for students.

Author

Dr. Bhaben Chandra Kalita has been first class throughout his career. He has served 37 years in the Department of Mathematics, Gauhati University, in capacity of assistant and associate professors and professor and head of the department since 1978. Dr. Kalita was granted the prestigious award “Professor Emeritus” by the University Grants Commission, Government of India on September 2015. He has published more than 50 papers in Physics of Fluids, Physics of Plasmas, Astrophysics and Space Science, Journal of Plasma Physics, Physical Society of Japan, Canadian Journal of Physics, IEEE Transaction on Plasma science, Communication in Theoretical Physics, and Plasma Physics Reports, besides some other papers of relativity and graph theory. He has acted as an invited speaker on astrophysics and particle physics in Dallas (2016) and San Antonio (2017). He has presented papers in Granada (Spain), Pissa (Italy), and Swansea (UK), and acted as a speaker in many local ­universities and institutions. He has also authored several textbooks on advanced ­mathematics and reference books of higher secondary level. Recently, he has served as ‘Keynote Speaker’ in Astrophysics and Particle Physics conference (2018) held in Chicago, Illinois, USA.

xiii

Part I

Formalism of Tensor Calculus

1 Prerequisites for Tensors

1.1 Ideas of Coordinate Systems Geometric ideas and entities can be well defined in various forms with ­reference to coordinate systems. In most of the cases for convenience, rectangular coordinate system is taken into account, but it is not applicable in all fields of physical system. Deviating from it, we may think of curvilinear coordinate system in the lowest level. To consider the geometry of space for dynamical scenario as “dynamics deals with the geometry of motion,” coordinate system should be suitably selected. For an essential entity in this sense, the author is tempted to give a brief idea of curvilinear coordinate system.

1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity) Consider the rectangular Cartesian coordinates (x, y, z) of any point P with  position vector r . For the coordinates (x, y, z) of P, a correspondence can be made with (u1, u2, u3) as

x = x(u1 , u2 , u3 ), y = y(u1 , u2 , u3 ), z = z(u1 , u2 , u3 ).

If these functions are single valued and have continuous partial derivatives, they can be solved as

u1 = u1  ( x , y , z ) , u2 = u2  ( x , y , z ) , u3 = u3 ( x , y , z ) .

Here, u1, u2, u3 are called the curvilinear coordinates of the point P. Consequently, the position vector r = ix + jy + kz can be expressed as r = r (u1, u2, u3).

3

4

Tensor Calculus and Applications

n^3

u2

^t3

^t2

n^2 ^t1

P

n^1

u3

u1

FIGURE 1.1 Vector reprersentations in Curvilinear system.

∴ A unit tangent vector tˆ1 to the curve u1 (i.e., the curve of intersection of u2 = c2, u3 = c3) at P (Figure 1.1) is

∂r ∂ u1 tˆ1 = . ∂r ∂u1

Similarly, the unit tangent vectors tˆ2 and tˆ3 along u2 and u3 curves, respectively, can be written as

∂r ∂r ∂u3 ∂u2 tˆ2 = and tˆ3 = ∂r ∂r ∂u2 ∂u3

Again the normal vectors to the surfaces u1 = c1 , u2 = c2 , u3 = c3 are given by the vectors ∇u1 , ∇u2 , ∇u3 , respectively. Hence, the unit normal vectors nˆ 1 , nˆ 2 , nˆ 3 to the direction of the vectors are

∇u1 ∇u2 ∇u3 , nˆ 2 = , nˆ 3 = . nˆ 1 = ∇ u1 ∇ u2 ∇ u3

Thus, at each point P of a curvilinear coordinate system, there exist two sets of unit vectors: (i) tˆ1 , tˆ2 , tˆ3 tangent to the coordinate curves and (ii) nˆ 1 , nˆ 2 , nˆ 3 normal to the coordinate surfaces. Of course, the two sets become identical if and only if the curvilinear coordinate system is orthogonal. In this case, the sets are similar to i, j, k of rectangular coordinate system but differ in the sense of changing directions  from point to point. Eventually, any vector A can be expressed in terms of the base vectors tˆi and nˆ j (i, j = 1, 2, 3) as  A = A1tˆ1 + A2 tˆ2 + A3tˆ3

    ∂ r (1.2.1) = a1 α 1 + a2 α 2 + a3 α 3 , where α j = . ∂u j

5

Prerequisites for Tensors

X2

S

P

N A O

R

M

X1

FIGURE 1.2 Vector representations in oblique Cartesian system.

These are not necessarily unit tangent base vectors. Also,  A = B1nˆ 1 + B2 nˆ 2 + B3 nˆ 3 (1.2.2)     = b1 β 1 + b 2 β 2 + b3 β 3 , where β i = ∇ui . These are not unit normal vectors. The quantities a1 , a2 , a3 are called the contravariant components, and  b1 , b2 , b3 are called the covariant components of the same vector A . Thus, based on the representative “base vectors,” a vector can have ­contravariant or covariant components. To illustrate this, let us consider the oblique Cartesian coordinate lines X1 and X2 (not rectangular) in two dimensions in a plane.   Let us consider the components OM and ON of any vector A = OP 1 2 ­measured parallel to the coordinate lines  OX and OX . They are called the contravariant components of the vector A (Figure 1.2). 2 Consider the perpendicular projections PR on OX1 and PSon  OX . OR and OS are called the covariant components of the same vector A . Obviously, if the coordinate lines are perpendicular, then there will not be any distinction between the contravariant and covariant components. Depending upon the parallel and perpendicular projections from the point P upon the coordinate axes, we assume the coordinates of P as (x1, x2) and (x1, x2), respectively. The contravariant components xi (i = 1, 2) of the vector A and its covariant components xi (i = 1, 2) are connected by the relations

x1 = x 1 + x 2 cos α x2 = x 2 + x 1 cos α

where α is the inclination between the coordinate axes OX1 and OX2.

 x1   1 ∴  =  x 2   cos α

cos α   x 1   2  1  x 

(1.2.3)

6

Tensor Calculus and Applications

  If e i and e j are the unit vectors stipulated along the coordinate axes, then   1−  A = x e1 + x 2 − e2 = x i ei so that the length OP is 2

A = A⋅A =

( )

= x1



2

(x

1−

e1 + x 2 − e2

)

2

    e1 ⋅ e1 + 2 x 1 x 2 e1 ⋅ e 2 + x 2

( )

2

  e2 ⋅ e2  .

= g ij x i x j Defining    1 g ij = ei ⋅ e j =   cos α



cos α   = g ji 1

∴ g ij = sin 2 α .

∴ The inverse of g ij =

1  1 sin 2 α  − cos α

− cos α    . 1 

∴ If we denote the inverse of g ij as g ij , then g ij =

1  1 sin 2 α  − cos α

− cos α    . 1

∴ Equation (1.2.3) can be written as xi = g ij x j Otherwise, x i = g ij x j  1 so that g ij g jk =   0

0  , and hence 1  2

A = g ij x i x j

= xi x i = x i g ij x j = g ij x i x j .

Definition Basis: A non-empty subset S = {α 1 , α 2 ,  , α n } of a vector space V(F) is said to be its basis if i. S is linearly independent of V.  ii. S generates V, i.e., if every vector α ∈V is expressible in terms of the basis set {α i }. But S is not unique for if S = {α 1 , α 2 , , α n } is the

7

Prerequisites for Tensors

basis, then {cα 1 , α 2 , … , α n } is also the basis. For classical example,  any vector d in three dimensions can be expressed in terms of three     noncoplanar vectors a, b , c as d = λ a + µb + υ c.  Symbolically, if b( i ) are three noncoplanar vectors, they can be taken as the   basis of vector a so that a = ai b( i ) , where ai ’s are the components. Definition Coordinate basis: The coordinates of a point in n-dimensional space are identified with reference to a set of axes, and every point xi (i = 1, 2,…, n) can be correlated with the position vector r (= xi eˆi ) = x1eˆ1 + x2 eˆ2 +  + xn eˆn ) where each term represents the displacement (vector) in the direction of the respective axis. The set {eˆi } is called the coordinate basis. For example, in the three ˆ ˆ ˆ is the position vector of the dimensional Euclidean space, E3 , r = ix + jy + kz ˆ ˆ ˆ point (x, y, z) with the coordinate  bases i , j , and k .  Orthogonality: Let α and β be any two vectors in an inner product  (or  dot product) space. The vector α is said to be orthogonal  to  the  vector β       if α ⋅ β (=<  α , β  >) = 0. any two vectors α i , α j (i ≠ j) of a set of vec If for     tors S = {α 1 , α 2 ,… , α n } , α i ⋅ α j = 0(= < α i , α j >), then the set is called the orthogonal set.  Orthonormal set: A set S = {αˆ 1 , αˆ 2 ,  , αˆ n } of vectors V ( F ) is said to be orthonormal if     α i ⋅ α j = 0 (= < α i , α j >), if i ≠ j = 1, if i = j . Norm: For any inner product space V, the norm (or length or magnitude)     of any vector α ∈V is defined by α = < α , α > = α ⋅ α α ≠ 0, and it is a ­nonnegative value. Multiplying each of the vectors of S by the reciprocal of its norm or length, S can be transformed to an orthonormal set. N.B.: The coordinate bases may not be orthogonal or orthonormal.

1.3 Quadratic Forms, Properties, and Classifications Definition A homogeneous second-degree polynomial in n variables (in general) x 1 , x 2 ,  , x n is called a quadratic form and is expressed as

ai j x i x j , (1.3.1)

8

Tensor Calculus and Applications

with double sum i, j = 1, 2,  , n. The form is said to be real if all the coefficients ai j are real, and it is called nonsingular if the corresponding determinant a11 a12  a1n aij = a21 a22  a2 n ≠ 0,



an1 an2  ann Otherwise, it is called singular. Further, the rank of the square matrix ( ai j ) represents the rank of the ­quadratic form. Again with the help of a nonsingular linear transformation x i = α ij y j , the quadratic form can be reduced to the form cij y i y j (i, j = 1, 2,  , n) in terms of the variables y i subject to the well-established result: rank of the matrix ( aij ) = rank of the matrix (cij ). Theorem If r is the rank of a real quadratic form (cij ) y i y j , there exists a nonsingular linear transformation of the variables which can reduce it to the form cr ( x r )2 where none of cr ’s is zero. The quadratic form can be expressed as c1 ( x 1 )2 + c2 ( x 2 )2 +  + c r ( x r )2 . (1.3.2)



Here, cr can be a positive or negative nonzero constant. Signature: The difference between the number of positive and negative coefficients or the excess of positive coefficients over the negative coefficients of (1.3.2) is called the signature of the real quadratic forms. Of course, according to the “Sylvester’s law of inertia,” the number of ­positive coefficients is invariant (remains unchanged). Normal form: If the real quadratic form cr ( x r )2 is transformable by means of a nonsingular linear transformation to a form in which the coefficients cr ’s take up values +1 and −1, it is called the normal form of it. If p is the number of positive coefficients (+1), q is the number of negative  coefficients (−1), and S is the signature, then the normal form can be written as

(x ) + (x ) 1 2

2 2

( ) − (x ) − (x )

+ + xp

2

where p + q = r and p − q = s, so that p =

p+1 2

1 (r + s). 2

p+2 2

( )

−  − xr

2

, (1.3.3)

9

Prerequisites for Tensors

Definite and indefinite quadratic forms: If all the signs of the quadratic normal form (1.3.3) are the same (positive or negative) or different, then the quadratic form is called definite or indefinite. Positive definite and negative definite quadratic forms: If all the signs of the normal quadratic form (1.3.3) are positive, it is called positive definite, and if all of them are negative, it is called negative definite.

1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials Definition A second-degree homogeneous polynomial of the differentials dxi of the variables x 1 , x 2 ,  , x n is called a quadratic differential form, e.g., Q( a) = aij dx i dx j is a quadratic differential form where aij’s are the functions of x i’s or may be constant. A quadratic differential form is of paramount importance, which will be demonstrated in the next few chapters. Considering n independent functions y i = y i ( x 1 , x 2 ,  , x n ) instead of the variables xi, we can get a homogeneous linear transformation of differentials: dy i =



This can also be written as dx i =

∂y i j dx . (1.4.1) ∂x j

∂xi dy j, since for nonsingular linear trans∂y j

formation of quadratic forms of the same rank



∂y i ≠ 0. ∂x j

∴ The quadratic differential form aij dx i dx j can be transformed to Q(b) = bij dy i dy j .

(1.4.2)

Noticeably at a given point or for a given x i , the relation (1.4.1) stands for a linear transformation of differentials with constant coefficient. Hence, it is analogous to quadratic forms discussed in Section 1.3. Contextually, (1.4.2) can also be reduced to the similar normal form:

( dy ) + ( dy ) 1 2

2 2

( ) − ( dy ) − ( dy )

+  + dy p

2

p+1 2

p+2 2

( )

−  − dy r

2

, (1.4.3)

where r is the rank of (bij). The notions called positive definite, negative definite, and signature of the normal form of this quadratic differential are exactly similar to those as ­discussed in Section 1.3.

10

Tensor Calculus and Applications

For simplicity, let us consider the position vector r of a point P (as in Figure  1.1) in curvilinear coordinates u1 , u2 , u3 as r = r (u1 , u2 , u3 ) in three dimensions (Figure 1.2). ∴ dr =



∂r ∂r ∂r du1 + du2 + du3. ∂u1 ∂u2 ∂u3

If ds is the element (length) between the adjacent points P( r ) and Q( r + dr ), then (in the limit) dr ⋅ dr = ds2 ∂r ∂r ∂r ∂r  ∂r   ∂r  du1 + du2 + du3  .  du1 + du2 + du3  =  ∂u1   ∂u1  ∂u2 ∂u3 ∂u2 ∂u3 2

 ∂r ∂r   ∂r   ∂r ∂r  du21 + 2  du1 du2 + 2  du1 du3 = . .  ∂u1   ∂u1 ∂u2   ∂u1 ∂u3 

2

2

 ∂r ∂r   ∂r   ∂r  du22 + 2  du2 du3 +  du32 + .    ∂u2   ∂u2 ∂u3   ∂u3  = a11 du12 + 2 a12 du1 du2 + 2 a13 du1 du3 + a22 du22 + 2 a23 du2 du3 + a33 du32 2

∴ ds =

3

3

i=1

j=1

∑ ∑a

ij

dui du j ,

where

aij =

∂r ∂r ≠0 . ∂ui ∂u j

and

ds2 = aij dui du j (1.4.4)

by dropping summation sign for summation convention (defined in Section 2.1). This is the quadratic differential form representing the elementary distance between two adjacent points in the oblique curvilinear coordinate system. This is called the metric or line element of the system. Of course, in oblique curvilinear coordinates (u, v , w), (1.4.4) can be written in explicit form as

ds2 = adu2 + bdv 2 + cdw 2 + 2 h dudv + 2 gdudw + 2 fdvdw.

11

Prerequisites for Tensors

But in rectangular Cartesian coordinates of Euclidean space of three dimensions, the elementary distance between the adjacent points ( x , y , z) and ( x + dx , y + dy , z + dz) is given by

ds2 = dx 2 + dy 2 + dz 2,

where a ij =

∂r ∂r . = 0 for i ≠ j ∂ui ∂u j

= 1 for i = j u1 = x , u2 = y , u3 = z .

To include almost all physical spaces, Riemann has generalized this concept (notion) to n dimensions. This will be discussed in detail in Chapter 3.

Exercises 1. If the quadratic form aijxixj transforms to a quadratic form bijyiyj, write down the corresponding form of nonsingular linear transformation of it and what is the rank of |bij| if the rank of |aij| is 5 under suitable transformation in the range of i and j. 2. If the quadratic form aijxixj (i, j = 1, 2, …, n) reduces to the form bi ( y i )2 , bi ≠ 0 ∀ i by means of a nonsingular linear transformation, what are the nonzero values of b’s if r ( 0, g < 0). ∂x j

(

)

Proof of (i) ij , k

i.

+

jk , i

=

1  ∂ g ik ∂ g jk ∂ g ij  1  ∂ g ji ∂ g ki ∂ g jk  + + − + − 2  ∂ x j ∂ x i ∂ x k  2  ∂ x k ∂ x j ∂ x i 

=

∂g 1 × 2 ikj , g ij = g ji , etc. 2 ∂x

=

∂ g ik . ∂x j

Hence, proved. Proof of (ii) Differentiating partially the elements of the determinant g = g ij row wise with respect to x j , and expanding each of the n determinants in terms of the elements of that particular differentiated row, it can be precisely written as a whole: ∂ g ∂ g ik ki = G , ∂x j ∂x j



where Gki are the cofactors of gik in g (i, k = 1, 2,…, n). ∂ g ik g g ik ∂x j



=



1 ∂g = g ik  g ∂x j

ij , k

+

jk , i

 , ( g ≠ 0) 

using the result (i) above. = g ik

=

i ij

ij , k

+

+ g ki

k jk ( k → i)

=2

jk , i i ij

g ik = g ki

43

Christoffel Three-Index Symbols

i ij

=

1 ∂g ∂ = log g j 2 g ∂x ∂x j

=

∂ log − g ∂x j



(

(

)

)

if g > 0

if g < 0.

Hence, proved.

4.3 Evolutionary Basis of Christoffel Symbols (Brackets) In the Cartesian coordinate system, the law of parallel displacement takes the form: ai , k δ x k = 0

i.e.,

∂ai δ xk = 0  ∂x k

α ik

=0

,

(4.3.1)

where δxk represents the infinitesimal displacement. Let us transform the vector components ai to a new coordinate system (ξi). Then, we have ai = ∴

∂ξ r ar′ ∂xi

∂ ai ∂  ∂ξ r  ∂ξ s = s  i ar′  k k ∂x ∂ξ  ∂ x  ∂ x = ∴0 =

∂ξ s ∂ξ r ∂ ar′ ∂ξ s ∂ x l ∂2 ξ r ar′ + ∂ x k ∂ x i ∂ξ s ∂ x k ∂ξ s ∂ x l ∂ x i

(4.3.2)

 ∂ξ r ∂ a ′ ∂ x l ∂2 ξ r  ∂ξ s ∂ ai δ x k =  i rs + s l i ar′  k δ x k k ∂x ∂ξ ∂ x ∂ x  ∂ x  ∂ x ∂ξ

  ∂ξ r ∂ a ′ ∂ x l ∂2 ξ r ∴ 0 = ai , kδ x k =  i rs + s l i ar′  δξ s ∂ξ ∂ x ∂ x   ∂ x ∂ξ

 ∂ξ s k  s δξ = δx .   ∂x k 

∂ ar′ s δξ is the actual increment of ar′ as a result of the displacement, and ∂ξ s it is denoted by δ ar′ .

(δ ar′ = )

44

Tensor Calculus and Applications

Multiplying the R.H.S. of (4.3.2) by

δ lrδ ar′ = −

∂xi , we get ∂ξ l

∂ xl ∂ x i ∂2 ξ r ar′δξ s ∂ξ s ∂ξ l ∂ x l ∂ x i

(4.3.3) ∂ xl ∂x i ∂2 ξ r s ⇒ δ al′ = − s l i l ar′δξ . ∂ξ ∂ξ ∂ x ∂ x



When no Cartesian coordinate system can be introduced, we shall retain the linear form of Equation (4.3.3) and assume that, because of a parallel displacement, the infinitesimal changes of the vector components are bilinear functions of the vector components, and the components of the infinitesimal displacement are defined by l



δ a i = − Γ i a kδξ l. (4.3.4)



δ ak = + Γ i aiδξ l . (4.3.5)

kl

ll

kl

l

ll

kl

kl

The coefficients Γ i and Γ i of these new tentative laws are, so far, entirely unknown quantities. But we can determine their transformation laws, δ a′ k as the difference between two vectors at two points, characterized by the coordinate values ξ1 and ξ1 + δξ1. In case of a coordinate transformation, the new δ a′ k is given by  ∂ξ ′ k   ∂ξ ′ k s   k ∂ξ ′ k s  − δ a′ k =  s a s  a a′ = a ∂ξ s   ∂ξ  ξ l +δξ l  ∂ξ s  ξ l  =

∂  ∂ξ ′ k s  l a δξ ∂ξ l  ∂ξ s 

=

∂ ξ′ ∂ξ ′ s l a sδξ l + a ,l δξ ∂ξ l ∂ξ s ∂ξ s



δ a′ k =

2



k

k

a s, l =

(4.3.6)

∂a ∂ξ l

s

∂2 ξ ′ k s l ∂ξ ′ k s a δξ + δa . ∂ξ l ∂ξ s ∂ξ s

Using (4.3.6) into the L.H.S. of the following equation corresponding to (4.3.4),

I

δ a′ k = Γ ′ k a′ mδξ ′ n mn

45

Christoffel Three-Index Symbols

and replacing a′ m and δξ ′ n by the expressions δξ ′ m s δξ ′ n l n = and δξ δξ , we get a′ m = a ′ δξ l ∂ξ s I ∂2 ξ ′ k s l ∂ξ ′ k s δξ ′ m s δξ ′ n l a δξ + δ a = − Γ′ k a δξ . l s s mn ∂ξ s δξ l ∂ξ ∂ξ ∂ξ



Substituting the value of δ a s from Equation (4.3.4) into it, we get m n I ∂2 ξ ′ k s l ∂ξ ′ k Is m n k ∂ξ ′ s ∂ξ ′ δξ δξ δξ l a a a − Γ = − Γ ′ mn mn ∂ξ s ∂ξ l ∂ξ s ∂ξ s ( m→ s)( n→ l ) ∂ξ l

( s→ r )



m n I  ∂2 ξ ′ k ∂ξ ′ k I r  s l k ∂ξ ′ ∂ξ ′ δξ = − Γ ∴ l s − a a sδξ l . Γ ′ mn ∂ξ s ∂ξ l ∂ξ r sl   ∂ξ ∂ξ

∂ξ ′ m ∂ξ ′ n I k ∂ξ ′ k I r ∂2 ξ ′ k Γ′ = Γ − , since a s and δξ l are arbitrary. ∂ξ s ∂ξ l mn ∂ξ r sl ∂ξ l ∂ξ s I ∂ξ s ∂ξ l r Multiplying by , we get the transformation formula Γ as sl ∂ξ ′ a ∂ξ ′ b

Hence,

l

Γ′ k =



ab

∂ξ s ∂ξ l  ∂ξ ′ k l r ∂2 ξ ′ k  Γ − . (4.3.7) ∂ξ ′ a ∂ξ ′ b  ∂ξ r sl ∂ξ l ∂ξ s 

The last term of the R.H.S. of (4.3.7) can be written as −

∂ξ s ∂ξ l ∂2 ξ ′ k ∂ξ s ∂  ∂ξ l ∂ξ ′ k  ∂ξ s ∂ξ ′ k ∂  ∂ξ l  + =− a b l s ∂ξ ′ ∂ξ ′ ∂ξ ∂ξ ∂ξ ′ a ∂ξ s  ∂ξ ′ b ∂ξ l  ∂ξ ′ a δξ l ∂ξ s  ∂ξ ′ b  =



∂ξ s ∂ ∂ξ ′ k ∂2 ξ l δ b′ k + a s ∂ξ ′ ∂ξ ∂ξ l ∂ξ ′ a ∂ξ ′ b

= 0+

( )

∂ξ ′ k ∂2 ξ l . ∂ξ l ∂ξ ′ a ∂ξ ′ b

( δ ba is constant)

∴−

∂ξ s ∂ξ 1 ∂2 ξ ′ k ∂ξ ′ k ∂2 ξ 1 = . ∂ξ ′ a ∂ξ ′ b ∂ξ l ∂ξ s ∂ξ l ∂ξ ′ a ∂ξ ′ b

46

Tensor Calculus and Applications

Therefore, Equation (4.3.7) becomes I

Γ′ k = ab

∂ξ s ∂ξ l ∂ξ ′ k I r ∂ξ ′ k ∂2 ξ l Γ + ∂ξ ′ a ∂ξ ′ b ∂ξ r sl ∂ξ l ∂ξ ′ a ∂ξ ′ b ( s→ r ) (l→ s ) ( r → l )



2 l k r s I   ∂ ξ ξ ξ ξ ∂ ∂ ∂ ′ ∴ Γ′ k = . Γl + ab ∂ξ l  ∂ξ ′ a ∂ξ ′ b rs ∂ξ ′ a ∂ξ ′ b 

(4.3.8)

I

II

I

ab

ab

Similarly, we can obtain the transformation law for Γ k . It is identical with Γ k . The transformation law consists of two terms. The first term depends on the I

Γ k in the old coordinate system, and the second term does not depend on ab I k

Γ and adds an expression which is symmetric in two subscripts. So, even ab

I

though the Γ k may vanish in one coordinate system, they do not vanish in ab

I

other systems. But if the Γ k were symmetric in their subscripts in one coorab

dinate system, they would be symmetric in every other coordinate system as well. I

This would be particularly true if the Γ k were to vanish in one system. I

k

II k

ab

ab

ab

Besides, if Γ were equal to the Γ in one coordinate system, this equality would be preserved by arbitrary coordinate transformations. We shall find I

that geometrical considerations of systems in which Γ k satisfies both these ab

conditions. Let us displace two vectors ai and b i parallel to themselves along an infinitesimal path δξ i. The change of their scalar product, ai b i , is given by

δ ( aib i ) = aiδ b i + b iδ ai II  I  = ai  − Γ i b kδξ l  + b i Γ k akδξ l il  kl  ( i↔ k )



I

II

= −Γ aib δξ + Γ aib δξ i

k

l

kl

i

k

l



(4.3.9a)

kl

 II I  = aib k  Γ i − Γ i  δξ l .  kl kl  When two vectors are displaced parallel to themselves, their scalar product I

II

kl

kl

always remains constant if and only if Γ i are equal to Γ i .

47

Christoffel Three-Index Symbols

If the law of parallel displacement of Equations (4.3.4) and (4.3.5) is extended from vectors to tensors, we can displace any tensor parallel to itself according to the rule: tikl = albi c k ( tensor of rank 3 ). (4.3.9b)



δ tikl = δ albi ck + alδ bi ck + albiδ ck

I

II

II

rs

is

ks

= − Γ l a r bi c kδξ s + Γ r br al c kδξ s + Γ r cr albiδξ s I II  II  =  Γ ris trkl + Γ rks tril − Γ lrs tikr  δξ s ,  

making use of (4.3.9b)

I II  II  ∴ δ tikl =  Γ ris trkl + Γ rks tril − Γ lrs tikt  δξ s . (4.3.10)  

Similarly, applying the law (4.3.10) to the parallel displacement of the Kronecker tensor, we get

I  II  δ lk = a k ai ∴ δ δ lk =  Γ ris − Γ ris  δξ s. (4.3.11a)  

( )

For

( ) (

δ δ ik = δ a k ai

)

 δ ik = a k ai

= δ a k a i + a k δ ai I

II

= −Γ rsk a r aiδξ s + a k Γ ris arδξ s II   I =  − Γ rsk δ ir + Γ ris δ rk  δξ s  



(4.3.11b)

I  II  =  Γ isk − Γ isk  δξ s   I  II  ∴ δ δ ik =  Γ isk − Γ isk  δξ s .  

( )

Now, we apply Equation (4.3.11a) to the parallel displacement of the product a iδ ik so that

48

Tensor Calculus and Applications

(

)

( ) = δ a + a δ (δ ) = δ a + a δ (δ ) ⇒ a δ (δ ) = 0 ⇒ δ (δ ) = 0 ( a

δ a iδ ik = δ ikδ a i + a iδ δ ik

∴ δ ak



k

i

k i

k

i

k i

i

k i

k i

i

)

is arbitrary .

I  II  Now (4.3.11b) reduces to  Γ isk − Γ isk  δξ s = 0   II

I

(δξ

∴ Γ isk = Γ isk



s

)

is arbitrary . I

Therefore, we shall omit the distinguishing marks I and II. The Γ isk are symmetric in their subscripts if it is possible to introduce a coordinate system in which they vanish at least locally. Henceforth, we shall consider only I

I

symmetric Γ isk . The Γ isk still to a high degree is arbitrary. They are, however, uniquely determined, if we connect them with the metric tensor gik by the  following condition. The result of the parallel displacement of a vector a shall not depend on whether we apply the law of parallel displacement to its contravariant or covariant representation. The two representations of a i and ak , the components a i + δ a i and ( ak + δ a k ) at the point ξ s + δξ s , respectively, where δ a i and δ a k are given by Equations (4.3.4) and (4.3.5). These two vectors are again the representations of the same vector ( ak + δ ak ) at the point ξ s + δξ s , expressed equivalently by the equation:

(

(

)

(

)

)



(

)

ak + δ ak = ( g ik + δ g ik ) a i + δ a i , (4.3.12)

where δ g ik is

δ g ik = gik , lδξ l .

Equation (4.3.12) must be satisfied up to linear terms in the differentials and for arbitrary ai and δξ s . If we multiply the R.H.S. of (4.3.12), we obtain

δ ak = g ikδ a i + δ g ik a i = g ikδ a i + g ik , lδξ i a i .

Substituting δ a i and δ a k from Equations (4.3.4) and (4.3.5), we get

49

Christoffel Three-Index Symbols

Γ ikl aiδξ l = − g ik Γ isl a sδξ l + g ik , lδξ l a i

(i → s)

g is a s Γ ikl δξ l + g ik Γ isl a sδξ l − g sk , l a sδξ l = 0

 ai = g is a s (4.3.13)

⇒ a sδξ l  g is Γ ikl + g ik Γ isl − g sk , i  = 0. Since a s and δξ l are arbitrary, the contents of the bracket must vanish.

4.4 Use of Symmetry Condition for the Ultimate Result We make use of the symmetry condition and write down the vanishing bracket three times with different index combinations:

Γ rik g rs + Γ rsk g ir − g is , k = 0 (i)



Γ rki g rs + Γ rsi g rk − g ks , i = 0 (ii)



Γ ris g rk + Γ rsk g ir − g ik , s = 0 (iii)

Now, (i) + (ii) − (iii) gives − g is , k − g ks , i + g ik , s + Γ rik g rs + Γ ikr g rs = 0

⇒ g is , k + g ks , i − g ik , s = 2 Γ rik g rs ⇒

1  g is , k + g ks , i − g ik , s  = Γ rik g rs . 2

(4.4.1)

Multiplying (4.4.1) by gsl, we get

δ rl

r ik

=



l ik

1 ls g  g is , k + g ks , i − g ik , s  2 

(4.4.2) 1 ls = g  g is , k + g ks , i − g ik , s  . 2

This expression is usually referred to as the Christoffel three-index symbol l  l  of the second kind, and it is denoted by   or ik . ik  

 l  1 ls   = g { g is , k + g ks , i − g ik , s } =  ik  2

l ik

.

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Tensor Calculus and Applications

The L.H.S. of Equation (4.4.1) is called the Christoffel symbol of the first kind. It is denoted by the sign [ik, s]:

1

[ ik , s ] = 2 g is  gis, k + g ks, i − gik , s  =

ik , s

.

In case of Cartesian coordinates, both kinds of Christoffel three-index symbols vanish, since gij = constant. Mathematically, the Christoffel symbols may be developed in this way from the concept of parallel displacement.

4.5 Coordinate Transformations of Christoffel Symbols 4.5.1 Transformation of the First Kind

ij, k

Let g ij and g ij′ be the fundamental tensors of the coordinate systems x i and x′ i, respectively. ∴ By tensor law of transformation,

g ij′ = g ab

∂x a ∂xb ∂ x′ i ∂ x′ j

i

j

k

a

b

c

.

Differentiating partially both sides with respect to x′ k , ∂ g ij′  ∂ g ab ∂ x c  ∂ x a ∂ x b ∂2 x a ∂ xb . = + g ab j ∂ x′ k  ∂ x c ∂ x′ k  ∂ x′ i ∂ x′ ∂ x′ i ∂ x′ k ∂ x′ j



∂ x a ∂2 xb . + g ab ∂ x′ j ∂ x′ j ∂ x′ k

(4.5.1)

(b ↔ a )

Also from g ′jk = gbc

∂xb ∂x c ∂ x′ j ∂ x′ k

∂ g ′jk  ∂ gbc ∂ x a  ∂ x b ∂ x c ∂2 xb ∂ x c g (4.5.2) = + ⋅ bc ∂ x′ i  ∂ x a ∂ x′ i  ∂ x′ j ∂ x′ k ∂ x′ j ∂ x′ i ∂ x′ k (b→ a , c → b )

+ gbc

∂ xb ∂2 x c . ∂ x′ j ∂ x′ k ∂ x′ i

( c → a )′′

(differentiating partially both sides with respect to x′ i)

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Christoffel Three-Index Symbols

Similarly, from g ′ki = g ca

∂xc ∂x a ∂ x′ k ∂ x′ i

   (4.5.3) ∂ g ′ki  ∂ g ca ∂ x b  ∂ x c ∂ x a ∂2 x c ∂ x a ∂ x c ∂2 x a . g g = + + ca ca ∂ x′ j  ∂ x b ∂ x′ j  ∂ x′ k ∂ x′ i ∂ x′ k ∂ x′ j ∂ x′ i ∂ x′ k ∂ x′ i ∂ x′ j ( c → a , a→ b )

( c → b )′′

(differentiating partially both sides with respect to x′ j ) (The indices are changed so as to convert all second-order derivatives in ∂2 x a terms of xa like and indices of g to gab.) ∂ x′ i ∂ x′ j (4.5.3) + (4.5.2) − (4.5.1) gives  ∂ g ′ki ∂ g jk ∂ g ij   ∂ x′ j + ∂ x i − ∂ x′ k 

∂g ∂ g  ∂x a ∂xb ∂x c ∂2 x a ∂ xb  ∂g =  cab + bca − abc  2 g + ab  ∂x ∂x ∂ x′ i ∂ x′ j ∂ x′ k ∂ x  ∂ x′ i ∂ x′ j ∂ x′ k ′ ij , k =

ab , c

∂2 x a ∂ xb ∂x a ∂xb ∂x c + g . ab ∂ x′ i ∂ x′ j ∂ x′ k ∂ x′ i ∂ x′ j ∂ x′ k



(4.5.4)

This is the transformation law of the Christoffel symbol of the first kind. Clearly, due to the presence of the second term in (4.5.4), it is not the transformation law of a tensor; hence, the Christoffel bracket of the first kind ij , k is not a tensor in general. But, if we consider a linear transformation of the type

x a = Aia x′ i + Ba

( a = 1, 2, , n),

∂x a ∂xb ∂x c ∂2 x a ′ ij , k = , which is the transab , c j = 0 for which i ∂ x′ i ∂ x′ j ∂ x′ k ∂ x′ ∂ x′ formation law of a third rank covariant tensor and subject to the linear ­transformation; Christoffel symbol (bracket) of the first kind is also a tensor. then

4.5.2 Transformation of the Second Kind

i jk

From the tensor law of transformation of the second fundamental tensor, we can write

g ′ pk = gαβ

∂ x′ P ∂ x′ k ∂ xα ∂ x β

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Tensor Calculus and Applications

Multiplying (4.5.4) by this relation, we can get



g ′ pk ′ ij , k = gαβ

ab , c

∂ x a ∂ x b ∂ x c ∂ x′ p ∂ x′ k ∂ 2 x a ∂ x b ∂ x′ p ∂ x′ k αβ + g g ab ∂ x ′ i ∂ x ′ j ∂ x ′ k ∂ xα ∂ x β ∂ x ′ i ∂ x ′ j ∂ x ′ k ∂ xα ∂ x β

= gαβ

ab , c

p ∂ x a ∂ x b c ∂ x′ p ∂2 x a b ∂ x′ αβ j δβ j δβ i i α + g ab g α ∂ x′ ∂ x′ ∂x ∂ x′ ∂ x′ ∂x

= gα c

ab , c

∂ x a ∂ x b ∂ x′ p ∂2 x a ∂ x′ p αb j 1⋅ j 1⋅ i i α + g ab g ∂ x′ ∂ x′ ∂x ∂ x′ ∂ x′ ∂ xα

p ∴ ij ′ =

α ab

′P = ij

∂ x a ∂ x b ∂ x′ p ∂ 2 x a ∂ x′ p α δ + a ∂ x ′ i ∂ x ′ j ∂ xα ∂ x ′ i ∂ x ′ j ∂ xα ∂ 2 xα ∂ x ′ p ∂ x a ∂ x b ∂ x′ p + , ∂ x ′ i ∂ x ′ j ∂ xα ∂ x ′ i ∂ x ′ j ∂ xα

α ab



(4.5.5)

P

which is the transformation law of the Christoffel symbol ij of the second kind, and it is also not the tensor due to the presence of the second term in general. For the linear transformation of the type, xα = Aiα x′ i + β α ,



∂ 2 xα = 0. ∂ x′ i ∂ x′ j

From (4.5.5), we get p a b ′ P = α ∂ x ∂ x ∂ x′ , which is the transformation law of a third rank ij ab ∂ x ′ i ∂ x ′ j ∂ xα mixed tensor. Hence, in this sense, the Christoffel symbol of the second kind is also a tensor. α a b P ∂x α ∂x ∂x ∂ 2 xα = + Note: (4.5.5) gives ij ′ , (Corollary 1) ab ∂ x′ p ∂ x′ i ∂ x′ j ∂ x′ i ∂ x′ j which will be of immense use in subsequent development. Example 1 If the metric in a Vn is such that g ij = 0 for i ≠ j, show that for all unequal i, j, k i

a. jk = 0. i

b. ij =

1 ∂ g ii . 2 g ii ∂ x j

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Christoffel Three-Index Symbols

i

c. jj = − i

d. ii =

1 ∂ g jj . 2 g ii ∂ x i

1 ∂ g ii . 2 g ii ∂ x i

Proof: Given g ij = 0 for i ≠ j in a space Vn; g ii ≠ 0, i.e., g ii exists. Therefore, l g ii = also exists. g ii By definition, i

iα a. jk = g

jk , α

= g ii

jk , i

=

1 2 g ii

 ∂ g ji ∂ g ki ∂ g jk  −  k + =0 ∂x j ∂x i   ∂x

 g ij = 0 for i ≠ j . i

iα b. ij = g

=

ij , α

ij , i

=

1 ∂ g ii ( i ≠ j) 2 g ii ∂ x j

i

iα c. jj = g

=

= g ii

jj , α

= g ii

1 2 g ii

 ∂ g ii ∂ g ji ∂ g ij   ∂x j + ∂xi − ∂xi   

( with two distinct indices ).

jj , i

1  ∂ g ji ∂ g ji ∂ g jj  + − ∂ x i  2 g ii  ∂ x j ∂ x j

=−

1 ∂ g jj ( with twodistinct indices ). 2 g ii ∂ x i

i

iα d. ii = g

ii , α

= g ii

ii , i

=

1 ∂ g ii ( with onedistinct index ). 2 g ii ∂ x i

Hence, proved. N.B.: These four results (a)–(d) will be of great help in determining nonvanishing Christoffel symbols of the second kind in future for many investigations. Example 2 Find the nonvanishing Christoffel symbols of the second kind for the metric:

(

)

ds 2 = a 2 dθ 2 + sin 2 θ dφ 2 .

In this case, it is a metric in ν 2 with two variables θ = x 1 (say) and φ = x 2 (say) so that

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Tensor Calculus and Applications

g11 = a 2 , g12 = g 21 = 0, g 22 = a 2 sin 2 θ and



0 = a 4 sin 2 θ ≠ 0 a 2 sin 2 θ

2 g= a 0

∂ g 22 ∂ g 22 ∂ g 22 ∂ g11 = 0 ν i, = = 2 a 2 sin θ cos θ , = 0. ∂x i ∂θ ∂x1 ∂φ

a. For distinct i, j, k (i, j, k = 1, 2), i



=0

jk

b. For two distinct indices, i



ij

1

For i = 1, j = 2

12





2 21

=

= 2

For i = 2, j = 1



 g ij = 0 for i ≠ j .

21

=

1 ∂ g ii ( i ≠ j ). 2 g ii ∂ x j

1 ∂ g11 =0. 2 g11 ∂ x 2 =

1 ∂ g 22 . 2 g 22 ∂ x 1

1 × 2 a 2 sin θ cos θ = cot θ . 2 a 2 sin 2 θ

c. For (another) two distinct indices, i



jj

1





22

=−

=

−1 ∂ g jj . 2 g ii ∂ x i

1 ∂ g 22 1 = − 2 × 2 a 2 sin θ cos θ 2 g11 ∂ x 1 2a

= − sin θ cos θ −1 ∂ g11 = 0. 2 g 22 ∂ x 2 d. With only one distinct index, and



2

11

=

1 11

=

1 ∂ g11 = 0 and 2 g11 ∂ x 1

2 22

=

1 ∂ g 22 = 0. 2 g 22 ∂ x 2

Hence, the nonvanishing Christoffel symbols of the second kind are

2 21

= cot θ ,

1 22

= − sin θ cos θ .

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Christoffel Three-Index Symbols

We have already come across with the entity called tensors born out of non-isotropic medium, a set of functions which obey the transformation laws (2.2.2) and (2.2.3). It has already been mentioned that there are many fields such as viscous fluids, elasticity, structures prone to deformations, general theory of relativity, and continuum mechanics, where tensors are used. To study the nature of changes from the mathematical point of view, some variation concept parallel to the derivatives of functions in ordinary calculus needs to be developed applicable for tensors. We intend to develop this concept in Section 4.6.

4.6 Covariant Derivative of Covariant Tensor of Rank One Let us consider a covariant tensor Ai of rank one, so by tensor law of transformation from xi to xʹ i: Ai′ = Aa



∂ x a . ∂ x′ i

Differentiating partially with respect to x′ j , we can get ∂2 x a ∂ Ai′  ∂ Aa ∂ x b  ∂ x a + Aa j = j b i ∂ x′ i ∂ x′ j ∂ x′  ∂ x ∂ x′  ∂ x′ ∂x λ ∂x µ  , using Corollary 1 ∂ x′ i ∂ x′ j  (4.6.1) a ∂x λ ∂x µ ∂ Aa ∂ x a ∂ x b  ∂x a  ′ p = + Aa ij − Aa λµ ∂ x′ i ∂ x′ j ∂ x b ∂ x′ i ∂ x′ j  ∂ x′ p 

=  

 ′ p ∂x a ∂ Aa ∂ x a ∂ x b A + a  ij ∂ x p − ∂ x b ∂ x′ i ∂ x′ j ′ 

a

λµ

λ ↔ a, µ→b

 ∂A =  ba − Aλ  ∂x

ab

p ∂x a ∂xb + A′p ′ ij  ∂ x′ i ∂ x′ j

λ

(The dummy indices are changed looking to the indices of the first term.)



∂ Ai′ ′ p =  ∂ Aa − A λ ij  b j − Ap′ ∂ x′ ∂x Ai′, j = Aa , b

ab

∂x a ∂xb  ∂ x′ i ∂ x′ j

λ



∂x a ∂xb ∂ x′ i ∂ x′ j

(4.6.2)

writing

Aa , b =

∂ Aa − Aλ ∂xb

λ ab

(4.6.3)

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Tensor Calculus and Applications

It we define the xb -covariant derivative of Aa with respect to the fundamental tensor gij by means of (4.6.3) and denote it as Aa , b , the relation (4.6.2) corresponding to it represents the transformation law of a covariant tensor of rank two. The covariant derivative defined by (4.6.3) contains the first∂ Aa order ­ordinary derivative following which the symbol Aa , b is used for ∂xb “Covariant Derivative.” N.B.: The first covariant derivative of covariant vector or tensor of rank one is found to increase the rank of the tensor by one.

4.7 Covariant Derivative of Contravariant Tensor of Rank One Consider the contravariant tensor Ai of rank one. Therefore, by the transformation law of tensors, Ai =



∂xi A′ a ∂ x′ a

Differentiating partially with respect to xj b ∂ A i ∂ A′ a ∂ x i ∂ x ′ b ∂2 x i a ∂ x′ j = j + A′ j b a ∂x ∂ x′ ∂ x′ ∂ x ∂ x ∂ x′ a ∂ x′ b



=

b ∂ A′ a ∂ x i ∂ x ′ b a ∂ x′  j + A′ j  b a ∂ x′ ∂ x′ ∂ x ∂x 

=

∂ A′ a ∂ x i ∂ x ′ b + A′ a ∂ x′ b ∂ x′ a ∂ x j

ab

∂ A′ a ∂ x i ∂ x ′ b + A′ p ∂ x′ b ∂ x′ a ∂ x j

pb

=

 ∂ A′ a + A′ p =  ∂ x′ b

pb

ab

i ′ p ∂x − ∂ x′ p

i

λµ

∂xλ ∂x µ   , using Corollary 1 ∂ x′ a ∂ x′ b 

λ µ i b b ′ p ∂ x ∂ x ′ − A′ a ∂ x ′ ∂ x ∂ x p j j a ∂ x′ ∂ x ∂ x ∂ x′ ∂ x′ b

i

λµ

(4.7.1)

( p ↔ a)

i b b λ µ ′ a ∂ x ∂ x′ − A′ a ∂ x  ∂ x′ ∂ x  j j a a  ∂ x′ ∂ x ∂ x′  ∂ x ∂ x′ b 

i b ′ a  ∂ x ∂ x′ − A λ  ∂ x′ a ∂ x j

i

λµ

i

λµ

δ jµ .





∂ Ai + Aλ ∂x j

i

λj

 ∂ A′ a = + A′ p  ∂ x′ b

pb

i b ′ a  ∂ x ∂ x′ (4.7.2)  ∂ x′ a ∂ x j

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Christoffel Three-Index Symbols

Writing A,i j =



∂ Ai + Aλ ∂x j

i

λj

(4.7.3)

in the relation (4.7.2), we can get A,i j = A,′ba



∂ x i ∂ x′ b , (4.7.4) ∂ x′ a ∂ x j

which is the transformation law of a mixed tensor of rank two, and (4.7.3) is called the xj-covariant derivative of the contravariant tensor Ai of rank one with respect to the fundamental tensor gij. As it satisfies the transformation law (4.7.4) of tensors, so it must be a tensor. For the presence of the ordinary ∂ Ai derivative in the covariant derivative, defined in (4.7.3), it is symbolically ∂x j denoted by A,i j . Note 1: The consideration of covariant derivative of Ai is also found to increase its rank by one. Note 2: Due to the presence of the second term in (4.7.1), the partial ∂ Ai ­derivative of the tensor is not a tensor in general. When can it be a ∂x j ­tensor? It is left for guessing.

4.8 Covariant Derivative of Covariant Tensor of Rank Two Consider the covariant tensor Aij of rank two. Therefore, by transformation law of tensors, Aij′ = Aab



∂x a ∂xb . ∂ x′ i ∂ x′ j

Differentiating partially with respect to x′ k, we get ∂ Aij′  ∂ Aab ∂ x c  ∂ x a ∂ x b ∂2 x a ∂ xb ∂x a ∂2 xb A A = ⋅ + + ab ab ∂ x′ k  ∂ x c ∂ x′ k  ∂ x′ i ∂ x′ j ∂ x′ i ∂ x′ k ∂ x j ∂ x′ i ∂ x′ j ∂ x′ k  

=

∂ Aab ∂ x a ∂ x b ∂ x c ∂xb  ′ p ∂x a + Aab − ik j c i k ∂ x ∂ x′ ∂ x′ ∂ x′ ∂ x′ j  ∂ x′ p + Aab

∂x a   ∂ x′ i 

jk

b ′ p ∂x − ∂ x′ p

b

λµ

∂x λ ∂x µ  . ∂ x′ j ∂ x′ k 

a

λµ

∂xλ ∂x µ  ∂ x′ i ∂ x′ k 

(4.8.1)

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Tensor Calculus and Applications

(Replacing the second-order derivatives by the corresponding differences of Christoffel’s brackets of the second kind from Corollary 1, α λ µ 2 α ′ p ∂ x = α ∂ x ∂ x + ∂ x .) ij λµ p j i ∂ x′ ∂ x′ ∂ x′ ∂ x′ i ∂ x′ j ∂ Aij′ ∂ Aab ∂ x a ∂ x b ∂ x c ∂xb ∂x a = + Aab j i k k c ∂ x′ ∂ x ∂ x′ ∂ x′ ∂ x′ ∂ x′ j ∂ x′ p

ik

′p − A

a

ab λµ

(λ ↔ a, µ → c)

+ Aab

∂x a ∂xb ∂ x′ i ∂ x′ p

jk

′p − A

b

ab λµ

∂xb ∂ x′ j

∂xλ ∂x µ ∂ x′ i ∂ x′ k

∂x a ∂xλ ∂x µ ∂ x′ i ∂ x′ j ∂ x′ k ( λ ↔ b , µ → c )

(The dummy indices are to be changed looking to the indices of the factors of the first term.) =

∂ Aab ∂ x a ∂ x b ∂ x c + Apj′ ∂xc ∂x′i ∂x′ j ∂x′ k + Aip′



jk

′p − A

∂ Aij′ − Apj′ ∂x′ k

λ

aλ bc

ik

′ p − A′

ik

′p − A

λ

λ b ac

∂x a ∂xb ∂x c ∂x′i ∂x′ j ∂x′ k

∂x a ∂xb ∂x c ∂x′i ∂x′ j ∂x′ k

ip jk

′ p =  ∂ Aab − A λb  ∂xc

λ ac

− Aaλ

bc

∂x a ∂xb ∂x c  × ∂x′i ∂x′ j ∂x′ k

λ

Writing

Aab , c =

∂ Aab − Aλ b ∂xc

λ ac

− Aaλ

λ bc

,

(4.8.2)

the above relation can be thrown to the form:

Aij′ , k = Aab , c

∂x a ∂xb ∂x c ∂ x′ i ∂ x′ j ∂ x′ k

(4.8.3)

The xc -covariant derivative of Aab with respect to the fundamental tensor gij is defined by (4.8.2), and (4.8.3) shows that it is a tensor of rank three. Hence,  covariant derivative is found to increase the rank of the tensor Aab by one. ∂ Aij′ Note: From (4.8.1), it is found that the partial derivative of the tensor ∂ x′ k Aij′ is not a tensor due to the presence of the second two terms. Search when can it be a tensor?

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Christoffel Three-Index Symbols

4.9 Covariant Derivative of Contravariant Tensor of Rank Two Consider the contravariant tensor Aij of rank two. By transformation law of tensors, A ij = A′ ab



 i   a

∂xi ∂x j ∂ x′ a ∂ x′ b

j

k

b

c

.

Differentiating partially with respect to x k , we get  ∂ 2 x i ∂ x′ c  ∂ x j ∂ A ij  ∂ A′ ab ∂ x′ c  ∂ x i ∂ x j = + A′ ab  k c k  a b ∂x  ∂ x′ a ∂ x′ c ∂ x k  ∂ x′ b  ∂ x′ ∂ x  ∂ x′ ∂ x′ + A′ ab =

∂ Aab ∂ x′ c ∂ x j  ′ ∂ x i ∂ x j ∂ x′ c + A′ ab × c a b k ∂ x′ ∂ x′ ∂ x′ ∂ x ∂ x k ∂ x′ b  + A′ ab

  =

∂ x i  ∂ 2 x j ∂ x′ c  ∂ x′ a  ∂ x′ b ∂ x′ c ∂ x k 

∂ x i ∂ x′ c  ∂ x′ a ∂ x k 

bc

j ′ p ∂x − ∂ x′ p

∂ A′ ab ∂ x i ∂ x j ∂ x′ c + A′ ab ∂ x′ c ∂ x′ a ∂ x′ b ∂ x k

ac

j

λµ

∂xλ ∂x µ   ∂ x′ a ∂ x′ c 

i

λµ

∂x λ ∂x µ  ∂ x′ b ∂ x′ c 

(4.9.1)

( p ↔ a)

λµ

∂ x λ ∂ x j  ∂ x′ c ∂ x µ  ∂ x′ a ∂ x′ b  ∂ x k ∂ x′ c 

+ A′ ab

bc

j c i ′ p ∂ x ∂ x ∂ x′ − A′ ab p k a ∂ x′ ∂ x′ ∂ x

( p ↔ b)

i ′ p ∂x − ∂ x′ p

j c i ′ p ∂ x ∂ x ∂ x′ p k b ∂ x′ ∂ x′ ∂ x

− A′ ab

i

ac

j

λµ

∂ x i ∂ x λ  ∂ x′ c ∂ x µ  . ∂ x′ a ∂ x′ b  ∂ x k ∂ x′ c 

(changing the dummy indices in light of the indices of the factors of the first term) ∂ A ij ∂ A′ ab ∂ x i ∂ x j ∂ x′ c = + A′ pb ∂ x k ∂ x′ c ∂ x′ a ∂ x′ b ∂ x k

+ A′ ap

pc

pc

j i c ′ a ∂ x ∂ x ∂ x′ − A′ ab a b k ∂ x′ ∂ x′ ∂ x

j i c ′ b ∂ x ∂ x ∂ x′ − A′ ab a b k ∂ x′ ∂ x′ ∂ x

 ∂ A′ ab + A′ pb =  ∂ x′ c

pc

′ a + A′ ap

j

λµ

i

λµ

∂xλ ∂x j µ δk ∂ x′ a ∂ x′ b

∂xi ∂xλ µ δk ∂ x′ a ∂ x′ b

j i c ′ b  ∂ x ∂ x ∂ x′ − A λ j pc   ∂ x′ a ∂ x′ b ∂ x k

i

λk

⋅ 1 − A iλ

j

λk

⋅ 1.

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Tensor Calculus and Applications



∂ A ij + Aλ j ∂x k

i

λk

+ A iλ

j

λk

 ∂ A′ ab = + A′ pb  ∂ x′ c

pc

′ a + A′ ap

pc

j i c ′ b  × ∂ x ∂ x ∂ x′  ∂ x′ a ∂ x′ b ∂ x k

∂ x i ∂ x j ∂ x′ c . ∂ x′ a ∂ x′ b ∂ x k (4.9.2) A,ijk = A,′cab

Writing A,ijk =



∂ A ij + Aλ j ∂x k

i

λk

+ A iλ

j

λk

. (4.9.3)

A,ijk of (4.9.3) is called the xk-covariant derivative of the second-order contravariant tensor Aij with respect to the fundamental tensor gij. The corresponding result (4.9.2) being the transformation law of a third-order mixed tensor must be a tensor. Hence, the covariant derivative thus defined by (4.9.3) is also a tensor, and due to consideration of covariant derivative, the rank of the tensor is found to increase by one.

4.10 Covariant Derivative of Mixed Tensor of Rank Two Consider the second-order mixed tensor A ij . ∴ By transformation law of tensors, A ij = Ab′ a



∂ x i ∂ x′ b . ∂ x′ a ∂ x j

Differentiating partially with respect to xk, we can get 2 i i ∂ A ij  ∂ Ab′ a ∂ x ′ c  ∂ x i ∂ x ′ b ∂2 x ′b ∂x′ c  ∂x′b a ∂ x a ∂x = j + Ab′  j + Ab′ k c k  a a c k  a ∂x ∂x′ ∂x j ∂x k  ∂x′ ∂x  ∂x′ ∂x  ∂x′ x′ ∂x  ∂x



=

∂ Ab′ a ∂ x i ∂ x ′ b ∂ x ′ c ∂x′b ∂x′ c  + Ab′ a j c a k ∂x′ ∂x′ ∂x ∂x ∂ x j ∂ x k  + Ab′ a

∂xi  ∂ x ′ a 

p jk

∂x′b − ∂x p

λµ

ac

i ′ p ∂x − ∂x′ p

i

λµ

∂xλ ∂x µ  ∂ x ′ a ∂ x ′ c 

λ µ ′ b ∂ x ′ ∂ x ′  , using Corollary 1 ∂ x j ∂ x k 

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Christoffel Three-Index Symbols

=

∂ Ab′ a ∂ x i ∂ x′ b ∂ x′ c + Ab′ a ∂ x′ c ∂ x′ a ∂ x j ∂ x k − Ab′ a



i

λµ

+A ∴

∂ A ij + A λj ∂x k

i

λk

∴ A ij , k = Ab′ ,ac 

∂ x λ ∂ x′ b − Ab′ ∂ x′ a ∂ x j a

− Api

p jk

p jk

( p ↔ a)

λ µ i ∂ x′ b ∂ x λ  ∂ x′ c ∂ x µ  a ∂ x ∂ x′ ∂ x′ A − ′ b ∂ x j ∂ x′ a  ∂ x k ∂ x′ c  ∂ x′ a ∂ x j ∂ x k

∂ A ij ∂ Ab′ a ∂ x i ∂ x′ b ∂ x′ c = + Ab′ p j k c a k ∂x ∂ x′ ∂ x′ ∂ x ∂ x p i p jk

i b c i b ′ p ∂ x ∂ x ′ ∂ x ′ + A′ a ∂ x ∂ x ′ b ∂ x′ p ∂ x j ∂ x k ∂ x′ a ∂ x p

ac

pc

i

′ b (λ ↔ b , µ → c )

i b c ′ a ∂ x ∂ x′ ∂ x′ j a k ∂ x′ ∂ x ∂ x i b c ′ ∂ x ∂ x′ ∂ x′ j a k ∂ x′ ∂ x ∂ x



λ a λ bc

µ k

δ − A′

λµ

λµ

 ∂ A′ a =  bc + Ab′ p  ∂ x′

pc

′ a − A′ a λ

bc

i b c ′ λ  × ∂ x ∂ x′ ∂ x′  ∂ x′ a ∂ x j ∂ x k

∂ x i ∂ x′ b ∂ x′ c . ∂ x′ a ∂ x j ∂ x k

(4.10.1)

Writing A ij , k =



∂ A ij + A λj ∂x k

i

λk

− Api

p jk

. (4.10.2)

In this case, A ij , k of (4.10.2) is called the xk-covariant derivative of the mixed tensor A ij with respect to the fundamental tensor gij; consequently, (4.10.1) represents the corresponding transformation law of a mixed tensor of rank three. Hence, the covariant derivative thus defined by (4.10.2) of the mixed tensor is also found to increase the rank by one. Note: In (4.10.2), when the covariant index j is associated with k (third term), the corresponding term is negative, and when the contravariant index i is associated with k, the corresponding term (second) is positive. 4.10.1 Generalization Following the above results, the covariant derivative of higher order tensors can be written as i. Aijk , l = ii. A,ijkl =

∂ Aijk − Aα jk ∂xl

∂ A ijk + A ajk ∂xl

α il i al

α

− Aiα k

+ A iak

jl j

al

α

− Aijα

+ A ija

kl k

al

.

.

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Tensor Calculus and Applications

ij k ,l

iii. A

α

j

i

− Aαij kl + Akα j α l + Akiα α l . ∂xl ∂ A ijk i α α = − Aαi k jl − A ijα kl + Aαjk α l . l ∂x

=

iv. A ijk , l

∂ A ijk

Similarly,



ij... k Alm ... n , r =

∂ ij... k aj... k Alm ... n + Alm... n ∂xr

(

)

ij... a + Alm ... n

k ar

ij... k − Abm ... n

i ar b lr

j

ia... k + Alm ... n

− Albij......kn

+

ar b mr

ij... k −  − Alm ...b

b nr

.

The generating factor of any space is the metric ds2 = g ij ( x i )dx i dx j, which ­contains the metric functions gij(xi). But it is proved to be a tensor called fundamental tensor, and the evolution of the branch tensor (the evolution of the branch tensor analysis) is dependent on this fundamental concept. This demands the consideration of covariant derivatives of all the fundamental tensors g ij ′ g ij and also g ij.

4.11 Covariant Derivatives of g ij ′ g ij and also g ij

i. Using the result (4.8.2), the xk-covariant derivative of gij, we can write g ij , k =



∂ g ij − gα j ∂x k

α ik

=

∂ g ij − ∂x k

=

∂ g ij ∂ g ij − , ∂x k ∂x k

(

ik , j

+

α

− g iα jk , i

)

jk

 gα j

α ik

∂ g ji ∂x k gij, k = 0. ii. It is already proved that g ij g jk = δ ik = 1 or 0. Differentiating partially with respect to xl, ∂ g ij jk ∂ g jk g + g ij = 0 for both the cases. l ∂xl ∂x using



jk , i

+

ik , j

=



(

jl , i

+

li , j

)g

jk

+ g ij

∂ g jk =0 ∂xl

=

ik , j

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Christoffel Three-Index Symbols

∴ g ij



∂ g jk + g jk ∂xl

jl , i

+

k li

= 0.

∂ g jk free from gij, we need to consider the inner product of ∂xl it with gmi:

To make



∴ g mi g ij



∴ δ jm

or 1 ⋅

∂ g mk + g jk ∂xl

m jl

∂ g jk + g mi g jk ∂xl

(

∂ g jk + g jk g mi ∂xl

+ g mi

k il

jl , i

jl , i

+ g mi

)+ g

mi

k

=0

k

=0

li

li

=0 ∴ g ,mkl = 0



This follows g ,ijk = 0. iii. Following the result (4.10.2), the xk-covariant derivative of g ij can be written as

g ij , k =

∂ g ij − gαi ∂x k

α jk

+ gαj

i

αk

= 0 − 1⋅

i jk

+ 1⋅

i jk

=0

gαi = 1 when i = α = 0 when i ≠ α . Thus, it is found that g ij , k = 0, g ,ijk = 0, and g ij , k = 0 . Since all the covariant derivatives of the fundamental tensors are zeros, they are called covariant constants.

4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors a. Two tensors of the same rank and similar characters are conformable for addition or subtraction. So, consider the two tensors Aij and Bij each of rank two so that Aij + Bij = Cij , a tensor of the same rank and character.

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Tensor Calculus and Applications

By the definition of xk-covariant derivative of Cij with respect to the fundamental tensor gij, Cij , k = =

  

∂Cij − Cα j ∂x k

α ik

− Ciα

α jk

∂ Aij + Bij − Aα j + Bα j ∂x k

(

) (

 ∂ Aij =  k − Aα j  ∂x

α ik

− Aiα

jk

)

α

− ( Aiα + Biα )

ik

 ∂Bij  +  ∂ x k − Bα j

α

α

α jk

(4.12.1)

− Biα

ik

jk

α



( Aij + Bij ), k = Aij , k + Bij , k . Hence, the xk-covariant derivative of the sum of two tensors is equal to the sum of their covariant derivatives. This result will hold good for the sum of two or more tensors (when conformable) of any character covariant, contravariant, or mixed. Similarly, it can be proved for the difference of two (or more) tensors: ( Aij − Bij ), k = Aij , k − Bij , k . (4.12.2)



b. Consider any two tensors Aij, Bk so that A ij Bk = Ckij, an outer product. Now the xm-covariant derivative of Ckij , can be written by virtue of (4.10.2) as Ckij, m =



∂Ckij − Cαij ∂xm

α km

+ Ckα j

i

αm

+ Ckiα

=

∂ A ij B k − A ij Bα ∂xm

=

∂ A ij ∂B Bk + A ij mk − A ij Bα ∂xm ∂x

(

)

 ∂ A ij =  m + Aα j  ∂x

i

αm

+A

α km



j

αm

+ Aα j B k α km

j

αm

i

αm

+ A iα Bk

+ Aα j Bk

i

αm

j

 A ij Bk = Ckij

αm

+ A iα Bk

 ij  ∂ Bk  Bk + A  ∂ x m − Bα

km

j

αm

α

(4.12.3)



= A,ijm Bk + A ij Bk , m

(

∴ A ij Bk

)

,m

= A,ijm Bk + A ij Bk , m

 Thus, the covariant derivative of the product (outer) of two tensors obeys the product rule of ordinary derivatives.

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Christoffel Three-Index Symbols

Example 3 If A,ijk is the xk-covariant differentiation of the second-order contravariant tensor A ij with respect to the fundamental tensor g ij , prove that 1 ∂ A ij g + A jk g ∂x j

(

A,ijj =



)

i jk

.

Also show that (i) the last term vanishes if Ajk is skew symmetric and k 1 ∂ (ii) Aij, j = Aij g − Akj ij . g ∂x j

(

)

Proof: By the definition of xj -covariant derivative, we have ∂ A ij + Aα j ∂x j

∴ A,ijj =



i

αj

+ A iα

j

αj

(considering a contraction setting j = k in A,ijk )



=

∂ A ij + Aα j ∂x j

=

1 ∂ ∂ A ij + A ij ∂x j g ∂x j

A,ijj =

i

+ A iα

αj

∂ log g ∂ xα

(

( g )+ A

1 ∂ A ij g + A jk g ∂x j

(

)

i jk

kj

i

)

(α → j ) , g > 0

(α → k in the last term)

kj

( j ↔ k ).

Hence, proved. A jk i.

i jk

∴ 2 A jk

= − A kj

jk

= − A jk

kj

= − A jk

jk

i jk

i

 A jk = − A kj

i

( j ↔ k)

i

(

= 0 ∴ A jk

i jk i jk

=

i kj

)

=0

ii. Also, from the definition of xk-covariant derivative, Aij, k =



∂ Aij − Aαj ∂x k

( )

α

+ Aiα

ik

j

αk

.

Considering a contraction with respect to j and k, i.e., setting j = k, we get Aij, j = =

∂ Aij − Aαj ∂x j

( )

α ij

+ Aiα

j

αj

α ∂ ∂ j j α (log g , g > 0 j Ai − Aα ij + Ai ∂x ∂ xα (α → j) (α → k )

( )

(

)

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Tensor Calculus and Applications

∴ Aij, j = =

∂ Aij − Akj ∂x j

( )

k ij

1 ∂ Aij g − Akj g ∂x j

(

)

1 ∂ g ∂x j

+ Aij

k ij

g

.

Hence, proved. Example 4 If, at a specified point, the derivatives of the gij ’s with respect to the coordinates are all zero, the components of covariant derivatives of a vector at the point are the same as ordinary derivatives. ∂ g ij Given that k = 0 at a point P (say) ∀ i, j, k. ∂x If A i is any covariant (may be contravariant also) vector, then its xj -covariant derivative is given by Ai , j =

= Ai , j =

∂ Ai − Aα ∂x j

α ij

=

∂ Ai − A α g αβ ∂x j

ij , β

∂ g jβ ∂ g ij  ∂ Ai αβ 1  ∂ g iβ − j − Aα g j +  2  ∂x ∂x ∂ x i ∂ x β  ∂ Ai ∂x j





∂ g ij = 0 ∀ i, j, k at the point P. ∂x β

4.13 Gradient of an Invariant Function The partial derivatives of an invariant function φ is defined as the components of a vector called grad φ or ∇φ . Theorem of Invariant Function If ϕ is invariant function, show that its covariant derivative is equal to its ordinary derivatives. Proof: Let Ai be an arbitrary vector so that (ϕAi) is also an arbitrary vector since ϕ is invariant. ∴ Its xj-covariant derivative with respect to gij can be written as

(φ Ai ), j =

∂ (φ Ai ) − (φ A )α ∂x j



α ij

∂A i ∂φ + Ai j − φ Aα ∂x j ∂x

α ij

67

Christoffel Three-Index Symbols

 ∂A φ, j Ai + φ Ai , j = φ  ji − Aα  ∂x



ij

∂φ  + Ai j ∂x

α

∂φ = φ Ai , j + Ai j . ∂x



∂φ =0 ∂x j



∴φ, j Ai − Ai



∂φ   ∴ Ai  φ, j − j  = 0  ∂x 

∂φ  Ai is arbitrary. ∂x j ∴ The xj-covariant derivative of ϕ, namely, ϕj, is equal to its ordinary par∂φ tial derivatives j . ∂x ∂φ Hence, these partial derivatives of the invariant function ϕ are the ∂x j components of grade ϕ vector. Otherwise, the covariant derivative ϕj of the invariant function is a vector which is nothing but the (∇ϕ) vector.

∴ φ, j =

4.14 Curl of a Vector The xj-covariant derivative of the covariant vector Ai with respect to the fundamental tensor gij is given by α ∂ Ai Ai , j = j − Aα ij , which is a second-order covariant tensor. ∂x The components Ai,j is interpreted as obtained from Ai due to the operation “covariant differentiation” with respect to the fundamental tensor gij. The curl of the vector A( Ai ) is defined as Curl A = Ai , j − A j , i .  ∂A =  ji − Aα  ∂x =

ij

 ∂ Aj  −  i − Aα ∂x

α

∂ Ai ∂ A j −  ∂x j ∂xi

α ij

=

ji

α



α ji

Hence, Curl A ( = Curl Ai ) = Ai , j − A j , i =

∂ Ai ∂ A j − . ∂x j ∂xi

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Tensor Calculus and Applications

Theorem: The Necessary and Sufficient Condition That the First Covariant Derivative of a Covariant Vector Will Be Symmetric If the Vector Is Gradient of Some Invariant Function Proof: By the definition of curl of a vector A, we have

Curl A = Ai , j − A j , i = 0 if Ai , j = A j , i (symmetric).

But curl ∇ϕ = 0, where ϕ is some scalar invariant. Otherwise, if A = ∇φ ,  Curl A = 0, ∴ Curl ∇φ = 0. ∴ Ai , j − A j , i = 0.



∴ Ai , j = A j , i , i.e., symmetric.

Hence, proved.

4.15 Divergence of a Vector

( )

The divergence of a contravariant vector uα ui is defined as the result of contraction with respect to its covariant derivative. i ∂ ui In, u,i j = j + uα α j ∂x Let us allow the contraction setting i = j, so that u,ii =

∂ ui + uα ∂xi

i

αi



(

)

=

∂ ui ∂ + uα α log g , ( g > 0 ) ∂xi ∂x (α → i )

=

∂ ui ∂ + ui i log g ∂xi ∂x

=

1 ∂ ∂ ui + ui i ∂x g ∂xi

=

1 ∂ ui g g ∂xi

(

(

∴ div u or div ui = u,ii = which is a scalar invariant.

)

(

1 ∂ g ∂xi

(

)

( g)

) )

gui ,

69

Christoffel Three-Index Symbols

4.16 Laplacian of a Scalar Invariant

( )

From the expression of divergence of the contravariant vector u ui (Section 4.15), we have

)

1 ∂ g ∂xi

(

gui

( )

1 ∂ g ∂xi

(

g g ij u j .

(

∴ div u = u,ii =

∴ div u = u,ii =

) )

(i)

∂φ (Section 2.13), which is the covari∂x j ant derivative of the scalar invariant ϕ and is the component of ∇ϕ; (i) can be written as But we have already proved that φ, j =



div ( ∇φ ) =

1 ∂ g ∂xi

(

g g ijφ, j

)

or

∇ 2φ =

1 ∂ g ∂xi

(

)

g g ijφ, j . (ii)

Hence, the divergence of the vector ∇ϕ, symbolically ∇ 2ϕ, is defined as the Laplacian of ϕ, and (ii) is its explicit expression. In deriving div ui from (i), we have replaced ui as

ui = g ij u j = g ijφ, j  grad φ = φ, j =

∂φ ∂x j

to recover the Laplacian of ϕ. ui = g ijφ, j , since g ij is symmetric. Therefore, div (∇ϕ) can also be interpreted as the outcome of contraction of the covariant derivative of gij𝜙,j: ∴∇ 2φ = g ijφ, ij



 φ, i =

∂φ and div ui = u,ii = ( g ijφ, j ), i ∂xi

 ∂2 φ ∂φ ∇ 2φ = g ij  i j − k ∂ x ∂ x ∂ x 

k ij

  

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Tensor Calculus and Applications

φ, ij =



∂  ∂φ  ∂φ  − ∂x j  ∂xi  ∂x k

k

 g ,iji = 0

ij

and div ui = u,ii = ( g ij u j ), i



= ( g ijφ, j ), i = g ijφ, ij ,

which is another form of the Laplacian of ϕ. Example 5 If Aij is the curl of a covariant vector, prove that Aij , k + A jk , i + Aki , j = 0; otherwise,

∂ Aij ∂ A jk ∂ Aki + + = 0. ∂x k ∂x i ∂x j

Let Bi be the covariant vector so that Aij = Bi , j − Bj , i =



∂Bi ∂Bj − . (i) ∂x j ∂xi

A ji = Bj , i − Bi , j interchanging i and j

(

= − Bi , j − Bj , i



)

= − Aij . Hence, Aij is an antisymmetric tensor. ∴ Aij + A ji = 0. (ii)

Now, Aij , k + A jk , i + Aki , j =

∂ Aij − Aα j ∂x k +



=

α ik

∂ A jk − Aα k ∂xi

α ji

α jk

− A jα

α ki

+

∂ Aki − Aα i ∂x j

∂ Aij ∂ A jk ∂ Aki + + − Aα j + A jα ∂x k ∂xi ∂x j

(

− ( Aiα + Aα i ) =

− Aiα

α jk

− ( Aα k + Akα )

∂ Aij ∂ A jk ∂ Aki + + ∂x k ∂x i ∂x j

)

α ki

α ij

α kj

− Akα

α ij

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Christoffel Three-Index Symbols

using (ii) and Also,

α ij

=

α ji

.

∂ Aij ∂ A jk ∂ Aki ∂  ∂Bi ∂Bj  ∂  ∂Bj ∂Bk  + + = k j − i+ i k − ∂x k ∂x i ∂x j ∂x  ∂x ∂x  ∂x  ∂x ∂ x j  + =

∂  ∂Bk ∂Bi  −  , ∂x j  ∂x i ∂x k 

using ( i )

∂2 Bj ∂2 Bj ∂ 2 Bi ∂ 2 Bk − k i + i k − i j j k ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x

+

∂ 2 Bi ∂ 2 Bk − j k = 0. j i ∂x ∂x ∂x ∂x

Hence, proved.



4.17 Intrinsic Derivative or Derived Vector of v

 If aˆ is any vector representing a direction and v is any vector with covariant components vi (or contravariant components vi), the intrinsic derivative or  derived vector of v in the direction of aˆ is defined by means of the covariant components vi , k a k or by contravariant components v,i k a k :



  For vi , k a k = δ ji v j 

(

)

  a k =  g iα g jα v j  a k g iα vα ,k  ,k

(

)

 a k = v,i k a k  . ,K 

It is usually denoted by the symbol a .∇v , which is nothing but the generalization of the derivative of a vector in Euclidean space E3 in the direction of a. Theorem: Show That a Vector of Constant Magnitude Is Orthogonal to Its Intrinsic Derivative Proof: Let u ( ui ) be a covariant vector of constant magnitude so that 

u2 = g ij ui u j.



Considering the xk-covariant differentiation with respect to the fundamental tensor gij, we can get

(

0 = g ij ui , k u j + ui u j , k

)

 g ,ijk = 0

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Tensor Calculus and Applications

( g u )u ij

i, k

j

(

)

+ g ij u j ui , k = 0 (i ↔ j in the second term) ui ui , k + ui ui , k = 0 ∴ 2 ui ui , k = 0



ui ui , k a k = 0 (considering the inner product with a k )

(

)

∴ ui ui , k a k = 0, which is of the form ui ui = 0. This shows that the intrinsic derivative of ui , k in the direction of aˆ a k is orthogonal to u itself. Hence, proved.

( )

4.18 Definition: Parallel Displacement of Vectors 4.18.1 When Magnitude Is Constant

 Let the unit vector tˆ t k represent the direction at any point of a vector field u. The vector u of constant magnitude (need not necessarily be of constant magnitude) is said to be parallel along a curve C with respect to a Riemannian Vn, if its intrinsic derivative (or derived vector) in the direction of the curve at all points of C vanishes; mathematically,

( )



u,i k t k = 0 or ui , k

dx k = 0. ds

This can be thrown to the form:

 ∂ ui α  ∂ x k + u

i

αk

 dx k =0  ds

or

∂ui dx k + uα ∂ x k ds

i

αk

dx k = 0, ds

so that

dui + uα ds

i

αk

dx k = 0 . (4.18.1) ds

Otherwise, the vector u is said to undergo parallel displacement (according to Levi-Civita) along the curve C of Vn if (4.18.1) is satisfied.

73

Christoffel Three-Index Symbols

Increment of u: k i dx dui From (4.18.1), + uα α k = 0. ds ds This can be said as the arc rate of change of the contravariant component ui. This can be put to the form: dui = − uα ment dxk.

i

αk

dx k , which is called the increment of ui due to the displace-

4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude If the direction of two vectors is the same or if their corresponding components are   proportional, they are said to be parallel. Let B be a vector of variable magnitude parallel to a vector A at each point of the curve C in Vn so that Bi = λ ( s ) A i. (4.18.2)



 If A A i is assumed to be avector of constant magnitude parallel to the curve C with respect to Vn, then B Bi must also be parallel with respect to Vn along the curve but of variable magnitude. As Ai undergoes parallel displacement,

( )

( )

A,i j



dx j = 0. (4.18.3) ds

Now, B,i j

dx j = λ Ai ds

(

= Ai

∴ B,i j

)

,j

dx j dx j dx j = λ, j A i + λ A,i j ds ds ds

∂λ dx j B i dλ i dλ , using ( 4.18.3 ), = = A ∂ x j ds ds λ ds

dx j = Bi f ( s) ds

(4.18.4)

d  log λ ( s)  . ds  i The vector B B of variable magnitude should be expressed in the form (4.18.4), if it undergoes parallel displacement with respect to Vn along the curve C. It can be promptly concluded that the intrinsic derivative of B (L.H.S. of 4.18.4) at each point on the curve C has the same direction as that of B Bi (R.H.S.). Otherwise, B is parallel to itself along the curve C subject to the condition (4.18.4). where f ( s ) =

( )

( )

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Tensor Calculus and Applications

Writing Ai = Bi μ(s), we can get A,i j

∂ µ  dx j dx j  i =  µB, j + Bi j  ds  ∂ x  ds dµ   , using ( 4.18.4 ). =  µBi f ( s) + Bi ds  



 dµ  = Bi  + µ f ( s)   ds  We can choose

dµ + µ f ( s) = 0 so that ds

dx j = 0, which is the condition of parallel displacement of the vector ds A A i of constant magnitude along the curve C in Vn. A,i j

( )

( )

(

)

Hence, B Bi also undergoes  A i = µBi parallel displacement along the curve C in Vn. Hence, proved. Note: From the condition of parallel displacement (4.18.4), B,i j



(

dx j = Bi f ( s) ds

Bk B,i j

dx j = Bk Bi f ( s) ds

( multiplying by B )

Bi B,kj

dx j = Bi Bk f ( s) ds

( k ↔ i ).

Subtracting, Bi B,kj − Bk B,i j

j

) dxds

k

= 0, (or) eliminating f ( s).

( )

This is a modified form of the condition of parallelism of the vector B Bi of variable magnitude. Theorem: If Two Vectors of Constant Magnitude Undergo Parallel Displacement along a Given Curve, They Incline at a Constant Angle

( )

( )

Proof: Let aˆ a k and bˆ b i be the two unit vectors of constant magnitudes, and θ be the angle between them:

75

Christoffel Three-Index Symbols

∴ cosθ = g ij a ib j ∴

d d dx k ∂ (cosθ ) = g ij a ib j = k g ij a ib j ds ds ds ∂x

(

)

(

= g ij a ib j

)

,k

(

dx k ds

)

 g ij a ib j

is invariant

dx k =  g ij a,i k b j + g ij a ib,jk    ds

(

(

)

)

 g ij , k = 0

  j dx k  dx k  a = bi  a,i k + j  b, k ds  ds   

(

(

)

)

= bi a,i k t k + a j b,jk t k . If aˆ and bˆ undergo parallel displacements along a curve C given by the direcdx k tion t k = , then ds a,i k t k = 0 = b,jk t k .



∴ The above relation reduces to − sin θ ∴



dθ =0 ds

dθ =0 ds

sin θ ≠ 0. Hence, the vectors aˆ and bˆ incline at a constant angle. Hence, proved. Theorem: If a Vector u Undergoes Parallel Displacement along a Given Curve, It Must Be of Constant Magnitude Proof: By definition, u2 = g ij ui u j = u j u j





d 2 d ∂ dx k u = uju j = k uju j ds ds ds ∂x

( )

(

 u j u j is invariant

)

(

)

76



Tensor Calculus and Applications

  dx k  dx k  = u j  uj, k + u j  u,j k  ds  ds    ⇒

d 2 u = 0. ds

( )

∴ For parallel displacement of u along the curve C of direction,

dx k dx k dx k = t k , uj, k = 0 = u,jk . ds ds ds

u2 = constant, i.e., u = constant. Hence, proved.

Exercises 1. If aˆ and bˆ are the two unit vectors, ϕ is a scalar invariant and the derivative of ϕ in the direction of aˆ is ϕ, i ai,, then show that the derivative of this quantity in the direction of bˆ is (ϕi ai),j bj = (ϕ,i ai,j + ai,ij)bj. 2. Find the nonvanishing Christoffel symbols for the metrics: i. ds2 = a2 dr2 + sin2𝜃d𝜃2. ii. ds2 = dr2 + r2 d𝜃2 + dz2. iii. ds2 = dr2 + r2 d𝜃2 + r2 sin2 𝜃 dϕ 2. iv. ds2 = e−2kt (dx2 + dy2 + dz2) − dt2. 3. If ϕ is a scalar and f(ϕ) is a function of ϕ, show that

∇ 2φ = f ′′(φ )(∇φ )2 + f ′(φ )∇ 2φ .

5 Properties of Curves in Vn and Geodesics Definition Curvature: The arc rate at which the tangent to a curve at a point P changes the direction as P moves along the curve is called the curvature (κ) of the curve.

5.1 The First Curvature of a Curve If the coordinates xi of the current point P of a curve C in a Vn is taken as function of the parameter arc length s, then the unit tangent vector tˆ t i to  dx i the curve is defined as t i = . The first curvature vector P( p i ) of the curve ds C relative to Vn is the derived vector of tˆ along the curve which is defined as

( )

p i = t,i k



dx k . (5.1.1) ds

The first curvature vector is of magnitude κ = g ij p i p j . From the definition (5.1.1),  ∂t i pi =  k +  ∂x



dt i = + ds

i jk

i jk

dx j  dx k ∂t i dx k = + ∂ x k ds ds  ds

dx j dx k d 2 x i = 2 + ds ds ds

i jk

i jk

dx j dx k ds ds

(5.1.2)

dx j dx k . ds ds

If the first curvature of the curve pi vanishes, then j k i dx dx d2 xi pi = + jk = 0 will represent a particular class of curves called 2 ds ds ds geodesics, which will be discussed in Section 5.2.  Principal normal: The direction of the first curvature vector P( p i ) is called the principal normal, and the unit vector nˆ in the direction of p is called the principal unit normal so that P = κ nˆ .

(

)

77

78

Tensor Calculus and Applications

5.2 Geodesics Definition A geodesic on a surface in Vn is a curve or path of extremum (or stationary) length joining any two given points on it. For example,

i. A line joining two points in a plane is the shortest distance called straight line which is geodesic. ii. The great circular arc joining two points on the surface of a sphere is the shortest distance between two points. Therefore, it is geodesic in three-dimensional spherical surface.

5.3 Derivation of Differential Equations of Geodesics Let A and B be the two points on a surface in an n-dimensional space Vn. The arc length joining the two points on the surface of Vn is given by ds2 = gij dxidxj. For stationary (or extremum) length, let us consider a small variation so that 2 dsδ ( ds ) =

∂ g ij k i j δ x dx dx + g ij δ dx i dx j + gij dx iδ dx j ∂x k

=

∂ g ij k i j δ x dx dx + g ij d δ x i dx j + gij dx i d δ x j ∂x k

=

∂ g ij k i j δ x dx dx + 2 g ij d δ x i dx j ∴ gij = g jt ∂x k

( )

( )

( )

( )



i↔ j

( )

 1 ∂ gij k dx i dx j d dx j  δx δ xi ∴ δ (ds) =  + gij ds. k ds ds ds ds   2 ∂x

( )

Applying variational principle for extremum values of the arc length joining the two arbitrarily chosen points A and B on Vn, we are to make use of



B

A

δ ( ds ) = 0. ∴



B A

j  1 ∂ g ij k dx i dx j d i dx   2 ∂ x k δ x ds ds + g ij ds δ x ds  ds = 0 (5.3.1)  

( )

79

Properties of Curves in Vn and Geodesics

Let us consider the second term of (5.3.1):



B A

B

 d  dx j  dx j  ⋅δ xi  − δ xi  ds =  g ij  g ij ds ds ds  A  

( )

=−



B A



 d  dx j   ⋅ δ x i  ds   g ij  ds   ds  

B A

 d  dx j  i  δ x  ds.   g ij ds   ds  

 For small variation, δ x i = 0 at both the ends A and B. ∴ Equation (5.3.1) reduces to





B A

   j j i  1 ∂ g ij dx dx δ x k − d  g ij dx  δ x i  ds = 0   2 ∂ x k ds ds  ds    ids →k   

or





B A

 1 ∂ g ij dx i dx j  ∂ g jk dx i dx j d2 x j  − i + g kj 2   δ x k ds = 0.  k ds    2 ∂ x ds ds  ∂ x ds ds

But for all δ x k, i.e., arbitrary value of δ x k if the above result is to hold good, then we must have ∂ g jk dx i dx j 1 ∂ g ij dx i dx j d2 x j − =0 1 − ⋅ g jk ∂ x i ds ds 2 ∂ x k ds ds ds 2



or

g jk

d 2 x j  1 ∂ g jk dx i dx j 1 dg ik dx j dx i  1 ∂ g ij dx i dx j + + =0 − ds2  2 ∂ x i ds ds 2 dx j ds ds  2 ∂ x k ds ds

(interchanging i and j in one term within the bracket). Considering the inner product of it with gmk and summing over k, we get or

g mk g jk

d2 x j + g mk ds 2

{ } dxds dxds i

ij , k

j

=0

80



Tensor Calculus and Applications

δ jm

d2 x j + ds2

m ij

dx i dx j =0 ds ds

g mk g jk = δ jm

or d2 xm + ds2

m ij

dx i dx j =0 ds ds

 δ jm = 1 for m = j = 0 for m ≠ j.

This can be written as

d2 xi + ds2

i jk

dx j dx k = 0 (5.3.2) ds ds

(changing m → i and i → k). The preceding equation is the differential equation of the curve called geodesis. These are the second-order differential equations in terms of n variables xi (i = 1, 2,…, n); each solution must have got two arbitrary constants. Therefore, second-order n differential equations should contain 2n arbitrary constants in their general solutions. Hence, 2n given conditions are necessary to know the complete solutions. The 2n coordinates of the points A and B are sufficient to determine the 2n arbitrary constants occurring in the solutions. Hence, the geodesic can be determined uniquely. Otherwise, through the two given points A and B on the surface in a Vn, one and only one geodesic can pass. Again, besides the coordinates A(xi), if the n components of the unit tan dx i  gent vector tˆ  t i = at the point A are known, then the geodesics can be ds   determined uniquely.

5.4 Aliter: Differential Equations of Geodesics as Stationary Length Let A and B be the two fixed points on a curve C of Vn, and t0 and t1 be the parametric values of A and B, respectively.

81

Properties of Curves in Vn and Geodesics

∴ The length of the arc joining A to B is given by t1



g ij

t0

dx i dx j dt dt dt

(

)

∵ ds 2 = g ij ( x i )dx i dx j . (5.4.1)



t1

=



g ij x i x j dt

t0

If the curve C is to be geodesic, the above length should be stationary (or extremum): t1

But the Euler’s equation for extremum value of the integral I = states that it must satisfy the differential equation:

∫ f ( x x ) dt i

t0

∂f d  ∂f  −  i  = 0 . (5.4.2) i ∂ x dt  ∂ x 

But for (5.4.1),

f = g ij x i x j = ∴

ds = s dt

∂ g jk j k 1 ∂ g jk j k ∂f 1 = x x = x x i i j k 2 s ∂ x i ∂x 2 g jk x x ∂ x

(i → k ).

∂f 1 = g ij x j . ∂ x i s ∴ Equation (5.4.2) reduces to

Also

1 ∂ g jk j k d  1  x x −  g ij x j  = 0  2 s ∂ x i dt  s



1 ∂ g jk j k  s 1 ∂ g ij k j 1    x x −  − 2 g ij x j + g ij x j  = 0 k x x +    2 s ∂ x i s s ∂ x s  

j

g ij x j +

∂ g ij j k 1 ∂ g jk j k s x x − x x − g ij x j = 0 ∂x k 2 ∂xi s

 1 ∂ g ij j k 1 ∂ g ik k j  1 ∂ g jk j k s g ij x j +  x x + x x  − x x − g ij x j = 0.  2 ∂x k  2 ∂xi 2 ∂x j s

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Tensor Calculus and Applications

g ij x j +

kj , i

g iα g ij x j + g iα



xα +

α jk

s g ij x j = 0 s

x j x k − kj , i

x j x k − g ij g iα x j

x j x k − x α

s =0 s

( taking the inner product with g ) iα

s =0 s

If the usual arc length parameter s is chosen instead of t, then the equation transforms to





d2 xi + ds 2

i jk

dx j dx k =0 ds ds

(∵ s = 1, s = 0) .

This is the differential equation of geodesics obtained from the notion of stationary arc length.

5.5 Geodesic Is an Autoparallel Curve Let tˆ be the unit tangent vector to a geodesic curve. The differential equations of geodesics are d2 xi + ds2



i jk

dx j dx k =0 ds ds

or



d  dx i  + ds  ds  ∴

dt i + ds

jk

i jk

dx j dx k =0 ds ds

i j k

t t = 0 ti =

dx i . ds

It can be written as



∂t i dx k + ∂ x k ds  ∂t i j  ∂ x k + t

jk

i jk

i j k

tt =0

 k i k  t = 0 ∴ t, k t = 0,

83

Properties of Curves in Vn and Geodesics

which is the condition of parallel displacement of the unit tangent vector ti to the curve geodesic but in the direction of itself. Hence, geodesics are autoparallel curves. Example 1 Determine the differential equations of geodesis in a space given by the metric

(

)

ds 2 = − e 2 kt dx 2 + dy 2 + dz 2 + dt 2 .



For the given metric, g11 = g 22 = g 33 = − e 2 kt , g 44 = 1. But g ij = 0 for i ≠ j





∂ g11 ∂ g 22 ∂ g 33 = = = − 2 ke 2 kt (i) ∂x 4 ∂x 4 ∂x 4 and

∂ g 44 = 0 for all “ i”. ∂x i

Also for the metric, gii exists only; therefore, g ii =

1 . g ii

Now the differential equations of geodesics are d2 xi + ds 2



i jk

dx j dx k = 0. (ii) ds ds

∴ We are to determine nonvanishing Christoffel symbols ­second kind: i

ii 1. jj = g

of the

1 ∂ g jj 1 −2 ke 2 kt = ke 2 kt =− 2 g ii ∂ x i 2.1

(

=−

jj , i

i jk

)

due to i = 4 only and j = 1, 2, 3(i ≠ j) 4



4

=

22

ii 2. ij = g

ij , i

11 i



1 14



,

3. But

i ii

= ke 2 kt 1 ∂ g ii . = 2 g ii ∂ x j 2

33

3

24

,



14

34 1

2

=

exists only due to (i) for i = 1, 2, 3, and j = 4 only. 1 ∂ g11 1 = × −2 ke 2 kt = k . 2 g11 ∂ x 4 2 − e 2 kt

(

3

= 34 = k. ∂ g 44 1 ∂ g ii = 0 ∴ 4 = 0. = 2 g ii ∂ x i ∂x

Similarly,

4

=

24

) (

)

84

Tensor Calculus and Applications

Now for i = 1, the differential equation (2) of the geodesic takes the form: d 2 x1 + ds 2



1 jk

dg j dg k =0 ds ds

or d 2 x1 + ds 2



1 14

dx 1 dx 4 + ds ds

1 41

dx 4 dx 1 =0 ds ds

or d2 x dx dt + 2k =0 ∴ ds 2 ds ds



1 14

1

=

41

=k

or d2 x ds 2 = −2 k dt . dx ds ds



dx Integrating, log   = −2 kt + log a.  ds  This can be written as dx = ae −2 kt . (iii) ds



Similarly, putting i = 2 and 3, respectively, we can get dy = +be −2 kt . (iv) ds dz = ce −2 kt . (v) ds



Again putting i = 4, Equation (ii) can be simplified as

d2 x4 + ds 2

jk

dx 1 dx 1 + ds ds

22

4

dx j dx k =0 ds ds

or

d2 x4 + ds 2

4 11

4

dx 2 dx 2 + ds ds

4 33

dx 3 dx 3 =0 ds ds

or

2 2 2  d2t  dy   dz   2 kt  dx  + +  =0 + ke      2  ds    ds  ds  ds 

85

Properties of Curves in Vn and Geodesics

or

d2t + ke 2 kt × a 2 + b 2 + c 2 e −4 kt = 0 using (iii), (iv), and (v) ds 2

(

)





(

d2t + kα 2 e −2 kt = 0 (vi) ds 2

)

putting α 2 = a 2 + b 2 + c 2 . Hence, Equations (iii)–(vi) are the required differential equations for the geodesics.

5.6 Integral Curve of Geodesic Equations The equations of geodesics are (Equation 5.3.2) d2 xi + ds2



i jk

dx j dx k = 0. (5.6.1) ds ds

If C is the curve geodesic through a point P and s is the arc length, then by Taylor’s theorem, we can write  dx i  1  d2 xi  2 1  d3xi  3 s s +  3  s +  . (5.6.2) x i = x0i +  + 2  ds 2  0 3  ds  0  ds  0



The second and higher derivatives of xi with respect to s can be determined from the equations of geodesics (5.6.1) as follows: d2 xi =− ds2

d d3 xi = − ds3  ds



( ) dxds dxds +

 ∂ = − l  ∂x 

j

i

k

jk

( ) i

jk

i jk

i jk

dx j dx k (5.6.3) ds ds

d 2 x j dx k + ds2 ds

dx j dx k dxl − ds ds ds

i

j

αβ

jk

i jk

dx j d 2 x k  ds ds2 

dxα dx β dx k − ds ds ds

(α →l ,

i

k

αβ

jk

β → j , j→α )

 dxα dx β dx j  ds ds ds  (α →l , β → k , k →α )  using ( 5.6.3 )

 ∂ =−  l  ∂x

dx − ( ) dxdsdxds ds  i

jk

j

k

l

i

αk

α lj ( j↔ k )

dx j dx k dx l − ds ds ds   

α

i jα

lk

dx j dx k dx l  ds ds ds    

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Tensor Calculus and Applications





d3xi + ds3

dx j dx k dx l = 0, (5.6.4) ds ds ds

i jkl

where i jkl

=

1  ∂ p 3  ∂ x l

( )−

=

1  ∂ p 3  ∂ x l

( )− 2



i

jk

lk

i

jk

α

i

αj

α

i

αj

kl

α

i





lk

 

 

and P indicates the sum of the terms obtained by permuting the subscripts cyclicly.* Using (5.6.3) and (5.6.4) in (5.6.2) for P(xo), we get  dx i  1 s− x i = xoi +  2  ds  o



( )  dxds   dxds  s − 13 ( ) j

i

jk

o

k

i

2

jkl

o

o

 dx j   dx k   dxl  3  ds   ds   ds  s + . o o o

 dx i  If we define ξ i =  , it can be thrown to the form:  ds  o x i = xoi + ξ i s −



1 2

( ) ξ ξ s − 13 ( ) ξ ξ ξ s +  (5.6.5) i

j

jk

k 2

o

i

j

jkl

l 3

k

o

i

The convergence of the series is dependent on ξ i and gij (for jk ). This, of course, represents an integral (solution) curve of the equations of geodesics (5.6.1) for small values of the parameter s.

5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates i. Riemannian coordinates: Consider yi = ξ is for the particular geodesic* (5.6.5) which passes through the point P(xo). It takes the form: x i = xoi + y i −



*

[2, p. 52].

1 2

( ) y y − 13 ( ) y y y +  . (5.7.1) i

jk

j

o

k

i

jkl

j

o

k

l

87

Properties of Curves in Vn and Geodesics

It represents all geodesics passing through Po(xo) given by various  dx i  directions ξ j =  . Also the equations yi = ξ is define a curve  ds  o otherwise geodesic for the given set of values of ξ i in terms of new coordinates yi. These coordinates yi ’s are called Riemannian coordinates as adopted by Riemann. Of course, these coordinates have got their own domain about Po. Let the fundamental form, Christoffel symbols in terms of the Riemannian coordinates yi, be represented by* g ij ( y i )dy i dy j ,



i

and

ij , k

jk

, respectively.

∴ The equations of geodesics are d2 y i + ds 2









i jk

dy j dy k = 0 (5.7.2) ds ds

i jk

ξ jξ k = 0  y i = ξ i s

i jk

j

.

y y =0 k

Clearly, subject to these conditions, the above equations of geodesics are satisfied, and hence, yi ’s are Riemannian coordinates. Following the method adopted in Section 5.6, but for Equation (5.7.2) in terms of the Riemannian coordinates yi, we can get

yi = ξ is −  This gives   ∴  



i jk

i ij , k

1  2

i jk

 j k 2 i  ξ ξ s +  , ( y )o = 0 for s = 0. o

 i i  = 0 if it is to reduce to y = ξ s . o

 = 0 ⇒  ∂ g ij  = 0   ∂ y k 

 ∂ g ij  g ij , k =  k  −  g α j  ∂y  o 

( i, j, k = 1, 2, , n)

o

α ik

+ gαi

α jk

 =  ∂ g ij  = 0.  o  ∂ y k  o

∴The first covariant derivative of the components of the fundamental tensor in these coordinates at the origin must vanish. This is the condition of yi ’s to be Riemannian coordinates.

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Tensor Calculus and Applications

ii. Geodesic coordinates: In an arbitrary Riemannian Vn, it is not possible to choose the Cartesian coordinate system, where the metric function gij ’s are constants. But it is possible to choose the coordinate system where gij ’s are locally constants so that ∂ g ij =0 ∂x k ≠0





at the point called the pole, elsewhere.

i

= 0 at the pole Po ; such a system of coordinates are called ∂ Ai geodesic coordinates with pole Po . Hence, for any tensor Ai , j = ∂x j which is a condition for geodesic coordinates. jk

5.7.1 Another Form of Condition for Geodesic Coordinates From the transformation law (Section 4.4), the resulting equation (4.4.5) of Christoffel symbol of the second kind, we have ij

∴−

λµ

α ′ p ∂x = ∂x′ p

∂xλ ∂x µ ∂ 2 xα j + i ∂x′ ∂x′ ∂x′ i ∂x′ j

λ µ α 2 ′α ∂ x ′ ∂ x ′ = ∂ x ′ − ∂xi ∂x j ∂xi ∂x j

=



α λµ

p ij

∂  ∂ x ′α  − ∂ x i  ∂ x j 

∂ x ′α interchanging x i and x ′ i systems ∂x p

(

p ji

 ∂ x ′α   ∂ x p 

)

(5.7.3)

 ∂ x ′α  = .  ∂ x j  , i

Taking a fixed value of x′α out of n independent values, we can entrust ∂ x ′α = x,′αj  x′α is a scalar invariant for the fixed value. ∂x j ∴ Equation (5.7.3) can be written as



λµ

λ µ ′α ∂ x′ ∂ x′ = x′α , (5.7.4) , ij ∂xi ∂x j

i.e., a second covariant derivative with respect to the metric of Vn. But if α x′α  are geodesic coordinates, then ′ = 0. λµ



∴ x,′ijα = 0.

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Properties of Curves in Vn and Geodesics

∴ The second covariant derivatives of the geodesic coordinates x′α must vanish. α Otherwise, if x′α = 0 , then from (5.7.4), ′ = 0 at pole. λµ

, ij

Hence, if a system of coordinates are geodesic coordinates with pole, then their second covariant derivatives with respect to the metric of the space must vanish at that point. From the above interpretation, it can be concluded that it is a necessary and sufficient condition.

5.8 If a Curve Is a Geodesic of a Space (Vm), It Is also a Geodesic of Any Space Vn in Which It Lies (Vn a Subspace) Let C(xi) be any non-minimal curve in a Vn at points xi(s) with fundamental dx i form ds2 = gij dxi dxj. If λ i = are the components of a unit vector field of Vn, ds then the derived vector of λ i in the direction of C is given by

λ ,ik



dx k = η i (say). (5.8.1) ds

But at points of C, if the vectors are parallel in the direction of C, then dx k ηi = λ ,ik = 0. ds Let the space Vn be immersed in a space Vm of coordinates yα and fundamental form ds2 = aαβ dyα dy β so that yα = f α ( x i ) (i = 1, 2, , n; α = 1, 2, , m)

∴ aαβ = g ij

∂xi ∂x j or ∂ yα ∂ y β

g ij = aαβ

∂ yα ∂ y β . (5.8.2) ∂xi ∂x j

Let the components in terms of y’s of the vector field in Vm be ξ α , and λ i’s be the corresponding components in x’s. ∴



dyα ∂ yα dx i = ds ∂ x i ds

∴ξα =

∂ yα i λ ∂xi

∂2 yα dx j dξ α dλ i ∂ yα ∴ = + λi i j i ds ds ∂ x ∂ x ∂ x ds ∴ ξ ,αβ

 ∂ξ β dyα β γ α +ξ i =η =  ∂ ds y 

(5.8.3)

( )a  dyds , β

γα

α

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Tensor Calculus and Applications

where

( ) β

γα

a

denotes the Christoffel (bracket) symbol of the second kind

with respect to the fundamental tensor aαβ . ∴ ηβ =

∂ξ β dyα + ∂ yα ds

( ) β

γα

dyα dξ β = + ds ds

( ) β

γα

2 β dλ i ∂ y β dx i j ∂ y + + λ ds ∂ x i ∂ x i ∂ x j ds

∴η β =



a

ξγ

( i→ j )

a

λi

( )

∂ yγ ∂ yα dx j , using (5.8.3) ∂ x i ∂ x j ds

β

γα

a

λi

∂ yγ ∂ yα dx j . (5.8.4) ∂ x i ∂ x j ds (i↔ j)

Now,

( ) ij , k

g

=

1  ∂ gik ∂ g jk ∂ gij  + − 2  ∂ x j ∂ x i ∂ x k 

=

∂2 yα ∂yγ ∂yα ∂yγ 1  ∂ aαγ ∂ y β ∂ yα ∂ yγ + aαγ  g ik = aαγ , etc.  β j j i k i k ∂x ∂x ∂x ∂x i ∂x k 2  ∂y ∂x ∂x ∂x ∂ a βγ ∂ y α ∂ y β ∂ y γ ∂yα ∂2 yγ ∂2 y β ∂yγ + + a βγ ∂x i ∂x j ∂x k ∂yα ∂x i ∂x j ∂x k ∂x i ∂x j ∂x k

+ aαγ

(β →α )

 i    α 



=

β

γ

 ∂ aαβ ∂ y γ ∂ y α ∂ y β ∂2 yα ∂ y β ∂yα ∂2 y β  ⋅ − a − a αβ αβ ∂yγ ∂x k ∂x i ∂x j ∂x i ∂x k ∂x j ∂x i ∂x j ∂x k  (α →γ ) ( β →γ ) 

g

(

αβ ,γ

= acγ

)

∂yα ∂y β ∂yγ ∂2 yα ∂yγ j i k + aαγ ∂x ∂x ∂x ∂x i ∂x j ∂x k

a

(α → c )

c αβ

) ∂∂yx

α

a

i

 ∂ yc = acγ  i j +  ∂x ∂x 2



( ) ij , k

g

    

k

1  ∂ aαγ ∂ aβγ ∂ aαβ  ∂ y α ∂ y β ∂ y γ ∂2 yα ∂ yγ  2 a +   β + α − αγ 2  ∂ y ∂y ∂ y γ  ∂ x i ∂ x j ∂ x k ∂ x i ∂ x j ∂ x k 

( ) =( ij , k

j

∂y β ∂yγ ∂2 y c ∂yγ j k + acγ ∂x ∂x ∂x i ∂x j ∂x k

( ) c

αβ

a

( )

∂yα ∂y β  ∂yγ  ∂x i ∂x j  ∂x k

β  ∂2 x β ∂y c ∂yα  ∂yγ . = aβγ  i j + α c j i  k a ∂x ∂x  ∂x  ∂x ∂x ( c ↔γ )

(5.8.5)

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Properties of Curves in Vn and Geodesics

Multiplying (5.8.4) by aβ c aβ c



∂y c , ∂x k

∂ y c β dλ j ∂y c ∂y β η = a β c ∂x k ∂ ds x k∂ xj  +λ j

∂y c  ∂2 x β dx i aβ c k  i j + ∂x  ∂x ∂x ds

= g jk

dλ j dx i + λj ds ds

= g jk

∂λ j dx i dx i + λj i ∂ x ds ds

= g jk

i dx i ∂λ j j dx λ + glk ds ∂ x i ds

( ) ij , k

ij , k

(

dx i ds

 ∂λ l j  ∂x i + λ 

β

a

∂yα ∂yγ   ∂x i ∂x j 

using ( 5.8.5 )

( )

( j→ l)

= glk

g

( ) αγ

g l

ij

)

g

( )  l

ij

g

 dx i l  λ, i  . = glk   ds  dyα If ξ β is parallel with respect to the curve C in Vm, then ξ ,βα = η β = 0; i ds dx ­therefore, λ,l i = 0. ds ∴ The vectors that are parallel to a curve C in Vm, they are also parallel with respect to Vn, a subspace of Vm. If λ i is a unit tangent vector to the curve C, then parallelism of the vectors along C means that the curve is a geodesic. Hence, if a curve is a geodesic in a space (here Vm), it is also a geodesic in the subspace Vn as it immerses in Vm. Hence, proved.

Exercises 1. Determine the differential equations of geodesics for the metric: i. ds2 = dr2 + r2 dθ 2 + r2sin2θdϕ 2. ii. ds2 = dr2 + r2 dθ 2 + dz2.

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Tensor Calculus and Applications



2. Find the differential equations of geodesics for the metric:



ds2 = f ( x ) dx 2 + dy 2 + dz 2 +









1 dt 2. f ( x)

3. Applying variational principles, derive the differential equations of geodesic: d2 xi + ds2

i jk

dx j dx k = 0. ds ds

4. If the coordinates xi of points on a geodesic are the functions of s, the arc length, show that d rφ dx i dx j dx p = φ, ij…p  , where ϕ is any scalar function of x’s. r ds ds ds ds 5. Obtain the differential equations of geodesic, if dt 2 1 − ds2 = (dx 2 + dy 2 + dz 2 ), k = constant. Hence, (1 − kx) c 2 (1 − kx)2 prove that along a geodesic, V2 − v2 = kc2x, where V is constant and 2 2 2  dx   dy   dz  v2 =   +   +   .  dt   dt   dt  6. Find the differential equations of geodesic for the metric: ds2 = − dx2 − dy2− dz2 + f(x, y, z) dt2. 7. Obtain the differential equations of geodesic as a means of stationary arc length. 8. Obtain the differential equations of geodesic for the metric:  g 2t 2  ds2 = dx 2 + dy 2 + dz 2 + 2 gt dx dt – c 2  1 − 2  dt 2 . c  

6 Riemann Symbols (Curvature Tensors)

6.1 Introduction The importance of intrinsic multidimensional differential geometry has found its place in the study of general theory of relativity. The general theory of relativity is inherently related to the geometry of curved space due to the effects of gravity. For this reason, Einstein needs to deal with four-­ dimensional space–time continuum in support of “absolute differential calculus” or tensor calculus. The Riemannian geometry associated with covariant differentiation through the fundamental tensor gij characterizing the space is properly suited for the curved space of general theory of relativity. This demands, though not general, an important notion (or concept), namely, Riemannian symbols or curvature tensors.

6.2 Riemannian Tensors (Curvature Tensors) The xj-covariant differentiation of the covariant vector Ai with respect to the fundamental tensor gij is given by Ai , j =



∂ Ai − Aα ∂x j

α ij

, (6.2.1)

which is again a second-order covariant tensor. Hence, considering its xkcovariant differentiation again, it can be written as

Ai , jk =



=

∂ Ai , j − Aa , j ∂x k

a ik

∂  ∂ Ai − Aα ∂ x k  ∂ x j

− Ai , a α ij

a jk

( treating A

  ∂ Aa  −  ∂ x j − Aα

ij

aj

α



= Ai , j a ik

)

 ∂A −  ai − Aα  ∂x

ia

α



a jk

using ( 6.2.1) 93

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Tensor Calculus and Applications

=

∂2 Ai − ∂x k ∂x j

α ij

∂ Aα ∂ − k k ∂x ∂x

 ∂2 Ai ∂ Aα Ai , jk =  k j − ∂x k  ∂x ∂x −

∂ ∂x k

( )A α

α

ij

α ij



( ) A − ∂∂Ax α

∂ Aa ∂x j α

+ Aα

a

aj

ik

a ik

a

a

α

ij

ik

j

∂ Ai ∂x a



a jk

α

+ Aα

a

aj

ik

α

+ Aα

a

ia

jk



∂ Ai ∂x a

a jk

+ Aα

α ia

a jk

  

. (6.2.2)

As covariant differentiation is not commutative, we can write (interchanging j and k)



 2 ∂ Aα ∂ A Ai , kj =  j i k −  ∂x ∂x ∂x j (α → a )  −

∂ ∂x j

( )A α

+ Aα

α

ik

α ik

∂ Aa ∂x k



a ij



( a→α )

α

a

ak

ij

∂ Ai ∂x a

a kj

α

+ Aα

ia

a kj

    (6.2.3)

.

Subtracting (6.2.3) from (6.2.2), we get ∂ + ( )+ A )A ∂x ( ∂  ∂ =A  − ( ) )+ − ∂x (  ∂x

Ai , jk − Ai , kj = − Aα

∂ ∂x k

α

ij

α

α aj

α

α

j

ik

α

a

ik

α

α

ij

k

aj

α

ik

j

α

a

ik

− Aα

ak

a ij

 

α ak

a ij

(6.2.4)

Ai , jk − Ai , kj = Aα Rαijk , where

α Rijk =

∂ ∂x j

( ) − ∂∂x ( ) + α

ik

α

k

ij

α aj

a ik



α ak

a ij

(6.2.5)

The left-hand side of (6.2.4) being the difference of two third-order covariant tensors must also be a third-order covariant tensor with real covariant indices, i, j, k. Hence, the right-hand side must also be a third-order covariant tensor with real indices i, j, k. But already there appears the first-order covariant tensor Aα on the right-hand side of (6.2.4). This necessitates the use of real α covariant indices i, j, k in the symbol Rijk with dummy index α in place of the bracket to yield a covariant tensor only. α ∴ By quotient law, Rijk must be a fourth-order mixed tensor with its explicit form:

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Riemann Symbols



∂ ∂x j

α Rijk =

( ) − ∂∂x ( ) + α

α

α

ik

ij

k

a

aj

ik



α ak

a ij

.

This is called the curvature tensor for the Riemannian metric ds 2 = g ij dx i dx j. α Also Rijk is referred to as the Riemannian symbols of the second kind.

6.3 Derivation of the Transformation Law α of Riemannian Tensor Rabc From the transformation law of Christoffel bracket (symbol) (4.4.5) of the second kind, we get

ij

α ′ p ∂x = ∂x′ p

α ab

∂x a ∂xb ∂ 2 xα . (6.3.1) j + i ∂x′ ∂x′ ∂x′ i ∂x′ j

 i   a



j b

k  . c 

Differentiating partially with respect to x ′ k , 2 α ∂  ′ p  ∂ xα ′p ∂ x + ij ij   ∂ x′ k  ∂ x′ p ∂ x′ k ∂ x′ p

=



∂ ∂xc +

( ) ∂∂xx′ α

ab

α ab

c k

∂x a ∂xb + ∂ x′ i ∂ x′ j

α ab

∂2 x a ∂ xb (6.3.2) ∂ x′ i ∂ x′ k ∂ x′ j

∂ 3 xα ∂ x a ∂2 xb + . ∂ x′ i ∂ x′ j ∂ x′ k ∂ x′ i ∂ x′ j ∂ x′ k

Interchanging j and k in the above expression, ∂   ∂ x′ j 

=

α ′ p  ∂x +  ∂ x′ p

ik

′p

∂ 2 xα ∂ x′ j ∂ x′ p

∂  α  ∂x c ∂x a ∂xb + ab ∂ x c  ( c ↔ b ) ∂ x′ j ∂ x′ i ∂ x′ k +

Eliminating

ik

α ab

α ab

∂2 x a ∂xb (6.3.3) ∂ x′ i ∂ x′ j ∂ x′ k

∂ 3 xα ∂x a ∂2 xb + . ∂ x′ i ∂ x′ k ∂ x′ j ∂ x′ i ∂ x′ k ∂ x′ j

∂ 3 xα from (6.3.3) and (6.3.2), ∂x′ ∂x′ j ∂x′ k i

(b→ c )

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Tensor Calculus and Applications

∂ xα ∂ x′ p

 ∂   j   ∂ x′ 

( )

∂ ∂xb

=

+



α

ac

α ac

ik

∂2 x a ∂x c − ∂ x′ i ∂ x′ j ∂ x′ k

∂ xα ∂ x′ p

 ∂   ∂ x′ j  

kp

 ∂ = b  ∂x

( )

α ab

 ∂   ∂ x′ j    ∂ = b  ∂x +

α ab

∴ Rijk ′p Rijk ′p

∂xλ ∂ x′ k

α λµ

( p ↔ m)

ac

∂ − c ∂x

( ) α

ab

( )



∂ xb ∂ x′ j

λµ

α

∂ ∂xc a

∂x a ∂xb ∂x c ∂ x′ i ∂ x′ j ∂ x′ k

∂2 x a ∂xb + ∂ x′ i ∂ x′ k ∂ x′ j

λµ

( ) α

ab

∂x a ∂2 xb − ∂ x′ i ∂ x′ k ∂ x′ j

α ab

 ′p  ik  

∂ xα − ∂ x′ m

m jp

α λµ

( p ↔ m)

∂xλ ∂ x′ j

α ab

∂x a ∂2 xb ∂ x′ i ∂ x′ j ∂ x′ k    ( µ → a) 

∂x µ ∂ x′ p

(λ →b)

   ( µ → a) 

∂x µ ∂ x′ p

(λ → c )

a

′p  − ∂  ′p  +  ∂ x′ k  ij 

ac

α

ab

a b c  ∂x ∂x ∂x j i  ∂ x′ ∂ x′ ∂ x′ k +

 ∂ xb  ′ m ∂ x a − ik ∂ x′ j  ∂ x′ m ( m→ p )  ik

2 α ∂ 2 xα ′p ∂ x j p − ij ∂ x′ k ∂ x′ p ∂ x′ ∂ x′

′p

∂  ′p   ′p  ij   + ik  −  ∂ x′ k  

α ′m ∂x − ∂ x′ m

α

α ab

ik

( )

∂x a ∂xb ∂x c ∂ − c j i k ∂ x′ ∂ x′ ∂ x′ ∂x

 p − ij ′   



′p  − ∂  ′p   +  ∂ x′ k  ij   

 ∂xc  ′m ∂x a − ij ∂ x′ k  ∂ x′ m ( m→ p ) 

α ac

a

λµ

 ∂xλ ∂x µ  ∂ x′ i ∂ x′ j  

 ∂xλ ∂x µ  ∂ x′ i ∂ x′ k   ′m

ik

jm

′ p − ′m ij

a b c  ∂x ∂x ∂x  ∂ x′ i ∂ x′ j ∂ x′ k −

km

α ′ p  ∂x  ∂ x′ p 

∂xc ∂ x′ k

α ac

a

λµ

∂xλ ∂x µ ∂ x′ i ∂ x′ j

(λ ↔ a , µ →b)

∂xλ ∂xµ ∂ x′ i ∂ x′ k

(λ ↔ a , µ → c)

∂ xα  ∂ = ∂ x′ p  ∂ xb

( ) − ∂∂x ( ) − α

α

ac

ab

c

α

λ

λc

ab

+

α

λ

λb

ac

a b c  ∂x ∂x ∂x × j i  ∂ x′ ∂ x′ ∂ x′ k

α ∂ xb ∂ x c ∂ xα α ∂x = R , abc ∂ x′ i ∂ x′ j ∂ x′ k ∂ x′ p

α = where Rabc

∂ ∂ xb

( ) − ∂∂x ( ) + α

ac

α

c

ab

α λb

λ ac



α λc

λ ab

.

(6.3.4)

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Riemann Symbols

Multiplying both sides by



Rijk ′p

∂x′ h , we get ∂ xα

a ∂ xα ∂ x ′ h ∂ xb ∂ x c ∂ x′ h α ∂x = R abc p ∂ x ′ ∂ xα ∂ x ′ i ∂ x ′ j ∂ x ′ k ∂ xα

or

Rijk ′ p δ p′ h = Rαabc



∴ Rijk ′ h = Rαabc

∂x a ∂ xb ∂ x c ∂ x′ h ∂ x ′ i ∂ x ′ j ∂ x ′ k ∂ xα

∂x a ∂ xb ∂ x c ∂ x′ h , ∂ x ′ i ∂ x ′ j ∂ x ′ k ∂ xα

which is the transformation law of the fourth-order mixed tensor. Rαabc in (6.3.4) is the similar expression for the Riemannian tensor or curvature tensor of the second kind.

α 6.4 Properties of the Curvature Tensor Rijk α α i. Rijk = − Rikj (antisymmetric with respect to the second pair of indices). α α α ii. Rijk + R jki + Rkij = 0. i iii. Contraction in two different ways: (a) Rijk = 0 and (b) Rijαα , a tensor called Ricci tensor.

Proof of (i) By definition,

α Rijk =

∂  ∂x j 

ik

α Rikj =

∂  ∂x k 

ij

α

− ∂   ∂x k 

α ij

α

+ 

bj

+ 

bk

b ik

α



bk



bj

b ij

and

α

− ∂   ∂x j 

α ik

α

b ij

α

b ik

.

α α On comparison, it is clear that Rijk = − Rikj , i.e., antisymmetric with respect to j and k. Hence, proved. Proof of (ii)

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Tensor Calculus and Applications

By definition, it can be written as α Rijk + Rαjki + Rαkij =



∂  ∂x j 

α ik

− ∂   ∂x k 

+

∂  ∂x k 

ji



∂  ∂x j 

ki

α

α

− ∂   ∂xi 

ij

α

b

bi

ji

b

bj

ik

bk

b

bk

α

α



α

+ 



kj b

=

α

+ 

jk

α

+  b

= 0 since

α ij

α



ji

b ij

bi

b jk

+

∂  ∂xi 

α kj

 

b

bj

ki

.

It is called cyclic property. Hence, proved. Proof of (iii) α a. Considering a contraction with the first index in Rijk , we get (i.e., α  = i) i Rijk =

=

∂  ∂x j 

i ik

− ∂   ∂x k 

i ij

+ 

i bj

b ik



i

b

bk

ij

(b ↔ i )

∂  ∂ ∂  ∂   log g  − k  j log g  , g > 0 j  k ∂x  ∂x  ∂x  ∂x 

(

)

(

)

∂2 ∂2 = j k log g − k j log g = 0 ∂x ∂x ∂x ∂x

(

i



ij

=

)

(

)



∂ log ± g . ∂x j

(

)

α b. Considering a contraction with the third index in Rijk , we get (i.e., α = k)

Rijαα = = ∴ Rij =

∂  ∂x j 

α iα

− ∂   ∂ xα 

α ij

α

+ 

b iα

bj

∂  ∂ ∂  log g  − α  j  i ∂x  ∂x  ∂x 

ij

∂2 ∂ log g − α  ∂xi ∂x j ∂x 

+ 

(

(

)

)

α ij

α



+ 

α ij

α

b

bj

α bj

b



b iα







ij

b

b ij

∂ log g , when g > 0 ∂xb

(

)

∂ log g . ∂xb

(

)

This contracted tensor is called the Ricci tensor. In future, it will have a great deal of application in the field equation of general theory of relativity. It can also be easily seen that Rij = Rji.

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Riemann Symbols

6.5 Covariant Curvature Tensor α The fourth-order covariant curvature tensor (by virtue of definition Rijk ) is defined as a Rhijk = g ha Rijk . (6.5.1)



The symbols Rhijk are known as the Riemannian symbols of the first kind. Now,

( ) − ∂∂x ( ) +

 ∂ Rhijk = g ha  j  ∂x

(

=

∂ g ha ∂x j

=

∂ ∂x j

Rhijk =

a

a

ik

a ik

ij

k

a bj

b ik

) − ∂∂x ( g ) − ∂∂gx a

( ) − ∂∂x ( ) − ( ik , h

k

ha

ha ij

k

ij , h

hj , a

+

j

aj , h

)



a

b

bk

a ik

a ik

ij

 

+

∂ g ha ∂x k

+

(

hk , a

a ij

+

+ g ha

ak , h

)

a bj

a ij

b ik

+

− g ha b

bj , h

ik

a bk



b ij

b bk , h

ij

1 ∂  ∂ g ih ∂ g kh ∂ g ik  1 ∂  ∂ g ih ∂ g jh ∂ g ij  + − h − + − 2 ∂ x j  ∂ x k ∂xi ∂ x  2 ∂ x k  ∂ x j ∂ x i ∂ x h  − gα a + gα h



α hj

a ik

− gα h

b α bj ik (b→ a )

− gα h

α aj

a ik

+ gα a

α hk

a ij

+ gα h

α ak

a ij

b α bk ij (b→ a )



∂2 g jh ∂2 g ij  ∂2 g ∂2 g ∂2 g 1  ∂2 g =  j ih k + j kh i − j ik h − k ih j − k i + k h  2  ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x  + gα a =

α hk

a ij

− gα a

α hj

a ik

∂2 g jh  ∂2 g kh ∂2 gik 1  ∂2 g ij  h k + i j − j h − i k 2  ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x  + gα a

α hk

a ij

− gα a

α hj

a ik

,

which is the expression of the curvature tensor of the first kind.

(6.5.2)

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Tensor Calculus and Applications

6.6 Properties of the Curvature Tensor Rhijk of the First Kind Rhijk = − Rihjk . i. ii. Rhijk = −Rhikj (antisymmetric with respect to the second pair of indices). iii. Rhijk = Rjkhi (symmetric with respect to the first and second pairs of indices). iv. Riijk = Rhikk = 0. v. Rhijk + Rhjki +Rhkij = 0. By definition (6.5.2), we get

Rhijk =

2 ∂2 g hj ∂2 g hk ∂2 g ik 1  ∂ g ij + − −  2  ∂x h ∂x k ∂xi ∂x j ∂x h ∂x j ∂xi ∂x k

  + gα a 

α hk

a ij

α

− gα a

ik

a hj

.

Interchanging h and i, Rihjk =

∂2 g ij  ∂2 g ik ∂2 g hk 1  ∂2 g hj  i k + h j − i j − h k 2  ∂x ∂x ∂x ∂x  ∂x ∂x ∂x ∂x + gα a

α ik

a hj

− gα a

α ij

a hk



(α ↔ a ).

Comparison gives the following result: i. Rhijk = −Rihjk, i.e., antisymmetric with respect to first pair of indices. ii. Similarly, it can be proved Rhijk =−Rhikj, i.e., antisymmetric with respect to second pair of indices. iii. Rhijk = Rjkhi, i.e., symmetric with respect to the first and second pair of indices. Proof From definition (6.5.2), it follows

Rhijk =

2 ∂2 g hj  ∂2 g hk ∂2 g ik 1  ∂ g ij  h k + i j − h j − i k  + gα a 2  ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 

α hk

a ij

α

− gα a

hj

and R jkhi =

∂2 g ji ∂2 g jh  ∂2 g ki 1  ∂2 g kh − + gα a + − j j i k h h 2  ∂ x ∂ x ∂ x ∂ x ∂ x k ∂ x i  ∂x ∂x − gα a

α ki

a jh

(α ↔ a).

α ij

a kh

a ik

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Riemann Symbols

∂2 g ik ∂2 g ki , i.e., symmetric j = h ∂x ∂x ∂x j ∂x h with respect to the two pairs of indices. Hence, proved. iv. Riijk = Rhikk = 0. Replacing k by i and j by k, it can easily be proved from definition. v. Rhijk + Rhjki +Rhkij = 0 α Using the definition of Rijk , it is proved in Section 6.4 (ii) that α α α Rijk + R jki + Rkij = 0, the cyclic property. Clearly, Rhijk = Rjkhi ∴

α

α

=

ij

ji

and

Multiplying it by g hα and summing over α , α g hα Rijk + g hα Rαjki + g hα Rαkij = 0



∴ Rhijk + Rhjki + Rhkij = 0.

Hence, proved.

6.7 Bianchi Identity Let

of a geodesic coordinates xi for which ∂ Ai , Ai is any tensor (vector). jk = 0 = ij , k and so Ai , j = ∂x j Now from definition of curvature tensor of the second kind, we have i

Po



be

the

pole

a Rijk =

∂ ∂x j

( ) − ∂∂x ( ) + a

a

ik

k

α

a

αj

ij

ik



α

a

αk

ij

. (6.7.1)

Considering xl-covariant differentiation of (6.7.1) with respect to gij, we get

a Rijk ,l =

∂2 ∂xl ∂x j

( ) − ∂x∂∂x ( )

at pole Po

a Rikl ,j =

∂2 ∂x j ∂x k

( ) − ∂x∂ ∂x ( )

at pole Po

Rilja , k =

∂2 ∂x k ∂xl

( ) − ∂ x∂ ∂ x ( )

at pole Po .

2

a

ik

l

a

k

ij

Similarly,

2

a

il

j

a

l

ik

and

2

a

ij

k

a

j

il

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Tensor Calculus and Applications

Addition of all these three relations gives a a a Rijk , l + Rikl , j + Rilj , k = 0 at pole Po . (6.7.2)



Each of the terms of the equation is a tensor, so it holds for all coordinate systems and at all points. Hence, it is an identity instead of an equation. It is known as Bianchi identity. The inner product of (6.7.2) with gha (summing over a) gives

(g

ha

a Rijk

) + (g ,l

ha

a Rikl

) + (g ,j

ha

Rilja

)

= 0  g ha , k = 0, etc.

,k

Rhijk ,l + Rhikl , j + Rhilj , k = 0 .



It is an alternative form of the Bianchi identity.

6.8 Einstein Tensor Is Divergence Free a a a The Bianchi identity Rijk , l + Rikl , j + Rilj , k = 0 can be written as



(

)

a a a a a Rijk , l − Rilk , j + Rilj , k = 0 Rikl , j = − Rilk , j .

Considering a contraction with respect to a and k,



a a a Rija , l − Rila , j + Rilj , a = 0

(R

a ija

)

= Rij , Rii is Ricci tensor .

An inner multiplication of it with gil gives

(g R ) − (g R ) + (g R ) il



ij

il

,l

il

il

,j

a ilj

,a

=0

 g il is covariant constant Rlj , l − R, j + R aj , a = 0.

It can be written as 2R ij , i − R, j = 0, changing l → i and a → i.

1   ∴  R ij − δ ji R  = 0 .   ,i 2

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Riemann Symbols

1 Hence, G ij , i = 0, where G ij = R ij − δ ji R is the Einstein tensor. 2 ∴ The Einstein tensor is divergence free ( contraction is with respect to covariant derivative index which is required for definition of divergence).

6.9 Isometric Surfaces The intrinsic geometry of a surface is based on the corresponding fundamental quadratic form or metric ds2 = aαβ duα uβ with surface coordinates uα = uα (u1 , u2 ). The intrinsic properties such as the lengths of curves and the angle between two intersecting curves primarily depend on the metric tensor of the surface and its derivatives. If there exists a coordinate system which characterizes the linear element of two surfaces S1 and S2 by the same metric aαβ , then they are called isometric. The corresponding parameters for the transformation is known as isometry. From the Euclidean plane, surfaces of cylinder and cone can be constructed by means of rolling without changing arc length, areas, and measurement of angles. Hence, they are isometric with the Euclidean plane.

6.10 Three-Dimensional Orthogonal Cartesian Coordinate Metric and Two-Dimensional Curvilinear Coordinate Surface Metric Imbedded in It In order to enter into the threshold of geometry of surfaces in a surrounding space, we need to consider two distinct coordinate systems. Let uα (u1 , u2 ) be the two curvilinear coordinates of the surface S imbedded in a three orthogonal Cartesian coordinate systems in E3. But the intrinsic property of geometry of a space is characterized by the metric or quadratic differential form. Let xi = xi (y1, y2, y3), i = 1, 2, 3, be the orthogonal Cartesian coordinates covering the space E3 and xi = xi (u1, u2) be the Gaussian surface coordinates of S imbedded in E3. The line element in E3 is given by ds 2 = g ij dx i dx j, where

dx i =

∂y k ∂y k ∂xi α = du and g ij ∂uα ∂xi ∂x j

= g ij

∂xi α ∂x j β du du ∂uα ∂ uβ

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Tensor Calculus and Applications

ds2 = g ij

∂xi ∂x j α β du du = aαβ duα duβ , ∂uα ∂uβ

where



(6.10.1)

aαβ = g i j

∂xi ∂x j , ∂uα ∂uβ

so that a = gJ2,|aαβ |= a, | g i j |= g. ∂xi α du , it can be concluded that dx i is a space vector and ∂uα is surface invariant, and duα is a surface vector and is space invariant. Looking at dx i =

6.11 Gaussian Curvature of the Surface S immersed in E 3

(

)

If the functions aαβ and bαβ = g ij in (6.10.1) related to some surface, then xi is to satisfy the condition of integrability: ∂2 xαi ∂2 xαi , (6.11.1) β γ = ∂u ∂u ∂uγ ∂uβ



∂xi is continuously differentiable of degree 2. ∂uα ∂xi Now denoting α by tαi which is tangent to the surface and considering ∂u the covariant (surface) derivative of it, we can get where xαi =



tαi , β =

∂tαi + ∂ uβ

i j jk α

t tβk −

γ i αβ tγ

=

∂2 x i + ∂uα ∂uβ

i j jk α

t tβk −

γ i αβ tγ .

Since tαi is tangent to the surface, its partial derivative tαi , β is normal to the surface; otherwise, it is proportional to the normal ni of the surface. Therefore, tαi , β = bαβ ni or

 ∂2 x i bαβ = tαi , β ni =  α β +  ∂u ∂u

i jk

tαj tβk −

γ αβ

 tγi  ni. (6.11.2) 

But for the use of Cartesian coordinates and geodesic surface coordinates, the Christoffel symbols can be made to zero at a particular point. Of course, i the derivatives of the Christoffel symbols in space will vanish but jk γ not the symbols with surface αβ .

( )

( )

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Riemann Symbols

∂3 x i ∂  ∈  i − αβ t∈.  ∂u ∂uβ ∂uγ ∂uγ  ∂3 x i ∂  ∈  i i Similarly, tα ,γβ = α γ β − β γα t∈.  ∂u ∂u ∂u ∂u 

Hence, tαi , βγ =



α

∈ ∴ tαi , βγ − tαi , γβ = Rαβγ t∈i . (6.11.3)

∈ Using (6.11.1) and (6.11.2) and making use of definition for Rαβγ i i From tα ,  β = bαβ n , we can get



tαi ,  βγ = bαβ , γ ni + bαβ ni , γ = bαβ , γ ni − bαβ bγδ aδε t i ε . ( ni , α =   −   bαβ a βγ t iγ  from Weingarten ′s formula due to the linear 



combination of nαi  and tβi , namely, n, i α = Cαβ tβi )



∴ tαi ,  βγ − tαi , γβ = ( bαβ , γ − bαγ ,  β ) ni + ( bαγ bβδ − bαβ bγδ ) aδε tεi .



ε ∴ Rαβγ tεi = ( bαβ , γ − bαγ ,  β ) ni + ( bαγ bβδ − bαβ bγδ ) aδε tεi , using (6.11.3)

∴ bαβ , γ − bαγ ,  β = 0, (6.11.4)



multiplying by ni and using nitεi = 0. This is known as Codazzi equation of surface. Putting this value, the above equation can be reduced to

ε Rαβγ tεi = ( bαγ bβδ − bαβ bγδ ) aδε tεi .

Therefore, for arbitrary tεi , it can be transformed to

Rλαβγ = bαγ bβλ − bαβ bγλ . (6.11.5)  aλ ∈aδ ∈ = δ λδ .

This is known as Gauss equation of surface. But for Riemannian curvature tensor Rλαβγ , Rααβγ = Rλαββ = 0, the only surviving component is R1212 = − R2112 = − R1221 = R2121. Since the surface x i = x i u1 ,  u2 in two-dimensional curvilinear coordinates is immersed in E3 with the metric aαβ  or  aαβ = a, the quantity defined R by κ = 1212 is called the total curvature or the Gaussian curvature. a 2 But R1212 = b11b22 − b12 b21 = b11b22 − b12 = b, from (6.11.5) so that

(

)

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Tensor Calculus and Applications



κ=



=

2 b11b22 − b12    =  a

b  a

2 b11b22 − b12 2 a11 a22 − a12

is the Gaussian curvature of two-dimensional surface. Example 1 Prove that the differential equation Ai,j = 0 is integrable only when the Riemann Christoffel tensor vanishes. a = 0, if the differential equation Ai,j = 0 is It needs to show that Rijk integrable. Now, ∂ Ai − Aa ∂x j

Ai , j = 0 gives



∂ Ai j dx = Aa ∂x j



a ij

dx j ∴ Ai =

a

= 0. (i)

ij

∫A

a

a ij

dx j . (ii)

This shows that the right-hand side of (ii) must be integrable, and hence, it should be a perfect differential of some function, say Bi, so that

a

Aa

ij

dx j = dBi

∂Bi j dx = Aa ∂x j







 ∂Bi  j − Aa ∂x

a ij

a ij

dx j

 j  dx = 0.

a ∂Bi = Aa ij  dx j is arbitrary. ∂x j Differentiating partially with respect to xk, we get

It gives

∂2 B ∂ Aa = ∂x j ∂x k ∂x k



a ij

+ Aa

∂ ∂x k

( ). (iii)

∂ ∂x j

( ). (iv)

a

ij

Interchanging j and k, it can be written as ∂2 B ∂ Aa = ∂x k ∂x j ∂x j



a ik

+ Aa

a

ik

From (iii) and (iv),



∂ Aa ∂x j

a ik

 ∂ + Aa  j  ∂x

b a aj ik ( a ↔ b)

Ab

+ Aa

∂ ∂x j

a ik

 ∂ Aa  − k ∂x

( )− A a

ik

a ij

 ∂ − Aa  k  ∂x

b a b ak ij ( a ↔ b)

− Aa

∂ ∂x k

a ij

  = 0

( ) = 0. a

ij

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Riemann Symbols

(making use of (i))  ∂ ∴ Aa  j  ∂x



( ) − ∂∂x ( ) + a

a

ik

k

ij

a bj

b ik



a bk

b ij

  = 0.

a a Aa Rijk = 0 ∴ Rijk = 0 Aa is arbitrary in the inner product. Hence, proved.

Example 2 Show that the number of independent components of the Riemannian curvature tensor of the first kind Rhijk in n-dimensional space Vn is 1 2 2 n n −1 . 2 In general, the number of independent components of the fourth-order tensor Rhijk in a Riemannian space Vn is n4. But due to the following properties, the number of independent components will be reduced from n4.

(

i Rhijk ii Rhijk iii Rhijk iv Rhijk

)

= − Rihjk (antisymmetric property with respect to h and i). = − Rhikj (antisymmetric property with respect to j and k). = R jkhi (symmetric with respect to the first and second pairs). + Rhjki + Rhkij = 0 (cyclic property).

Case I: When there is only one distinct index of the type Rhhhh

Rhhhh = − Rhhhh ( due to (i)) ∴ Rhhhh = 0.

Hence, there is no component of Rhijk with one distinct index. Case II: When there are two distinct indices of the type Rhihi ( h ≠ i ) The two distinct indices from n values can be selected in n(n − 1) ways which correspond to n(n − 1) numbers of independent components of Rhijk. But Rhihi = − Rihhi = Rihih, due to (i) and (ii). After interchanging h and i in the first, the last one can be recovered. 1 1 ∴ The number n(n − 1) is reduced only by so that it becomes n(n − 1). 2 2 Also, Rhihi = Rhihi (due to (iii)), so there is no reduction due to this property. Interestingly, the cyclic property

Rhihi + Rhhii + Rhiih = Rhihi − Rhihi = 0

is identically satisfied due to (ii). Hence, there is no reduction due to the cyclic property.

1 n(n − 1). 2 Case III: When there are three distinct indices of the type Rhihj ∴ The values of h, i, j can be selected from n values in n(n − 1)(n − 2) ways, and hence, the number of independent components of Rhijk in this case is n(n − 1)(n − 2). But this number will be reduced for the properties (i)–(iv). ∴ The number of independent components in this case is

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Tensor Calculus and Applications

Now, Rhihj = Rhjhi (due to (iii)) which can be recovered by merely interchanging i and j  = −Rjhhi (due to (i))  = −Rhijh (due to (iii)) which is nothing but (ii). Hence, due to symmetric properties, the number of independent com1 1 ponents is finally reduced by only to yield n(n − 1)(n − 2). 2 2 The cyclic property Rhihj + Rhhji + Rhjih = Rhihj + 0 + Rihhj = Rhihj − Rhihj = 0 is identically satisfied, and hence, there is no reduction of the above number. Case IV: When all the four indices are distinct of the type R hijk Clearly, the values of h, i, j, k can be selected in n(n − 1)(n − 2)(n − 3) ways, and hence, the number of independent components in this case is n(n − 1)(n − 2)(n − 3). But due to the three symmetric properties, (i)–(iii), it is reduced to 1 1 1 × × n(n − 1)(n − 2)(n − 3). 2 2 2 But from (iv), we get Rhijk + Rhjki = −Rhkij. This shows that, knowing two components, the third can be readily determined. Hence, due to (iv), the number of independent components 2 is reduced by . 3 ∴ In this case, the number of independent components of R hijk is 1 2 × n(n − 1)(n − 2)(n − 3). 8 3 Hence, the total number of independent components of the Rieman­ nian curvature tensor R hijk is 0+ =

1 1 1 n(n − 1) + n(n − 1)(n − 2) + n(n − 1)(n − 2)(n − 3) 2 2 12

1 n(n − 1) [ 6 + 6(n − 2) + (n − 2)(n − 3)] 12

1 n(n − 1) 6n − 6 + n2 − 5n + 6  = 12 =



1 2 2 1 n(n − 1)n(n + 1) = n (n − 1). 12 12

Hence, proved.

Exercises 1 1. Show that the divergence of the tensor R ij − δ ji R is identically zero. 2 2. If the metric of V2 formed by the surface of a sphere of radius r is ds2 = r2 (dθ 2 + sin2θ dϕ 2) in spherical polar coordinates, show that R1212 = r2sin2 θ.



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Riemann Symbols



a 3. Derive the expression of the curvature tensor of the second kind Rijk .



4. Derive the expression of the curvature tensor of the first kind Rhijk . 5. Discuss the properties of Rhijk . 6. For a V3 referred to a triply orthogonal coordinate system, prove that 1 Rikkj (where i ≠ j ≠ k) Rij = g kk 1 1 Rhiih + Rhjjh . and Rhh = g ii g jj 7. Calculate the Ricci tensor Rij for the metric on the sphere ds2 = a2 (dθ 2 + sin2 θ dϕ 2), where i, j = 1, 2, x1 = θ and x2 = ϕ, and a is constant. α i ∂ [Hint: Find ij and use ij = j (log g ).] ∂x 8. Show that on a two-dimensional surface, the curvature tensor is completely defined by a single component, say R1212. 9. For a surface with the metric ds2 = (du)2 + λ 2 (dv)2 , show that the 1 ∂2 λ Gaussian curvature is − . λ ∂u2 2 2 2 2 10. Show that the surface with the metric ds2 = u2 du1 + u1 du2 is isometric or developable. [Hint: Show κ  = 0.] 11. Find the conditions that the surfaces S1 : y 1 = v 1 cos v 2 ,  v1  y 2 = v 1 sin v 2 , y 3 = a cosh −1   ; S2 : y 1 = u1 cos u2, y 2 = u1 sin u2 , y 3 = au2  a are isometric. [Hint: Show that the two metrics are the same subject to some conditions.]

( )( ) ( )( )

Part II

Application of Tensors

7 Application of Tensors in General Theory of Relativity As mentioned in Chapter 1, the knowledge of the geometry of space is important for Newtonian (classical) as well as Einstein’s relativistic mechanics which is reflected in the statement “Dynamics deals with the geometry of motion.” To develop geometry of a space, the paramount importance is the assumption of coordinate systems to suitably describe the space concerned based on the corresponding metric. Deformation is an essential notion to invite the concept of “tensors” in non-isotropic medium from an applicable point of view in mathematical science. Tensors being independent of any coordinate system possess the intrinsic property of the geometry of space.

7.1 Introduction The general theory of relativity is known as the theory of gravitation. For applicability of principle of relativity [5, p. 17] to preserve fully the privileged position among all conceivable frames of reference, the concept of special theory of relativity (STR) based on Lorentz transformation [5, p. 39] is developed. To develop the new theory of gravitation, the idea of privileged position, namely, inertial frames is destroyed to include the most general form of transformation applicable to any positive integral number of dimensions. Setting aside the Galileo’s view of “law of inertia” and the amount of gravitational action of one mass point on another (great mass) of Newton (Kepler), Einstein gave a different interpretation. He has concluded that gravity is not a force (as Newton believed), but the curvature of space–time, and the matter is the source for it and material objects create the gravitational field, which distorts (deforms) or curves the surrounding space–time as the magnet sets up the magnetic field. So, the generating space due to the presence of material objects demands the use of non-Euclidean geometry of curved space for its true description. Otherwise, curvilinear coordinates are essential at large as rectilinear coordinates cannot be set up in the curved region of space– time. Hence, curvature of the space–time continuum is the fundamental ingredient for characteristic representation of gravitational theory or the general theory of relativity. Eventually, tensors applicable to all coordinate 113

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Tensor Calculus and Applications

systems developed in Riemannian space of n dimensions (a manifold) are the best tool (as detected by Einstein) to study general theory of relativity [5,6]. There is no gravity and no curvature, so free particles follow geodesics (shortest path), i.e., straight lines, when the space–time is flat in special theory of relativity. Therefore, to enter into the threshold of general theory of relativity, it is essential to know the curvature of the space–time, a four-dimensional manifold which is a clear deviation from flat space of special theory of relativity.

7.2 Curvature of a Riemannian Space Riemann adopted the Gaussian curvature (Section 6.11) of a geodesic surface   S at a point P determined by the orientation of the unit vectors p( p i ) and q(q i ) at P as the definition of (Riemannian) curvature of Vn at that point. The pencil of directions can be expressed as

(

)

y i = α p i + β q i s (7.2.1)



 dy i  with unit tangent ti =  of geodesics at P, where α and β given by α s = u1  ds  and β s = u2 are the current coordinates of points on S generated through y i = u1 p i + u2 q i . (7.2.2)



If the metric of Vn is denoted by gij dyi dyj and that of the surface S by aαβ duα duβ (α, β = 1, 2 ) , then dy i dy j . (7.2.3) duα duβ

aαβ = g ij



∴ Using Equation (5.8.5), we can write

( ) ij , k



g

 ∂2 y β = aβ c  i j +  ∂x ∂x = aβ c

( ) β

αγ

∂y c ∂2 y β + ∂ u k ∂ ui ∂ u j

a

∂ yα ∂ yγ  ∂ y c  ∂xi ∂x j  ∂x k

( ) αγ , c

a

∂ yα ∂ yγ ∂ y c ∂ ui ∂ u j ∂ u k

Changing variables of the present systems; in this assumption αγ , c for aαβ (uα) and ij , k for gij(yi). ∂ yα ∂ yγ ∂ y c ∴ ij , k = αγ , c , using (7.2.2) where pi, qi are constants. g a ∂ ui ∂ u j ∂ u k

( ) ( )

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Application of Tensors

Hence,

(



αβ ,γ

)

g

= g kl

∂y i ∂y j ∂y k ∂uα ∂uβ ∂ yγ

( ) (7.2.4) l

ij

a

i ↔α

changing



γ



β↔ j

c



γ ↔k

on both sides.

But yi ’s are the Riemannian coordinates of geodesics of Vn for the metric gij dyi dyj and Christoffel symbol αβ ,γ = 0. ∴

( )

(

l

ij

a

)

g

= 0 at the origin P, from (7.2.4).

Hence, the Riemannian curvature tensor Rh𝛼𝛽𝛾 for the surface S can take nonzero values R1212 for two values α,β = 1,2 subject to the properties Rhαβγ = − Rα hβγ , Rhαβγ = − Rhαγβ , Rhαβγ = Rβγ hα  Rhhβγ = Rhαββ = 0. ∴ The nonvanishing Riemannian curvature tensors are R1212 , − R2112 , − R1221 , and R2121



 In two-dimensional space, the number of independent components is 1  N 2 ( N 2 − 1)  = 1 only, namely, R1212. 12  ∴ From transformation laws of tensors, R1212 ′ = Rhαβγ

∂uh ∂uα ∂uβ ∂uγ ∂u′ 1 ∂u′ 2 ∂u′ 1 ∂u′ 2 2

2

 ∂u1   ∂u2  ∂u2 ∂u1 ∂u1 ∂u2 − = R1212  R 1212 ∂u′ 1 ∂u′ 2 ∂u′ 1 ∂u′ 2  ∂u′ 1   ∂u′ 2 

− R1221

∂u1 ∂u2 ∂u2 ∂u1 ∂u′ 1 ∂u′ 2 ∂u′ 1 ∂u′ 2

+ R2121

∂u2 ∂u1 ∂u2 ∂u1 ∂u′1 ∂u′ 2 ∂u′1 ∂u′ 2

(7.2.5)

2

 ∂u1 ∂u2 ∂u1 ∂u2  ∴ R1212 − . ′ = R1212  1 2 ∂u′ 2 ∂u′1   ∂u′ ∂u′ The Gaussian curvature κ (Section 6.11) which is invariant for coordinate transformations is defined as

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κ= =

R1212 R1212 ′ = where a = aαβ = a a′ R1212 R1212 ′ = a a a11a22 − a 212

(

a11 a21

a12 a22

 a′ = a J 2 ,

)

∂u and a = aαβ , a′ = aαβ ′ . ∂u′ Now, by definition,

where J =

 ∂ a Rhijk = g ha Rijk = g ha  j  ∂x



∴ R1212 

(

αβ ,γ

)

a

∂ = 1 ∂u

( ) 22,1

( ) − ∂∂x ( ) + a

a

ik

∂ − 2 a ∂u

k

ij

a bj

b ik



a bk

( ). 21,1

b ij

 

(7.2.6)

a

= 0 at the origin P for Riemannian coordinates.

If L, M, N are the second-order and E, F, G are the first-order magnitudes of Gaussian surface* characterized by ds 2 = Ldu2 + 2 Mdudv + Ndv 2 and ds 2 = Edu2 + 2 Fdudv + Gdv 2, respectively, then R1212 LN − M 2 = a EG − F 2



a = EG − F 2 = a11a 22 − a 212 ,

where E = a11, F = a12 = a21, G = a22. cofactor of a11 in aαβ = a a But a11 = = 22 . a a a12 22 a11 12 Similarly, a = − ,a = a a and



a11 = g ij

∂y i ∂y j = g ij p i p j = g hj p h p j ∂u1 ∂u1

a22 = g ij

∂y i ∂y j = g ij q i q j = g ik q i q k ∂ u2 ∂ u2

a12 = g ij

∂y i ∂y j = g ij p i q j = g hk q k p h = g ji p j q i ∂u1 ∂u2

∴ a = a11a22 − a 212 = g hj g ik p h p j q i q k − g hk g ij p h p j q i q k = p h p j q i q k ( g hj g ik − g hk g ij ). * Willmore [4].

(7.2.7)

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Now, from (7.2.4),

(

22,1

)

a

= g hl

∂y i ∂y j ∂y h ∂u2 ∂u2 ∂u1

= g hl q i q k p h

∂ ∴ 1 ∂u

(

22,1

)

l ik

∂ = g hl q q p ∂y j i k

a

( )

h

= g hl q i q k p h p j

( ) l ij

( j → k)

g

( ) l ik

∂ ∂y j

( k → h)

g

g

(7.2.8)

∂y j ∂u1

( ). l ik

g

Also,

(

21,1

)

a

= g hl

∂y i ∂y j ∂y h ∂u2 ∂u1 ∂u1

= g hl q i p j p h

∂ ∴ 2 ∂u

(

21,1

)

l ij

∂ = g hl q p p ∂y k i

a

( )

j

h

= g hl q i p j p h q k

( ) l ij

g

g

( )

∂ ∂y k

l ij

g

∂y k ∂ u2

(7.2.9)

( ). l ij

g

Hence, (7.2.6) can be written as (using 7.2.8 and 7.2.9)



 ∂ R1212 = g hl p h p j q i q k  j  ∂y

( ) − ∂∂y ( ) l ik

l ij

k

l = p h p j q i q k g hl Rijk = p h p j q i q k Rhijk .

∴ The curvature

κ=

p h p j q i q k Rh ijk R1212 = h j i k (7.3.10) a p p q q g hj g ik − g hk g ij

(

)

is invariant. This is the mathematical expression of curvature of the Riemannian space Vn with Rhijk. This ascertains the nomenclature “covariant curvature tensor” for Rhijk. Hence, Rhijk (or Rhijk) characterizes the behavioral properties of the curve space of general theory of relativity.

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7.3 Flat Space and Condition for Flat Space Definition If the curvature κ of the Riemannian space Vn, namely Rhijk vanishes, it is called flat space. Condition for Flat Space From the definition of curvature shown in (7.3.10), where Rhijk is the covariant curvature tensor, and pi, qj are the unit vectors showing orientation at the origin P of geodesics.

κ=



p h p j q i q k Rhijk . p p q q ( g hj g ik − g ij g hk ) h

j i k

For flat space κ = 0 ⟹ Rhijk = 0, the bracket in the denominator must not be zero. a a ∴ If Rhijk = g ha Rijk = 0, or if all the components of Rhijk or Rijk are zero, then the curvature will be zero, and the space will be flat. This is the required condition for flat space. a b a b ∂2 g hj  ∂2 g ∂2 g 1  ∂2 g ij But Rhijk =  h k + i hk j − h ik j − i k  + g ab hk ij − g ab hj ik . 2  ∂x ∂x ∂x ∂x  ∂x ∂x ∂x ∂x

(

)

b

If gij ’s are constants, ij , k = ij = 0, ∴ Rhijk = 0. Hence, gij = constants are basically the conditions for flat space. Note: STR is restricted to flat space only.

7.4 Covariant Differential of a Vector Let Ai be the components of a vector at the point xi and Ai + dAi be the components at a neighboring point xi + dxi in a vector field. Therefore, the difference dAi of the two vectors Ai + dAi and Ai being ordinary differential is a vector. ∂ x′ i k The transformation of the coordinates gives dx′ i = dx , otherwise, ∂x k ∂ x′ i k i A stands for the transformation of a vector. A′ = ∂x k But

dA′ i =

∂ x′ i ∂ 2 x′ i k dA + dx j A k , ∂x k ∂x j ∂x k

(7.4.1)

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which is not a vector for the presence of the second term. Of course, for linear transformation belonging to rectilinear coordinate systems, this ­transformation characteristically behaves like a vector since the second term vanishes in that case. On the other hand, the difference of two vectors needs to be a vector in a general coordinate system. Hence, to get this difference to be a vector in a curvilinear coordinate system, it necessitates to translate a vector Ai at xi to the location of xi + dxi of the other vector Ai + dAi so that they are located at the same point. This translation is related to parallel translation to itself. In a general curvilinear coordinate system, this translation by itself changes the components of the vector, and the changes are denoted by δAi different from ordinary differential dAi. From (7.4.1), it is observed that this change δAi should be proportionate to both the vectors Ak (i.e., Ai) and  ∂ 2 x′ i  the displacement dxj  if j k ≠ 0 .   ∂x ∂x Hence, the difference between the original vector Ai + dAi and the transported vector Ai + δAi at the point (xi + dxi) is given by

( DA ) = dA i



i

– δ Ai . (7.4.2)

∴ In consistent with the above changes in the components Ai of the vector due to translation and the components of the displacement dxi, the changes δAi in (7.4.2) can be written as†

δ Ai = −

(

i jk

A k dx j . (7.4.3)

i jk

are some functions of coordinates xi) Thus, (7.4.2) is called the covariant differential of the given vector Ai.

7.5 Motion of Free Particle in a Curvilinear Co-Ordinate System for Curved Space dx i (i = 1, 2, 3, 4) are the four vectors (velocity) tangential to the time ds track of a free particle, it is represented in a rectilinear coordinate system by the equations dui = 0. This gives xi = As + B which is the intrinsic equation of a straight line in a four-dimensional continuum. Eventually, dui = 0 characterizes the uniform rectilinear motion in three-dimensional physical space also. But to write the corresponding equations of motion in a curvilinear coordinate system, we If ui =



Section 6.4 of Ref. [5].

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are to make use of covariant differential instead of ordinary differential. Hence, the equations of motion of free particle in the curvilinear coordinate system are given by

( Du = ) du – δ u i

i.e., dui +



i jk

i

i

= 0 (7.5.1)

uk dx j = 0 from Equations (7.4.2) and (7.4.3),

i.e. ,

d2 xi + ds2

i jk

dx j dx k = 0 (7.5.2) ds ds

which are the differential equations of geodesics, and contextually, they characterize the world line (track) of free particle in the general coordinate system. j k i dx dx d2 xi If we compare Equation (7.5.2) or 2 = − jk with Newtonian gravds ds ds 2 i i d x dφ = −δ ij j , then jk can be itational equation with potential ϕ, namely, 2 dt dx i attributed to “forces” that arise inherently in the system. Hence, jk ≠ 0 or g ij ≠ constant is the generating factor of the potential (ϕ) and responsible for the curvature of the path of free particle caused by gravitational field or mass energy of matter in curved space.

7.6 Necessity of Ricci Tensor in Einstein’s Gravitational Field Equation In classical (Newtonian) mechanics, the field equation in the presence of matter according to Newton’s law of gravitation is described by Poisson’s equation: ∇ 2φ = 4πγρ ,

i.e. ,

(7.6.1) ∂2 φ ∂2 φ ∂2 φ + 2 + 2 = 4πγρ , 2 ∂x ∂y ∂z

where ϕ is the gravitational potential, ρ is the density of distributed matter, and γ is the gravitational constant. So, to get its general relativistic analogue, we need to correlate these quantities suitably under physical system introducing general relativistic concepts.

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1. It is discussed is Section 7.1 that the material energy creates the gravitational field to turn the space–time continuum into a curved space. Therefore, the counterpart of Equation (7.6.1) should be made applicable to curved space. 2. For the “principle of covariance,”‡ an essence of general relativity, all natural laws must be expressed in tensor forms (Covariant) for their validity in all coordinate systems including non-inertial frames of curved space required for general theory of relativity. Hence, Equation (7.6.1) needs to be expressed completely in tensor form. 3. It is shown in Section 7.5 that the equations of motions of free fall of a particle in curvilinear coordinates of space–time are characterized j k i dx dx d2 xi by the geodesic equations + jk = 0 , which is reducible 2 ds ds ds 2 i d x ∂φ to Newton’s equation of motions 2 = − i (= −∇φ ), where ϕ is the dt dx gravitational potential. It has already been mentioned in Section 7.5 that gij (≠ constant) 2φ   is responsible to generate gravitational potential φ  g 44 = 1 ± 2  c   [Section 7.3 of Ref. (5, p. 191)]. Hence, in relativistic theory of gravitation, ϕ is to be replaced by the metric tensor g µυ (≠ constant) or by some relation with g µυ . Moreover, the left-hand side (L.H.S.) of Equation (7.6.1) does not involve the derivatives of ϕ higher than two (second), so the replacement tensor (for covariant form) must also contain the second-order derivatives of g µυ , but the Ricci tensor Rµυ is the tensor with second-order derivatives of g µυ as β α α α β α ∂ ∂ Rµυ = υ µα − α µυ + βυ µα − βα µυ ∂x ∂x α ∂g  ∂g 1  ∂g with µυ = gα a µυ , a = gα a  µυa + υµa − µυa  2 ∂x  ∂x  ∂x ∂  α  ∂2 and υ µα = υ µ (log ± g ).  ∂x ∂x ∂x  Hence, the suitable relativistic analogue of the L.H.S. of Equation (7.6.1) is the Ricci tensor, if it satisfies some other conditions in conformity with the right-hand side (R.H.S.). 4. In agreement with the L.H.S. of (7.6.1), we need a second-order tensor for the R.H.S. in place of density ρ of matter. The gravitational field is the outcome (effect) of mass distribution or mass–energy distribution of matter, and energy momentum tensor Tµυ or T µυ characterizes the cause of the distributions. Hence, the density ρ of material particles generating the force of gravity is

( )





Section 6.1 of Ref. [5, p. 155].

( )

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to be represented by the second-order “energy momentum” tensor T µυ >(or Tµυ ). Also, the energy momentum tensor (T µυ or Tµυ ) of a closed system (like one comprising the material distribution and the force, or the field together) requires to be bounded by the conservation laws Tυµ, µ = 0 or gαυ Tυµ = 0 or T,αµµ = 0, ,µ which is divergence free and T αµ = T µα , i.e., symmetric. 5. In light of conservation laws Tυµ, µ = 0 as satisfied by the energy momentum tensor, it is desirable to make it applicable to the Ricci 1   tensor Rµυ also. But the Einstein tensor Gυµ =  Rυµ − δ υµ R  in terms of   2  µ 1 µ  µ Ricci tensor is divergence free, i.e., Gυ , µ =  Rυ − δ υ R  = 0.   ,µ 2

(

)

1 1     Otherwise,  gαυ Rυµ − gαυδ υµ R  = 0 or  Rαµ − gαµ R  = 0 .   ,µ   ,µ 2 2 µα αµ Also, R = R , which is symmetric. Hence, the Ricci tensor R µα (or Rµυ ) also satisfies the required other conditions like T µυ . ∴ Based on the aforesaid requirements (1)–(5), Einstein adopted the 1 field equations for general theory of relativity as R µυ − g µυ R = − KT µυ 2 1 or Rµυ − g µυ R = − KTµυ . 2 These are the essential field equations for general theory of relativity, which were expanded subsequently after various cosmological considerations. This is purely a subject matter of “general theory of relativity,” and it is beyond the scope of the book, to discuss it completely. Of course, in the absence of matter (or gravitational field),

T µυ = T µυ = 0, g µυ = constant.

The field equations reduce to R µυ = 0 = Rµυ for empty space which is identical to ∇ 2φ = 0, which is Newton’s vacuum equation. N.B.: The importance of Ricci tensor in general theory of relativity is beyond description. In this pursuit, it is amplified only the use of tensors (primarily the Ricci tensor) in general theory of relativity.

8 Tensors in Continuum Mechanics

8.1 Continuum Concept In the investigation of material behavior of a body or medium, the bulk of the matter is considered as a whole but not the individual molecule. For this reason, the observed macroscopic behavior is counted in general by assuming that the material is continuously distributed throughout its volume and completely fills the space it occupies instead of considering its molecular distributions. This continuum concept of matter is the fundamental postulate of continuum mechanics.

8.2 Mathematical Tools Required for Continuum Mechanics The physical quantities related to continuum mechanics for its in-depth description are independent of any particular coordinate system for reference. Generally, from a mathematical point of view, these physical quantities conveniently need to describe by means of referring them to some coordinate system. Eventually, tensors which are independent of any particular coordinate system are the appropriate tools to adopt in this consideration. Hence, physical laws of continuum mechanics are expressed in terms of tensor equations. Usually, tensor transformations are linear and homogeneous, and if they (laws of continuum mechanics) are expressed in the form of tensor equations in one coordinate system, they remain valid in any other coordinate system. This invariance of tensor equations under coordinate transformations is one of the principal reasons (similar to general theory of relativity) for the utility of tensors in continuum mechanics. Of course, Cartesian tensors are sufficient to deal with continuum mechanics, and hence, it can be developed with reference to Cartesian tensors. In Euclidean space of three dimensions, the number of components of a tensor of order n is 3n. A vector is a tensor of order one with 31 components. 123

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Tensor Calculus and Applications

The stress and strain are the second-order tensors having 32 = 9 components in general. The physical quantities (mentioned earlier) involved in the study of continuum mechanics are the stresses and strains, which are invariably related to the deformation of media and bodies. Second-order tensors are also known as “dyadics,” and the quantities in continuum mechanics are represented by dyadics [11]. Definition Stress: The forces occurring in a bulk of the material medium proportional to the mass of the substance (e.g., gravity, magnetic force, centrifugal force) are known as body forces, which are measured per unit volume. The forces acting over the volume of bounding surface of a body and measured in units of force per unit area are called surface forces. This force per unit area is called stress which gives the measure of the intensity of the reaction of the material lying on one side of the material or that which lies on the other side. For this inherent property, the tendency of deformation of the state of a body or medium is bound to be acted by stress. Stress being a force per unit area must have two directions those of the force and normal to the area associated with it. The stress tensor components perpendicular to the surface (or plane face) are called normal stress (σ) and tangential to the surface (or plane face) are called shearing stress (τ). Definition Strain: The deformation caused by stress which may be dilated resulting in change in volume or distortion with changes in form or both is called strain. Otherwise, a distortion, deformation of change in the position of particles relative to each other, is known as strain. The change in confining pressure can change in volume but in shape for isotropic bodies where mechanical properties are uniform in all directions. With increasing/decreasing confining pressure, the volume of the body decreases/increases, and dilation is negative/positive accordingly. Mathematically, if Δfi (fi body force) is the resultant force exerted across the surface element Δs of S enclosing a volume V, then the Cauchy’s stress prin∆f ciple states that the average force per unit area on Δs, namely, i , tends to ∆s dfi a finite limit as Δs → 0 at a point P of the surface. Symbolically, the stress ds  df ∆f vector is written as t( nˆ )i = lim i = i when the moment of Δfi at the point P ∆s → 0 ∆s ds vanishes in the limiting process. The stress vectors are different at the same time for different surfaces containing the same point P. The stress principle

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is necessary to know the state of stress at a point in a medium in motion or a body subjected to deformation.

8.3 Stress at a Point and the Stress Tensor

 In a continuum, the Cauchy’s stress principle associates a stress vector t( nˆ i ) with each unit normal nˆ i representing the orientation of an infinitesimal surface element having an arbitrary internal point P.  The totality of all possible pairs of such combined vectors t( nˆ i ) and nˆ at P defines the state of stress at that point. Every pair of stresses and normal vectors are essential to represent the state of stress at that point. Eventually, the stress vectors on each of the three mutually perpendicular planes (for Cartesian system) through P can give the state of stress. Of course, coordinate transformation equations can serve to relate the stress vector on any other plane at that point with the given three planes. The state of stress is perfectly homogeneous under the application of a force without rotation. The stress vectors in the coordinate plane surfaces of a cubic element with mutually perpendicular axes 1(X1), 2(X2), 3(X3) can be written as   t( nˆ 1 ) = t( nˆ 1 ) nˆ 1 + t( nˆ 1 ) nˆ 2 + t( nˆ 1 ) nˆ 3 1



2

3

= t( nˆ 1 ) j nˆ j

  Similarly, t( nˆ 2 ) = t( nˆ 2 ) j nˆ j and t( nˆ 3 ) = t( nˆ 3 ) j nˆ j The nine components are t( nˆ i ) j ≡ σ ij or τ ij which is called the second-order (Cartesian) stress tensor. Here, t( nˆ i ) is the stress vector in the direction nˆ i perpendicular to the surface, and t( nˆ i ) j is its resolved part in the jth direction. The stress tensor can be expressed in matrix form as



 σ 11  σ ij =  σ 21  σ 31

σ 12 σ 22 σ 32

σ 13 σ 23 σ 33

  σ xx    or  σ yx   σ zx 

σ xy σ yy σ zy

σ xz   σ yz   σ zz 

with reference to the coordinate planes (shown in Figure 8.1). The components in the diagonal σ 11 = σxx, σ 22 = σyy, σ 33 = σzz are called normal stresses perpendicular to planes, and the components σij (i ≠ j) or σxy, σyx, σxz, σzx, σyz, σzy tangential to the coordinate planes are called shear stresses (τij) in other notations.

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3(X3)

σ33

τ32

τ23

τ31 τ13

σ22

τ21 σ11

τ12

2(X2)

1(X1) FIGURE 8.1 Stress components on the plane surfaces of a cubic element (for simplicity).

8.4 Deformation and Displacement Gradients Let (X1, X2, X3) be the material coordinates of the point Po in undeformed configuration of a material continuum at t = 0, with respect to O−X1X2X3, and P(x1, x2, x3) be the corresponding position in the deformed configuration at t = t1 with respect to the system O′ − x1x2 x3 so that  Cartesian   OP o = X = X 1Iˆ1 + X 2 Iˆ2 + X 3 Iˆ3 and O′P = x = x1eˆ1 + x2 eˆ2 + x3 eˆ3. The particles of the continuum undergoing deformation can move along different paths in space. If the particle Po initially at t = 0 is assumed to move to the position P at t = t1, then functionally, it can be represented by xi = xi(X1, X2, X3, t) = xi (X, t) or conversely Xi = Xi(x1, x2, x3, t) = Xi (x, t).

(

)



(

∴ dxi =

)

∂ xi dX j (8.4.1) ∂X j

∂ xi is called the material deformation gradient tensor, ∂X j and t is the absolute time. Similarly, from Xi = Xi(x, t),

where the tensor



It can be seen that

dX i =

∂Xi j dx (8.4.2) ∂x j

∂Xi is also a tensor and is called spatial deformation ∂x j

gradient tensor. Definitely, the material and spatial deformation tensors are interrelated by means of the chain rule of partial differential:

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Tensors in Continuum Mechanics

x3 X3

P(x1,x2,x3)

u

ˆI 3

X

O

O/ X2

ˆI 2

X1

x2

P o(X 1

b ˆI 1

x

eˆ3

) ,X 2,X 3

eˆ1

eˆ2 t = t1

t=0

x1

FIGURE 8.2 Deformation graph.



∂ xi ∂ X j ∂Xi ∂x j δ = (8.4.3) ik ∂X j ∂x k ∂x j ∂X k

From the displacement vector ui = xi − Xi + bi (Figure 8.2), the partial differentiation with respect to coordinates gives the material displacement gradient ∂ui ∂u and the material displacement i as ∂X j ∂x j

∂ui ∂ xi ∂ X i ∂ xi = − = − δ ij (8.4.4) ∂X j ∂X j ∂X j ∂X j

and

∂ui ∂ xi ∂X i ∂X i (8.4.5) = − = δ ij − ∂x j ∂x j ∂x j ∂x j

8.5 Deformation Tensors and Finite Strain Tensors Let us consider two superimposed rectangular Cartesian coordinate systems O–X1X2X3 for initial configuration and O–x1x2x3 for final configuration

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after deformation of a material continuum. To extract the characteristic difference of the measure of material deformation, it is justified to refer to a different coordinate system (instead of referring to the same system) but from the same position. The neighboring particles Po and Qo before deformation are supposed to move to the points P and Q, respectively, in the deformed configuration (Figure 8.3). The square of the differential element of length between Po and Qo dX 2 = dX i dX i = δ ij dX i dX j (8.5.1)

where dX i =

∂X i dx j (from 8.4.2). ∂x j

( dX )2 = dX k dX k =



∂X k ∂X k dxi dx j ∂ xi ∂ x j (8.5.2)

= Cij dxi dx j ∂X k ∂X k in which the second-order tensor Cij = is called the Cauchy’s defor∂xi ∂x j mation tensor. Also, the square of the differential element of length between P and Q for the deformed configuration is

( dx )2 = dxi dxi = δ ij dxi dx j , (8.5.3)

where dxi =

∂ xi dX j (from 8.4.1). ∂X j X3 (x3) Q

dX

u

u+d

Qo

X+

dX

dX

Po

u

P

x

O

x

X1 (x1)

FIGURE 8.3 Graph of deformation tensor and strain tensor.

X2 (x2)

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Tensors in Continuum Mechanics

It can be written as

( dx )2 =



∂ xk ∂ xk dX i dX j ∂X i ∂X j (8.5.4)

= Gij dX i dX j , where the second-order tensor Gij =

∂ xk ∂ xk is called the Green’s deforma∂X i ∂X j

tion tensor. 2 2 The difference ( dx ) − ( dX ) corresponding to two neighboring particles between the initial and final configurations of a material continuum is taken as the measure of deformation. For all neighboring particles, if this difference vanishes for a continuum, the corresponding displacement is called rigid displacement. Now, (dx)2 − (dX)2 = (Gij − δij)dXidXj = 2LijdXidXj, using (8.5.4) and (8.5.1) where Lij =



 1 1  ∂ x ∂ xk Gij − δ ij =  k − δ ij  (8.5.5)  2 2  ∂X i ∂X j

(

)

which is called the Lagrangian (or Green’s) finite strain tensor. Also, (dx)2 − (dX)2 = (δij − Cij)dxidxj using (8.5.3) and (8.5.2) = 2Eijdxidxj, using (8.5.3) and (8.5.2) where Eij =



1 1 ∂X k ∂X k  (8.5.6) δ ij − Cij =  δ ij − 2 2 ∂ xi ∂ x j 

(

)

which is called the Eulerian finite strain tensor. Making use of the relation of displacement vector, ui   = xi – X i + bi

Lij can be written as Lij =



    ∂u 1  ∂uk + δ ki   k + δ kj  − δ ij     ∂X j 2  ∂X i  

=

 1  ∂uk ∂uk ∂u j ∂ui + + + δ ij − δ ij  (8.5.7)  2  ∂X i ∂X j ∂X i ∂X j 

=

1  ∂ui ∂u j ∂uk ∂uk  + +   2  ∂X j ∂X i ∂X i ∂X j 

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Similarly,

Eij =

1  ∂ui ∂u j ∂uk ∂uk  + −   (8.5.8) 2  ∂ x j ∂ xi ∂ xi ∂ x j 

Now, if we impose the condition for small deformation theory on the displacement gradients, which are very small compared to unity, then the product term in (8.5.7) can be ignored. Hence, the expression corresponding to Lij can be written as

lij =

1  ∂ui ∂u j  (8.5.9) + 2  ∂X j ∂X i 

It is called the Lagrangian infinitesimal strain tensor. ∂uk Also, for  1, ignoring the product term, (8.5.8) can be equivalently ∂ xi written as



eij =

1  ∂ui ∂u j  (8.5.10) + 2  ∂ x j ∂ xi 

which is called the Eulerian infinitesimal strain tensor.

8.6 Linear Rotation Tensor and Rotation Vector in Relation to Relative Displacement

  Let u( Po ) and u(Qo ) represent the displacement vectors of the two neighboring particles in a material medium.



  ∴ du = u (Qo ) − u( Po ) ∴ dui = ui (Qo ) − ui( Po )

which is the measure of relative displacement vector of the particle originally at Qo with respect to the particle originally at Po. Subject to continuous displacement, ui( Po ) the expansion about by Taylor’s series, it can be written  ∂u  as dui =  i  dX j (neglecting higher order terms for small displacement).  ∂X j  Po

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Tensors in Continuum Mechanics

∂ui can be decomposed into a symmetric part ∂X j and an antisymmetric part so that The material displacement

 1  ∂u ∂u j  1  ∂ui ∂u j   dui =   i + − +    dX j  2  ∂X j ∂X i  2  ∂X j ∂X i   Po



∂u ∴ i = lij + Wij ∂X j



(8.6.1)

( using 8.59) ,

where Wij =



1  ∂ui ∂u j  (8.6.2) − 2  ∂X j ∂X i 

which is called the linear Lagrangian rotation tensor (Figure 8.4). For infinitesimal rigid body rotation, corresponding to relative displacement at point Po, the Lagrangian strain tensor lij vanishes. Hence, the infinitesimal rotation vector is thus given by wi =



1 ∈ijk Wkj (8.6.3) 2 Q

u du Qo

dX

Po u FIGURE 8.4 Graph for rotation tensor and vector.

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Tensor Calculus and Applications

Again, according to Eulerian description of the relative displacement vector, it can be written as

dui =

∂ui dx j ∂x j

so that



∂ui  1  ∂ui ∂u j  1  ∂ui ∂u j   = + + −  ∂ x j  2  ∂ x j ∂ xi  2  ∂ x j ∂ xi   (8.6.4) = eij + wij ,

where

wij =

1  ∂ui ∂u j  (8.6.5) − 2  ∂ x j ∂ xi 

which is called the Eulerian rotation tensor with Eulerian rotation vector

wi =

1 ∈ijk wkj . (8.6.6) 2

It has already been discussed that deformation is connected with displacements. Therefore, motion of any continuum such as fluid or gas requires the use of tensors (or vectors), namely, stress–strain tensors, rotation tensor, and displacement vector. Ironically, there is no alternative but to use tensors in non-isotropic medium to arrive at accurate results of an investigation.

9 Tensors in Geology

9.1 Introduction The structural investigations of spasmodic deformation [it means the change in shape of a body from the initial (undeformed) configuration to a subsequent (deformed) configuration] caused by nature and origin of some forces need application of some mathematical concepts such as tensors. The mathematical entities such as stress and strain (Section 8.2) are largely responsible to study any kind of deformation. A second-order tensor called “stress” is the essence of structural geology. Rheology is closely related to the study of deformation of the structures of the earth and any kind of material structure ranging from the order of seconds (seismic-wave propagation) to hundreds of millions of years (geodynamics). The aforesaid “stress and strain” are the fundamental ingredients to deal with the analysis of continuum mechanics of deformation (discussed in Section 8.2) of extended bodies in the context of rheology. Newtonian and non-Newtonian viscosity, linear rheological bodies, plasticity, and brittle failure can be investigated with these mathematical entities—the second-order tensors. From a rheological standpoint, properties of lithosphere and the mantle, temperature distribution of lithosphere, thermal convection in the mantle, flexure of the lithosphere, stresses on it, and viscosity of the ­mantle from surface loading data are some of the features within this domain. Of course, atomic basic deformation and flow in polycrystalline m ­ aterials ­covering hydrolytic weakening dynamic recrystallization and pressure and temperature effects may attract the involvement of both geologists and ­geophysicists [Ref. 7]. The displacement gradient tensor (one of many tensors related to strain) relates the position vector of a point to the displacement of the point during a displacement. Knowing displacement gradient tensor, it is possible to calculate how all points within a body are displaced as a function of position during deformation. If we know the stress tensor, we can calculate the stress vector on a plane of any orientation within a body, which is

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largely important in any study of earthquakes, induced seismicity, faulting, etc. There are three stages of deformation (mentioned earlier) when direct forces act on a body extending to a short period of time, minutes, or hour. They are as follows: i. Elastic ii. Plastic iii. Rupture Elastic: If the body returns to the original shape and size after the withdrawal of the stress, the deformation is called elastic. If the body does not return to the original shape when the stress exceeds a certain stage, it is called elastic limit. The strain is proportional to stress when it remains always less than the elastic limit, and the deformation obeys Hooke’s law. Plastic: The deformation is said to be plastic if the stress exceeds the elastic limit. N.B.: The difference between the external force applied to a body and the corresponding outcomes of internal actions and reactions generates stress. Rupture: If the specimen is subjected to continuous increase of stress, one or more fractures based on several factors can develop, which eventually fails by rupture. Rupture is responsible for “brittle” in substances before certain stage of plastic deformation. From a geometrical point of view, strain causing distortion of a body can also be classified as homogeneous and inhomogeneous. Homogeneous deformation: After deformation, i. Straight lines remain straight lines. ii. Parallel lines remain parallel. iii. In the strained body, all lines in the same direction have constant value of e (extension—change in unit in length), λ (quadratic elongation), ψ (angular strain), and γ (shear strain). Inhomogeneous deformation: After deformation, i. Straight lines change to curves. ii. Parallel lines turn nonparallel. iii. The values of the above four parameters e, λ, ψ, and 𝛾 in any one given direction of the body become variable, not constant.

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Tensors in Geology

9.2 Equation for the Determination of Shearing Stresses on Any Plane Surface Let the coordinate axes x, y, z be rotated in the directions of three mutually perpendicular axes of principal stress, σ 1, σ 2, σ 3 acting on any plane surface. Let ABC be the plane of unit area which inclines to these coordinate axes. The normal stress σ is supposed to act along the normal to this plane having direction cosines l, m, n and shearing stress τ on it. If S (Sx, Sy, Sz) is the resultant stress on the plane ABC (Figure 9.1) considered at the point P, then S2   = σ 2 + τ 2 , (9.2.1)

where Sx = σ 1l, Sy = σ 2m, Sz = σ 3n so that

S2 = σ 12l 2 + σ 22 m2 + σ 32 n2 . (9.2.2)



Also, the measure of the normal stress σ in the directions l, m, n is given by σ.

σ = Sxl + Sy m + Sz n



σ = σ 1l 2 + σ 2 m2 + σ 3 n2 .

(9.2.3)

Z σ

Sz

C

σ3

S P

Sy σ2 B Y

O

Sx σ1 A X

FIGURE 9.1 Normal stress σ and shearing stress τ in any plane ABC of unit area.

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Tensor Calculus and Applications

Using (9.2.2) and (9.2.3) in (9.2.1), we get

(

) n ( 1 − n ) − 2σ σ l m

τ 2 = σ 12l 2 + σ 22 m2 + σ 32 n2 − σ 1l 2 + σ 2 m2 + σ 3 n2

(

)

(

)

= σ 12l 2 1 − l 2 + σ 22 m2 1 − m2 + σ 32



− 2σ 1σ 3 l 2 n2

(

)

(

2

2

2

)

1

(

= σ 12l 2 m2 + n2 + σ 22 m2 l 2 + n2 + σ 32 n2 m2 + l 2

(∴ l



2

2

2

2

− 2σ 2σ 3 m2 n2

)

)

+ m 2 + n2 = 1

−2σ 1σ 2l 2 m2 − 2σ 2σ 3 m2 n2 − 2σ 1σ 3l 2 n2



∴ τ 2 = l 2 m2 (σ 1 − σ 2 ) + m2 n2(σ 2 − σ 3 ) + l 2 n2 (σ 3 − σ 1 ) . 2

2

2

(9.2.4)

This is the required equation for shearing stress τ in terms of principal stress in the direction of the normal stress σ to the plane surface ABC. To determine the maximum and minimum stresses, we are to study the stationary values of τ from this equation replacing n2 by n2 = 1 − l2 − m2.

(

)

∴ τ 2   = l 2 m2 (σ 1 – σ 2 ) + m2 1 – l 2 – m2 (σ 2 – σ 3 ) 2



(

)

2

+ l 2 1 – m2 – l 2 (σ 3 – σ 1 ) . 2

Differentiating it partially with respect to l and m successively, we get 2τ

∂τ 2 2 2 2 = 2lm2 (σ 1 − σ 2 ) − 2lm2 (σ 2 − σ 3 ) + 2l l − l 2 − m2 (σ 3 − σ 1 ) − 2l 3 (σ 3 − σ 1 ) ∂l

(

)

2 2 2 = 2lm2 (σ 1 − σ 2 ) − (σ 2 − σ 3 )  − 2l (σ 3 − σ 1 ) 1 − l 2 − m2 − l 2 



(

= 2lm2 (σ 1 − σ 3 )(σ 1 − 2σ 2 + σ 3 ) − 2l (σ 3 − σ 1 ) 1 − 2l 2 − m2 2

τ



)

∂τ = l (σ 1 − σ 3 )[m2 (σ 1 − 2σ 2 + σ 3 ) + (σ 1 − σ 3 ) 1 − 2l 2 − m2 , ∂l

(

)

(9.2.5)

and similarly,

τ

∂τ = m (σ 2 − σ 3 ) l 2 (σ 2 − 2σ 1 + σ 3 ) + (σ 2 − σ 3 ) 1 − 2 m2 − l 2  . (9.2.6) ∂m

But for stationary values,

(

∂τ ∂τ = = 0 simultaneously. ∂l ∂m

)

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Tensors in Geology

Hence, from (9.2.5), l = 0 for unequal values of σ 1, σ 2, σ 3 (in general) gives when substituted in (9.2.6), 1 m = 0 and m = ± . 2 Thus,



l = 0,

m=0

l = 0,

m=±

l = 0,

gives n = 1

1 2 1 m=± 2

1 (9.2.7) 2 1 gives n =  . 2 gives n = ±

Similarly, eliminating l and m successively from (9.2.4), we get for other ­stationary values





l = 0, 1 , =± 2 1 , l= 2

m = 1,

n=0

m = 0,

n= ±

l = 0, 1 =± , 2 1 =± , 2

m = 1,

m = 0,

1 (9.2.8) 2 1 n= ± . 2

1 , 2 1 , m= 2 m=±

n=0 n=0

(9.2.9)

n = 0.

The first three stationary values of the three sets of (9.2.7)–(9.2.9) are related to principal planes which correspond to minimum values of the shearing stress. The remaining stationary values represent the maximum shearing stresses with respective planes orienting to contain one of the principal axes of stress, which makes 45° angle with the other two axes [7, p. 38] in the plane.

9.3 General Transformation and Maximum and Minimum Longitudinal Strains Let P(x, y) be any point before deformation which is displaced to the position Q(x1, y1) with the general displacement.

x1 = ax + by and y1 = cx + dy .

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Tensor Calculus and Applications

Solving for x and y (a > 0, d > 0),

x=

ay − cx1 dx1 − by1 . (9.3.1) , y= 1 − ad − bc ad bc

Let the rectangle joining the points O(0, 0), M(x, 0), P(x, y), and N(0, y) be deformed with the same point P into the parallelogram ORQS with points (0,0), (ax, cx), (ax + by, cx + dy), and (by, dy). It is to be noted that a and d represent the components of longitudinal strains parallel to the x and y coordinates, respectively. Moreover, b and c represent the part shear components characterizing the angular displacements of the initial sides of the rectangle so that ax = l cosθ , cx = l sinθ ,

c = tan θ a c = a tanθ

and dy = l cosφ , by = l sinφ

b = tan φ d b = d tanφ .

Now let us investigate what will be the change in a general line y = mx + p after deformation subject to the above general displacement with x, y given by (9.3.1). ∴ Putting the values of x, y in the equation of the line, we get

ay1 − cx1 dx − by1 =m 1 +p ad − bc ad − bc



y1 ( a + bm ) = ( c + dm ) x1 + p ( ad – bc )



∴ y1 =

p( ad − bc) c + dm x1 + , (9.3.2) a + bm a + bm

which is also a straight line. Hence, the general transformation representing Figure 9.2 characterizes the homogeneous strain.

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Tensors in Geology

Y Q(x1, y1) S

P(x, y)

N (o,y)

.dy .Θφ

R

O(o,o)

.Θθ

M(x,o)

X

.ax FIGURE 9.2 General strain translation.

Again the points lying on the circle (inscribed rectangle) x2 + y2 = 1 are subjected to change due to the general transformation, and the circle takes the form: 2





2

 dx1 − by1   ay1 − cx1   =1  +  ad − bc   ad − bc  (c 2 + d 2 )x12 − 2(bd + ac)x1y1 + ( a 2 + b 2 )y12 = ( ad − bc)2, (9.3.3)

which is an ellipse called strain ellipse. The major and minor axes of this strain ellipse identify the positions of maximal and minimal longitudinal strains. By means of a rotation ψ of axes, the term with x1y1 of the equation of strain ellipse (9.3.3) can be removed to reduce it to a standard form, and ψ is given by

ψ =

  −2( ac + bd) 1 . (9.3.4) tan −1  2 2 2 2  2  (c + d ) − ( a + b ) 

The lengths of the semimajor axis λ1 and minor axis λ2 or maximum and minimum strains are given by 1 + e1 = λ1 and 1 + e2 = λ2 , respectively, x2 y 2 where the strain ellipse is + = 1. λ1 λ 2

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Tensor Calculus and Applications

9.4 Determination of the Two Principal Strains in a Plane Let P(x, y) be a point with origin O and OP be one of the principal axes of strains so that OP = 1. Let θ be the angle made by OP with x-axis. Q (x1, y1) is the deformed position of P without rotation, and hence, for the strain ellipse, the length of one principal strain is OQ = 1 + e (Figure 9.3). ax + by bx + dy Now from the Figure 9.3, cosθ = x = and sin θ = y = . 1+ e 1+ e ( for irrotational strain c = b) ∴ x  a – ( 1 + e )  + by = 0 (9.4.1)

and

y  d – ( 1 + e )  + bx = 0. (9.4.2)

Dividing,

b a − (1 + e) =+ d − (1 + e) b ∴  a – ( 1 + e )   d – ( 1 + e )  – b 2 = 0



(1 + e )2 – ( 1 + e )( a + d ) + (ad – b 2 ) = 0, (9.4.3)



which gives the two principal irrotational strains (the two roots of it), (1 + e1) and (1 + e2). From (9.4.1) and (9.4.2), replacing x by cos θ and y by sin θ, we can get Y Q(x1, y1)

. bx

by P (x , y

. .dy φ 1 .θ O

FIGURE 9.3 Graph for principal strains.

.ax

)

X

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Tensors in Geology



a – ( 1 + e ) + b tan θ = 0 and d – ( 1 + e ) + b cot θ = 0

so that

tan 2 θ +

a−d tan θ − 1 = 0, (9.4.4) b

which is quadratic in tanθ. This gives the directions of the two mutually perpendicular axes of strain ellipse. Hence, with reference to the chosen reference axes, the finite displacements (two dimensional) can be instrumental to determine the principal elongation 1 + e1 = λ1 and 1 + e2 = λ2 with lengths of principal axes λ1, λ2 of x2 y 2 + = 1 derived from the deformation of the initial unit the strain ellipse λ1 λ 2 circle x2 + y2 = 1. To speak the truth, the measures of the two principal strains (magnitudes of strain tensor) can be completely determined from Equations (9.4.3) and (9.4.4), and the angle of rotation ψ by the lines turning to axes of the ellipse. Finally, to determine the displacements of the points in space resulting in the strain ellipse, significantly no particular coordinate system is chosen in the deformation state. There could have been the same distortions or strains for the displacements with reference to different coordinate systems (Cartesian, of course, in this case) though the equations would have been slightly different. It is discussed in the beginning of the book that the independence of reference frame to study some mathematical concepts is essentially required to use tensors. Quantities such as stresses and strains related to geological context are some second-order tensors or general tensors. For three-dimensional cases, the mathematical entities such as stress will require (32 − 3) = 6 components (for a symmetric tensor (σxy = σyx)) and 32 = 9 components (for anti-symmetric case). Of course, there are some quantities which remain invariant irrespective of reference frames but need full knowledge of tensor calculus. The geometrical processes of determination of the measures of stress and strain are discussed here (instead of using truly tensors but magnitudes) to help in its geological context. N.B.: Though actually stress and strain are three dimensional, it is beyond the scope of the book to discuss the general case, and only classical result is reported.

10 Tensors in Fluid Dynamics

10.1 Introduction Modern scientists and physicists believe that matter is composed of elementary particles, and in most of the scientific fields, it is not looked into the individual molecules, which is regarded as an entity of small but infinite dimensions interacting with its fellows according to certain laws. So matter is not continuous but discrete, and its gross properties are taken as averages over a large number of molecules. The equations of fluid motion have been formulated from this viewpoint, though they are considered at first sight as much more fundamental one. The average velocity of the molecules is taken in the neighborhood of a point, but how large this neighborhood should be is a questionable one. In the investigation of ordinary fluid motion, the variations in the medium is considered isotropic (uniform in all directions), and hence, pressure (can be said as classical stress) is a constant quantity. But in case of viscous fluids, it is to be replaced by stress tensor. Eventually, the corresponding body and the surface forces will occur in the system in appropriate forms. The Navier– Stokes equations can suitably meet all the requirements to study the motion of fluids when viscosity is present in the medium.

10.2 Equations of Motion for Newtonian Fluid Let us consider an arbitrary volume V within the fluid medium enclosed by a surface S. If ρ is the density of the fluid particles moving with velocity ν ν i , then by Reynolds’ transport theorem,

( )

143

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Tensor Calculus and Applications

δ δt

 δρ



∫∫∫ ρ dV = ∫∫∫  δ t + ρv  dV  ∂ρ

i ,i



=

∫∫∫  ∂t + ρ v + ρv  dV

=

∫∫∫  ∂t + (ρv )  dV.

,i

 ∂ρ

i

i

i ,i

,i



But for conservation of mass within the arbitrary volume dV, ∂ρ + ( ρ v i ), i = 0. (10.2.1) ∂t



This is the equation of continuity characterizing the conservation of mass. If T ij is the contravariant stress tensor on the element ds of the surface with normal nj, then the force on this element is Tijnjds. Also let the contravariant vector f i represent the external body forces per unit mass of the fluid. ∴ The total net force on the arbitrary volume V enclosed by the surface S when resolved in any direction li (say) to it is

∫ ∫∫ ρ f l dV + ∫∫ T n l ds = ∫ ∫∫ ρ f l dV + ∫∫ ∫ T l dV i

V



ij

i

i

j i

S

V

=

∫ ∫∫ (ρ f

ij ,j i

i

V

i

+ T,ijj )li dV

V

using Gauss’s divergence theorem to the second term. But the rate of change of linear momentum is equal to the net external force in the direction li. δ ρ v ili dV = Hence, ( ρ f i + T,ijj )li dV δt

∫ ∫∫

∫ ∫∫

V



=

∫ ∫∫ V

ρ

V

δ vi li dV = δt

∫ ∫∫ (ρ f

i

+ T,ijj )li dV .

V

But for arbitrary volume V and direction li it holds, only when  (for the velocity vector (tensor) v , v,i j means covariant derivative)

 ∂vi i  ρ ∂t = ρ a

   ∂vi + v i v,i j  = ρ f i + T,ijj = ρ     ∂t

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Tensors in Fluid Dynamics

so that ij



(a i =)

T, j ∂vi + v i v,i j = f i + , (10.2.2) ∂t ρ

which is the equation for conservation of linear momentum. Thus, the equations of motion of Newtonian fluids are given by Equations (10.2.1) and (10.2.2).

10.3 Navier–Stokes Equations for the Motion of Viscous Fluids In the case of ordinary fluids, the stress tensor Tij corresponding to ­hydrostatic pressure p is Tij = − pgij, and the fluid medium is isotropic. But in the presence of viscosity in the medium, it is anisotropic and is related to deformation. Let vi and v i + dv i be the velocities of two neighboring points xj and xj + dxj.

dv i =

∂vi j dx = v,i j dx j . ∂x j

But 1 1 v,kj = g ik vi , j = g ik  vi , j + v j , i + vi , j − v j , i  2 2 

(



)

(

 1  ∂v ∂v j  1  ∂v ∂v j   = g ik   ij + i  +  ij − i   ∂x  2  ∂x ∂x    2  ∂x

)

(10.3.1)

= g ik  (eij + wij ), where

eij =

1  ∂ vi ∂ v j  + (10.3.2) 2  ∂ x j ∂ x i 

which is the deformation symmetric strain tensor (8.6.4) and

wij =

1  ∂ vi ∂ v j  − (10.3.3) 2  ∂ x j ∂ x i 

which is the rotation tensor indicating rigid body rotation of the element. Now the equation of motion with acceleration ai, stress tensor Tij, and body force fi is (from Equation (10.2.2))

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Tensor Calculus and Applications

ρ a i = ρ f i + T,ijj .



(10.3.4)

But the stress tensor Tij = −pgij for viscous fluids is to be supplemented by the viscous stress tensor pij to result for deformation T ij = – pg ij   + p ij .



This relationship must be linear and isotropic (same for all coordinate ­systems) when Newtonian fluid is isotropic. ∴ For the isotropic fourth-order symmetric tensor Gijmn with respect to i, j and m, n, pij can be written as pij = Gijmn emn, in terms of the deformation strain tensor emn. But for generalization of isotropic fourth-order symmetric tensors, it must be a linear combination of gijgmn and (gimgjn + gingjm) so that p ij   = λ g ij g mnemn   + µ( g im g jn + g in g jm )emn



= λ g ij emm   + 2 µe ij T ij = (– p + λ emm )g ij  + 2 µeij . (10.3.5)



In the equations of motion (10.3.4), it is necessary to employ stress and strain relation for viscous fluid motion. This requires to express Tij in the following form: T,ijj = [(− p + λ emm ) g ij + 2 µe ij ], j = (− p, j + λ emm, j ) g ij + 2 µe ij, j = (− p, j + λ emm, j ) g ij + 2 µ( g ik e kj ), j = (− p, j + λ emm, j ) g ij + 2 µ( g ik g jm e km ), j

1 = (− p, j + λ emm, j ) g ij + 2 µ  g ik g jm ( vk , m + vm , k )  2 , j

(

= (− p, j + λ emm, j ) g ij + µ g jm v,imj + g ik v,jkj

)

( j ↔ k)

= (− p, j + λ emm, j ) g ij + µ g jm v,imj + µ g ij v,kjk T,ijj = (− p, j + λ emm, j + µ v,kjk ) g ij + µ g jm v,imj . But emi = g ki ekm =

(

1 ki 1 g ( vk , m + vm , k ) = v,im + g ki vm , k 2 2

)

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Tensors in Fluid Dynamics



∴ emm =

1 m 1 ( v, m + g km vm , k ) = ( v,mm + v,kk ) = v,mm 2 2

( k → m).

Hence,

T,ijj = (− p, j + λ v,mmj + µ v,mmj ) g ij + µ g jm v,imj = [− p, j + (λ + µ )v,mmj ] g ij + µ g jm v,imj .

Equation (10.3.4) for viscous fluids takes the form:

ρ a i = ρ f i − g ij p, j + (λ + µ )v,kkj g ij + µ g jk v,i jk ( v, jk = v, kj ),

which is the Navier–Stokes equations for viscous fluids.

(10.3.6)

Appendix Some Standard Integrals in Connection with Applications In many investigations of dynamical scenarios, different kinds of integrals appear to characterize the development of physical situations inviting transformations. The transformation from a volume integral of certain physical property to an integral over bounding surface is a veryimportant frequent measure of classical or tensor analysis. For example, if F is the flux of some   physical quantity such as flux of fluids or charges, then the integral F ⋅ n ds  over the whole surface S, where n is the outward drawn normal to it, is equal to the total flux out of the closed volume. The Green’s theorem (or Gauss’s divergence theorem) in connection with such entity can be mathematically stated as follows:

∫∫

Green’s Theorem  Statement: If F is any continuously differentiable vector field in volume V and bounded closed surfaces which may have piecewise smooth boundary with outward drawn normal n, then



 

∫∫∫ ∇ ⋅ F dv = ∫∫ F ⋅ n ds,

In tensor form,

∫∫∫

F, ii dv =

v

For covariant formalism,

where ∇ is the vector operator .

s

V

∫∫∫ g v

ij

∫∫

 F i ni ds, for contravariant vector F F i .

( )

s

⋅ Fi , j dv =

∫∫ g F n ds = ∫∫ F n ds. ij

s

j

j i

j

s

The partial derivatives in Cartesian coordinates are replaced by covariant derivatives, since they are identical. Stoke’s Theorem  Statement: If F is any continuous vector field with continuous partial ­derivatives, then for any two-sided piecewise smooth surface S spanning 149

150

Appendix

a closed curve C,

∫

  F ⋅ t ds =

c



∫∫ curl F ⋅ n dS, where t is the tangent vector to s

 the curve C and n is the normal right handedly orienting the direction of the curve C. But in tensor form,

∫∫ e

ijk

Fk , j ni dS =

s

∫ F t k

k

ds, where ∈ijk is the permutation tensor.

c

Reynold’s Transport Theorem If f ( x , t) is any function and V(t) is a closed volume with the fluid consisting of the same fluid particles, then

 d  dt  

Let F (t) =

∫∫∫ v(t )

 f ( x , t) dv  =  

∂ f



∫∫∫  ∂t + ∇ ⋅ ( f ν ) dv

x = x ( x1 , x2 , x3 ).

v(t )

dF

∫∫∫ f (x , t) dv, so dt

is the material derivative which needs to be

v(t )

determined. ∴



dF (t) d = dt dt

∫∫∫ f (x , t) dv. v(t )

Since V(t) is variable volume, the differentiation with respect to time c­ annot be taken under the integral sign. But if we consider the integration with respect to volume in ξ − space, ξ being material Cartesian coordinate, and change ξ (ξ1 , ξ 2 , ξ 3 ) coordinates to x ( x1 , x2 , x3 ), so that ∂( x1 , x2 , x3 ) dV = dV0 = JV0 with dV0 = dξ1 .dξ 2 .dξ 3 at t = 0, ∂(ξ1 , ξ 2 , ξ 3 ) then differentiation and integration can be interchanged. In this case, V(t) can be taken as the moving material volume coming from fixed initial volume V0 at t = 0 due to the transformation x = x ξ , t and ξ can be treated as constant.

( )



 d  dt  

∫∫∫ v(t )

  d f ( x , t) dv  =   dt    =

∫∫∫ v0

 df

 f { x ( ξ , t), t} J dv0    dJ 

∫∫∫  dt J + f dt  dv v0

0

151

Appendix

=

 df



∫∫∫  dt + f (∇ ⋅ v) J dv

0

v0





dJ ∂ J = + ( v ⋅∇) J dt ∂t =

 df



∫∫∫  dt + f ∇ ⋅ v  dv v(t )

Associating the gradient terms after making use of material derivatives, it can be arranged to the form:  d  dt  



∫∫∫ v(t )

 f ( x , t) dv  =  

∂ f



∫∫∫  ∂t + ∇ ⋅ ( f v) dv, v(t )

which is the Reynold’s transport theorem. The function f can be some scalar or component of tensor. Corollary 1 Now

∫∫∫ ∇ ⋅ ( f v) dv = ∫∫ f v ⋅ nˆ ds, v(t )

s

(applying Green’s theorem) where S is the surface enclosing the volume V(t). ∴ The Reynolds’ transport theorem can be put to the form:

d dt

∂f

∫∫∫ f ( x , t) dv = ∫∫∫ ∂t dv + ∫∫ f v ⋅ nˆ ds. v(t )

v(t )

s( t )

Corollary 2 The Jacobian of the transformation from material to spatial coordinates in dv = J dv0 is a scalar, and the divergence of the velocity field v,ii is also scalar. δ J dJ ∴ (The intrinsic derivative =) = = v,ii J . δ t dt ∴ The Reynolds’ transport theorem in tensor form can be written as



 δ  δt  

∫∫∫ v(t )

 f ( x , t) dv  =   =

δ f



∫∫∫  δ t + f v  dv i ,i

v(t )

∂ f

∫∫∫  ∂t + f v(t )

,i

 v i + f v,ii  dv. 

152

Appendix

 Intrinsic derivative of scalar invariant



=

∂ f

δ f ∂f = + f, i v i δt ∂t 

∫∫∫  ∂t + ( fv )  dv (i) i

,i

v(t )



=

∂f

∫∫∫ ∂t dv + ∫∫ fv ⋅ n ds, v(t )

i

i

s

where f is a scalar or component of tensor. Replacing f by ρ (density), the equation of continuity can be recovered from (i).

Remarks In Chapters 7–10, importance is laid on the theoretical aspects only, showing the use of tensors in the respective branches and that too in restricted senses. Only the fundamental aspects required for initial approach and to pave the way for studying each of the branches which are included in this book are discussed. For this reason, problems are not discussed which would be possible for books of individual subject. The fundamental objective of the author in the book is to report only the glimpses of applications of tensors in some known branches so that readers can in reality enter into the threshold of these branches.

153

Bibliography

1. P. G. Bergmann, Introduction to the Theory of Relativity. Prentice Hall, New Delhi (1962). 2. L. P. Eisenhart, Riemannian Geometry. Princeton University Press/Oxford University Press, Princeton, NJ/London (1949). 3. C. E. Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus. Cambridge University Press, Cambridge (1963). 4. T. J. Willmore, An Introduction to Differential Geometry. Oxford University Press, Oxford (1959). 5. R. K. Pathria, The Theory of Relativity. Dover Publications, New York (2003). 6. J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity. Pearson, New Delhi (2007). 7. J. G. Ramsay, Folding and Fracturing of Rocks. McGraw-Hill, New York (1967). 8. G. Ranalli, Rheology of the Earth. Allen & Unwin, Boston, MA (1982). 9. M. P. Billings, Structural Geology. Prentice Hall, New Delhi (1987). 10. M. R. Spiegel, Schaum’s Outline of Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis, SI edition. McGraw-Hill, New York (1959). 11. G. E. Mase, Continuum Mechanics. Tata McGraw-Hill, New York (2005). 12. R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, New York (1989).

155

Index A Absolute tensor, 16 Angular Strain, 134 Anisotropic, 145 Antisymmetric, 20, 21, 22, 31, 70, 100 Associate vector, 32, 33 Autoparallel curve, 82, 83

symmetric tensors 30 tensor, 31, 55 Curvature, 77, 114, 120 tensors, 93, 95, 99, 100, 109 Curved space, 93, 113, 119–121 Curvilinear coordinantes 3, 4, 103, 113, 119 Cyclic property, 98

B

D

Bianchi Identity, 103,104 Body force, 124, 143, 145 Bounding surface, 124 Brittle, 134 failure, 133

Definite quadratic form, 9 Deformation, 113, 124–125, 133, 134, 145, 146 configuration, 126 strain tensor, 146 tensor, 127 Dialation (distrotion), 124 Discrete, 143 Displacement gradients, 126, 133 Distorts, 113 Divergence of a vector, 68, 108 Dummy index, 13 Dynamical scenario, 3

C Cartesian coordinate, 3, 22, 27, 29, 30, 43, 104 basis, 6 orthogonal coordinates, 34 tensors, 16, 123 Cauchy’s deformation tensor, 128 stress principle, 124, 125 Centrifugal force, 124 Christoffel symbols (brackets), 41–43, 50, 51, 53, 58, 87, 88, 90, 104, 115 Codazzi equation, 105 Conjugate, 27 Conservation of linear momentum 145 mass, 144 Continuum concept, 123 mechanics, 16, 123, 133 Contraction, 18, 98 Contravariant, 3–5, 14–15,17–18, 21–22, 24, 27, 32, 33, 56, 59, 60, 68, 70, 71 stress tensor, 144 Covariant constants, 63 curvature tensor, 99, 117 derivatives, 55, 57–61, 64, 67, 68 differential, 118, 119, 120 differentiations, 63, 65, 71, 93,

E Elastic, 134 limit, 134 Einstein, 93, 113, 114 tensor, 122 Empty space, 122 Energy momentum, 121, 122 Entity, 3 Equation of continuity, 144 Euclidean coordinates, 30 geometry, 30 metric, 30 plane, 103 space, 11, 30, 71 Eulerian finite strain tensor, 129 infinitesemal strain tensor 130 rotation tensor, 132 Evolutionary basis, 43

157

158

F Faulting, 134 Finite strain tensor, 127, 129 First curvature of a curve, 77 Flat, 114 space, 118 Flexure, 133 Fundamental tensor, 33, 56–58, 60–62, 65, 67, 71,87, 90, 93 contravariant tensor, 40 covariant tensor, 32 ingradients, 133 quadratic form, 103 G Galileo’s view, 113 Gauss equation, 105 Gaussian curvature, 104, 105, 109, 114, 115 surface, 103, 116 divergence theorem, 144, 149 General theory of relativity, 22, 93, 113, 114, 117, 122 Generating factor, 120 Geodesics, 77–78, 80, 82–83, 91–92, 114 coordinates, 86, 88 Geodynamics, 133 Geometry of motion, 113 Gravitational field equation, 120 potential, 120 Green’s deformation tensor, 129 theorem, 149, 151 H Homogeneous deformation, 134 strain, 138 Hooke’s law, 134 Hydrolytic, 133, 138 Hydrostatic pressure, 145 I Imbedded, 103 Immersed, 89 Increment, 73 Indefinite quadratic forms, 9

Index

Induced secismicity, 134 Infinitesemal, 14 Inherent property, 124 Inner product, 7, 19 Integral curve of geodesic equation 85 Intensity, 124 Intrinsic concept, 30, 93 derivative, 71, 151 equation, 119 geometry, 103 property, 113 Invariance, 123 Invariant, 8, 15, 16 Irrotational strain, 140 Isometric, 109 surfaces, 103 Isometry, 103 Isotropic, 143, 145, 146 bodies, 124 J Jacobian, 14–16 K Kepler, 113 Kronecker delta, 14, 27 L Lagrangian (Green’s) finite strain tensor 129 infinite strain tensor, 130 Laplacian, 69 Law of inertia, 113 Levi-Civita, 72 Linear Lagrangian rotation tensor 130, 131 strain tensor, 131 Linear rheological bodies, 133 transformation, 9, 119 Linearly independent, 3 Lithosphere, 133 Loci, 11, 7 Longitudinal strains, 137 Lorentz transformation, 113 Lower down, 32

159

Index

M Macroscopic, 123 Magnetic field, 113 force, 124 Mantle, 133 Material cartesian coordinates, 150 deformation, 128 deformaton gradient tensor, 126 derivative, 151 volume, 150 Measure of deformation, 129 Metric, 9, 10 functions, 30 space, 9, 83, 92, 103 Mixed tensor, 60 Molecular distribution, 123 N Navier’s stokes, 143, 145 Negative definite, 9 Newton, 113 Newton’s law of gravitation, 120 vacuum equation, 122 Newtonian, 113 fluid, 143, 145, 146 gravitational equation, 120 Non-Euclidean geometry, 113 Non-inertial frames, 121 Non-Newtonian viscosity, 133 Non-singular, 8 Non-isotropic medium, 113 Norm, 7 Normal form, 8, 9 stress, 124, 125 Notion of stationary arc length, 82 N-tuple, 14 O Oblique cartesian coordinates, 5 curvilinear coordinates, 10 Orientation, 133 Orthogonal, 4, 7 cartesian coordinates, 103 condition, 32 coordinate system, 29, 30

Orthogonality, 39 Order, 18 Orthonormal, 7 Orthonormal condition, 32 coordinate system, 29, 30 set, 7 Outer or open product, 18 P Parallel displacement, 43, 72–76, 83 translation, 119 Permutation or pseudo tensor, 16 Plastic, 133, 134 Plasticity, 133 Poission’s equation, 120 Positive definite, 9, 29 fundamental form, 39 Principal elongation, 141 irrotational strain, 140 normal, 77 strain, 141 Principle of covariance, 121 relativity, 113 Q Quadratic differential forms, 9, 103 elongation, 134 forms, 7 Quotient law, 25, 27, 33 Questionable one, 143 R Raise the indices, 32 Rank, 8, 18 Real index, 13 Recrystallisation, 133 Rectangular cartesian coordinates, 3, 11, 26 coordinates, 45 Rectilinear coordinates, 119 motion 119 Relative displacement, 130 tensor, 116 vector, 130 Reynold’s transport theorem, 143, 151

160

Index

Rheology, 133 Rheological standpoint, 133 Ricci tensor, 98, 109, 120, 122 Riemann, 11 Riemannian, 39, 87 Christoffel tensor, 106 coordinates, 86, 87, 115, 116 curvature tensor, 107, 108, 115 geometry, 29, 93 metric, 29, 40 space, 27, 29, 39, 114 symbols, 93, 95, 99 Rotation tensor, 145 vector, 130 Rupture, 134

Strain ellipse, 139, 141 Stress and strain, 124, 133, 134 tensor, 125, 133, 145, 146 vectors, 124 vector on a plane, 133 Structural geology, 133 Stylistic, 13 Subspace, 89 Supplement, 25 Summation convention, 13, 15 Surface invariant, 104 Sylvester law of inertia, 8 Symmetric, 20, 22, 100 stain tensor, 145 tensors, 20, 21, 23, 26, 31, 32, 46

S

T

Scalar invariant, 69 product, 31 Seismic wave propagation, 133 Seismicity, 134 Shear strain, 134 Shearing stress, 124, 125, 135 Signature, 8, 9, 11 Single value, 3 Singular, 8 Skew symmetric, 39 Space invariant, 104 Space time continuum, 113, 114, 121 Spasmodic deformation, 133 Spatial deformation gradient tensor 126 Special theory of relativity, 113, 114 State of stress, 125, 133 Stationary length, 78, 80, 81 Stipulated, 6 Stoke’s theorem, 149

Tangential, 124 Tensor density, 16 Transformation of coordinates, 14 V Variable magnitude, 73, 74 principle, 78, 92 Variation, 79 Vector of constant magnietude 73, 75 Viscosity, 143 Viscous fluids, 143 stress tensor, 146 W Weight, 27 Weight of tensors, 16 World line (track), 120 lines of light, 38