Tensors: the mathematics of relativity theory and continuum mechanics

300 Pages·2007·2.562 MB·English

by  Anadi Jiban Das

Tensors
Anadijiban Das
Tensors
The Mathematics of Relativity Theory
and Continuum Mechanics
AnadijibanDas
DepartmentofMathematicsand
PacificInstitutefortheMathematicalSciences
SimonFraserUniversity
8888UniversityAvenue
Burnaby,V5A1S6
BC,Canada
ISBN978-0-387-69468-9 e-ISBN978-0-387-69469-6
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(cid:2)c 2007SpringerScience+BusinessMedia,LLC
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Dedicated To Sri Sarada Devi
Preface
Tensor algebra and tensor analysis were developed by Riemann, Christoffel,
Ricci, Levi-Civita and others in the nineteenth century. The special theory
of relativity, as propounded by Einstein in 1905, was elegantly expressed by
Minkowski in terms of tensor fields in a flat space-time.
In 1915, Einstein formulated the general theory of relativity, in which the
space-time manifold is curved. The theory is aesthetically and intellectually
satisfying. The general theory of relativity involves tensor analysis in a pseudo-
Riemannian manifold from the outset. Later, it was realized that even the
pre-relativistic particle mechanics and continuum mechanics can be elegantly
formulatedintermsoftensoranalysisinthethree-dimensionalEuclideanspace.
In recent decades, relativistic quantum field theories, gauge field theories, and
various unified field theories have all used tensor algebra analysis exhaustively.
This book develops from abstract tensor algebra to tensor analysis in vari-
ousdifferentiablemanifolds inamathematically rigorousandlogically coherent
manner. The material is intended mainly for students at the fourth-year and
fifth-year university levels and is appropriate for students majoring in either
mathematical physics or applied mathematics.
The first chapter deals with tensor algebra, or algebra of multilinear map-
pings in a general field F. (The background vector space need not possess an
inner product or dot product.). The second chapter restricts the algebraic field
to the set of real numbers R. Moreover, it is assumed that the underlying real
vector space is endowed with an inner product (or dot product). Chapter 3
defines and investigates a differentiable manifold without imposing any other
structure. Chapter 4 discusses tensor analysis in a general differentiable man-
ifold. Differential forms are introduced and investigated. Next, a connection
formindicatingparalleltransportisbroughtforward. Asalogicalconsequence,
the fourth-order curvature tensor is generated. Chapter 5 deals with Rieman-
nianandpseudo-Riemannianmanifolds. Tensoranalysis,intermsofcoordinate
componentsaswellasorthonormalcomponents,isexhaustivelyinvestigated. In
Chapter 6, special Riemannian and pseudo-Riemannian manifolds are studied.
Flat manifolds, spaces of constant curvature, Einstein spaces, and conformally
flat spaces are explored. Hypersurfaces and submanifolds embedded in higher-
dimensional manifolds are discussed in chapter 7. Extrinsic curvature tensors
vii
viii Preface
are defined in all cases. Moreover, Gauss and Codazzi-Mainardi equations are
derived.
We would like to elaborate on the notation used in this book. The let-
ters i, j, k, l, m, n, etc., are used for the subscripts and superscripts of
a tensor field in the coordinate basis. However, we use the letters a, b, c,
d, e, f, etc., for subscripts and superscripts of the same tensor field rela-
tive to an orthonormal basis. The numerical enumeration of coordinate com-
ponents vi of a vector field is given by v1,v2,...,vN. However, numerical
elaboration of orthonormal components of the same vector field is furnished by
v(1),v(2),...,v(N) (toavoidconfusion). Similardistinctionsaremadefortensor
fieldcomponents. Theflat metriccomponentsaredenotedeitherby d or d .
ij ab
(The usual symbol η.. is reserved only for the totally antisymmetric pseudo-
tensorofLevi-Civita.) ThegeneralizedLaplacianinthe N-dimensionisdenoted
by Δ.
I would like to acknowledge my gratitude to several people for various rea-
sons. During my stay at the Dublin Institute for Advanced Studies from 1958
to 1961, I learned a lot of classical tensor analysis from the late Professor
J. L. Synge, F. R. S.. Professor W. Noll, a colleague of mine at Carnegie-
Mellon University from 1963 to 1966, introduced me to the abstract tensor
algebra, or the algebra of multilinear mappings. My research projects and
teachings on general relativity for many years have consolidated the under-
standingoftensors. Dr.AndrewDeBenedictishaskindlyreadtheproof, edited
and helped with computer work. Mrs. Judy Borwein typed from chapter 1
to chapter 5 and edited the text diligently and flawlessly. Mrs. Sabine Leb-
hart typed the difficult chapter 7 and appendices. She also helped in the
final editing. Mr. Robert Birtch drew thirty-four figures of the book. Last
but not least, my wife, Mrs. Purabi Das, was a constant source of encourage-
ment.
Contents
Preface vii
List of Figures xi
1 Finite-Dimensional Vector Spaces and Linear Mappings 1
1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Finite-Dimensional Vector Spaces . . . . . . . . . . . . . . . . . . 3
1.3 Linear Mappings of a Vector Space . . . . . . . . . . . . . . . . . 9
1.4 Dual or Covariant Vector Spaces . . . . . . . . . . . . . . . . . . 11
2 Tensor Algebra 16
2.1 Second-Order Tensors . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Higher-Order Tensors . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Exterior or Grassmann Algebra . . . . . . . . . . . . . . . . . . . 31
2.4 Inner Product Vector Spaces and the
Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Tensor Analysis on a Differentiable Manifold 52
3.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Tangent Vectors, Cotangent Vectors, and
Parametrized Curves . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Tensor Fields over Differentiable Manifolds . . . . . . . . . . . . 69
3.4 Differential Forms and Exterior Derivatives . . . . . . . . . . . . 80
4 Differentiable Manifolds with Connections 92
4.1 The Affine Connection and Covariant Derivative . . . . . . . . . 92
4.2 Covariant Derivatives of Tensors along a Curve . . . . . . . . . . 101
4.3 Lie Bracket, Torsion, and Curvature Tensor . . . . . . . . . . . . 107
ix
x Contents
5 Riemannian and Pseudo-Riemannian Manifolds 121
5.1 Metric Tensor, Christoffel Symbols,
and Ricci Rotation Coefficients . . . . . . . . . . . . . . . . . . . 121
5.2 Covariant Derivatives and the Curvature
Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3 Curves, Frenet-Serret Formulas,
and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4 Special Coordinate Charts . . . . . . . . . . . . . . . . . . . . . . 181
6 Special Riemannian and Pseudo-Riemannian Manifolds 200
6.1 Flat Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.2 The Space of Constant Curvature . . . . . . . . . . . . . . . . . . 205
6.3 Einstein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.4 Conformally Flat Spaces . . . . . . . . . . . . . . . . . . . . . . . 216
7 Hypersurfaces, Submanifolds, and Extrinsic Curvature 225
7.1 Two-Dimensional Surfaces Embedded
in a Three-Dimensional Space . . . . . . . . . . . . . . . . . . . . 225
7.2 (N −1)-Dimensional Hypersurfaces . . . . . . . . . . . . . . . . 233
7.3 D-Dimensional Submanifolds . . . . . . . . . . . . . . . . . . . . 245
Appendix 1 Fibre Bundles 257
Appendix 2 Lie Derivatives 263
Answers and Hints to Selected Exercises 271
References 277
List of Symbols 280
Index 285
List of Figures
3.1 A chart (χ,U) and projection mappings. . . . . . . . . . . . . . 53
3.2 Two charts in M and a coordinate transformation. . . . . . . . . 54
3.3 Spherical polar coordinates. . . . . . . . . . . . . . . . . . . . . . 57
3.4 A function from U ⊂M into R. . . . . . . . . . . . . . . . . . . 57
3.5 A Ck-diffeomorphism F. . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Tangent vector in E and R3. . . . . . . . . . . . . . . . . . . . 60
3
3.7 A curve γ into M. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 Reparametrization of a curve. . . . . . . . . . . . . . . . . . . . . 67
3.9 The Jacobian mapping of tangent vectors. . . . . . . . . . . . . . 74
3.10 A star-shaped domain in R2. . . . . . . . . . . . . . . . . . . . . 85
3.11 A star-shaped domain D∗ in R3. . . . . . . . . . . . . . . . . . . 87
2
4.1 Parallel propagation of a vector along a curve.. . . . . . . . . . . 102
4.2 Parallel transport along a closed curve.. . . . . . . . . . . . . . . 117
5.1 Deformation of a cubical surface into a spherical one. . . . . . . . 154
5.2 A circular helix in R3. . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Two-dimensional surface generated by geodesics. . . . . . . . . . 172
5.4 Geodesic deviation between two neighboring longitudes. . . . . . 175
5.5 An incomplete manifold R2−{(0,0)}. . . . . . . . . . . . . . . . 177
5.6 The exponential mapping. . . . . . . . . . . . . . . . . . . . . . . 183
5.7 A normal coordinate x(cid:2)N. . . . . . . . . . . . . . . . . . . . . . . 189
6.1 The normal section of M along (cid:4)t(x). . . . . . . . . . . . . . . . 206
2
6.2 (i) A plane, (ii) a sphere, and (iii) a saddle-shaped surface. . . . 206
(cid:3)
7.1 A two-dimensional surface embedded in R3. . . . . . . . . . 226
2
7.2 A smooth surface of revolution. . . . . . . . . . . . . . . . . . . . 231
(cid:3)
7.3 The image of a parametrized hypersurface ξ. . . . . . . . 234
N−1
7.4 Coordinate transformation and reparametrization
of hypersurface ξ. . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.5 Change of normal vector due to extrinsic curvature. . . . . . . . 241
xi