Tensores - P Norrington - Tensor Calculus

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AMA303: Tensor Field Theory
Patrick Norrington
The School of Mathematics and Physics
Queens University Belfast
2003
1
CONTENTS
Contents
1 Introduction and suggested reading 13
1.1 What are tensors? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Historical aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Recommended reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Notation 18
2.1 Components of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Summation convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Dummy index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Free index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Range convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Coordinate derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Summary of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Symmetry and skew-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Kronecker delta, permutation symbol and determinants 21
3.1 Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Permutation symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.3 Cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.5 Worked examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Generalised Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Tensor algebra 27
4.1 Vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Transformation of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Transformation of coordinate differentials . . . . . . . . . . . . . . . . . . . 28
4.4 Contravariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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4.5 Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Transformation of the gradient of a scalar field . . . . . . . . . . . . . . . . 29
4.7 Covariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.8 Definition of a tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.9 Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.10 Tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.11 Linear combination of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.12 Outer product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.13 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.14 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.15 Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.16 Tensor equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.17 Symmetry and skew-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Relative tensors 36
5.1 Transformation rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Transformation of determinant . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.4 Transformation of permutation symbol . . . . . . . . . . . . . . . . . . . . . 37
6 Riemannian space 38
6.1 Line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Local Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.3 Spherical surface in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 39
6.4 Raising and lowering indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.5 Length and direction of a vector . . . . . . . . . . . . . . . . . . . . . . . . 41
6.6 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.7 Geodesics on the surface of a sphere . . . . . . . . . . . . . . . . . . . . . . 44
6.8 Christoffel symbols from the geodesic equations . . . . . . . . . . . . . . . . 45
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7 Tensor calculus 46
7.1 Gradient of a scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2 Covariant derivative of a covariant vector . . . . . . . . . . . . . . . . . . . 46
7.3 Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.4 Rules for parallel displacement . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.5 Covariant derivative of a contravariant vector . . . . . . . . . . . . . . . . . 50
7.6 Covariant derivative of tensors . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.7 Covariant derivative of fundamental tensor . . . . . . . . . . . . . . . . . . 51
7.8 Product rule for covariant differentiation . . . . . . . . . . . . . . . . . . . . 51
7.9 Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.10 Symmetric affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8 Metric affinity 54
8.1 Ricci theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 Formula for the metric affinity . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.3 Condition for a flat space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9 Constant vector fields and geodesics 57
9.1 Constant vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10 Covariant derivative of relative tensors 60
11 Properties of the curvature tensor 63
11.1 Geodesic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
11.2 Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
11.3 Symmetries of curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 65
11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12 Ricci and Einstein tensors 68
12.1 Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
12.2 Divergence and Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
12.3 Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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13 Special spaces 73
13.1 Two-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13.2 Spherical surface in two-dimensions . . . . . . . . . . . . . . . . . . . . . . . 73
13.3 Spaces with constant curvature . . . . . . . . . . . . . . . . . . . . . . . . . 75
13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
14 Examples of curved two-dimensional surfaces 77
14.1 Paraboloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.1.1 Paraboloidal coordinates . . . . . . . . . . . . .. . . . . . . . . . . . 77
14.1.2 Properties of a parabola . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.1.3 Coordinate surface — paraboloid . . . . . . . . . . . . . . . . . . . . 77
14.1.4 Line-element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
14.1.5 Determination of R1212 and K . . . . . . . . . . . . . . . . . . . . . 80
14.1.6 Curvature of paraboloid . . . . . . . . . . . . . . . . . . . . . . . . . 80
14.2 Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
14.2.1 Prolate spheroidal coordinates . . . . . . . . . . . . . . . . . . . . . 81
14.2.2 Properties of an ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . 81
14.2.3 Coordinate surface — ellipsoid . . . . . . . . . . . . . . . . . . . . . 82
14.2.4 Line-element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
14.2.5 Determination of R1212 and K . . . . . . . . . . . . . . . . . . . . . 84
14.2.6 Curvature of ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . 85
14.3 Hyperboloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
14.3.1 Properties of a hyperbola . . . . . . . . . . . . . . . . . . . . . . . . 86
14.3.2 Coordinate surface — hyperboloid . . . . . . . . . . . . . . . . . . . 87
14.3.3 Line-element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
14.3.4 Determination of R1212 and K . . . . . . . . . . . . . . . . . . . . . 89
14.3.5 Curvature of hyperboloid . . . . . . . . . . . . . . . . . . . . . . . . 90
15 Cartesian tensors 91
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
15.2 Orthogonal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
15.2.1 Orthogonality conditions . . . . . . . . . . . . . . . . . . . . . . . . 92
15.2.2 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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15.2.3 Orthogonal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
15.2.4 Rotations in 2 dimensions . . . . . . . . . . . . . . . . . . . . . . . . 93
15.2.5 Rotations in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . 94
15.2.6 Eigenvalues of orthogonal matrices . . . . . . . . . . . . . . . . . . . 94
15.2.7 Summary of properties of orthogonal matrices . . . . . . . . . . . . . 95
15.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
15.4 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15.5 Tensor densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15.6 Isotropic tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15.6.2 Isotropic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
15.6.3 Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
15.6.4 Levi-Civita tensor density . . . . . . . . . . . . . . . . . . . . . . . . 98
15.6.5 Isotropic tensors in E3 . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.7 Tensor fields and calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.8 Vectors in E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.8.1 Dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.8.2 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.8.3 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
15.8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
15.8.5 Vector identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
15.9 Rigid body motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
15.9.1 Space and body axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
15.9.2 Euler’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
15.9.3 Rotating frames of reference . . . . . . . . . . . . . . . . . . . . . . . 104
15.9.4 Coriolis and centrifugal forces . . . . . . . . . . . . . . . . . . . . . . 105
15.9.5 Angular momentum of rigid body . . . . . . . . . . . . . . . . . . . 105
15.9.6 Kinetic energy of rigid body . . . . . . . . . . . . . . . . . . . . . . . 106
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16 Special relativity 107
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
16.2 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
16.3 Galilean transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
16.4 Principle of special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 109
16.5 Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
16.6 Lorentz factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
16.7 Vector form of Lorentz transformation . . . . . . . . . . . . . . . . . . . . . 111
16.8 Transformation of time- and space-intervals . . . . . . . . . . . . . . . . . . 112
16.9 Velocity transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
16.10Time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
16.11Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
16.12Lorentz-Fitzgerald contraction . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.13Length paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
17 Minkowski space-time 117
17.1 Line-element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
17.2 4-position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
17.3 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
17.4 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
17.5 Transformation of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
17.6 Raising and lowering indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
17.7 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
17.9 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
17.10Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
17.11Space-rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
17.12Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
17.13Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
17.14Boost in hyperbolic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
17.15Objective of tensor formulations . . . . . . . . . . . . . . . . . . . . . . . . 127
17.16Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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18 Relativistic mechanics 130
18.1 4-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
18.2 Transformation of 4-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
18.3 4-acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
18.4 Transformation of 4-acceleration . . . . . . . . . . . . . . . . . . . . . . . . 133
18.5 4-momentum and 4-force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
18.6 Mass-energy relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
18.7 Energy-momentumrelationship . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.8 Transformation of 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . 138
18.9 Conservation of 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 138
18.10Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
18.11Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
19 Collision examples 141
19.1 Elastic collision of two equal particles . . . . . . . . . . . . . . . . . . . . . 141
19.2 Elastic collision of two unequal particles . . . . . . . . . . . . . . . . . . . . 146
19.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
20 Energy-momentum tensor 155
20.1 Volume elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
20.2 4-force density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
20.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
20.4 External force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
20.5 Incoherent cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
20.6 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
20.7 Perfect fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
21 Electromagnetism 159
21.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
21.2 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
21.3 Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
21.4 Scalar and vector potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
21.5 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
21.6 Lorentz gauge and wave-equations . . . . . . . . . . . . . . . . . . . . . . . 162
21.7 Coulomb gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
21.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
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22 Tensor formulation of electromagnetism 164
22.1 4-current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
22.2 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
22.3 4-potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
22.4 Electromagnetic tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
22.5 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
22.6 Dual electromagnetic tensor density . . . . . . . . . . . . . . . . . . . . . . 166
22.7 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
22.8 Transformation of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
22.9 Fields produced by a moving charge . . . . . . . . . . . . . . . . . . . . . . 169
22.10Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
22.11Electromagnetic energy tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 176
22.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
23 Wave motion 179
23.1 Wave-equation and solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
23.2 Transformation of 4-propagation . . . . . . . . . . . . . . . . . . . . . . . . 179
23.3 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
24 Quantum theory 183
24.1 de Broglie waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
24.2 The quantum recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
24.3 Schro¨dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
24.4 Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
24.5 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
24.6 Charged particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
25 Introduction to General Relativity 188
25.1 Absolute space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
25.2 Mach’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
25.3 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
25.4 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
25.5 Weak Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
25.6 Einstein Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . 190
25.7 Metric theory of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
25.8 Tests of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9 May 11, 2004 5:16pm
CONTENTS
26 Equations of General Relativity 193
26.1 Newtonian gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
26.2 Vacuum field equations of General Relativity . . . . . . . . . . . . . . . . . 194
26.3 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
26.4 Gravitational red-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
26.5 Examples of gravitational red-shift . . . . . . . . . . . . . . . . . . . . . . . 199
26.6 Gravitational red-shift using Einstein Equivalence Principle . . . . . . . . . 200
26.7 Energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
26.8 Field equations of General Relativity . . . . . . . . . . . . . . . . . . . . . 202
26.9 Alternative forms for the field equations . . . . . . . . . . . . . . . . . . . . 203
26.10Identification of the constant κ . . . . . . . . . . . . . . . . . . . . . . . . . 203
27 Black-holes 204
27.1 Schwarzschild solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
27.2 Time-like geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
27.3 Orbit equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
27.4 Advance in the perihelion of planets . . . . . . . . . . . . . . . . . . . . . . 208
27.5 Null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
27.6 Deflection of a light ray near the sun . . . . . . . . . . . . . . . . . . . . . . 211
27.7 Black-holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
27.8 Eddington form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
27.9 Radial motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
28 Cosmology 214
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
28.2 A little astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
28.3 Copernican principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
28.4 Red-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
28.5 Background microwave radiation . . . . . . . . . . . . . . . . . . . . . . . . 217
28.6 Age of Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
28.7 Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
28.8 Spatial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
28.9 Scale factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10 May 11, 2004 5:16pm
CONTENTS
28.10Field equations of General Relativity . . . . . . . . . .. . . . . . . . . . . 219
28.11Energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
28.12Evaluation of the Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 221
28.13Conservation of energy-momentum . . . . . . . . . . . . . . . . . . . . . . . 221
28.14Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
28.15Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
28.16Summary of formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
28.17Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
28.18‘Big bang’ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
28.19Matter-dominated model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A 3-dimensional vectors 229
A.1 Cartesian vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.2 Vector products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.3 Gradient operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.4 Vector identities involving the gradient . . . . . . . . . . . . . . . . . . . . . 232
A.5 Vector theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.6 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
A.7 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . 235
A.8 Cylindrical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 235
A.9 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B Odds and ends 239
C Biography of Maxwell 246
D Biography of Einstein 251
E Simultaneity 258
F Electromagnetism using SI units 262
G Global Positioning System 267
G.1 Einstein’s Relativity and Everyday Life by Clifford M. Will . . . . . . . . . 267
G.2 General relativity in the global positioning system by Neil Ashby . . . . . 268
G.3 Some Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
11 May 11, 2004 5:16pm
LIST OF FIGURES
List of Figures
1 Parabola y2 = 4ax with a = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2 Family of confocal parabolas labelled by β . . . . . . . . . . . . . . . . . . . 79
3 Ellipse x
2
a2
+ y
2
b2
= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Family of confocal ellipses labelled by α . . . . . . . . . . . . . . . . . . . . 83
5 K as a function of β for various α . . . . . . . . . . . . . . . . . . . . . . . 85
6 Hyperbola x
2
a2
− y2
b2
= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Family of confocal hyperbolas labelled by β . . . . . . . . . . . . . . . . . . 89
8 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9 Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10 Ratio of relativistic and non-relativistic kinetic energy (energy in units of the
rest energy). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11 Collision between two equal particles . . . . . . . . . . . . . . . . . . . . . . 142
12 Collision between equal masses, v/c = .8. Dashed curves are non-relativistic,
solid curves are relativistic. Quantities are plotted as a function of α in the
range 0 to pi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
13 Collision between equal masses, v/c = .99. Dashed curves are non-relativistic,
solid curves are relativistic. Quantities are plotted as a function of α in the
range 0 to pi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
14 Collision between unequal masses (m2/m1 = 2), v/c = .8. Dashed curves
are non-relativistic, solid curves are relativistic. Quantities are plotted as a
function of α in the range 0 to pi. . . . . . . . . . . . . . . . . . . . . . . . 153
15 Collision between unequal masses (m2/m1 = 2), v/c = .99. Dashed curves
are non-relativistic, solid curves are relativistic. Quantities are plotted as a
function of α in the range 0 to pi. . . . . . . . . . . . . . . . . . . . . . . . 154
16 Moving charge in frame S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
17 Field lines for a charge at rest . . . . . . . . . . . . . . . . . . . . . . . . . . 172
18 Field lines for a moving charge . . . . . . . . . . . . . . . . . . . . . . . . . 173
19 Plot of f against w = ct for b = 1 and a range of v/c–values . . . . . . . . . 174
20 Plot of g against w = ct for b = 1 and a range of v/c–values . . . . . . . . . 175
21 Cylindrical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 236
22 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
12 May 11, 2004 5:16pm
1 INTRODUCTION AND SUGGESTED READING
1 Introduction and suggested reading
1.1 What are tensors?
In dynamics we are familiar with the notion of scalar and vector quantities. Examples of
scalars are the mass m of a particle, the speed v and the kinetic energy 12mv
2. A scalar
has a single numerical value. In contrast a vector has three numbers associated with it.
Examples from dynamics are a particle’s position r, velocity v and acceleration f . Force
F is also a vector. Newton’s 2nd law is a vector equation that relates the vector force to
the vector acceleration.
F = mf (1)
A particle’s position r is often given as 3 components (x, y, z) referred to a right-handed
rectangular set of axes. These are the well-known Cartesian 1 components. Equally well the
position can be expressed in terms of either spherical polar coordinates (r, θ, φ) or cylindrical
polar coordinates (ρ, φ, z). These are the most familiar (and useful) coordinates but there
is no limit to the possibilities. We are at liberty to choose the coordinates and of course
one should select those that make the given problem easiest.
The motion does not depend on the choice of reference axes. Thus the equations eq.(1) that
describe the motion are independent of coordinate system. The vector components in one
coordinate system will transform in a prescribed way to another coordinate system so as to
ensure that this independence is preserved.
The key idea is that of invariance with respect to coordinate transformations.
A tensor is a set of quantities that transform in a prescribed manner when a coordinate
transformation is made. It is a generalisation of the notion of a vector. A scalar, which
is unchanged by coordinate transformation, is a tensor of rank 0. A vector is a tensor of
rank 1. Higher rank tensors occur. An example of a tensor of rank 2 from dynamics (rigid
body motion) is the inertia tensor. The inertia tensor can be written as a matrix with 9
components. A tensor of rank r in a space of dimension N has N r components. Tensors
are the appropriate objects for describing many physical phenomena, such as solid and fluid
mechanics, elasticity, special and general relativity. They allow coordinate transformations
of any type.
A special case is to restrict the type of coordinates to being Cartesian, i.e. rectangular axes.
The possible transformations are then rotations and/or reflections so that the reference axes
remain orthogonal. Cartesian tensors are defined by their transformation properties under
orthogonal transformations. While the theory of Cartesian tensors is simpler than that of
general tensors, it still has wide application in physics.
In general we consider any type of coordinate transformation. This allows us to treat
curvilinear coordinates, such as spherical polar, and to analyse the properties of more
general spaces than Euclidean. Tensors can be defined in quite general vector spaces but
our attention willbe restricted to Riemannian 2 spaces.
1See Appendix B ‘Odds and ends’ note 1.
2See Appendix B ‘Odds and ends’ note 2.
13 May 11, 2004 5:16pm
1 INTRODUCTION AND SUGGESTED READING
1.2 Historical aspect
The following brief historical introduction is taken from ‘Elementary Vector Analysis’ by
CE Weatherburn, 1955, Bell & Sons.
During the last sixty or seventy years there has appeared a broad generalisation
of vector analysis under the name of Tensor Analysis, which sprang from the
study of differential geometry of multidimensional space. The history of differ-
ential geometry of spaces of more than three dimensions may be said to have
begun with a paper by Bernhard Riemann (1826-1866) on the hypotheses which
lie at the foundation of geometry, read before the Philosophical Faculty of the
University of Go¨ttingen in 1854, but not published till 1868, after Riemann’s
death. Riemannian geometry is based on the assumption that the square of the
linear element is represented by a quadratic differential form, usually called a
Riemannian metric. The corresponding Riemannian space, which may be of any
number of dimensions, is a generalisation of Euclidean space of three dimensions.
During the twenty years immediately following Riemann’s death several math-
ematicians contributed to the development of the subject; but the year 1887 is
especially memorable for the publication by Gregorio Ricci (1853-1924) of his
first short note dealing with the calculus which is now known variously as the
Ricci calculus, the Absolute Differential Calculus or the Calculus of Tensors. In
this calculus scalars appear as tensors of order zero; covariant and contravari-
ant vectors, which are a generalisation of the vectors of Euclidean 3-space, are
tensors of order one. The dyads and dyadics of vector analysis are examples of
tensors of order two. The order of a tensor may be any positive integral num-
ber. During the ensuing decade Ricci ‘elaborated the theory, and worked out
the elegant and comprehensive notation which enables it to be easily adapted
to a wide variety of questions of analysis, geometry and physics’ (quote from
the text ‘The Absolute Differential Calculus’ by T Levi-Civita, 1927, Blackie &
Son). Not only did this new analysis greatly simplify Riemannian geometry, but
it also led to a wider extension of the field of research; and, during the thirty
years from 1887, Ricci and the Italian school of mathematicians contributed
very largely to this branch of mathematics.
For some years Ricci’s labours and his new method attracted the attention of
only a small number of mathematicians; but the publication in 1913 and 1916
of Albert Einstein’s first papers in general relativity focused the attention of
mathematicians on Ricci’s calculus. Einstein assumed a Riemannian space of
four dimensions as the basis of his general theory, and found in the absolute
differential calculus the best instrument for formulating his ideas. Since then
the tensor calculus has been used extensively by mathematicians and physicists,
and has proved itself a useful and powerful instrument of research.
The following quote is taken from ‘Space–Time–Matter’ by H Weyl, Dover 1952 reprint of
1922 4th edition.
14 May 11, 2004 5:16pm
1 INTRODUCTION AND SUGGESTED READING
The study of tensor calculus is, without doubt, attended by conceptual difficul-
ties — over and above the apprehension inspired by indices — which must be
overcome. From the formal aspect, however, the method of reckoning used is
of extreme simplicity; it is much easier than, e.g., the apparatus of elementary
vector calculus. There are two operations, multiplication and contraction; i.e.,
putting the components of two tensors with totally different indices alongside
one another; the identification of an upper index with a lower one, and, finally
summation (not expressed) over this index. Various attempts have been made
to set up a standard terminology in this branch of mathematics involving only
the vectors themselves and not their components, analogous to that of vectors
in vector analysis. This is highly expedient in the latter, but very cumbersome
for the much more complicated framework of the tensor calculus. In trying to
avoid continual reference to the components we are obliged to adopt an end-
less profusion of names and symbols in addition to an intricate set of rules for
carrying out calculations, so that the balance of advantage is considerably on
the negative side. An emphatic protest must be entered against these orgies of
formalism which are threatening the peace of even the technical scientist.
Lord Kelvin, whose statue dominates the entrance to Botanic Gardens, had an interesting
view on the value of vectors 3.
1.3 Recommended reading
This module provides an introduction to tensor calculus and applies it to special and general
relativity.
The module was originally based on the text ‘An Introduction to Tensor Calculus, Relativity
and Cosmology’ by DF Lawden (3rd edition, Wiley, 1982). The most important change is
the notation used for special relativity. However this remains an excellent introduction to
the subject.
Our treatment of tensor calculus is an old-fashioned one that concentrates on the coordinate
representation of the tensor quantities. This is a constraint imposed by lack of time and
student background.
When Einstein published his classic 1916 paper on general relativity, he devoted a large
section to outlining tensor calculus. It is a clear and concise exposition. This was necessary
because the mathematics was unfamiliar to his audience of physicists. A translation of the
paper (Annalen der Physik, 49, 1916) can be found in ‘The Principle of Relativity’ by HA
Lorentz, A Einstein, H Minkowski and H Weyl (Dover, 1952). This text is a reprint of the
papers that founded the theory of relativity and as such represent one of mankind’s great
cultural achievements. A proper test of your understanding of this module is the facility
with which you can read these original papers.
Einstein’s problem with his audience is reflected in all texts on relativity. Tensor calculus
is described in either the introductory chapters or as an appendix. While the notation and
3See Appendix B ‘Odds and ends’ note 3.
15 May 11, 2004 5:16pm
1 INTRODUCTION AND SUGGESTED READING
terminology is fairly standard, there are differences in how it is applied to relativity. The
authors must establish their conventions and the readers must take proper note of them.
The text ‘Tensor Calculus’ by JL Synge and A Schild (University of Toronto, 1949 and
Dover, 1978), adopts the coordinate representation with applications in classical mathe-
matical physics but avoiding relativity. The treatment of tensor calculus is particularly well
done.
‘Space-Time Structure’ by Erwin Schro¨dinger (Cambridge University Press, 1950) concen-
trates on general relativity. This is quite short and very well-written.
The text by Lovelock and Rund is excellent for the theory of tensors and how it generalises
to manifolds and differential forms.
The modern treatment of tensors is based on the differential geometry due to Cartan 4.
Tensors represent geometric entities in the study of manifolds. A global study of the man-
ifolds is essentially coordinate independent. This introduces the notions of tangent and
cotangent spaces, forms, Lie derivatives and bundles.
This approach can be found in the text ‘Gravitation’ by CW Misner, KS Thorne and JA
Wheeler (Freeman, 1973). This is widely regarded as the best book on relativity. It is not
for the faint-hearted, extending to almost 1300 pages. But as a reference it is unsurpassed.
The style is vivid with plenty of illustrations. Any difficulty with the basic concepts could
be resolved by studying this book.
A gentler introduction to modern differential geometry is ‘Geometrical Methods of Mathe-
matical Physics’by BF Schutz (Cambridge University, 1980). A complementary book is ‘A
First Course in General Relativity’ by BF Schutz (Cambridge University, 1985).
Other texts include ‘An Introduction to General Relativity’ by LP Hughston and KP
Tod (Cambridge University, 1991) and ‘Introducing Einstein’s Relativity’ by RA d’Inverno
(Clarendon, 1992). The text ‘General Relativity’ by H Stephani (2nd edition, Cambridge
University, 1990) can also be recommended. The recent texts contain a discussion of more
up-to-date experimental evidence in support of relativity.
Chandrasekhar’s book on black-holes is a classic but hard going. This is an advanced text
suitable for postgraduates.
The concepts of relativity are difficult and background reading is important to complement
the more terse presentation of the material in the lectures. There is insufficient time to go
into lengthy explanations and you can get this in these texts. Certainly you should at least
browse some of these books in the library.
4See Appendix B ‘Odds and ends’ note 4.
16 May 11, 2004 5:16pm
1 INTRODUCTION AND SUGGESTED READING
text QUB library availability on
www.amazon.co.uk
‘An Introduction to Tensor Calculus, Relativity QA433/LAWD no
and Cosmology’ by DF Lawden
(3rd edition, Wiley, 1982)
‘The Principle of Relativity’ by HA Lorentz, QC11/LORE £5
A Einstein, H Minkowski and H Weyl (Dover, 1952)
‘Tensor Calculus’ by JL Synge and A Schild QA433/SYNG £7.80
(University of Toronto, 1949 and Dover, 1978)
‘Space-Time Structure’ by Erwin Schro¨dinger QC11/SCHR £16.95
(Cambridge University Press, 1950)
‘Tensors, Differential Forms, and Variational Principles’ no £8.35
by D Lovelock and H Rund (Dover, 1975)
‘Gravitation’ by CW Misner, KS Thorne and QC178/MISN £67.99
JA Wheeler (Freeman, 1973)
‘Geometrical Methods of Mathematical Physics’ QC20.7.D52/SCHU £20.95
by BF Schutz (Cambridge University, 1980)
‘A First Course in General Relativity’ QC173.6/SCHU £21.95
by BF Schutz (Cambridge University, 1985)
‘General Relativity’ by H Stephani QC173.55/STEP £25.95
(2nd edition, Cambridge University, 1990)
‘An Introduction to General Relativity’ QC173.6/HUGH £16.95
by LP Hughston and KP Tod
(Cambridge University, 1991)
‘Introducing Einstein’s Relativity’ QC173.55/DINV £29.95
by RA d’Inverno (Clarendon, 1992)
‘The Mathematical Theory of Black Holes’ QB843.B55/CHAN £22.95
by S Chandrasekhar (Clarendon, 1983)
Note:
• Availability on AMAZON can be deceptive. It is only after you actually order the
book that you learn its true availability. Having said that I have always found it good.
• These prices were checked in January 2003.
17 May 11, 2004 5:16pm
2 NOTATION
2 Notation
2.1 Components of a vector
You should be familiar with the properties of 3-dimensional vectors, see Appendix A.
In an N -dimensional space a vector will have N components. We will label these by an
index. Normally one would use a subscript: vector A has components (A1, A2 · · ·AN ).
However in general tensor theory both subscripts and superscripts are used to identify the
components. So vector A also has components (A1, A2 · · ·AN ).
Ai, (i = 1 · · ·N) are the up-components, called contravariant.
Ai, (i = 1 · · ·N) are the down-components, called covariant.
In general Ai 6= Ai but there may be a relationship between them.
One problem with using superscripts as an index is that they can be confused with powers.
You must be careful with this. For example, use brackets,
(A1)2 = A1 ×A1 (2)
2.2 Summation convention
Tensor analysis is characterised by lots of indices and summations. To enable the easy
handling of these we introduce the summation convention due to Einstein.
An equation is an expression set to zero. An expression is the sum of a number of terms. In
a term a repeated index, one a subscript and the other a superscript, implies a summation.
For example
Aij X
j ≡
N∑
j=1
Aij X
j (3)
where the repeated index is j. The upper limit of the summation is the dimension of the
space we are considering i.e. in an N -dimensional space j = 1, 2 · · ·N .
This is a very compact notation and care should be taken with it. At times it is inconvenient
and then it can be suspended. This should be stated explicitly.
2.3 Dummy index
A repeated index is known as a dummy index and in any term it can only occur twice. A
dummy index (like an integration variable) can be replaced by another index provided that
index does not already occur in the term.
Aij X
j = Aik X
k (4)
18 May 11, 2004 5:16pm
2 NOTATION
2.4 Free index
Other indices are known as free indices.
In tensor equations it is important to ensure that the free covariant and contravariant indices
match in the various terms.
Aji = B
j
ikX
k +Dji (5)
= BjimX
m +Dji
Here k,m are used as dummy indices, i is a covariant free index and j is a contravariant
free index.
2.5 Range convention
When we write
Bi = Aij X
j (6)
then we mean any of the equations i = 1, 2 · · ·N .
2.6 Coordinate derivatives
Tensor quantities are often functions of the coordinates xi. The coordinates are always
written with a superscript.
Partial derivatives with respect to xi are written as ∂i. For example
∂Ai
∂xj
= ∂jA
i (7)
2.7 Summary of notation
• Tensors use both subscripts and superscripts
• A subscript is called covariant
• A superscript is called contravariant
• A repeated (dummy) index, one a subscript and the other a superscript, implies a
summation
• A dummy index can only occur twice in any term
• A dummy index can be replaced by another index provided that index does not already
occur in the term
• Free indices match in the terms of an equation
19 May 11, 2004 5:16pm
2 NOTATION
2.8 Greek alphabet
Greek letters are often used in relativity. You should expect to meet these in indices. Just
in case . . .
alpha α beta β gamma γ Γ
delta δ ∆ epsilon � or ε zeta ζ
eta η theta θ or ϑ Θ iota ι
kappa κ lambda λ Λ mu µ
nu ν xi ξ Ξ o o
pi pi or $ Π rho ρ or % sigma σ or ς Σ
tau τ upsilon υ Υ phi φ or ϕ Φ
chi χ psi ψ Ψ omega ω Ω
2.9 Symmetry and skew-symmetry
A quantity is symmetric in two indices of the same type if it is unchanged when the indices
are interchanged i.e. Tijk is symmetric in i and j if Tijk = Tjik.
By completely symmetric we mean that the quantity is symmetric in all pairs of indices.
Similarly a quantity is skew-symmetric in two indices of the same type if it changes sign
when the indices are interchanged i.e. Tijk is skew-symmetric in i and j if Tijk = −Tjik.
If Tij is skew-symmetric then Tij = 0 when i = j.
We can write any Tij as the sum of a symmetric part and a skew-symmetric part
Tij =
1
2 (Tij + Tji) +
1
2(Tij − Tji) (8)
Symmetry and skew-symmetry are invariant properties of tensors. Invariant means that
the property is preserved when any coordinate transformation is made.
It is important to exploit any such properties as they reduce the number of non-zero in-
dependent components. For example if N = 3 a skew-symmetric quantity Tij has only 3
non-zero independent components, namely T12, T13 and T23.
20 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
3 Kronecker delta, permutation symbol and determinants
3.1 Kronecker delta
The Kronecker 5 delta is defined by
δij =
{
1 if i = j
0 otherwise
(9)
A similar definition holds for δij and δij .
• δii = N
• δijδjk = δik
• δijAj = Ai
• δij is an isotropic, type (1, 1) tensor
Isotropic means that the components are unchanged by any coordinate transformation.
3.2 Permutation symbol
A permutation of N objects is an arrangement or a rearrangement of the N objects. The
number of possible permutations is N !.
Starting from arrangement A then we can obtain arrangement B by the repeated procedure
of swapping two objects. If the number of swaps required is aneven number then we say
that B is an even permutation of A. If the number is odd then we have an odd permutation.
For example, if arrangement A is 1234 then 3124 is an even permutation since it can be
obtained by two swaps : 1234 → 2134 → 3124. The permutation 4123 is odd since you
require three swaps : 1234 → 4231 → 4213 → 4123.
The permutation symbol is defined by
ei1···iN =


1 if i1 · · · iN is an even permutation of 1 · · ·N
−1 if i1 · · · iN is an odd permutation of 1 · · ·N
0 otherwise
(10)
A similar definition holds for ei1···iN .
• It has N indices
• e1···N = 1 and e1···N = 1
• It is completely skew-symmetric
5See Appendix B ‘Odds and ends’ note 5.
21 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
• It vanishes if an index is repeated
• When N = 2, e12 = −e21 = 1 and e11 = e22 = 0
• When N = 3, there are 6 non-zero values e123 = e231 = e312 = 1 ( i.e. indices are
cyclic in 123), and e132 = e213 = e321 = −1
• ei1···iN−2iN−1iN ei1···iN−2jN−1jN = (N − 2)!
[
δ
iN−1
jN−1
δiNjN − δ
iN−1
jN
δiNjN−1
]
i.e. N − 2 dummy
indices
• ei1···iN−1iN ei1···iN−1jN = (N − 1)! δiNjN i.e. N − 1 dummy indices
• ei1···iN ei1···iN = N ! i.e. N dummy indices
• ei1···iN is an isotropic, type (N, 0) relative tensor of weight 1 (or tensor density)
• ei1···iN is an isotropic, type (0, N) relative tensor of weight −1
3.3 Determinants
3.3.1 Definition
If (aij) is an N ×N matrix (i labels the row and j labels the column) then its determinant
is given by
a = det(aij) (11)
=
∣∣∣∣∣∣∣∣∣∣
a11 a12 · · · a1N
a12 a22 · · · a2N
...
... · · · ...
a1N a2N · · · aNN
∣∣∣∣∣∣∣∣∣∣
=
∑
j1j2···jN
ej1j2···jN a1j1a2j2 · · · aNjN
The summation sign is retained for the present.
3.3.2 Properties
If A,B are square matrices and T denotes matrix transpose, then the following are properties
of determinants:
• det(AT) = det(A)
• det(AB) = det(A) det(B)
• det(A−1) = 1/det(A)
• det(A) changes sign if two rows (or columns) are interchanged
22 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
• det(A) = 0 if two rows (or columns) are the same
• det(λA) = λN det(A) where A is an N ×N matrix
• If a row (column) is multiplied by λ then the determinant is multiplied by λ
• The determinant is unchanged if a multiple of a row (or column) is added to another
row (or column)
• We can add/subtract determinants by the decomposition of a row (or column)
3.3.3 Cofactors
Let k be a fixed index with 1 ≤ k ≤ N . Then since each term in this expansion has only
one factor from row k, we can write the determinant as
a = ak1A
1k + ak2A
2k + · · ·+ akNANk (12)
=
∑
j
akjA
jk
where Ajk is the cofactor of element akj. The summation convention is not applied here.
Ajk is the coefficient of akj in the expansion of the determinant. Since k is held fixed, this
formula corresponds to an expansion by row k of the determinant. Ajk is (−1)k+j times
the (N − 1)× (N − 1) determinant obtained from a by deleting row k and column j.
Analogous formulae are
a =
∑
j1j2···jN
ej1j2···jN aj11aj22 · · · ajNN (13)
= a1kA
k1 + a2kA
k2 + · · ·+ aNkAkN
=
∑
j
ajkA
kj
corresponding to an expansion by column k of the determinant. The summation convention
is not applied here.
Consider ∑
j
akjA
jh (14)
when k 6= h. This is the determinant of a matrix in which two rows are the same. It is
therefore zero.
Therefore we can write
aδhk = akjA
jh (15)
= ajkA
hj
using the summation convention.
23 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
3.3.4 Results
• a = ej1j2···jN a1j1a2j2 · · · aNjN = ej1j2···jN aj11aj22 · · · ajN N
• ei1i2···iN a = ej1j2···jN ai1j1ai2j2 · · · aiN jN = ej1j2···jN aj1i1aj2i2 · · · ajN iN
• aδhk = akjAjh = ajkAhj
• Ajk = ∂a
∂akj
• If the elements depend on coordinates xi, then ∂ia = Ajk ∂iakj.
• If a 6= 0, then the inverse of (aij) is (aij) with aij = Aij/a.
• If N = 2, then
(aij) =
(
a11 a12
a21 a22
)
−→ (aij) = 1
a
(
a22 −a12
−a21 a11
)
(16)
where a = a11a22 − a12a21.
• If (aij) is symmetric, then the inverse (aij) is symmetric.
• If aij = BiBj , then a = 0.
• If (aij) is skew-symmetric and N is odd, then a = 0.
• det(Aij) = aN−1
• If aij is a type (0, 2) tensor, then aij is type (2, 0) tensor.
• If aij is a type (0, 2) tensor, then a is relative scalar of weight 2.
3.3.5 Worked examples
We will choose N = 3 but the results can be easily generalised to N > 3
• To show that a determinant changes sign when two rows are interchanged
det(A) = eijk a1i a2j a3k (17)
= −ejik a1i a2j a3k
= −ejik a2j a1i a3k
= −det(A′)
where A′ is obtained from A by interchanging rows 1 and 2.
24 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
• To show that det(AB) = det(A) det(B), let C = AB
det(C) = eijk c1i c2j c3k (18)
= eijk (a1l δ
lm bmi)(a2n δ
no boj)(a3p δ
pq bqk)
= (eijk bmi boj bqk) δ
lmδnoδpq a1l a2n a3p
= emoq det(B) δ
lmδnoδpq a1l a2n a3p
= elnp det(B) a1l a2n a3p
= det(A) det(B)
• To show that det(AT) = det(A)
det(AT) = eijk
(
AT
)
1i
(
AT
)
2j
(
AT
)
3k
(19)
= eijk ai1 aj2 ak3
= det(A)
3.4 Generalised Kronecker delta
The generalised Kronecker delta is defined by
δi1i2···irj1j2···jr =
∣∣∣∣∣∣∣∣∣∣
δi1j1 δ
i1
j2
· · · δi1jr
δi2j1 δ
i2
j2
· · · δi2jr
...
... · · · ...
δirj1 δ
ir
j2
· · · δirjr
∣∣∣∣∣∣∣∣∣∣
(20)
• When r = 1 we have the Kronecker delta.
• When r = 2 we have
δi1i2j1j2 = δ
i1
j1
δi2j2 − δi1j2δi2j1 (21)
• When r = 3 we have
δi1i2i3j1j2j3 = δ
i1
j1
δi2i3j2j3 − δi2j1δi1i3j2j3 + δi3j1δi1i2j2j3 (22)
= δi1j1δ
i2
j2
δi3j3 − δi1j1δi3j2δi2j3 + δi3j1δi1j2δi2j3 − δi2j1δi1j2δi3j3 + δi2j1δi3j2δi1j3 − δi3j1δi2j2δi1j3
• In general δi1 ···irj1···jr is the sum of r! terms, each of which is the product of r Kronecker
deltas.
• δi1 ···irj1···jr is skew-symmetric under interchange of any two subscripts (or superscripts).
• δ12···r12···r = 1
• If any two subscripts (or superscripts) are the same, then δ i1···irj1···jr = 0.
25 May 11, 2004 5:16pm
3 KRONECKER DELTA, PERMUTATION SYMBOL AND DETERMINANTS
• If r > N , then δi1i2···irj1j2···jr = 0. We assume below that r ≤ N .
• δi1 ···ir−1irj1···jr−1ir = (N − r + 1) δ
i1 ···ir−1
j1···jr−1
• δi1 ···isis+1···irj1···jsis+1···ir =
(N−s)!
(N−r)! δ
i1···is
j1···js
• δi1 ···itit+1···iri1···itjt+1···jr =
(N−r+t)!
(N−r)! δ
it+1···ir
jt+1···jr
• Taking t = r gives δi1···iri1···ir = N !(N−r)!
• ei1···iN = δ1···Ni1···iN and ei1···iN = δi1···iN1···N
• ei1···iN ej1···jN = δi1···iNj1···jN
• ei1···itit+1···iN ei1···itjt+1···jN = t! δit+1···iNjt+1···jN
• det(aij) = 1N ! δi1···iNj1···jN a
j1
i1
· · · ajNiN
• If Ai1···ir is completely skew-symmetric, then δi1 ···irj1···jrAj1···jr = r!Ai1···ir .
• If Bi1···ir is completely skew-symmetric, then δi1···irj1···jrBi1···ir = r!Bj1···jr .
• δi1 ···irj1···jr is an isotropic, type (r, r) tensor.
• If Ti1···ir is a type (0, r) tensor field, then δi1···ir+1j1···jr+1 ∂ir+1Ti1···ir is a type (0, r+1) tensor
field. It is skew-symmetric.
• When r = 1
Sj1j2 = δ
i1i2
j1j2
∂i2Ti1 (23)
= (δi1j1δ
i2
j2
− δi1j2δi2j1) ∂i2Ti1
= ∂j2Tj1 − ∂j1Tj2
• When r = 2
Sj1j2j3 = δ
i1i2i3
j1j2j3
∂i3Ti1i2 (24)
=
[
δi1j1δ
i2
j2
δi3j3 − δi1j1δi3j2δi2j3 + δi3j1δi1j2δi2j3 − δi2j1δi1j2δi3j3 + δi2j1δi3j2δi1j3 − δi3j1δi2j2δi1j3
]
∂i3Ti1i2
= ∂j3Tj1j2 − ∂j2Tj1j3 + ∂j1Tj2j3 − ∂j3Tj2j1 + ∂j2Tj3j1 − ∂j1Tj3j2
= ∂j3(Tj1j2 − Tj2j1) + ∂j1(Tj2j3 − Tj3j2) + ∂j2(Tj3j1 − Tj1j3)
26 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
4 Tensor algebra
4.1 Vector space
A point isN ordered real numbers (anN -tuple) written with theN coordinates as (x1, x2, . . . xN )
or simply xi.
The coordinatesare independent so that
∂xi
∂xj
= δij (25)
The totality of all points, in specified ranges, constitutes a space of N dimensions, denoted
VN .
A curve is the totality of points given by:
xi = f i(u) (26)
where u is a parameter and f i are N functions.
A subspace VM is given by
xi = f i(u1, u2, . . . uM ) (27)
where uj are M parameters and M < N . VN−1 is a surface. A surface divides the adjacent
space into two parts.
4.2 Transformation of coordinates
Consider a general transformation of coordinates
xi = f i(x1, x2, . . . xN ) (28)
The functions f i are single-valued, continuous and differentiable. It is more convenient to
replace f i by xi so that
xi = xi(x1, x2, . . . xN ) (29)
For the transformation to be useful it is important that the inverse exists
xi = xi(x1, x2, . . . xN ) (30)
A point in the space can be uniquely represented either by (x1, x2, . . . xN ) or (x1, x2, . . . xN ).
The condition for this is that the Jacobian 6 is non-zero.
If we define
J ij =
∂xi
∂xj
(31)
6See Appendix B ‘Odds and ends’ note 6.
27 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
then the Jacobian is
J = det(J ij) (32)
=
∣∣∣∣∣∣∣∣∣∣
∂x1
∂x1
. . . ∂x
1
∂xN
...
...
∂xN
∂x1
. . . ∂x
N
∂xN
∣∣∣∣∣∣∣∣∣∣
The Jacobian for the inverse transformation is
J = det(J
i
j) (33)
where
J
i
j =
∂xi
∂xj
(34)
Consider
J iaJ
a
j =
∂xi
∂xa
∂xa
∂xj
(35)
=
∂xi
∂xj
by the chain rule
= δij
Therefore the matrix (J ij) is the inverse of the matrix (J
i
j) and taking determinants JJ = 1.
We consider only transformations for which J 6= 0 and J 6= ∞.
For a given transformation the Jacobian will be a function of the coordinates. Because
of the continuity (smoothness) of the coordinates and the condition that the Jacobian is
non-zero, it follows that the Jacobian will either be always positive or always negative.
4.3 Transformation of coordinate differentials
The coordinate differentials transform as
dxi =
∂xi
∂xa
dxa (36)
The coefficients are the elements of the Jacobian J .
The transformation is affine i.e. it is linear and homogeneous.
Also it is transitive. By this we mean that the transformation from xi to yi followed by
the transformation from yi to zi gives the same result as the transformation from xi to zi.
Given
dyi =
∂yi
∂xa
dxa and dzi =
∂zi
∂ya
dya (37)
28 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
then substitute for dya
dzi =
∂zi
∂ya
dya (38)
=
∂zi
∂ya
∂ya
∂xb
dxb
=
∂zi
∂xb
dxb
giving the transformation from xi to zi.
4.4 Contravariant vectors
A contravariant vector is a set of N quantities that transforms like the coordinate differen-
tials. If
T
i
=
∂xi
∂xa
T a (39)
then T i is a contravariant vector. We use a superscript for the index.
The coefficients ∂x
i
∂xj
in general depend on the coordinates xi.
The inverse transformation is
T i =
∂xi
∂xa
T
a
(40)
While the coordinate differentials form a contravariant vector, the coordinates xi do not in
general.
The tangent vector dx
i
du
to a curve with parameter u is contravariant.
4.5 Scalar
A scalar quantity T remains invariant under a coordinate transformation.
T (x1, x2, . . . xN ) = T (x1, x2, . . . xN ) (41)
The functional forms of T and T are different. It is the numerical value at a given point in
the space that is unchanged.
A function of position is called a field.
4.6 Transformation of the gradient of a scalar field
The gradient of a scalar field is
∂T
∂xi
(42)
This transforms as follows (again using the chain rule)
∂T
∂xi
=
∂xa
∂xi
∂T
∂xa
(43)
29 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
4.7 Covariant vectors
A covariant vector is a set of N quantities that transforms like the gradient of a scalar field.
If
T i =
∂xa
∂xi
Ta (44)
then Ti is a covariant vector. We use a subscript for the index.
The inverse transformation is
Ti =
∂xa
∂xi
T a (45)
4.8 Definition of a tensor
A tensor of type (p, q) in a space of dimension N has rank t = p + q and N t components.
It is written as
T
i1i2···ip
j1j2···jq
(46)
The components are labelled by t indices. There are p contravariant and q covariant indices.
The general transformation law is
T
i1i2···ip
j1j2···jq
=
∂xi1
∂xa1
∂xi2
∂xa2
· · · ∂x
ip
∂xap
∂xb1
∂xj1
∂xb2
∂xj2
· · · ∂x
bq
∂xjq
T
a1a2···ap
b1b2···bq
(47)
The position of the index indicates how the tensor transforms with respect to that index. A
subscript implies a covariant index and the transformation is similar to a covariant vector. A
superscript implies a contravariant index and the transformation is similar to a contravariant
vector.
Tensors may be covariant in all indices, contravariant in all indices or mixed.
• If Tab is a covariant (0, 2) tensor of rank 2 then it transforms as
T ij =
∂xa
∂xi
∂xb
∂xj
Tab (48)
• If Tabc is a covariant (0, 3) tensor of rank 3 then it transforms as
T ijk =
∂xa
∂xi
∂xb
∂xj
∂xc
∂xk
Tabc (49)
• If T ab is a contravariant (2, 0) tensor of rank 2 then it transforms as
T
ij
=
∂xi
∂xa
∂xj
∂xb
T ab (50)
30 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
• If T abc is a contravariant (3, 0) tensor of rank 3 then it transforms as
T
ijk
=
∂xi
∂xa
∂xj
∂xb
∂xk
∂xc
T abc (51)
• If T cab is a mixed (1, 2) tensor of rank 3 then it transforms as
T
k
ij =
∂xa
∂xi
∂xb
∂xj
∂xk
∂xc
T cab (52)
A zero rank tensor, known as a scalar, is invariant T = T .
A vector is a first rank tensor and may be covariant (0, 1) or contravariant (1, 0).
4.9 Kronecker delta
An isotropic tensor is one whose components are the same in all coordinate systems.
The Kronecker delta δji is a mixed (1, 1), isotropic tensor of rank 2. It is known as the
fundamental mixed tensor. To show this suppose T ji is a mixed tensor of rank 2 and
T ji = δ
j
i in coordinate system x
i. Then in any other coordinate system
T
j
i =
∂xj
∂xb
∂xa
∂xi
T ba (53)
=
∂xj
∂xb
∂xa
∂xi
δba
=
∂xj
∂xa
∂xa
∂xi
=
∂xj
∂xi
= δji
We can still use δij as the Kronecker delta but its definition only holds in the particular
coordinate system being considered. Once we transform to another coordinate system then
the components change i.e. it is not isotropic.
4.10 Tensor field
A tensor is a set of quantities defined at a point in the space. As you vary the point, the
components of the tensor vary. This gives rise to a tensor field. When we combine tensors,
through algebra, all the tensor quantities refer to the same point in the space.
31 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
4.11 Linear combination of tensors
The linear combination of tensors of the same rank and type is a tensor of that rank and type.
Rank is the number of indices, type is how the indices are divided between contravariant
and covariant. This follows from the affine nature of the transformation.
If Rij and Sij are covariant tensors of rank 2 then Tij = α Rij + β Sij, where α and β are
real numbers, is a covariant tensor of rank 2. To see this consider how Tij transforms
T ij = α Rij + β Sij (54)
= α
∂xa
∂xi
∂xb
∂xj
Rab + β
∂xa
∂xi
∂xb
∂xj
Sab
=
∂xa
∂xi
∂xb
∂xj
(α Rab + β Sab)
=
∂xa
∂xi
∂xb
∂xj
Tab
which is the correct transformation rule.
4.12 Outer product
If T is a tensor of rank t and S is a tensor of rank s then the outer product TS is a tensor
of rank t+ s.
The outer product of two covariant vectors Ai and Bi is Tij = AiBj, a covariant tensor of
rank 2. You can see this from the transformation rule.
T ij = AiBj (55)
=
∂xa
∂xi
Aa
∂xb
∂xj
Bb
=
∂xa
∂xi
∂xb
∂xj
(AaBb)
=
∂xa
∂xi
∂xb
∂xj
Tab
whichis the correct transformation rule.
Similarly T ji = AiB
j is a mixed tensor of rank 2 and T ij = AiBj is a contravariant tensor
of rank 2.
The outer product allows the construction of tensors of higher rank from vectors. However,
in general, given a tensor of higher rank it is not possible to write it as the outer product
of vectors.
4.13 Inner product
If T is a tensor of rank t and S is a tensor of rank s then the inner product T ·S is obtained
by first taking the outer product and then setting a contravariant index of either S or T
32 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
equal to a covariant index of the other tensor. There is then a summation over that index.
This gives a tensor of rank t+ s− 2.
The inner product of two vectors Ai and B
i is a scalar i.e. T = AiB
i. You can see this
from the transformation rule.
T = AiB
i
(56)
=
∂xa
∂xi
Aa
∂xi
∂xb
Bb
=
∂xa
∂xb
(
AaB
b
)
= δab AaB
b
= AaB
a
= T
which is the correct transformation rule. Notice that the contraction removes two coefficients
by the chain rule.
If Aij is a second rank covariant tensor and B
i is a contravariant vector then Ci = AijB
j,
the inner product is a covariant vector.
4.14 Contraction
If T is a mixed tensor of rank t > 1 then the process of equating a covariant index and a
contravariant index (summing over them) is called contraction. This gives a tensor of rank
t− 2.
If T ji is a mixed tensor of rank 2 then T
i
i , the trace, is a scalar. You can see this from the
transformation rule.
T ii =
∂xa
∂xi
∂xi
∂xb
T ba (57)
=
∂xa
∂xb
T ba
= δab T
b
a
= T aa
which is the correct transformation rule.
Contraction of the fundamental mixed tensor gives the dimension N of the space.
δii = δ
1
1 + · · ·+ δNN = N (58)
Starting from T ijkl we can form 3 different contravariant tensors of rank 2.
T ljkl and T
ilk
l and T
ijl
l (59)
33 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
4.15 Quotient Rule
Suppose that you have a set of quantities whose inner product with an arbitrary tensor
produces a tensor. Then the set of quantities form a tensor of the appropriate rank i.e. if
S = T ·X (60)
where S is a tensor and X is an arbitrary tensor, then T must be a tensor.
Suppose Si = TijX
j where Si is a covariant vector and X
i is an arbitrary contravariant
vector. Then by the quotient rule Tij should be a covariant tensor of rank 2. To confirm
this consider how these quantities transform
Si = T ij X
j
(61)
∂xa
∂xi
Sa =
∂xj
∂xb
T ij X
b
∂xa
∂xi
Tab X
b =
∂xj
∂xb
T ij X
b
Therefore [
∂xa
∂xi
Tab − ∂x
j
∂xb
T ij
]
Xb = 0 (62)
Now because Xb are arbitrary the expression in the square brackets must be zero.
∂xa
∂xi
Tab − ∂x
j
∂xb
T ij = 0 (63)
Now take the inner product with ∂x
b
∂xk
∂xa
∂xi
∂xb
∂xk
Tab =
∂xj
∂xb
∂xb
∂xk
T ij (64)
=
∂xj
∂xk
T ij
= δjk T ij
= T ik
which gives the required transformation formula.
4.16 Tensor equations
Due to the affine (linear and homogeneous) nature of the transformation a tensor whose
components are zero in one coordinate system are zero in all coordinate systems. It is a
numerically invariant tensor. Any tensor equation can be expressed as:
T = 0 (65)
34 May 11, 2004 5:16pm
4 TENSOR ALGEBRA
where the left-hand side would be in general a linear combination of outer or inner products
of tensors. Since the right-hand side is numerically invariant it follows that the left-hand
side is numerically invariant. Therefore the tensor equation is independent of the coordinate
system.
4.17 Symmetry and skew-symmetry
Symmetry properties are invariant only if they exist between indices of the same type.
If Tij is symmetric ( i.e. Tij = Tji) then T ij is symmetric.
If Tij is skew-symmetric ( i.e. Tij = −Tji) then T ij is skew-symmetric.
To verify this you can consider the transformation rule
T ij =
∂xa
∂xi
∂xb
∂xj
Tab (66)
= ±∂x
a
∂xi
∂xb
∂xj
Tba where +/− for symmetry/skew-symmetry
= ±∂x
b
∂xj
∂xa
∂xi
Tba
= ±T ji
A tensor Tij can always be written as the sum of symmetric and skew-symmetric parts
Tij =
1
2 (Tij + Tji) +
1
2 (Tij − Tji) (67)
35 May 11, 2004 5:16pm
5 RELATIVE TENSORS
5 Relative tensors
5.1 Transformation rule
A relative tensor transforms in the same way as a tensor except for an outside factor JW
where J is the Jacobian
J = det
(
∂xi
∂xj
)
(68)
and W is known as the weight.
A tensor, sometimes known as an absolute tensor, is a relative tensor of weight 0.
If T cab is a mixed relative tensor of weight 2 and rank 3, with 2 covariant indices and 1
contravariant index, then it transforms as
T
k
ij = J
2 ∂x
a
∂xi
∂xb
∂xj
∂xk
∂xc
T cab (69)
A relative tensor of weight 1 is known as a tensor density.
5.2 Summary
We characterise tensor quantities by their behaviour under coordinate transformations. In
general they have rank, type and weight. Relative tensors represent the most general form
we shall consider. Let
J
i
j =
∂xi
∂xj
and J ij =
∂xi
∂xj
(70)
Rank is the number of coefficients J ij or J
i
j which occur in the transformation rule.
Type refers to the coefficient type where J ij indicates covariant and J
i
j indicates contravari-
ant. The overall type can be covariant (all coefficients are covariant), contravariant or
mixed.
Weight refers to W in a factor JW in the transformation rule.
For any coordinate transformation such that J 6= 0 and J 6= ∞:
1. a relative tensor has rank, type and weight.
2. a tensor density has rank and type with W=1.
3. a tensor has rank and type with W=0.
It is only possible to add and subtract relative tensors of the same rank, type and weight.
36 May 11, 2004 5:16pm
5 RELATIVE TENSORS
tensor rank covariant contravariant weight
indices indices
T r1 c1 d1 w1
S r2 c2 d2 w2
outer product TS r1 + r2 c1 + c2 d1 + d2 w1 + w2
inner product T · S r1 + r2 − 2 c1 + c2 − 1 d1 + d2 − 1 w1 + w2
contracted T r1 − 2 c1 − 1 d1 − 1 w1
5.3 Transformation of determinant
Consider the transformation of gij , a covariant tensor of rank 2,
gij =
∂xa
∂xi
∂xb
∂xj
gab = J
a
i J
b
j gab (71)
Writing this in terms of matrices (g), (J) and (g) gives
(g) = (J)T(g)(J) (72)
where (J)T is the transpose of the matrix (J). Taking determinants we obtain
g = J2 g (73)
Then g is a relative scalar of weight 2. Clearly the sign of g is invariant.
5.4 Transformation of permutation symbol
In a 3-dimensional space (result generalises easily), let uijk be a covariant relative tensor
of weight −1 and rank 3. In coordinate system xi we have uijk = eijk. Then is any other
coordinate system
uijk = J
−1 ∂x
a
∂xi
∂xb
∂xj
∂xc
∂xk
uabc (74)
= J−1
∂xa
∂xi
∂xb
∂xj
∂xc
∂xk
eabc
= J−1 J eijk
= eijk
Therefore eijk is an isotropic covariant relative tensor of weight −1 and rank 3.
If g > 0 and J > 0, it follows that
√
g is a scalar density. Therefore
εijk =
√
g eijk (75)
is a covariant tensor of rank 3. Similarly we can show that
εijk =
eijk√
g
(76)
is a contravariant tensor of rank 3.
37 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
6 Riemannian space
6.1 Line element
In an N -dimensional space any point is specified by the N coordinates (x1, x2, . . . xN ).
If the line element ds between two neighbouring points, xi and xi+dxi, satisfies the quadratic
form
(ds)2 = gij dx
i dxj (77)
where the gij are functions of the x
i, then we have a Riemannian space, denoted RN .
Eq.(77) is known as the metric of the space and gij is called the metric tensor.
The metric tensor is positive definite, i.e. gijx
ixj > 0 for all non-zero xi. This is consistent
with an interpretationof ds as a distance.
A Euclidean space EN is a special type of Riemannian space for which gij = δij , the
Kronecker delta.
(ds)2 = δij dx
i dxj (78)
In 3 dimensions ds is the familiar distance between the points.
The line element is an invariant. We can use s to parameterise a curve in the space.
The tensor nature of gij can be seen by applying the quotient rule to eq.(77). (ds)
2 is an
invariant and dxi is an arbitrary contravariant vector. However dxidxj is not arbitrary due
to its symmetry in i and j. Provided we take gij to be symmetric gij transforms like a
covariant tensor of rank 2.
The distance between two points s1 and s2 on a curve is
∫ s2
s1
ds =
∫ u2
u1
√
gij
dxi
du
dxj
du
du (79)
where u parameterises the curve.
In general, it is not possible to obtain a transformation to a new coordinate system such
that eq.(77) transforms into eq.(78) over the whole space. If this can be done the space is
Euclidean. The geometry of a Euclidean space is described as flat. Otherwise the geometry
is said to be curved.
6.2 Local Cartesian coordinates
Now consider the metric eq.(77) at a particular point. The metric is a positive definite
quadratic form. The theorem of inertia due to Sylvester 7 states that a linear transformation
exists that will transform the metric to Euclidean form. Note that it can only be done locally.
In the neighbourhood of each point we can obtain local Cartesian coordinates.
7See Appendix B ‘Odds and ends’ note 7.
38 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
6.3 Spherical surface in two dimensions
Our starting point is the Cartesian metric in E3
(ds)2 = (dx)2 + (dy)2 + (dz)2 (80)
Transform this to spherical polar coordinates using
x = r sin θ cosφ (81)
y = r sin θ sinφ
z = r cos θ
The differentials are
dx = (sin θ cosφ) dr + (r cos θ cosφ) dθ − (r sin θ sinφ) dφ (82)
dy = (sin θ sinφ) dr + (r cos θ sinφ) dθ + (r sin θ cosφ) dφ
dz = (cos θ) dr − (r sin θ) dθ
Substitute into eq.(80) to give
(ds)2 = (dr)2 + r2
(
(dθ)2 + sin2 θ (dφ)2
)
(83)
Now let r = a, a constant, to obtain
(ds)2 = a2
(
(dθ)2 + sin2 θ (dφ)2
)
(84)
Since we cannot express this as (ds)2 = (dX)2 + (dY )2, the spherical surface is not a
Euclidean space of 2 dimensions. Euclidean spaces are flat. This result just expresses our
understanding that a spherical surface is curved. This is an example of a curved Riemannian
space R2 embedded in a Euclidean space E3 of higher dimension.
6.4 Raising and lowering indices
In a Riemannian space gij is the metric tensor, a symmetric covariant tensor of rank 2. The
tensor is positive definite i.e. g = det(gij) > 0 and the inverse exists. Let g
ij be the inverse
matrix of gij , then
gij gjk = gkj g
ji = δik (85)
Since gij is symmetric it follows that g
ij is also symmetric.
If Ak is an arbitrary contravariant vector then Bj = gjkA
k is an arbitrary covariant vector.
Then
gij Bj = g
ij gjk A
k (86)
= δik A
k
= Ai
39 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
By the quotient rule it follows that gij is a contravariant tensor of rank 2. From the isotropy
of δik we see that g
ij is the inverse of gij in all coordinate systems.
gij is said to be conjugate to gij .
By taking the inner product of a tensor with gij and g
ij we can lower and raise indices
respectively. Consider Xi a covariant vector. The associated contravariant vector X
i is
Xi = gij Xj (87)
We use the same symbol X for the vectors because they refer to the same object and we
have a rule for relating the covariant and contravariant components. The rule is consistent
because we can lower the index of X i to give
gij X
j = gij g
jk Xk (88)
= δki Xk
= Xi
which gives the original covariant vector as it should.
When raising and lowering indices it is important to remember the position of the index
being moved. This is sometimes done by putting a ‘.’ in the old position of the index.
Alternatively the indices are offset.
Consider Aijkl.
A ji kl = gia A
aj
kl lower index i
Aijkl = gja A
ia
kl lower index j
Aijkl = g
ka Aijal raise index k
Aijkl = gka glb Aijab raise indices k and l
A ji kl, A
i
jkl, A
ijk
l and A
ijkl are all said to be tensors associated with Aijkl.
Suppose Rij is a symmetric covariant tensor. Then the associated mixed tensor is given by
Rij = g
ia Raj (89)
= gia Rja
= R ij
= Rij
i.e. we do not need to worry about the position of the raised index. This depends on the
symmetry of the tensor. However even though the covariant tensor is symmetric it does not
imply symmetry in the mixed tensor e.g. in 2 dimensions
R12 = g
1aRa2 = g
11R12 + g
12R22
R21 = g
2aRa1 = g
21R11 + g
22R21
(90)
40 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
which are clearly not the same.
The associated contravariant tensor is symmetric since
Rij = giagjbRab (91)
= giagjbRba
= Rji
6.5 Length and direction of a vector
The length squared of a contravariant vector Ai is given by
a2 = gij A
i Aj (92)
= gij A
i gjk Ak lowering index
= δki A
i Ak
= Ai Ai (93)
= gij Ai Aj (94)
If gij is positive definite then a
2 > 0 for non-zero vectors and we can take the square root
to obtain a real length.
The angle θ between two vectors Ai and Bi of lengths a and b respectively is given by the
scalar product
ab cos θ = gij A
i Bj (95)
= Ai Bi (96)
= Ai B
i (97)
= gij Ai Bj (98)
We require | cos θ| ≤ 1 so that the concept of angle (direction) is meaningful. Let X i = 1
a
Ai
and Y i = 1
b
Bi so that they are unit vectors. Then
cos θ = gij X
i Y j (99)
Consider Z i = Xi + kY i where k is a real number.
gij Z
i Zj = gij (X
i + kY i) (Xj + kY j) (100)
= 1 + k2 + 2k gij X
i Y j
=
(
k + gij X
i Y j
)2
+
[
1−
(
gij X
i Y j
)2]
≥ 0 due to positive definite
This holds for any k. Let k = −gij Xi Y j
1−
(
gij X
i Y j
)2 ≥ 0 (101)
41 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
giving ∣∣∣gij Xi Y j∣∣∣ ≤ 1 −→ | cos θ| ≤ 1 (102)
Two vectors Ai and Bi are orthogonal if gij A
i Bj = 0.
If we parameterise a curve by the distance s then the tangent vector dx
i
ds
is a unit vector.
(ds)2 = gij dx
i dxj −→ 1 = gij dx
i
ds
dxj
ds
(103)
6.6 Geodesics
A geodesic is a generalisation of the straightline in a Euclidean space. A straightline is the
shortest distance between two points and this suggests a variational definition of geodesic
through the Euler-Lagrange equations. We need to define distance of course and this is
done through the metric in a Riemannian space.
A curve in RN can be represented by the equation
xi = xi(u) (104)
and we define x˙i = dx
i
du
.
From the calculus of variations we find that for fixed endpoints u1 and u2
∆
∫ u2
u1
L(xi, x˙i, u) du = 0 (105)
when the Euler-Lagrange equations are satisfied
d
du
(
∂L
∂x˙i
)
− ∂L
∂xi
= 0 (106)
L is the Lagrangian function. ∆ denotes weak variations in the integral so that we are
requiring the integral to be stationary for small variations in the curve.
The distance along a curve joining two points is given by
I =
∫ u2
u1
ds
du
du (107)
This curve is a geodesic if I is stationary for small variations in the curve. Thus the geodesic
is a solution of the Euler-Lagrange equations with
L =
ds
du
(108)
=
√
gij x˙i x˙j
42 May 11, 2004 5:16pm
6 RIEMANNIAN SPACE
We can now apply the Euler-Lagrange equations eq.(106). However the algebra is compli-
cated by the presence of the square root. Due to the positive definite nature of the metric
we get the same result by considering
L = gij x˙
i x˙j (109)
We need the following
∂L
∂xk
=
∂gij
∂xk
x˙i x˙j and
∂L
∂x˙k
= 2gik x˙
i (110)
Therefore
d
du
(
∂L
∂x˙k
)
=