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A long set of lecture-style notes by Nikodem J. Poplawski (Indiana University, July 15, 2011), apparently a web download kept as tensor support material. It covers tensors, affine connection, curvature, metric, tetrads, the Lorentz group and spinors; then fields (action principle, Einstein and Einstein-Cartan gravity, Dirac and electromagnetic fields); then particle and rigid-body mechanics and ideal fluids. The text seen is mainly the table of contents.
AI-written summary; may contain errors.
Extracted text (machine-read; may contain errors)
Classical physics: Spacetime, elds and particles
Nikodem J. Pop lawski
Department of Physics, Indiana University, Bloomington, IN 47405, USA
July 15, 2011
Contents
1 Spacetime 5
1.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 Kronecker and Levi-Civita symbols . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.6 Dual densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.7 Covariant integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.8 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Ane connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Covariant di erentiation of tensors . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Torsion tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Covariant di erentiation of densities . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.6 Partial integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.7 Geodesic frame of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.8 Ane geodesics and four-velocity . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.9 In nitesimal coordinate transformations . . . . . . . . . . . . . . . . . . . . . 15
1.2.10 Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Integrability of connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Parallel transport along closed curve . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.4 Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.5 Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.6 Geodesic deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Christo el symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.4 Properties of Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.5 Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.6 Metric geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.7 Galilean frame of reference and Minkowski tensor . . . . . . . . . . . . . . . . 26
1.4.8 Intervals, proper time and distances . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.9 Spatial vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Tetrad and spin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.1 Tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1
1.5.2 Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.3 Tetrad transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.4 Spin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.5 Tetrad representation of curvature tensor . . . . . . . . . . . . . . . . . . . . 34
1.6 Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.1 Subgroups of Lorentz group and principle of relativity . . . . . . . . . . . . . 35
1.6.2 In nitesimal Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.3 Generators and Lie algebra of Lorentz group . . . . . . . . . . . . . . . . . . 36
1.6.4 Rotations and boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.5 Poincar e group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.6 Invariants of Lorentz and Poincar e group . . . . . . . . . . . . . . . . . . . . 41
1.6.7 Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.6.8 Four-acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.7 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.7.1 Spinor representation of Lorentz group . . . . . . . . . . . . . . . . . . . . . . 46
1.7.2 Spinor connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.7.3 Curvature spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 Fields 49
2.1 Principle of least action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Action for gravitational eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Metric dynamical energy-momentum density . . . . . . . . . . . . . . . . . . 51
2.3.2 Tetrad dynamical energy-momentum density . . . . . . . . . . . . . . . . . . 52
2.3.3 Canonical energy-momentum density . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.4 Spin density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.5 Belinfante-Rosenfeld relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.1 Noether theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.2 Conservation of spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.3 Conservation of metric energy-momentum . . . . . . . . . . . . . . . . . . . . 56
2.4.4 Conservation of tetrad energy-momentum . . . . . . . . . . . . . . . . . . . . 57
2.4.5 Conservation laws for Lorentz group . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.6 Components of energy-momentum tensor . . . . . . . . . . . . . . . . . . . . 59
2.4.7 Mass and Papapetrou equations of motion . . . . . . . . . . . . . . . . . . . . 63
2.4.8 Spin tensor for particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.9 Energy-momentum tensor for particles . . . . . . . . . . . . . . . . . . . . . . 71
2.5 Gravitational eld equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.1 Einstein-Hilbert action and Einstein equations . . . . . . . . . . . . . . . . . 73
2.5.2 Einstein pseudotensor and principle of equivalence . . . . . . . . . . . . . . . 74
2.5.3 Landau-Lifshitz energy-momentum pseudotensor . . . . . . . . . . . . . . . . 76
2.5.4 Utiyama action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.5.5 M ller pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.6 Einstein-Cartan action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.7 Sciama-Kibble action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5.8 Einstein-Cartan pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5.9 Palatini variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5.10 Gravitational potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.11 Relativistic ideal
uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.5.12 Relativistic spin
uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.13 Raychaudhuri equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5.14 Event horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.6 Spinor elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.6.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2
2.6.2 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.6.3 Spinors in Einstein-Cartan-Sciama-Kibble gravity . . . . . . . . . . . . . . . . 92
2.6.4 Discrete symmetries of spinors . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.7 Electromagnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.7.1 Gauge invariance and electromagnetic potential . . . . . . . . . . . . . . . . . 95
2.7.2 Electromagnetic eld tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.7.3 Lagrangian density for electromagnetic eld . . . . . . . . . . . . . . . . . . . 98
2.7.4 Electromagnetic current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.7.5 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.7.6 Energy-momentum tensor for electromagnetic eld . . . . . . . . . . . . . . . 102
2.7.7 Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Particles 105
3.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.1 Coordinates, velocities and accelerations . . . . . . . . . . . . . . . . . . . . . 105
3.1.2 Hamilton principle and Lagrange equations . . . . . . . . . . . . . . . . . . . 106
3.1.3 Action for particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.1.4 Conservation laws and integrals of motion . . . . . . . . . . . . . . . . . . . . 110
3.1.5 Nonrelativistic mechanics and Galileo principle of relativity . . . . . . . . . . 114
3.1.6 Momentum, force and Newton equations of motion . . . . . . . . . . . . . . . 115
3.1.7 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.1.8 Center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.1.9 Angular momentum and torque . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.1.10 Mechanical similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.1.11 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2 Rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2.1 Angular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2.2 Inertia tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.2.3 Eulerian angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.2.4 Newton and Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2.5 Noninertial frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2.6 Constraints and d'Alembert principle . . . . . . . . . . . . . . . . . . . . . . 130
3.2.7 Maggi and Appell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3 Ideal
uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.1 Lagrange formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.2 Euler formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.4 Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.1 Legendre transformation, Hamilton and Routh equations . . . . . . . . . . . 132
3.4.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.3 Maupertuis principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.5 Liouville theorem and distribution functions . . . . . . . . . . . . . . . . . . . 132
3.4.6 Adiabatic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Hamilton-Jacobi mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5.1 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5.2 Canonical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 Applications 133
4.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1 Orthonormal systems of coordinates . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.2 Uniformly accelerated motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1.3 Uniformly rotating frame of reference . . . . . . . . . . . . . . . . . . . . . . 136
4.1.4 Unidimensional nonrelativistic motion . . . . . . . . . . . . . . . . . . . . . . 138
4.1.5 Central eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3
4.1.6 Kepler motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.1.7 Decays and collisions of particles . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1.8 Scattering of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1.9 Small oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.1.10 Motion of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.1.11 Motion of ideal
uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1.12 Motion in ideal
uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.1 Electrostatic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.2 Magnetostatic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.2.3 Motion in constant uniform electromagnetic eld . . . . . . . . . . . . . . . . 160
4.2.4 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.2.5 Retarded potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.2.6 Electromagnetic eld in second approximation . . . . . . . . . . . . . . . . . 170
4.2.7 Electromagnetic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.8 Radiation reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.2.9 Scattering of electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . 178
4.2.10 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.3.1 Constant gravitational eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.3.2 Synchronous system of reference . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.3.3 Nonrelativistic gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.3.4 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.3.5 Motion in Schwarzschild eld . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.3.6 Other coordinates for Schwarzschild eld . . . . . . . . . . . . . . . . . . . . 194
4.3.7 Weyl isotropic metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.3.8 Interior Schwarzschild solution . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.3.9 Tolman solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.3.10 Kerr metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.3.11 Motion in Kerr eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.3.12 Reissner-Nordstr om metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.3.13 Kerr-Newman metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.3.14 Weak gravitational eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.3.15 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.3.16 Kottler-de Sitter metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.3.17 Friedmann-Lema^ tre-Robertson-Walker metric . . . . . . . . . . . . . . . . . 204
4.4 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.4.1 Free spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.4.2 Dirac equation in central electric eld . . . . . . . . . . . . . . . . . . . . . . 204
4.4.3 Schr odinger equation in central electric eld . . . . . . . . . . . . . . . . . . . 204
4.4.4 Dirac equation in uniform magnetic eld . . . . . . . . . . . . . . . . . . . . . 204
4
1 Spacetime
Einstein's principle of general covariance states that all physical laws do not change their form (are
covariant) under continuous coordinate transformations in four-dimensional spacetime.
1.1 Tensors
1.1.1 Vectors
We consider a coordinate transformation from old (unprimed) to new (primed) coordinates in a
four-dimensional manifold:
xi!x0j(xi); (1.1.1)
wherex0jare di erentiable and nondegenerate functions of xiand the index ican be 0,1,2,3. Thus
the matrix@x0j
@xihas the nonzero determinant j@x0j
@xij6= 0, soxiare di erentiable and nondegenerate
functions of x0j. The matrix@xi
@x0jis the inverse of@x0j
@xi:
X
i@x0i
@xj@xk
@x0i=k
j; (1.1.2)
where
i
k=1i=k
0i6=k
: (1.1.3)
The di erentials and derivatives transform according to
dx0j=@x0j
@xidxi; (1.1.4)
@
@x0j=@xi
@x0j@
@xi: (1.1.5)
Ascalar (invariant) is de ned as a quantity that does not change:
0= : (1.1.6)
Acontravariant vector is de ned as a quantity that transform like a di erential:
A0j=@x0j
@xiAi: (1.1.7)
Acovariant vector is de ned as a quantity that transforms like a derivative:
B0j=@xi
@x0jBi: (1.1.8)
Therefore a derivative of a scalar is a covariant vector. The coordinates xido not form a vector.
1.1.2 Tensors
A product of several vectors transforms such that each coordinate index transforms separately:
A0iB0j:::C0kD0l=@x0i
@xm@x0j
@xn@xp
@x0k@xq
@x0lAmBn:::CpDq::: : (1.1.9)
Atensor is de ned as a quantity that transforms like a product of vectors:
T0ij:::
kl:::=@x0i
@xm@x0j
@xn@xp
@x0k@xq
@x0lT0mn:::
pq:::: (1.1.10)
5
A tensor is of rank ( k;l) if it haskcontravariant and lcovariant indices. A scalar is a tensor of
rank (0,0), a contravariant vector is a tensor of rank (1,0), and a covariant vector is a tensor of rank
(0,1). A linear combination of two tensors of rank ( k;l) is a tensor of rank ( k;l). The product of
two tensors of ranks ( k1;l1) and (k2;l2) is a tensor of rank ( k1+k2;l1+l2). Tensor indices (all
contravariant or all covariant) can be symmetrized :
T(ij:::k )=1
n!X
permutationsTfij:::kg; (1.1.11)
orantisymmetrized :
T[ij:::k ]=1
n!X
permutationsTfij:::kg( 1)m; (1.1.12)
wherenis the number of symmetrized or antisymmetrized indices and mis the number of per-
mutations that bring Tij:::k intoTfij:::kg. For example, for two indices: T(ik)=1
2(Tik+Tki) and
T[ik]=1
2(Tik Tki), and for three indices: T[ijk]=1
3(Tijk+Tjki+Tkij). Ifn>4 thenT[ij:::k ]= 0.
Symmetrized and antisymmetrized tensors or rank ( k;l) are tensors of rank ( k;l). Symmetrization
of an antisymmetric tensor or antisymmetrization of a symmetric tensor bring these tensors to zero.
Any tensor of rank (0,2) is the sum of its symmetric and antisymmetric part,
T(ik)+T[ik]=Tik: (1.1.13)
The number 0 can be regarded as a tensor of arbitrary rank. Therefore all covariant equations of
classical physics must be represented in the tensor form: Tij:::
kl:::= 0.
1.1.3 Densities
The element of volume in four-dimensional spacetime transforms according to
d4x0= @x0i
@xk d4x: (1.1.14)
Ascalar density is de ned as a quantity that transforms such that its product with the element of
volume is a scalar, s0d4x0=sd4x:
s0= @xi
@x0k s: (1.1.15)
Atensor density , which includes a contravariant and covariant vector density, is de ned as a quantity
that transforms like a product of a tensor and a scalar density:
T0ij:::
kl:::= @xi
@x0k @x0i
@xm@x0j
@xn@xp
@x0k@xq
@x0lT0mn:::
pq:::: (1.1.16)
For example, the square root of the determinant of a tensor of rank (0 ;2) is a scalar density of weight
1:q
jT0
ikj=r
j@xl
@x0i@xm
@x0kTlmj=r
j@xj
@x0nj2jTik=j@xj
@x0njp
jTikj: (1.1.17)
The above densities are said to be of weight 1. One can generalize this de nition of densities
by introducing densitites of weight w, which transform like normal densities except that @xi
@x0k is
replaced by @xi
@x0k w. For example, d4xis a scalar density of weight -1. A linear combination of two
densities of weight wis a density of weight w. The product of two densities of weights w1andw2is
a density of weight w1+w2. Symmetrized and antisymmetrized densities of weight ware densities
of weightw. Densities of weight 1 are simply referred to as densities. Tensors are densities of weight
0.
6
1.1.4 Contraction
We adopt Einstein's convention: if the same coordinate index iappears twice (as a contravariant
index and covariant index) then we perform the summationP
iover a given tensor or density. Such
a tensor or density is said to be contracted over indexi. A contracted tensor of rank ( k;l) transforms
like a tensor of rank ( k 1;l 1):
T0ij:::
il:::=@x0i
@xm@x0j
@xn@xp
@x0i@xq
@x0lT0mn:::
pq:::=@x0j
@xn@xq
@x0lp
mT0mn:::
pq:::=@x0j
@xn@xq
@x0lT0mn:::
mq:::: (1.1.18)
For example, the contraction of a contravariant and covariant vector AiBiis a scalar ( scalar product ).
A contracted tensor density of rank ( k;l) and weight wtransforms like a tensor density of rank
(k 1;l 1) and weight w:
T0ij:::
il:::= @xi
@x0k w@x0i
@xm@x0j
@xn@xp
@x0i@xq
@x0lT0mn:::
pq:::= @xi
@x0k w@x0j
@xn@xq
@x0lp
mT0mn:::
pq:::
= @xi
@x0k w@x0j
@xn@xq
@x0lT0mn:::
mq:::: (1.1.19)
Contraction of a symmetric tensor with an antisymmetric tensor (over indices with respect to which
these tensors are symmetric or antisymmetric) gives zero. If contraction of two tensors gives zero,
these tensors are said to be orthogonal . Two orthogonal vectors (one contravariant and one covariant)
are said to be perpendicular .
1.1.5 Kronecker and Levi-Civita symbols
The Kronecker symbol i
k(1.1.3) is a tensor with constant components:
0i
k=@x0i
@xj@xl
@x0kj
l=@x0i
@xj@xj
@x0k=i
k: (1.1.20)
A totally antisymmetric tensor of rank (4 ;0),Tijkl=T[ijkl]has 1 independent component T:Tijkl=
Tijkl, whereijklis the totally antisymmetric, contravariant permutation Levi-Civita symbol :
0123= 1; ijkl= ( 1)m; (1.1.21)
andmis the number of permutations that bring ijklinto0123. The determinant of a matrix Si
k,
det(Si
k) =jSi
kj, is de ned through the permutation symbol:
jSr
sjijkl=Si
mSj
nSk
pSl
qmnpq: (1.1.22)
TakingSi
k=@x0i
@kgives
ijkl= @xr
@x0s @x0i
@xm@x0j
@xn@x0k
@xp@x0l
@xqmnpq: (1.1.23)
This equation looks like a transformation law for a tensor density with constant components: 0ijkl=
ijkl. Accordingly, Tis a scalar density of weight -1. We also introduce the covariant Levi-Civita
symbolijklthrough:
ijklmnpq = i
mi
ni
pi
q
j
mj
nj
pj
q
k
mk
nk
pk
q
l
ml
nl
pl
q : (1.1.24)
Thus the covariant Levi-Civita symbol is a tensor density of weight -1 and its product with a scalar
density is a tensor. The covariant Levi-Civita symbol is given by
0123= 1; ijkl= ( 1)m; (1.1.25)
7
wheremis the number of permutations that bring ijklinto0123, and satis es
jSr
sjijkl=Sm
iSn
jSp
kSq
lmnpq: (1.1.26)
Contracting (1.1.24) gives the following relations:
ijklmnpl = i
mi
ni
p
j
mj
nj
p
k
mk
nk
p ;
ijklmnkl = 2(i
mj
n i
nj
m);
ijklmjkl = 6i
m;
ijklijkl= 24: (1.1.27)
1.1.6 Dual densities
A contracted product of a covariant tensor and the contravariant Levi-Civita symbol gives a dual
contravariant tensor density:
iklmAm=Aikl; iklmBlm=Bik; iklmCklm=Ci: (1.1.28)
A contracted product of a contravariant tensor and the covariant Levi-Civita symbol gives a dual
covariant tensor density:
iklmAm=Aikl; iklmBlm=Bik; iklmCklm=Ci: (1.1.29)
Therefore there exists an algebraic correspondence between covariant tensors and contravariant den-
sities, and between contravariant tensors and covariant densities.
1.1.7 Covariant integrals
A covariant line integral is an integral of a tensor contracted with the line di erential dxi:R
Tj:::
i:::dxi.
A covariant surface integral is an integral of a tensor contracted with the surface di erential dfik=
dxidx0k dxkdx0i(which can be geometrically represented as a parallelogram spanned by the vectors
dxianddx0i):R
Tj:::
ik:::dfik. A covariant hypersurface (volume) integral is an integral of a tensor
contracted with the volume di erential dSikl= dxidx0idx\i
dxkdx0kdx\k
dxldx0ldx\l (which can be geometrically
represented as a parallelepiped spanned by the vectors dxi,dx0i) anddx\i:R
Tj:::
ikl:::dSikl. A covariant
four-volume integral is an integral of a tensor contracted with the four-volume di erential dSijkl,
de ned analogously to dSikl. The dual density corresponding to the surface element is given by
df?
ik=1
2lmikdflm: (1.1.30)
The dual density corresponding to the hypersurface element is given by
dSi=1
6klmidSklm: (1.1.31)
The dual density corresponding to the four-volume element is given by
d
=1
24iklmdSiklm=dx0dx1dx2dx3: (1.1.32)
8
Covariant integrands that include the above dual densities of weight -1 must be multiplied by a scalar
density, for example, by the square root of the determinant of a tensor of rank (0 ;2). According to
Gau' and Stokes' theorems, there exists relations between integrals over di erent elements:
dxi$dfik@
@xk; (1.1.33)
df?
ik$dSi@
@xk dSk@
@xi; (1.1.34)
dSi$d
@
@xi: (1.1.35)
1.1.8 Derivatives
A derivative of a covariant vector does not transform like a tensor:
@A0
k
@x0i=@xl
@x0i@
@xl@xm
@x0kAm
=@xl
@x0i@xm
@x0k@Am
@xl+@2xm
@x0i@x0kAm; (1.1.36)
because of the second term which is linear and homogeneous in Ai, unlessxiare linear functions
ofx0j. This term is symmetric in the indices i;kso the antisymmetric part of@Ak
@xiwith respect to
these indices is a tensor:
@[iA0
k]=@xl
@x0[i@xm
@x0k]@lAm=@xl
@x0i@xm
@x0k@[lAm]; (1.1.37)
where we denote @i=@
@xi. The curlof a covariant vector Aiis de ned as twice the antisymmetric
part of@iAk:@iAk @kAi, and is a tensor. We will also use ;i=@
@xito denote a partial derivative
with respect to xi. Similarly, totally antisymmetrized derivatives of tensors of rank (0 ;2) and (0;3),
@[iBkl]and@[iCklm], are tensors. If Bkl=A[k;l]then@[iBkl]= 0, or conversely, if @[iBkl]= 0 then
there exists a vector Aisuch thatBkl=A[k;l]. The divergence of a tensor (or density) is a contracted
derivative of this tensor (density): @iT:::il:::
jk:::. Because of the correspondence between tensors and
dual densities, divergences of (totally antisymmetric if more than 1 index) contravariant densities
are densities, dual to totally antisymmetrized derivatives of tensors:
@iCi=iklm@[iCklm]; @kBik=iklm@[kBlm]; @lAikl=iklm@[lAm]: (1.1.38)
For example, the equations Fik
;i=jkandF[ik;l]= 0, that describe Maxwell's electrodynamics, are
tensorial.
References: [1, 2].
1.2 Ane connection
1.2.1 Covariant di erentiation of tensors
An ordinary derivative of a covariant vector Aiis not a tensor, because its coordinate transformation
law contains an additional noncovariant term, linear and homogeneous in Ai. We consider the
expression
Ai;k=Ai;k l
ikAl; (1.2.1)
where the quantity l
ik(in the second term which is linear and homogeneous in Ai) transforms such
thatAi;kis a tensor:
A0
i;k=@xl
@x0i@xm
@x0kAl;m=@xl
@x0i@xm
@x0k(Al;m n
lmAn): (1.2.2)
On the other hand (1.1.36) gives
A0
i;k=A0
i;k 0l
ikA0
l=@xm
@x0k@xl
@x0iAl;m+@2xn
@x0k@x0iAn @xn
@x0l 0l
ikAn; (1.2.3)
9
so we obtain
@xn
@x0l 0l
ik=@xl
@x0i@xm
@x0k n
lm+@2xn
@x0k@x0i: (1.2.4)
Multiplying this equation by@x0j
@xngives the transformation law for l
ik:
0j
ik=@x0j
@xn@xl
@x0i@xm
@x0k n
lm+@x0j
@xn@2xn
@x0k@x0i: (1.2.5)
The algebraic object l
ik, which equips spacetime in order to covariantize a derivative of a vector,
is referred to as the ane connection , anity or simply connection. The connection has generally
64 independent components. The tensor Ai;kis the covariant derivative of a vector Aiwith respect
toxi. We will also use ri=;ito denote a covariant derivative. The contracted ane connection
transforms according to
0i
ik=@xm
@x0k l
lm+@x0i
@xn@2xn
@x0k@x0i: (1.2.6)
The ane connection is not a tensor because of the second term on the right-hand side of (1.2.5).
A derivative of a scalar is a covariant vector. Therefore a covariant derivative of a scalar is equal
to an ordinary derivative:
;i= ;i: (1.2.7)
If we also assume that a covariant derivative of the product of two tensors obeys the same chain rule
as an ordinary derivative:
(TU);i=T;iU+TU;i; (1.2.8)
then
(AkBk);i= (AkBk);i=Ak;iBk+AkBk
;i=Ak;iBk k
liAkBl+AkBk
;i: (1.2.9)
Therefore we obtain a covariant derivative of a contravariant vector:
Bk
;i=Bk
;i+ k
liBl: (1.2.10)
The chain rule (1.2.8) also implies that a covariant derivative of a tensor is equal to the sum of the
corresponding ordinary derivative of this tensor and terms with the ane connection that covari-
antize each index:
Tij:::
kl:::;m=Tij:::
kl:::;m+ i
nmTnj:::
kl:::+ j
nmTin:::
kl:::+ n
kmTij:::
nl::: n
lmTij:::
kn::: ::: : (1.2.11)
A covariant derivative of the Kronecker symbol vanishes:
k
l;i= k
jij
l j
lik
j= 0: (1.2.12)
The second term on the right-hand side of (1.2.5) does not depend on the ane connection, but
only on the coordinate transformation. Therefore the di erence between two di erent connections
transforms like a tensor of rank (1,2). Consequently, the variation j
ik, which is an in nitesimal
di erence between two connections, is a tensor of rank (1,2).
1.2.2 Parallel transport
We consider two in nitesimally separated points in spacetime, P(xi) andQ(xi+dxi), and a vector
eldAwhich takes the value AkatPandAk+dAkatQ. BecausedAk=Ak
;idxiandAk
;iis not
a tensor, the di erence dAkis not a vector, which is related to subtracting of two vectors at two
points with di erent coordinate transformation laws. In order to calculate the covariant di erence
between two vectors at two di erent points, we must bring these vectors to the same point. Instead of
subtracting from the vector Ak+dAkatQthe vectorAkatP, we must subtract a vector Ak+Akat
Qthat corresponds to AkatP, so the resulting di erence (covariant di erential) DAk=dAk Ak
is a vector. The vector Ak+Akis the parallel-transported or parallel-translated AkfromPtoQ.
A parallel-transported linear combination of vectors must be equal to the same linear combination
10
of parallel-transported vectors. Therefore Akis a linear and homogeneous function of Ak. It is also
on the order of a di erential, thus a linear and homogeneous function of dxi. The most general form
ofAkis
Ak= k
liAldxi; (1.2.13)
so
DAk=dAk+ k
liAldxi=Ak
;idxi: (1.2.14)
BecauseAkis not a vector, k
liis not a tensor. Because DAkis a vector, Ak
;iis a tensor. The
expressions for covariant derivatives of a covariant vector and tensors result from
= 0; (TU) =TU +TU: (1.2.15)
1.2.3 Torsion tensor
The second term on the right-hand side of (1.2.5) is symmetric in the indices i;kso the antisymmetric
part of the connection with respect to these indices,
Sj
ik= j
[ik]; (1.2.16)
is a tensor:
S0j
ik=@x0j
@xn@xl
@x0i@xm
@x0kSn
lm: (1.2.17)
This tensor is called the Cartan torsion tensor . The torsion tensor has generally 24 independent
components. The contracted torsion tensor,
Sk
ik=Si; (1.2.18)
is called the torsion trace or torsion vector .
1.2.4 Covariant di erentiation of densities
A derivative of a scalar density of weight w,s, does not transform like a covariant vector density:
@is0=@xl
@x0i@l @xj
@x0k w
s
=@xl
@x0i @xj
@x0k w
@ls+w@xl
@x0i @xj
@x0k w 1
@l @xr
@x0s s
=@xl
@x0i @xj
@x0k w
@ls+w@xl
@x0i @xj
@x0k w 1 @xr
@x0s @x0n
@xm@
@xl@xm
@x0ns
=@xl
@x0i @xj
@x0k w
@ls+w @xj
@x0k w@x0n
@xm@2xm
@x0n@x0is: (1.2.19)
We consider the expression
s;i=s;i w is; (1.2.20)
where the quantity itransforms such that s;iis a vector density of weight w:
s0
;i=@xl
@x0i @xj
@x0k w
s;l=@xl
@x0i @xj
@x0k w
(s;l w ls): (1.2.21)
On the other hand (1.2.19) gives
s0
;i=s0
;i w 0
is0=@xl
@x0i @xj
@x0k w
@ls+w @xj
@x0k w@x0n
@xm@2xm
@x0n@x0is w @xj
@x0k w
0
is; (1.2.22)
so we obtain the transformation law for i:
0
i=@xl
@x0i l+@x0n
@xm@2xm
@x0n@x0i; (1.2.23)
11
which is the same as the transformation law for k
ki(1.2.6). Therefore the di erence i k
kiis
some covariant vector Vi.
If we assume that parallel transport of the product of a scalar density of any weight and a tensor
obeys the chain rule:
(sT) =sT+sT; (1.2.24)
so a covariant derivative of such product behaves like an ordinary derivative:
(sT);i=s;iT+sT;i; (1.2.25)
then a covariant derivative of a tensor density of weight wis equal to the sum of the corresponding
ordinary derivative of this tensor, terms with the ane connection that covariantize each index, and
the term with i:
Tij:::
kl:::;m=Tij:::
kl:::;m+ i
nmTnj:::
kl:::+ j
nmTin:::
kl:::+:::
n
kmTij:::
nl::: n
lmTij:::
kn::: w mTij:::
kl:::: (1.2.26)
A covariant derivative of the contravariant Levi-Civita density is
ijkl
;m= i
nmnjkl+ j
nminkl+ k
nmijnl+ l
nmijkn mijkl: (1.2.27)
In the summations over nonly one term does not vanish for each term on the right-hand side of
(1.2.27), so
ijkl
;m= i
n=ijmn=ijjkl+ j
n=jjmijn=jjkl+ k
n=kjmijjn=kjl+ l
n=ljmijkjn=l mijkl
= ( n
nm m)ijkl= Vmijkl: (1.2.28)
The Levi-Civita symbol is a tensor density with constant components, so it does not change under
a parallel transport, = 0. Therefore ijkl
;m= 0, soVi= 0 and
i= k
ki: (1.2.29)
1.2.5 Covariant derivatives
Totally antisymmetrized ordinary derivatives of covariant tensors, A[i;k],B[ik;l]andC[ikl;m], are
tensors because of antisymmetrization. Totally antisymmetrized covariant derivatives of tensors are
clearly tensors because riis a covariant operation, and are given by direct calculation using the
de nition of a covariant derivative:
A[i;k]=A[i;k] Sl
ikAl; B [ik;l]=B[ik;l] 2Sm
[ikBl]m: (1.2.30)
Divergences of (totally antisymmetric if more than 1 index) contravariant densities, Ci
;i,Bik
;iand
Aikl
;i, are densities because of the correspondence between tensors and dual densities. Covariant
divergences of contravariant densities are clearly densities, and are given by direct calculation:
Ci
;i=Ci
;i+ 2SiCi;Bik
;i=Bik
;i Sk
ilBil+ 2SiBik: (1.2.31)
1.2.6 Partial integration
If the product of two quantities (tensors or densities) TUis a contravariant density Ckthen
Z
TU;kd
=Z
(TU);kd
Z
T;kUd
=Z
(TU);kd
+ 2Z
SkTUd
Z
T;kUd
: (1.2.32)
The rst term on the right-hand side can be transformed into a hypersurface integralR
TUdSk. If
the region of integration extends to in nity and Ckcorresponds to some physical quantity then the
boundary integralR
TUdSkvanishes, giving
Z
TU;kd
= 2Z
SkTUd
Z
T;kUd
: (1.2.33)
12
IfT=k
ithenU=CiandZ
Ci
;id
= 2Z
SiCid
: (1.2.34)
1.2.7 Geodesic frame of reference
We consider a coordinate transformation
xk=x0k+1
2ak
lmx0lx0m; (1.2.35)
whereak
lmis symmetric in the indices l;m. Substituting this transformation to (1.2.5) and calcu-
lating it at xk=x0k= 0 gives
@xi
@x0k=i
k (1.2.36)
and
0j
ik= j
ik+aj
ik: (1.2.37)
Putting
aj
ik= j
(ik)jxl=0 (1.2.38)
gives
0j
(ik)= 0: (1.2.39)
Therefore there always exists a coordinate frame of reference in which the symmetric part of the
connection vanishes locally (at one point). If the ane connection is symmetric in the covariant
indices, j
ik= j
ki(the torsion tensor vanishes) then (1.2.39) gives
0j
ik= 0: (1.2.40)
The coordinate frame of reference in which the connection vanishes (locally) is referred to as geodesic .
1.2.8 Ane geodesics and four-velocity
We consider a point in spacetime P(xk) and a vector dxkat this point. Construct a point P0(xk+dxk)
and nd the vector d0xkwhich is the parallel-transported dxkfromPtoP0. Then construct a point
P00(xk+dxk+d0xk) and nd the vector d00xkwhich is the parallel-transported d0xkfromP0to
P00. The next point is P000(xk+dxk+d0xk+d00xk) etc. Repeating this step constructs a polygonal
line which in the limit dxk!0 becomes a curve such that the vectordxk
d(whereis a parameter
along the curve) tangent to it at any point, when parallely translated to another point on this curve,
coincides with the tangent vector there. Such curve is referred to as an autoparallel curve or ane
geodesic . Ane geodesics can be attributed with the concept of length, which, for the polygonal
curve, is proportional to the number of parallel-transport steps described above.
The condition that parallel transport of a tangent vector be a tangent vector is
dxi
d+dxi
d
=dxi
d i
kldxk
ddxl=Mdxi
d+d2xi
d2d
; (1.2.41)
where the proportionality factor Mis some function of , or
Md2xi
d2+ i
kldxk
ddxl
d=1 M
ddxi
d; (1.2.42)
from which it follows that Mmust di er from 1 by the order of d. In the rst term on the left-hand
side of (1.2.42) we can therefore put M= 1, and we denote 1 Mby ()d, so
d2xi
d2+ i
kldxk
ddxl
d= ()dxi
d: (1.2.43)
13
If we replace by a new variable s() then (1.2.43) becomes
d2xi
ds2+ i
kldxk
dsdxl
ds= s0 s00
s02dxi
ds; (1.2.44)
where the prime denotes di erentiation with respect to . Requiring s0 s00= 0, which has a
general solution s=Rdexp[ R (x)dx], brings (1.2.44) into
d2xi
ds2+ i
kldxk
dsdxl
ds= 0; (1.2.45)
where the scalar variable sis the ane parameter. The autoparallel equation (1.2.45) is invariant
under linear transformations s!as+bsince the two lower limits of integration in the expression
fors() are arbitrary. De ning the four-velocity vector
ui=dxi
ds(1.2.46)
brings (1.2.14) into
DAk
ds=Ak
;iui;dAk
ds=Ak
;iui; (1.2.47)
so
Dui
ds=dui
ds+ i
klukul=ui
;juj= 0: (1.2.48)
The relations (1.2.47) can be generalized to any tensor density T:
DT
ds=T;iui;dT
ds=T;iui; (1.2.49)
The vectordxi
dsjQis a parallel translation ofdxi
dsjP. Because dsis a scalar, it is invariant under
parallel transport, dsjQ=dsjP. Therefore the vector dxijQis a parallel translation of dxijP, sods
measures the length of an in nitesimal section of an ane geodesic.
Only the symmetric part i
(kl)of the connection enters the autoparallel equation (1.2.45) because
of the symmetry ofdxk
dsdxl
dswith respect to the indices k;l; ane geodesics do not depend on torsion.
At any point, a coordinate transformation to the geodesic frame (1.2.35) brings all the components
i
(kl)to zero, so the autoparallel equation becomesdui
ds= 0. The autoparallel equation is also
invariant under a projective transformation
i
kl! i
kl+i
kAl; (1.2.50)
whereAiis an arbitrary vector. Substituting this transformation to (1.2.48) gives
dui
ds+ i
klukul= uiukAk: (1.2.51)
If we replace sby a new variable ~ s(s) then (1.2.51) becomes
dUi
d~s+ i
klUkUl= ukAk~s0+ ~s00
~s02dxi
d~s; (1.2.52)
where
Ui=dxi
d~s(1.2.53)
and the prime denotes di erentiation with respect to s. Requiring ukAk~s0+ ~s00= 0, which has a
general solution ~ s= Rsdsexp[RsAkuk(x)dx], brings (1.2.52) into
dUi
d~s+ i
klUkUl= 0: (1.2.54)
14
1.2.9 In nitesimal coordinate transformations
We consider a coordinate transformation
x0i=xi+i; (1.2.55)
wherei=xiis an in nitesimal vector (variation of xi). For a tensor or density Tde ne
T=T0(x0i) T(xi); (1.2.56)
T=T0(xi) T(xi) =T kT;k: (1.2.57)
For a scalar we nd
= 0; = k ;k: (1.2.58)
For a covariant vector
Ai=@xk
@x0iAk Ai k
;iAk; (1.2.59)
Ai k
;iAk kAi;k: (1.2.60)
The variation (1.2.59) is not a tensor, but (1.2.60) is:
Ai= k
;iAk kAi;k 2Sj
ikkAj: (1.2.61)
We refer to Tas a Lie derivative ofT,LT. For a contravariant vector
Bi=@x0i
@xkBk Bii
;kBk; (1.2.62)
Bii
;kBk kBi
;k=i
;kBk kBi
;k+ 2Si
jkkBj: (1.2.63)
For a scalar density
s= @xi
@x0i 1
s i
;is; (1.2.64)
s i
;is ks;k= i
;is ks;k+ 2Siis: (1.2.65)
The chain rule for implies that for a tensor density of weight w(which includes tensors as densities
of weight 0)
Tij:::
kl:::i
;mTmj:::
kl:::+j
;mTim:::
kl:::+ m
;kTij:::
ml::: m
;lTij:::
km::: :::
wm
;mTij:::
kl:::; (1.2.66)
Tij:::
kl:::i
;mTmj:::
kl:::+j
;mTim:::
kl:::+ m
;kTij:::
ml::: m
;lTij:::
km::: :::
wm
;mTij:::
kl::: mTij:::
kl:::;m+ 2Si
nmmTnj:::
kl:::+ 2Sj
nmmTin:::
kl:::+:::
2Sn
kmmTij:::
nl::: 2Sn
lmmTij:::
kn::: + 2wSmmTij:::
kl:::: (1.2.67)
A Lie derivative of a tensor density of rank ( k;l) and weight wis a tensor density of rank ( k;l) and
weightw.
The formula for a covariant derivative of Tcan be written as
T;k=T;k+ j
ik^Ci
jT; (1.2.68)
where ^Cis an operator acting on tensor densities:
^Ci
j = 0;^Ci
jAk= i
kAj;^Ci
jBk=k
jBi;^Ci
js= i
js; (1.2.69)
or generally
^Cm
nTij:::
kl:::=i
nTmj:::
kl:::+j
nTim:::
kl:::+ m
kTij:::
nl::: m
lTij:::
kn::: wm
nTij:::
kl::::(1.2.70)
Such de ned operator also enters the formula for T:
T=^Ck
iTi
;k: (1.2.71)
15
1.2.10 Killing vectors
A vectorithat satis es
(i;k)= 0 (1.2.72)
is referred to as a Killing vector . Along an ane geodesic
D
ds(uii) =uk(uii);k=uiuki;k+iukui
;k= 0: (1.2.73)
The rst term in the sum in (1.2.73) vanishes because of the de nition of iand the second term
vanishes because of the ane geodesic equation. Therefore, to each Killing vector ithere corre-
sponds a quantity uiiwhich is constant along an ane geodesic.
References: [1, 2, 3].
1.3 Curvature
1.3.1 Curvature tensor
The commutator of covariant derivatives of a contravariant vector is a tensor:
[rj;rk]Bi= 2r[jrk]Bi= 2@[jrk]Bi 2 l
[kj]rlBi+ 2 i
l[jrk]Bl
= 2@[j( i
jmjk]Bm) + 2Sl
jkrlBi+ 2 i
l[j@k]Bl+ 2 i
l[j l
jmjk]Bm
= 2(@[j i
jmjk]+ i
l[j l
jmjk])Bm+ 2Sl
jkrlBi=Ri
mjkBm+ 2Sl
jkrlBi; (1.3.1)
wherejjan index which is excluded from symmetrization or antisymmetrization. Therefore Ri
mjk,
de ned as
Ri
mjk=@j i
mk @k i
mj+ i
lj l
mk i
lk l
mj; (1.3.2)
is a tensor, referred to as the curvature tensor . The curvature tensor Ri
mjkis antisymmetric in the
indicesj;kand has generally 96 independent components. The commutator of covariant derivatives
of a covariant vector is
[rj;rk]Ai= Rm
ijkAm+ 2Sl
jkrlAi; (1.3.3)
and the commutator of covariant derivatives of a tensor is
[rj;rk]Tin:::
lp:::=Ri
mjkTmn:::
lp:::+Rn
mjkTim:::
lp:::+ Rm
ljkTin:::
mp::: Rm
pjkTin:::
lm:::
+ 2Sl
jkrlTin:::
lp:::: (1.3.4)
A change in the connection
i
jk! i
jk+Ti
jk; (1.3.5)
whereTi
jkis a tensor, results in the following change of the curvature tensor:
Ri
klm!Ri
klm+Ti
km;l Ti
kl;m+Tj
kmTi
jl Tj
klTi
jm: (1.3.6)
For a projective transformation (1.2.50), Ti
jk=i
jAk, so
Ri
klm!Ri
klm+i
k(Am;l Al;m): (1.3.7)
The variation of the curvature tensor is
Ri
klm= ( i
km);l ( i
kl);m+ i
jl j
km+ i
jl j
km i
jm j
kl i
jm j
kl
= ( i
km);l i
jl j
km+ j
kl i
jm+ j
ml i
kj ( i
kl);m+ i
jm j
kl j
km i
jl
j
lm i
kj+ i
jl j
km+ i
jl j
km i
jm j
kl i
jm j
kl
= ( i
km);l ( i
kl);m 2Sn
lm i
kn: (1.3.8)
16
1.3.2 Integrability of connection
The ane connection is integrable if parallel transport of a vector from point Pto pointQis inde-
pendent of a path along which this vector is parallelly translated, or equivalently, parallel transport
of a vector around a closed curve does not change this vector. For an integrable connection, we can
uniquely translate parallelly a given vector hiat pointPto all points in spacetime:
hi=dhi; (1.3.9)
or
hi
;k= i
jkhj: (1.3.10)
Therefore
( i
jkhj);l ( i
jlhj);k= i
jk;lhj i
jk j
mlhm i
jl;khj+ i
jl j
mkhm=Ri
jlkhj= 0;(1.3.11)
so, because hiis arbitrary,
Ri
klm= 0: (1.3.12)
Spacetime with a vanishing curvature tensor Ri
klm= 0 is
at. We consider 4 linearly independent
vectorshi
a, whereais 1,2,3,4, and vectors inverse to hi
a:
X
ahi
ahka=i
k: (1.3.13)
If the ane connection is integrable then (1.3.10) becomes
hi
a;k= i
lkhl
a: (1.3.14)
Multiplying (1.3.14) by hjagives
i
jk= hjahi
a;k=hja;khi
a: (1.3.15)
An integrable connection has thus 16 independent components. If the connection is also symmetric,
Si
jk= 0, then
hja;k hka;j= 0; (1.3.16)
which is the condition for the independence of the coordinates
ya=ZQ
Phiadxi(1.3.17)
of the path of integration PQ. Adopting yaas the new coordinates (with point P= (0;0;0;0) in
the center) gives
@ya
@xi=hia;@xi
@ya=hi
a; (1.3.18)
so (1.3.15) becomes
i
jk(xi) =@xi
@ya@2ya
@xk@xj: (1.3.19)
The transformation law for the connection (1.2.5) gives (with yacorresponding to x0j)
i
jk(ya) = 0: (1.3.20)
A torsionless integrable connection can be thus transformed to zero; one can always nd a system
of coordinates which is geodesic everywhere. If a connection is symmetric but nonintegrable then a
geodesic frame of reference can be constructed only at a given point (or along a given world line).
17
1.3.3 Parallel transport along closed curve
We consider parallel transport of a covariant vector around an in nitesimal closed curve. Such
transport changes this vector by
Ak=I
Ak=I
i
klAidxl=1
2Z@( i
kmAi)
@xl @( i
klAi)
@xm
dflm
1
2Z@ i
km
@xl @ i
kl
@xm
Ai+ ( i
km n
il i
kl n
im)An
dflm
=1
2Ri
klmAiflm; (1.3.21)
where we use Stokes' theorem (1.1.33) and Ak;l= i
klAiwhich is valid along the curve and thus is
approximately valid (to terms of rst order in flm) inside this curve. The change of a contravariant
vector due to parallel transport around an in nitesimal closed curve results from ( AkBk) = 0:
Bk= 1
2Rk
ilmBiflm; (1.3.22)
and the corresponding change of a tensor results from the chain rule for parallel transport:
Tik:::
np:::= 1
2(Ri
jlmTjk:::
np:::+Rk
jlmTij:::
np:::+ Rj
nlmTik:::
jp::: Rj
plmTik:::
nj::: :::)flm:(1.3.23)
1.3.4 Bianchi identities
We consider
rjr[krl]Bi=1
2rj(Ri
mklBm) +rj(Sm
klrmBi) (1.3.24)
and
r[jrk]rlBi= 1
2Rm
ljkrmBi+1
2Ri
mjkrlBm+Sm
jkrmrlBi= 1
2Rm
ljkrmBi
+1
2Ri
mjkrlBm+Sm
jkrlrmBi+Sm
jkRi
nmlBn+ 2Sm
jkSn
mlrnBi: (1.3.25)
Total antisymmetrization of the indices j;k;l in (1.3.24) and (1.3.25) gives
r[jrkrl]Bi=1
2r[jRi
jmjkl]Bm+1
2Ri
m[kl]rj]Bm+r[jSm
kl]rmBi+Sm
klrj]rmBi(1.3.26)
and
r[jrkrl]Bi= 1
2Rm
[ljk]rmBi+1
2Ri
m[jkrl]Bm+Sm
[jkrl]rmBi
+Sm
[jkRi
jnmjl]Bn+ 2Sm
[jkSn
jmjl]rnBi; (1.3.27)
so
1
2r[jRi
jmjkl]Bm+r[jSm
kl]rmBi= 1
2Rm
[ljk]rmBi+Sm
[jkRi
jnmjl]Bn
+2Sm
[jkSn
jmjl]rnBi: (1.3.28)
Comparing terms in (1.3.28) with Bigives the rst Bianchi identity or simply Bianchi identity :
Ri
n[jk;l]= 2Ri
nm[jSm
kl]; (1.3.29)
while comparing terms with rkBigives the second Bianchi identity or cyclic identity :
Rm
[jkl]= 2Sm
[jk;l]+ 4Sm
n[jSn
kl]: (1.3.30)
18
For a symmetric connection, Si
jk= 0, these identities reduce to
Ri
n[jk;l]= 0; (1.3.31)
Rm
[jkl]= 0: (1.3.32)
The cyclic identity (1.3.32) imposes 16 constraints on the curvature tensor, so the curvature tensor
with a vanishing torsion has 80 independent components.
1.3.5 Ricci tensor
Contraction of the curvature tensor with respect to the contravariant index and the second covariant
index gives the Ricci tensor :
Rik=Rj
ijk= j
ik;j j
ij;k+ l
ik j
lj l
ij j
lk: (1.3.33)
Contraction of the curvature tensor with respect to the contravariant index and the third covariant
index gives the Ricci tensor with the opposite sign due to the antisymmetry of the curvature tensor
with respect to its last indices. Contraction of the curvature tensor with respect to the contravariant
index and the rst covariant index gives the homothetic or segmental curvature tensor:
Qik=Rj
jik= j
jk;i j
ji;k; (1.3.34)
which is a curl. A change in the connection (1.3.5) results in the following changes of the Ricci
tensor and segmental curvature tensor:
Rik!Rik+Tl
ik;l Tl
il;k+Tj
ikTl
jl Tj
ilTl
jk; (1.3.35)
Qik!Qik+Tj
jk;i Tj
ji;k: (1.3.36)
For a projective transformation (1.2.50)
Rik!Rik+Ak;i Ai;k; (1.3.37)
Qik!Qik+ 4(Ak;i Ai;k): (1.3.38)
Therefore the symmetric part of the Ricci tensor is invariant under projective transformations. The
variation of the Ricci tensor is
Rik= ( l
ik);l ( l
il);k 2Sj
lk l
ij; (1.3.39)
while the variation of the segmental curvature tensor is
Qik= ( j
jk);i ( j
ji);k: (1.3.40)
1.3.6 Geodesic deviation
We consider a family of ane geodesics characterized by the ane parameter sand distinguished
by a scalar parameter t. We de ne the separation vector
vi=dxi
dt; (1.3.41)
so
vi
;kuk ui
;kvk=vi
;kuk ui
;kvk 2Si
klukvl=dui
dt dvi
ds 2Si
klukvl= 2Si
klukvl:(1.3.42)
19
Therefore
D2vi
ds2= (vi
;juj);kuk= (ui
;jvj);kuk 2(Si
klukvl);juj
=ui
;jkvjuk+ui
;jvj
;kuk 2(Si
klukvl);juj
=ui
;kjvjuk Ri
ljkulvjuk 2Sl
jkui
;lvjuk+ui
;jvj
;kuk 2(Si
klukvl);juj
=ui
;kjvjuk Ri
ljkulvjuk 2Sl
jkui
;lvjuk+ui
;j(uj
;kvk 2Sj
klukvl)
2(Si
klukvl);juj= (ui
;kuk);jvj+Ri
jklujukvl 2(Si
klukvl);juj
=Ri
jklujukvl 2D
ds(Si
klukvl) (1.3.43)
or
D
dsDvi
ds+ 2Si
klukvl
=Ri
jklujukvl: (1.3.44)
This is the equation of geodesic deviation . If we replace ane geodesics by arbitrary curves then
ui
;kuk6= 0 and (1.3.44) becomes
D
dsDvi
ds+ 2Si
klukvl
=Ri
jklujukvl+ (ui
;kuk);jvj: (1.3.45)
References: [1, 2, 3, 4].
1.4 Metric
1.4.1 Metric tensor
An ane parameter sis a measure of the length only along an ane geodesic. In order to extend
the concept of length to all points in spacetime, we equip spacetime with an algebraic object gik,
referred to as the covariant metric tensor and de ned as
ds2=gikdxidxk: (1.4.1)
The metric tensor is a symmetric tensor of rank (0,2):
gik=gki: (1.4.2)
The ane parameter s, whose di erential is given by (1.4.1), is referred to as the interval . Because
dsdoes not change under parallel transport along an ane geodesic from point P(xi) to point
Q(xi+dxi),dsjQ=dsjP, anddxijQis a parallel translation of dxijP,gikjQ=gikjP+gik;jdxjis a
parallel translation of gikjP:
gikjQ=gikjP+gik; (1.4.3)
so
Dgik=gik;jdxj=dgik gik=gik;jdxj gik= 0: (1.4.4)
Therefore a covariant derivative of the covariant metric tensor vanishes:
Njik= gik;j= 0 (1.4.5)
or
gik;j l
ijglk l
kjgil= 0; (1.4.6)
whereNijkis the nonmetricity tensor . The symmetric contravariant metric tensor gik=gkiis
de ned as the inverse of gik:
gijgik=k
j: (1.4.7)
20
A covariant derivative of the contravariant metric tensor also vanishes:
gik
;j= 0: (1.4.8)
The metric tensor allows to associate covariant and contravariant vectors:
Ai=gikAk; (1.4.9)
Bi=gikBk; (1.4.10)
because such association works for the covariant di erentials of these vectors which are vectors:
DAi=D(gikAk) =gikDAi; DBi=D(gikBk) =gikDBk(1.4.11)
(raising and lowering of coordinate indices commutes with covariant di erentiation with respect to
). For covariant and contravariant indices of tensors and densities this association is
gimTij:::
kl:::=Tj:::
m kl:::; (1.4.12)
gkmTij:::
kl:::=Tijm:::
l:::: (1.4.13)
The contravariant and covariant components of a two-dimensional vector are shown in Fig. 1. The
square root of the absolute value of the determinant
g=jgikj (1.4.14)
of the metric tensor is a scalar density, which we can use to multiply covariant integrands that
contain dual densities of weight -1, since
eiklm=p
jgjiklm; eiklm=1p
jgjiklm(1.4.15)
are tensors. Thus the relations (1.1.27) are also valid if we replace bye. The variation of the
determinant of the metric tensor is
g=ggikgik= ggikgik: (1.4.16)
A covariant derivative of the determinant of the metric tensor vanishes:
g;j= 0: (1.4.17)
A Lie derivative of the metric tensor is
Lgik= 2(i;k) 4S(ik)
ll; (1.4.18)
where;i=;kgik. The four-velocity vector (1.2.46) is normalized due to (1.4.1):
uiui= 1; (1.4.19)
thus having 3 independent components.
The commutator of covariant derivatives (1.3.4) of the metric tensor gives
R(ij)
kl= Nij
[k;l] Sm
klNij
m= Nij
[k ;l]; (1.4.20)
so the segmental curvature tensor (1.3.34) is
Qkl= Nij
[k ;l]gij: (1.4.21)
Because the nonmetricity tensor (1.4.5) vanishes, the curvature tensor is antisymmetric in its rst
two indices:
Rijkl= Rjikl: (1.4.22)
Thus the segmental curvature tensor also vanishes, and
Rijklgjl=Rik; (1.4.23)
so there is only one independent way to contract the curvature tensor, which gives the Ricci tensor
up to the sign.
21
Figure 1: Contravariant and covariant components of a vector.
1.4.2 Christo el symbols
The condition (1.4.5) is referred to as metricity or metric compatibility of the ane connection, and
imposes 40 constraints on the connection:
gik;j+gkj;i gji;k=gik;j l
ijglk l
kjgil+gkj;i l
kiglj l
jigkl gji;k+ l
jkgli
+ l
ikgjl=gik;j+gkj;i gji;k 2 l
(ij)gkl 2Sl
kjgil 2Sl
kigjl= 0: (1.4.24)
Multiplying (1.4.24) by gkmgives
m
(ij)=fm
ijg+ 2Sm
(ij); (1.4.25)
where
fm
ijg=1
2gmk(gki;j+gkj;i gij;k) (1.4.26)
are the Christo el symbols , symmetric in their covariant indices:
fk
ijg=fk
jig: (1.4.27)
Because k
ij= k
(ij)+Sk
ij, the metric-compatible ane connection equals
k
ij=fk
ijg+Ck
ij; (1.4.28)
where
Ck
ij= 2Sk
(ij)+Sk
ij (1.4.29)
is the contortion tensor , antisymmetric in its rst two indices:
Cijk= Cjik: (1.4.30)
The inverse relation between the torsion and contortion tensor is
Si
jk=Ci
[jk]: (1.4.31)
The di erence between two ane connections is a tensor, so the sum of a connection and a
tensor of rank (1,2) is a connection. Therefore the Christo el symbols form a connection, referred
to as the Levi-Civita connection . We de ne the covariant derivative with respect to the Levi-Civita
connection analogously to (1.2.11), with k
ijreplaced byfk
ijg, and denote it :iinstead of ;i, orrfg
i
instead ofri. A covariant derivative with respect to the Levi-Civita connection of the metric tensor
vanishes due to the de nition of the Christo el symbols:
gik:j=gik;j fl
ijgglk fl
kjggil= 0; (1.4.32)
which gives the inverse relation between ordinary derivatives of the metric tensor and the Christo el
symbols. The variation of the Levi-Civita connection is a tensor:
fk
ijg=1
2gkl
(gli):j+ (glj):i (gij):l
: (1.4.33)
22
The covariant derivative over sof a tensor density with respect to the Levi-Civita connection is,
analogously to (1.2.49),
DfgT
ds=T:iui: (1.4.34)
One can show that the following formulae hold:
fk
kig= (lnp
jgj);i; (1.4.35)
fk
ijggij= 1p
jgj(p
jgjgik);i; (1.4.36)
Bi
:i=1p
jgj(p
jgjBi);i; (1.4.37)
Fik
:i=1p
jgj(p
jgjFik);i; (1.4.38)
Ai:k Ak:i=Ai;k Ak;i; (1.4.39)I
Bip
jgjdSi=Z
Bi
:ip
jgjd
; (1.4.40)
whereFik= Fki. The Christo el symbols satisfy all formulae that are satis ed by k
ijin which
Si
jk= 0. Because the Levi-Civita connection is a symmetric connection, it can be brought to zero
by transforming the coordinates to a geodesic frame. In a geodesic frame, the covariant derivative
with respect to the Levi-Civita connection, rfg
i, coincides with the ordinary derivative @i. A Lie
derivative of the metric tensor (1.4.18) can be written as
Lgik= 2(i:k);Lgik= 2(i:k); (1.4.41)
where:i=:kgik. A Killing vector (1.2.72) for the Levi-Civita connection satis es
(i:k)= 0; (1.4.42)
thus becomes a generator of isometries , transformations that do not change the metric tensor.
If the nonmetricity tensor does not vanish, the general formula for the ane connection (1.4.28)
is
k
ij=fk
ijg+Ck
ij Nk
ij+1
2Nk
(i j): (1.4.43)
1.4.3 Riemann curvature tensor
The curvature tensor constructed from the Levi-Civita connection is referred to as the Riemann
tensor :
Pi
mjk=@jfi
mkg @kfi
mjg+fi
ljgfl
mkg fi
lkgfl
mjg: (1.4.44)
The commutator of covariant derivatives of the metric tensor vanishes:
[rfg
j;rfg
k]glp= Pm
ljkgmp Pm
pjkglm= 0; (1.4.45)
so the covariant Riemann tensor Pimjk is also antisymmetric in the indices i;m. Substituting (1.4.26)
in (1.4.44) gives
Piklm=1
2(gim;kl +gkl;im gil;km gkm;il) +gjn(fj
imgfn
klg fj
ilgfn
kmg); (1.4.46)
which explicitly shows the following symmetry and antisymmetry properties:
Piklm= Pikml; (1.4.47)
Piklm= Pkilm; (1.4.48)
Piklm=Plmik: (1.4.49)
23
Accordingly, the Riemannian Ricci tensor is symmetric:
Pik=Pj
ijk=Pki: (1.4.50)
Substituting (1.4.28) in (1.3.5) and (1.3.6) gives the relation between the curvature and Riemann
tensors:
Ri
klm=Pi
klm+Ci
km:l Ci
kl:m+Cj
kmCi
jl Cj
klCi
jm: (1.4.51)
Contracting (1.4.51) with respect to in the indices i;lgives
Rkm=Pkm+Ci
km:i Ci
ki:m+Cj
kmCi
ji Cj
kiCi
jm: (1.4.52)
Consequently, the Ricci or curvature scalar ,
R=Rikgik; (1.4.53)
is given by
R=P gik(2Cl
il:k+Cj
ijCl
kl Cl
imCm
kl); (1.4.54)
wherePis the Riemannian curvature scalar,
P=Pikgik: (1.4.55)
The variation of the Riemann tensor is, analogously to (1.3.8),
Pi
klm= (fi
kmg):l (fi
klg):m; (1.4.56)
and the variation of the Riemannian Ricci tensor is
Pik= (fl
ikg):l (fl
ilg):k: (1.4.57)
The Bianchi identities (1.3.29) and (1.3.30) contracted with respect to one contravariant and one
covariant index give
Ri
n[ik;l]= 2Ri
nm[iSm
kl]; (1.4.58)
Rk
[jkl]= 2Sk
[jk;l]+ 4Sk
n[jSn
kl]: (1.4.59)
Contracting these equations with the metric tensor gives
Rnk;l Rnl;k+Ri
nkl;i= 2RnmSm
kl 2Ri
nmkSm
il+ 2Ri
nmlSm
ik (1.4.60)
and the contracted cyclic identity :
Rjl Rlj= 2Sj;l+ 2Sl;j 2Sk
lj;k+ 4SnSn
lj: (1.4.61)
Further contraction of (1.4.60) with the metric tensor gives the contracted Bianchi identity :
Ri
l;i 1
2R;l= 2RkmSmk
l Rik
mlSm
ik: (1.4.62)
The Bianchi identities (1.3.31) and (1.3.32) for the Riemann tensor are
Pi
n[jk:l]= 0; (1.4.63)
Pm
[jkl]= 0: (1.4.64)
Contracting these equations with the metric tensor gives
Pnk:l+Pi
nkl:i Pnl:k= 0; (1.4.65)
Pjl Plj= 0; (1.4.66)
in agreement with (1.4.50). Further contraction of (1.4.65) with the metric tensor gives the covariant
conservation,
Gi
k:i= 0; (1.4.67)
of the symmetric Einstein tensor ,
Gik=Pik 1
2Pgik: (1.4.68)
24
1.4.4 Properties of Riemann tensor
In two dimensions there is only 1 independent component of the Riemann tensor, P1212. The
Riemann scalar is
P=2P1212
s; (1.4.69)
where sis the determinant of the two-dimensional metric tensor
ik:
s=j
ikj=
11
22
2
12: (1.4.70)
A surface near point x= 0;y= 0 is given by
z=x2
21+y2
22; (1.4.71)
where1and2are the radii of curvature. Substituting (1.4.71) to
dl2=dx2+dy2+dz2=
ikdxidxk(1.4.72)
gives
ik(x;y), which then gives
P
2
x=y=0=K=1
12; (1.4.73)
whereKis the Gaucurvature.
In three dimensions there are 3 independent pairs, 12, 23, and 31, so the Riemann tensor has 6
independent components: 3 with identical pairs and32
2= 3 with di erent pairs (the cyclic identity
does not reduce the number of independent components). The Ricci tensor has also 6 components,
which are related to the components of the Riemann tensor by
P
=P
P
+P
P
+P
2(
): (1.4.74)
Choosing the Cartesian coordinates at a given point, de ned by the condition
g = diag(1;1;1); (1.4.75)
and diagonalizing P , which is equivalent to 3 rotations, brings P to the canonical form with
6 3 = 3 independent components. Consequently, the Riemann tensor in three dimensions has 3
physically independent components. The Gaucurvature of a surface perpendicular to the x3axis is
given by
K=P1212
11
22
2
12: (1.4.76)
In four dimensions there are 6 independent pairs, 01, 02, 03, 12, 23, and 31, so there are 6
components with identical pairs and65
2= 15 with di erent pairs. The cyclic identity reduces the
number of independent components by 1, so the Riemann tensor in four dimensions has generally
20 independent components. Choosing the Cartesian coordinates at a given point and applying 6
rotations brings Pijklto the canonical form with 20 6 = 14 physically independent components.
1.4.5 Weyl tensor
In four dimensions the Weyl tensor is de ned as
Wiklm=Piklm 1
2(Pilgkm+Pkmgil Pimgkl Pklgim) +1
6P(gilgkm gimgkl): (1.4.77)
This tensor has all the symmetry and antisymmetry properties of the Riemann tensor, and is also
traceless (any contraction of the Weyl tensor vanishes).
25
1.4.6 Metric geodesics
We consider two points in spacetime, PandQ. Among curves that connect these points, one curve
has the minimal value of the interval s=R
ds, and is referred to as a metric geodesic . The equation
of a metric geodesic is given by the condition thatR
dsbe an extremum with the endpoints of the
curve xed:
Z
ds=Z
(gikdxidxk)1=2=Zdxigijdxj
ds+1
2Zgijdxidxj
ds=Z
gijujdxi
+1
2Z
gij;kxkuiujds=Z
d(uixi) Z
duixi+1
2Z
gij;kxkuiujds
= Zdui
dsxids+1
2Z
gjk;ixiujukds= 0; (1.4.78)
where we omit the total di erential termR
d(uixi) becausexi= 0 at the endpoints. Since xiis
arbitrary, we obtain
d
ds(gijuj) 1
2Z
gjk;iujukds=gijduj
ds+ukgij;kuj 1
2Z
gjk;iujukds
=gijduj
ds+fm
jkggimujuk= 0 (1.4.79)
or, after multiplying (1.4.79) by gil:
Dfgul
ds=dul
ds+fl
jkgujuk=uiul
:i= 0: (1.4.80)
The metric geodesic equation (1.4.80) can be written as
d2xi
ds2+fi
klgdxk
dsdxl
ds= 0: (1.4.81)
Using (1.4.28) and (1.4.29), the ane geodesic equation (1.2.45) can be written as
d2xi
ds2+fi
klgdxk
dsdxl
ds+ 2Si
kldxk
dsdxl
ds= 0: (1.4.82)
If the torsion tensor is completely antisymmetric then the last term in (1.4.82) vanishes and the ane
geodesic equation coincides with the metric geodesic equation. The equation of geodesic deviation
with respect to the Levi-Civita connection is, analogously to (1.3.44),
Dfg2vi
ds2=Pi
jklujukvl: (1.4.83)
1.4.7 Galilean frame of reference and Minkowski tensor
At a given point, the nondegenerate ( g6= 0) metric tensor can be brought to a diagonal (canonical)
formgik= diag(1;1;1;1).Physical systems are described by the metric tensor with g<0.
Without loss of generality, we assume that the canonical form of the metric tensor is
gik=ik= diag(1; 1; 1; 1); gik=ik= diag(1; 1; 1; 1): (1.4.84)
A frame of reference in which gikhas the canonical form is referred to as Galilean . The spatial
coordinates in the Galilean frame of reference are Cartesian (1.4.75). The transformation (1.2.35)
with (1.2.38) brings a symmetric ane connection, thus the Christo el symbols, to zero at a given
point without changing the components of the metric tensor because of (1.2.36). Therefore a frame
of reference can be both geodesic and Galilean. In such locally inertial frame rst derivatives of the
26
metric tensor vanish because of (1.4.32). The corresponding metric tensor (1.4.84) is referred to as
theMinkowski tensor . The interval for this metric is
ds2=c2dt2 dx2 dy2 dz2: (1.4.85)
In a locally inertial frame the coordinates xi, not only the di erentials dxi, are components of a
contravariant vector.
In the absence of torsion, spacetime with a vanishing Riemann tensor Pi
klm= 0 is
at. In the
new coordinates ya(1.3.17), (1.3.18) gives
gab(y) =gik(x)@xi
@ya@xk
@yb=gik(x)hiahkb=ab: (1.4.86)
Therefore in a
at spacetime without torsion one can always nd a system of coordinates which is
Galilean everywhere.
1.4.8 Intervals, proper time and distances
The form of the Minkowski tensor distinguishes the coordinate x0from the rest of the coordinates
x , where the index can be 1,2,3. The temporal coordinate x0=ct, wheretis referred to as time
andcis referred to as the velocity of propagation of interaction . The coordinates x arespatial and
span space . The set of 4 coordinates xidescribe an event and span spacetime . The curve xi(),
whereis a parameter, is referred to as a world line of a given point. The quantitites
v =dx
dt(1.4.87)
are the components of a three-dimensional vector, the velocity of this point. An in nitesimal interval
dsistimelike ifds2>0,spacelike ifds2<0, and nullifds2= 0. In the Galilean frame, the interval
between two in nitesimally separated points (events) is
ds2=ikdxidxk=c2dt2 dx dx ; (1.4.88)
wheredxiare in nitesimal coordinate di erences between the two points. The interval between two
nitely separated points is
s2=ikxixk=c2t2 x x ; (1.4.89)
where xiare nite coordinate di erences between the two points. If sis timelike, one can always
nd a frame of reference in which the two events occur at the same place, x = 0. A frame of
reference in which dx = 0 describes a point at restand is referred to as the rest frame or the
comoving frame . In this frame t= ,
ds2=c2d 2; (1.4.90)
where is the proper time . Ifdx 6= 0 along a world line then the point moves or is in motion . The
proper time for a moving point is equal to the time measured by a clock moving with this point.
If sis spacelike, one can always nd a frame of reference in which the two events occur at the
same time (are synchronous ), x0= 0. Ifds= 0 along a world line, this world line describes the
propagation of a signal ( interaction ), withv= (v v )1=2=c. Equations (1.4.88) and (1.4.90) give
d 2=dt2 1
cdx dx ; (1.4.91)
so the proper time goes more slowly than the coordinate time t. If sis timelike, the two events
occur at di erent times: t16=t2. Ift2>t1thent2is in the future with respect to t1andt1is in the
pastwith respect to t2. The time of a measurement t0is called the present time.
All events for which t<t 0form the absolute past relative to the event Oat the present (events in
this region occur beforeOin all systems of reference). All events for which t>t 0form the absolute
27
future relative to the event Oat the present (events in this region occur afterOin all systems
of reference). Such a division into the absolute past and the absolute future with respect to Ois
possible only for events for which their intervals with respect to Oare timelike, as shown in Fig. 2.
ForO= (0;0;0;0), these events ( ct;x;y;z ) lie within a cone ( ct)2 x2 y2 z2= 0 which is called
the null cone or light cone . All events for which their intervals with respect to Oare spacelike are
absolutely remote relative toO. The principle of causality states that any event Ocan be a ected
only by events in the absolute past relative to O.
Figure 2: Light cone.
In the rest frame dx = 0 givesu = 0. At each point in space, the condition dx = 0 gives the
relation between the proper time and the coordinate time:
d =1
cpg00dx0; (1.4.92)
which requires
g000: (1.4.93)
The relation (1.4.19) gives
u0= (g00) 1=2: (1.4.94)
The distance between two in nitesimally separated points cannot be obtained by imposing dx0
becausex0transforms di erently at these points. Instead, consider a signal that leaves point B(x +
dx ) atx0+dx0
, reaching point A(x ) atx0and coming back to point Batx0+dx0
+, as shown
in Fig. 3. Therefore
ds2=g00(dx0)2+ 2g0 dx0dx +g dx dx = 0 (1.4.95)
gives
dx0
=1
g00( g0 dx q
(g0 g0 g00g )dx dx ): (1.4.96)
The di erence in the time coordinate between emitting and receiving the signal at point Bis equal
to the di erence between dx0
+anddx0
timespg00=c, and the distancedlbetween points AandB
is equal to this di erence times c=2:
dl2=
dx dx ; (1.4.97)
where
= g +g0 g0
g00(1.4.98)
is the symmetric spatial metric tensor of spacetime, i.e. the metric tensor of space. The event
at pointAatx0issynchronized with the event at point Bat the arithmetic mean of the time
coordinates of emitting and receiving the signal, i.e. at
x0+1
2(dx0
+dx0
+) =x0+g dx ; (1.4.99)
28
where
g = g0
g00: (1.4.100)
Therefore
x0=g x ; (1.4.101)
which is equivalent to x0= 0, is the di erence in x0between two synchronized in nitesimally
separated points.
Figure 3: Distance.
1.4.9 Spatial vectors
The spatial components of a contravariant four-vector Aiform a three-dimensional, spatial vector
A:
Ai= (A0;A ) = (A0;A): (1.4.102)
The contravariant four-vector index is also the contravariant spatial-vector index. The covariant
components of a spatial vector are related to the contravariant components by the spatial metric
tensor (1.4.98) which raises and lowers indices of spatial vectors analogously to the metric tensor
acting on four-vectors:
A =
A ; (1.4.103)
B =
B ; (1.4.104)
where
is the inverse of
:
=
: (1.4.105)
One can show that the following formulae hold:
= g ; (1.4.106)
g= g00s; (1.4.107)
g = g0 ; (1.4.108)
g00=1
g00 g g ; (1.4.109)
where
s= det
: (1.4.110)
The components g form a spatial vector g.
The scalar product of two spatial vectors is
AB=
A B : (1.4.111)
The square of a spatial vector Ais
A2=AA (1.4.112)
and its norm is
A=p
A2: (1.4.113)
29
The angle between two spatial vectors is de ned through
AB=ABcos: (1.4.114)
In three-dimensional space, the permutation symbol is de ned as
= 0
: (1.4.115)
The three-dimensional equivalent of (1.4.15) is
e
=ps
; e
=1ps
: (1.4.116)
The cross product of two three-dimensional vectors AandB,C=ABis
C =e
A B
; C =e
A B
: (1.4.117)
The permutation symbol (1.4.115) satis es
=
; (1.4.118)
= 2
; (1.4.119)
= 6: (1.4.120)
The spatial covariant derivative r acts on spatial vectors analogously to the metric covariant
derivative acting on four-vectors:
r A =@ A +f
g
A
; (1.4.121)
r A =@ A f
g
A
; (1.4.122)
wheref
g
are the three-dimensional, spatial Christo el symbols :
f
g
=1
2
(
; +
;
;
): (1.4.123)
The gradient operator is given by
(grad) = (r) =
r : (1.4.124)
The spatial components of a covariant-vector operator @iacting on a scalar form the gradient of
:
@i =@
c@t;@
@x
=@
c@t;r
: (1.4.125)
The divergence of a spatial vector Ais, analogously to (1.4.37),
divA=rA=1ps@ (psA ): (1.4.126)
The curl of a spatial vector is
(curlA) = (rA) =e
@ A
: (1.4.127)
The Laplace-Beltrami operator or Laplacian is the divergence of the gradient,
4=r2=rr=1ps@ (ps
@ ): (1.4.128)
The d'Alembert operator or d'Alembertian is de ned as
=1
c2@2
@t2 4: (1.4.129)
30
In a locally Galilean frame of reference, the covariant and contravariant three-dimensional com-
ponents of a vector are identical because
= ; (1.4.130)
where is the Cartesian metric tensor ,
= diag(1;1;1); = diag(1;1;1): (1.4.131)
In this frame we refer to the coordinates x1;x2;x3, which are Cartesian, as x;y;z . One can show
that the following formulae hold:
AB= BA; (1.4.132)
A(BC) =B(CA) =C(AB); (1.4.133)
A(BC) =B(AC) C(AB); (1.4.134)
(AB)2+ (AB)2=A2B2; (1.4.135)
curl grad = 0; (1.4.136)
div curl A= 0; (1.4.137)
grad( ) = grad + grad ; (1.4.138)
grad(AB) = (Ar)B+ (Br)A+AcurlB
+BcurlA; (1.4.139)
div( A) = grad A+ divA; (1.4.140)
curl( A) = grad A+ curlA; (1.4.141)
div(AB) =BcurlA AcurlB; (1.4.142)
curl(AB) = (Br)A (Ar)B+AdivB BdivA; (1.4.143)
curl curl A= grad div A 4A; (1.4.144)
where
(Ar)B=A @ B: (1.4.145)
The cross product of two spatial vectors AandBsatis es
AB=ABsinn; (1.4.146)
where nis a unit vector perpendicular to both AandB, in the direction given by the right-handed
corkscrew rule.
References: [1, 2, 3, 4].
1.5 Tetrad and spin connection
1.5.1 Tetrad
In addition to the coordinate systems, at each spacetime point we set up four linearly independent
vectorsei
asuch that
ei
aeib=ab; (1.5.1)
wherea;b= 0;1;2;3 are Lorentz indices andab= diag(1; 1; 1; 1) is the coordinate-invariant
Minkowski metric tensor in a locally geodesic frame of reference at this point. This set of four vectors
is referred to as a tetrad . The inverse tetrad eaisatis es
ei
aeb
i=b
a; (1.5.2)
ei
aea
k=i
k: (1.5.3)
The coordinate metric tensor gikis related to the Minkowski metric tensor through the tetrad:
gik=ea
ieb
kab: (1.5.4)
31
Accordingly, the determinant gof the metric tensor gikis related to the determinant of the tetrad
e=jea
ijbyp
jgj=e: (1.5.5)
Any vector Vcan be speci ed by its components Viwith respect to the coordinate system or by
the coordinate-invariant projections Vaof the vector onto the tetrad eld:
Va=ea
iVi; Va=ei
aVi; (1.5.6)
Vi=ei
aVa; Vi=ea
iVa; (1.5.7)
and similarly for tensors and densities with more indices. We can use aband its inverse abto lower
and raise Lorentz indices, as we use gikand its inverse gikto lower and raise coordinate indices.
1.5.2 Lorentz transformation
The relation (1.5.4) imposes 10 constraints on the 16 components of the tetrad, leaving 6 components
arbitrary. If we change from one tetrad ei
ato another, ~ ei
b, then the vectors of the new tetrad are
linear combinations of the vectors of the old tetrad:
~ei
a= b
aei
b: (1.5.8)
The relation (1.5.4) applied to the tetrad eld ~ ei
b,
gik= ~ea
i~eb
kab; (1.5.9)
imposes on the matrix b
athe orthogonality condition:
c
ad
bcd=ab: (1.5.10)
We refer to b
aas a Lorentz matrix , and to a transformation of form (1.5.8) as the Lorentz trans-
formation .
1.5.3 Tetrad transport
A natural choice for the zeroth component of a tetrad at a given point is
ei
0=ui: (1.5.11)
Along a world line this tetrad should be transported such that the zeroth component always coincides
with the four-velocity. The Fermi-Walker transport of a tetrad is de ned as
rei
a
ds= uiej
aDuj
ds+Dui
dsej
auj: (1.5.12)
Puttinga= 0 in (1.5.12) gives
rui
ds=Dui
ds; (1.5.13)
so the Fermi-Walker transport of the four-velocity is equivalent to its covariant change and thus
(1.5.11) is valid at all points. This transport does not change the orthogonality relation for tetrads
(1.5.1) since (1.5.12) gives
r
ds(ei
aeib) = 0: (1.5.14)
32
1.5.4 Spin connection
We de ne
!i
ak=ei
a;k=ei
a;k+ i
jkej
a: (1.5.15)
The quantities
!a
bi=ea
j!j
bi(1.5.16)
transform like vectors under coordinate transformations. We can extend the notion of covariant dif-
ferentiation to quantities with Lorentz coordinate-invariant indices by regarding !ab
ias a connection,
referred to as Lorentz or spin connection . For a contravariant Lorentz vector
Va
ji=Va
;i+!a
biVb; (1.5.17)
wherejiis a covariant derivative of such a quantity with respect to xi. The covariant derivative of
a scalarVaWacoincides with its ordinary derivative:
(VaWa)ji= (VaWa);i; (1.5.18)
which gives a covariant derivative of a covariant Lorentz vector:
Waji=Wa;i !b
aiWb: (1.5.19)
The chain rule implies that a covariant derivative of a Lorentz tensor is equal to the sum of the
corresponding ordinary derivative of this tensor and terms with spin connection corresponding to
each Lorentz index:
Tab:::
cd:::ji=Tab:::
cd:::;i +!a
eiTeb:::
cd:::+!b
eiTae:::
cd:::+ !e
ciTab:::
ed::: !e
diTab:::
ce::: ::: : (1.5.20)
We assume that the covariant derivative jiis total, that is, also recognizes coordinate indices, acting
on them like ;i. For a tensor with both coordinate and Lorentz indices
Taj:::
bk:::ji=Taj:::
bk:::;i+!a
eiTej:::
bk:::+ j
liTal:::
bk:::+ !e
biTaj:::
ek::: l
kiTaj:::
bl::: ::: : (1.5.21)
A total covariant derivative of a tetrad is
ei
ajk=ei
a;k+ i
jkej
a !b
akei
b= 0; (1.5.22)
due to (1.5.15). Therefore total covariant di erentiation commutes with converting between coor-
dinate and Lorentz indices. Equation (1.5.22) also determines the spin connection !a
biin terms of
the ane connection, tetrad and its ordinary derivatives:
!a
bi=ea
k(ek
b;i+ k
jiej
b): (1.5.23)
Conversely, the ane connection is determined by the spin connection, tetrad and its derivatives:
j
ik=!j
ik+ea
i;kej
a: (1.5.24)
The torsion tensor is then
Sj
ik=!j
[ik]+ea
[i;k]ej
a; (1.5.25)
and the torsion vector is
Si=!k
[ik]+ea
[i;k]ek
a: (1.5.26)
Metric compatibility of the ane connection leads to
gik;j=gikjj=ea
ieb
kabjj= ea
ieb
k(!c
ajcb+!c
bjac) = (!kij+!ikj) = 0; (1.5.27)
so the spin connection is antisymmetric in its rst two indices:
!a
bi= !a
b i: (1.5.28)
33
Accordingly, the spin connection has 24 independent components. The contortion tensor is
Cijk=!ijk+ ijk; (1.5.29)
where
ijk=eiaea
[j;k] ejaea
[i;k] ekaea
[i;j] (1.5.30)
are the Ricci rotation coecients . The rst term on the right-hand side in (1.5.29) is expected
because both the contortion tensor and spin connection are antisymmetric in their rst two indices.
The quantities
$i
ak=ei
a:k=ei
a;k+fi
jkgej
a (1.5.31)
form the Levi-Civita spin connection and are related to the Ricci rotation coecients by (1.5.29)
withCijk= 0,
$ijk= ijk; (1.5.32)
so
Cijk=!ijk $ijk: (1.5.33)
1.5.5 Tetrad representation of curvature tensor
The commutator of the covariant derivatives of a tetrad with respect to the ane connection is
2ek
a;[ji]=Rk
lije
a+ 2Sl
ijek
a;l: (1.5.34)
This commutator can also be expressed in terms of the spin connection:
ek
a;[ji]=!k
a[j;i]= (ek
b!b
a[j);i]=!ba[j!kb
i]+!b
a[j;i]ek
b
=!ba[j!kb
i]+!b
a[j;i]ek
b+Sl
ij!k
al: (1.5.35)
Consequently, the curvature tensor with two Lorentz and two coordinate indices depends only on
the spin connection and its ordinary derivatives:
Ra
bij=!a
bj;i !a
bi;j+!a
ci!c
bj !a
cj!c
bi: (1.5.36)
Because the spin connection is antisymmetric in its rst two indices, the tensor (1.5.36) is antisym-
metric in its rst two (Lorentz) indices, like the Riemann tensor. The contraction of the curvature
tensor (1.5.36) with a tetrad gives the Ricci tensor with one Lorentz and one coordinate index:
Rbj=Ra
bijei
a: (1.5.37)
The contraction of the tensor Ra
iwith a tetrad gives the Ricci scalar,
R=Ra
iei
a=Rab
ijei
aej
b: (1.5.38)
The Riemann tensor with two Lorentz and two coordinate indices depends on the Levi-Civita
connection (1.5.31) the same way the curvature tensor depends on the ane connection:
Pa
bij=$a
bj;i $a
bi;j+$a
ci$c
bj $a
cj$c
bi: (1.5.39)
The contraction of (1.5.39) with a tetrad gives the Riemannian Ricci tensor with one Lorentz and
one coordinate index:
Pbj=Pa
bijei
a: (1.5.40)
The contraction of the tensor Pa
iwith a tetrad gives the Riemann scalar,
P=Pa
iei
a=Pab
ijei
aej
b: (1.5.41)
References: [3, 4, 5, 6, 7].
34
1.6 Lorentz group
1.6.1 Subgroups of Lorentz group and principle of relativity
A composition of two Lorentz transformations 1and 2,
a
b= a
(1)cc
(2)b; (1.6.1)
satis es (1.5.10), so it is a Lorentz transformation. The Kronecker symbol a
balso satis es (1.5.10),
so it can be regarded as the identity Lorentz transformation. Therefore Lorentz transformations
form a group, referred to as the Lorentz group . Taking the determinant of the relation (1.5.10) gives
ja
bj=1: (1.6.2)
A Lorentz transformation with ja
bj= 1 is proper and withja
bj= 1 is improper . Proper
Lorentz transformations form a group because the determinant of the product of two proper Lorentz
transformations is 1. Improper Lorentz transformations include the parity transformation P
a
b(P) = diag(1; 1; 1; 1); t!t;x! x; (1.6.3)
and the time reversal T
a
b(T) = diag( 1;1;1;1); t! t;x!x: (1.6.4)
The relation (1.5.10) gives 0
00
0 0
0
= 1, so
j0
0j1: (1.6.5)
Lorentz transformations with 0
01 are orthochronous and form a group. If xiis a timelike vector,
xixi>0, then for an orthochronous transformation x00= 0
0x0+ 0
x ,
j0
x jq
0
0
x x <q
(0
0)2(x0)2=j0
0x0j: (1.6.6)
Thus the time component of a timelike vector does not change the sign under orthochronous trans-
formations. Einstein's principle of relativity states that all physical laws are invariant under trans-
formations within the orthochronous proper subgroup of the Lorentz group.
Under the parity transformation, the spatial components of contravariant and covariant vectors
(three-dimensional vectors) change the sign, while the spatial components of dual vectors (such as
cross products of vectors) do not change the sign. Similarly, the scalar contraction of the Levi-Civita
symbol and a tensor changes the sign, while a scalar does not. Quantities that transform under
proper Lorentz transformations like vectors and do not change the sign in their spatial components
under parity are referred to as axial vectors or pseudovectors . Quantities that transform under
proper Lorentz transformations like scalars and change the sign under parity are referred to as
pseudoscalars .
1.6.2 In nitesimal Lorentz transformations
We consider an in nitesimal Lorentz transformation
=
+
; (1.6.7)
where
are in nitesimal quantities. The relation (1.5.10) gives
= ; (1.6.8)
where the indices are raised and lowered using the Minkowski metric tensor. Therefore Lorentz
transformations are given by 6 independent antisymmetric parameters . The corresponding
transformation of a contravariant vector Ais
A0=A+
A=A+1
2(
)A=A+1
2J
A; (1.6.9)
35
where
J
=
: (1.6.10)
We de ne matrices Jsuch that
(J)
=J
: (1.6.11)
Therefore, in the matrix notation (with Atreated as a column),
A0=
1 +1
2J
A: (1.6.12)
The 6 matrices Jare the in nitesimal generators of the vector representation of the Lorentz group.
The explicit form of the generators of the Lorentz group in the vector representation is
J01=0
BB@0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 01
CCA; J02=0
BB@0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 01
CCA;
J03=0
BB@0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 01
CCA; J12=0
BB@0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 01
CCA;
J23=0
BB@0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 01
CCA; J31=0
BB@0 0 0 0
0 0 0 1
0 0 0 0
0 1 0 01
CCA: (1.6.13)
1.6.3 Generators and Lie algebra of Lorentz group
The commutator of the generators of the Lorentz group in the vector representation is, using (1.6.10)
and (1.6.11),
[J ;J]
= (J )
(J)
(J)
(J )
= ( J J +J +J )
;(1.6.14)
so
[J ;J] = J J +J +J : (1.6.15)
The relation (1.6.15) constitutes the Lie algebra of the Lorentz group . If a set of qantitites
transforms under a Lorentz transformation with a matrix D()
!D() ; (1.6.16)
thenDis a representation of the Lorentz group if
D(I) =I; D (12) =D(1)D(2); (1.6.17)
whereIdenotes the identity transformation, and 1and 2are two Lorentz transformations. There-
fore
D( 1) =D 1(); (1.6.18)
where 1is the Lorentz transformation to : 1=I. For an in nitesimal Lorentz transforma-
tion in any representation,
D() =I+1
2J; (1.6.19)
according to (1.6.12). The relation
D(12 1
1) =D(1)D(2)D 1(1) (1.6.20)
gives (1.6.15), valid for any representation of the Lorentz group.
36
If 1and 2are two group transformations then 3= 12 1
1is a group transformation. If
2=I+2G2is an in nitesimal group transformation with generator G2then 3=I+21G2 1
1
is an in nitesimal group transformation with generator G3= 1G2 1
1. If 1=I+1G1is an
in nitesimal group transformation with generator G1then, neglecting terms in 1of higher order,
G3=G2+1[G1;G2], so [G1;G2] is a generator. For a nite number Nof linearly independent
generators, a general in nitesimal group transformation is = I+ N
a=1aGa. Because [ Ga;Gb] is
a generator, it is a linear combination of the Ngenerators: [ Ga;Gb] = N
c=1fabcGc, wherefabcare
structure constants of the Lie algebra of the given group. For the Lorentz group, aGa=D() I,
whereD() is given by (1.6.19).
1.6.4 Rotations and boosts
Rotations are proper orthochronous Lorentz transformations with
0
=
0= 0;0
0= 1: (1.6.21)
Rotations act only on the spatial coordinates x and form a group, referred to as the rotation group .
Boosts are proper orthochronous Lorentz transformations with
= 0: (1.6.22)
We de ne
J =1
2e
J
; (1.6.23)
K =J0 ; (1.6.24)
and
# =1
2e
; (1.6.25)
=0 (1.6.26)
(for the Lorentz group g= 1, so the tensors eand densities are numerically identical). The explicit
form of the generators of the rotation group J in the vector representation is
J1=0
@0 0 0
0 0 1
0 1 01
A; J2=0
@0 0 1
0 0 0
1 0 01
A; J3=0
@0 1 0
1 0 0
0 0 01
A: (1.6.27)
For an in nitesimal Lorentz transformation (1.6.19)
D= 1 +#J+K: (1.6.28)
A nite Lorentz transformation can be regarded as a composition of successive identical in nites-
imal Lorentz transformations:
D= limn!1(1 +J=n+K=n)n=eJ+K: (1.6.29)
The nite parameters ,are the canonical parameters for a given Lorentz transformations. For a
nite Lorentz transformation, (1.6.19) gives
D() =e1
2J; (1.6.30)
so
J=@D()
@
=I: (1.6.31)
37
The explicit form of a nite Lorentz transformation in the vector representation is
R1=eJ1=0
BB@1 0 0 0
0 1 0 0
0 0 cos sin
0 0 sincos1
CCA; R 2=eJ2=0
BB@1 0 0 0
0 cos0 sin
0 0 1 0
0 sin0 cos1
CCA;
R3=eJ3=0
BB@1 0 0 0
0 cos sin0
0 sincos0
0 0 0 11
CCA; B 1=eK1=0
BB@coshsinh0 0
sinhcosh0 0
0 0 1 0
0 0 0 11
CCA;
B2=eK2=0
BB@cosh0 sinh0
0 1 0 0
sinh0 cosh0
0 0 0 11
CCA; B 3=eK3=0
BB@cosh0 0 sinh
0 1 0 0
0 0 1 0
sinh0 0 cosh 1
CCA;
(1.6.32)
whereR denotes a rotation about the x -axis andB denotes a boost along this axis. The canonical
parametersandare respectively referred to as the angle of rotation andrapidity . The parameters
#andin (1.6.25) and (1.6.26) are thus respectively in nitesimal values of the angle of rotation
and rapidity. A rotation about any axis, say z, by an angle turns the two other axes, xandy, into
new axes,x0andy0, such that the angle between xandx0(oryandy0) (1.4.114) is . The rotation
group is compact : 2[0;2] and= 2,= 0. The explicit form of a nite rotation in the
three-dimensional vector representation is
R1() =0
@1 0 0
0 cos sin
0 sincos1
A; R 2() =0
@cos0 sin
0 1 0
sin0 cos1
A;
R3() =0
@cos sin0
sincos0
0 0 11
A: (1.6.33)
For instance,0
@Vx
Vy
Vz1
A!0
@V0
x
V0
y
V0
z1
A=R30
@Vx
Vy
Vz1
A=0
@Vxcos Vysin
Vxsin+Vycos
Vz1
A: (1.6.34)
The relation (1.6.31) gives
J =@R ()
@
=0: (1.6.35)
The orthogonality relation (1.5.10) applied to any of the rotation matrices (1.6.33) shows that a
rotation matrix Ris orthogonal, that is, its transpose RTis equal to its inverse R 1:
RT
=R 1
; R RT
=RT
R =I; (1.6.36)
whereIis the identity matrix.
The commutation relation (1.6.15) gives
[J ;J ] =e
J
; (1.6.37)
[J ;K ] =e
K
; (1.6.38)
[K ;K ] = e
J
: (1.6.39)
Therefore rotations do not commute and form a nonabelian group, rotations and boosts do not
commute, and boosts do not commute. Changing the order of two nonparallel boosts is equivalent
38
to applying a rotation, referred to as the Thomas-Wigner rotation . The structure constants of the
Lie algebra of the rotation group are fabc=eabc. Moreover, the square of the generators of rotation,
J2=J J ; (1.6.40)
commutes with J :
[J2;J ] = [J ;J ]J +J [J ;J ] =e
(J
J +J J
) = 0: (1.6.41)
De nining
L=1
2(J+iK); (1.6.42)
Q=1
2(J iK); (1.6.43)
gives
[L ;L ] =e
L
; (1.6.44)
[Q ;Q ] =e
Q
; (1.6.45)
[L ;Q ] = 0; (1.6.46)
so the Lorentz group is isomorphic with the product of two complex rotation groups. Accordingly,
the Lorentz group can be regarded as the group of four-dimensional rotations in the Minkowski
space, or the group of tetrad rotations .
1.6.5 Poincar e group
Under the in nitesimal coordinate transformation (1.2.55) in a locally
at spacetime, (1.4.41) gives
ik!ik i;k k;i: (1.6.47)
Thus the tensor ikis invariant under (1.2.55) (isometric) if iis a Killing vector,
(i;k)= 0; (1.6.48)
which has the solution
i=ikxk+i; (1.6.49)
whereikandiare constant. The rst term on the right-hand side of (1.6.49) corresponds to a
Lorentz rotation described by 6 parameters ik. The second term on the right-hand side of (1.6.49)
corresponds to a translation . A combination of two translations does not change if their order is
reversed, so translations commute:
[T;T] = 0; (1.6.50)
whereTis the generator of translation. The relations (1.6.37) and (1.6.38) mean that J andK
are spatial vectors under rotations. Spatial translations are spatial vectors under rotations, while a
time translation is a scalar:
[J ;T ] =e
T
; (1.6.51)
[J ;T0] = 0: (1.6.52)
The last relation indicates that the generators of rotations, like generators of spatial translation,
correspond to conserved quantities, which are quantities that do not change in time. The covariant
generalization of (1.6.51) and (1.6.52) is
[J;T] =T T: (1.6.53)
39
The relations (1.6.15), (1.6.50) and (1.6.53) constitute the Lie algebra of the inhomogeneous Lorentz
orPoincar e group . In particular,
[K ;T ] = T0 ; (1.6.54)
[K ;T0] = T : (1.6.55)
The last relation indicates that the generators of boosts do not correspond to conserved quantities.
For an in nitesimal rotation about the z-axis,
(1 +Jz)f(ct;x) =D(Rz())f(ct;x) =f(ct;Rz()x)f(ct;x y;x +y;z)
=f(ct;x) y@f
@x+x@f
@y; (1.6.56)
or
Jz=x@
@y y@
@x; (1.6.57)
which gives the di erential representation of rotations:
J =e
x @
: (1.6.58)
For an in nitesimal boost along the z-axis,
(1 +Kz)f(ct;x) =D(Bz())f(ct;x) =f(Bz()(ct;x))f(ct+z;y;z +ct)
=f(ct;x) +z@f
c@t+ct@f
@z; (1.6.59)
or
Kz=z@
c@t+ct@
@z; (1.6.60)
which gives the di erential representation of boosts:
K =x @
c@t+ct@
@x : (1.6.61)
The relation for an in nitesimal translation, analogous to (1.6.19), is
D(t) =I+T; (1.6.62)
so a nite translation is given by
D(t) =eT: (1.6.63)
Translation in (1.6.49) can also be written as
t()x=x+
: (1.6.64)
The relation analogous to (1.6.35) is
T=@t()
@
=0: (1.6.65)
The di erential representation of a translation is thus
T=@
@x: (1.6.66)
40
1.6.6 Invariants of Lorentz and Poincar e group
Analogously to (1.6.41),
[L2;L ] = 0; (1.6.67)
[Q2;Q ] = 0; (1.6.68)
soL2andQ2commute with all 6 generators of the Lorentz group. Consequently, J2+K2andJK
commute with all generators of the Lorentz group, that is, are the invariants or Casimir operators
of the Lorentz group. The Casimir operators of Lorentz group do not commute with the generators
of translation T, so they are not the invariants of the Poincar e group. Instead, the mass operator
m2= TT (1.6.69)
and
W2=WW; (1.6.70)
whereWis the Pauli-Luba nski pseudovector
W=1
2eJT; (1.6.71)
commute with all generators of the Poincar e group, so they are the Casimir operators of the Poincar e
group. The Pauli-Luba nski pseudovector obeys the commutation relations
[T;W] = 0; (1.6.72)
[J;W] =W W; (1.6.73)
[W;W] =eWT: (1.6.74)
The relation (1.6.73) is analogous to (1.6.53) because Wbehaves like a vector under proper Lorentz
transformations.
We de ne the four-momentum operator
P=iT; (1.6.75)
whose time component is the energy operator P0=iT0and spatial components form the momentum
operatorP =iT . We de ne the angular four-momentum operator
M=iJ; (1.6.76)
whose spatial components form the angular momentum operator
M =iJ : (1.6.77)
Therefore the following relations are satis ed:
[M;M] = i(M+M M M); (1.6.78)
[P;P] = 0; (1.6.79)
[M;P] =i(P P); (1.6.80)
m2=PP; (1.6.81)
W= 1
2MP; (1.6.82)
[P;W] = 0; (1.6.83)
[M;W] =i(W W); (1.6.84)
[W;W] = ieWP; (1.6.85)
[M ;M ] =ie
M
: (1.6.86)
41
1.6.7 Relativistic kinematics
We consider a boost in the direction of the z-axis
x0i=e K3xi; (1.6.87)
wherexiandx0ihave a form of a column (4 1 matrix), and eK3is given by (1.6.32). Therefore
the coordinates in an inertial K-system (unprimed) are related to the coordinates in an inertial
K0-system (primed) by
ct=ct0cosh+z0sinh;
x=x0; y=y0;
z=z0cosh+ct0sinh: (1.6.88)
We consider the origin of the K0-system,x0=y0=z0= 0, in the K-system. Therefore
ct=ct0cosh;
z=ct0sinh; (1.6.89)
which gives the relation between the rapidity and velocity V=dz
dtofK0relative toK:
tanh= ; (1.6.90)
where
=V
c: (1.6.91)
Accordingly, cosh =
and sinh=
, where
=
1 V2
c2 1=2
: (1.6.92)
The relations (1.6.88) become
t=
t0+V
c2z0
;
x=x0; y=y0;
z=
(z0+Vt0); (1.6.93)
and are referred to as a special Lorentz transformation in thez-direction. The reverse transformation
is
t0=
t V
c2z
;
x0=x; y0=y;
z0=
(z Vt): (1.6.94)
For a boost along an arbitrary direction, the spatial vector x= (x;y;z ) transforms such that its
component parallel to the velocity V=c ofK0relative to K,xk= (xV)V=V2(similarly for
primed), behaves like zin (1.6.93) and its component perpendicular to V,x?=x xk, behaves
likexin (1.6.93):
t=
t0+Vx0
c2
;
x?=x0
?;
xk=
(x0
k+Vt0); (1.6.95)
so
x=
(x0
k+Vt0) +x0
?=
Vt0+x0+(
1)(Vx0)V
V2: (1.6.96)
42
Therefore the transformation law for the coordinates in two inertial frames of reference is
ct
x
=
1 +(
1)
2 !ct0
x0
; (1.6.97)
or equivalentlyct0
x0
=
1 +(
1)
2 !ct
x
: (1.6.98)
The matrix in (1.6.98) is called a boost matrix . In the local Minkowski spacetime, contravariant
vectors transform like xi, according to (1.6.95) and (1.6.97),
W0
W
=
1 +(
1)
2 !
W00
W0
; (1.6.99)
covariant vectors transform such that they remain related to contravariant vectors by the Minkowski
metric tensor, and tensors transform like products of vectors. For example, if V=c ^zis parallel to
thez-axis, a tensor of rank (0,2) transforms according to
T00=
(T000+ T030) =
2(T0000+ T3000+ T0030+ 2T3030);
T0?=
(T00?0+ T30?0);
T03=
(T030+ T000) =
2(T0030+ T3030+ T0000+ 2T3000);
T??=T?0?0;
T3?=
(T30?0+ T00?0);
T33=
(T330+ T300) =
2(T3030+ T0030+ T3000+ 2T0000); (1.6.100)
where the index?denotes either 1 or 2, and the transposed components TT
ik=Tkitransform like
the transpositions of the right-hand sides in (1.6.100). If Tikis antisymmetric then T03=T0030.
The relations (1.6.93) can be written as
dt=
dt0+V
c2dz0
;
dx=dx0; dy =dy0;
dz=
(dz0+Vdt0); (1.6.101)
which gives
vx=v0
x
(1 +Vv0z=c2);
vy=v0
y
(1 +Vv0z=c2);
vz=v0
z+V
1 +Vv0z=c2; (1.6.102)
where
v=dx
dt;v0=dx0
dt0: (1.6.103)
Two special Lorentz transformations in the same direction commute because of (1.6.39). If a Lorentz
transformation from K0toKhas parameters 1and
1, and a Lorentz transformation from K00to
K0has parameters 2and
2, then a Lorentz transformation from K00toKhas parameters 3and
3such that
3= 1+ 2
1 + 1 2;
3=
1
2(1 + 1 2): (1.6.104)
43
For a boost along an arbitrary direction, (1.6.97) gives the Lorentz transformation of velocities:
v=v0+
V+ (
1)(v0V)V=V2
(1 +v0V=c2): (1.6.105)
Ifv0=jV0j=cthenv=jVj=c, in agreement with the constancy of the velocity of propagation of
interaction.
We consider two points at rest in the inertial frame of reference Kwith positions z1andz2, so
the distance between them is z=z2 z1. In the inertial frame K0, moving relative to Kin the
z-direction with velocity V,z1=
(z0
1+Vt0
1) andz2=
(z0
2+Vt0
2), so ift0
1=t0
2is the time at
which we measure (simultaneously) the positions of the two points then z=
(z0
2 z0
1) =
z0.
Therefore the length of an object in K0, whose length in the rest frame Kisl(proper length ), is
l0=l
<l; (1.6.106)
which is referred to as the Lorentz-FitzGerald contraction . The volume of an object in K0, whose
volume in the rest frame KisV(proper volume ), is
V0=V
: (1.6.107)
Suppose that there are two rods of equal lengths, moving parallel relative to each other. From the
point of view of an observer moving with the rst rod, the second one is shorter, and from the point
of view of an observer moving with the second rod, the rst one is shorter. There is no contradiction
in this statement because the positions of both ends of a rod must be measured simultaneously and
the simultaneity is not invariant: from the transformation law (1.6.93) it follows that if t= 0 then
t06= 0 and ift0= 0 thent6= 0.
We consider a clock (any mechanism with a periodic or evolutionary behavior) at rest in K0with
positionz0; the time di erence between two events with t0
1andt0
2, as measured by this clock, is
t0=t0
2 t0
1. In the frame K,t1=
(t0
1+Vz0=c2) andt2=
(t0
2+Vz0=c2), so
t=t2 t1=
t0>t0: (1.6.108)
Thus the rate of time is slower for moving clocks than those at rest ( time dilation ), in agreement
with (1.4.91) and (1.4.97), from which c2d 2=c2dt2 dl2and
d =1
dt: (1.6.109)
Suppose that there are two clocks linked to the inertial frames KandK0, and that when the clock
inKpasses by the clock in K0the readings of the two clocks coincide. From the point of view
of an observer in Kclocks inK0go more slowly, and from the point of view of an observer in K0
clocks inKgo more slowly. There is no contradiction in this statement because to compare the
rates of the two clocks in KandK0we must compare the readings of the same moving clock in
K0with di erent clocks in K; we require several clocks in one frame and one in the other, thus the
measurement process is not symmetric with respect to the two frames of reference. The clock that
goes more slowly is the one which is being compared with di erent clocks in the other frame. The
time interval measured by a clock is equal to the integral
t=1
cZ
ds (1.6.110)
along its world line. Since the world line is a straight line for a clock at rest and a curved line for a
clock moving such that it returns to the starting point, the integralR
dstaken between two world
points has its maximum value if it is taken along the straight line connecting these two points.
For a Lorentz transformation with velocity V=jVj, (1.6.105) gives
tan=v0sin0
(v0cos0+V); (1.6.111)
44
whereis the angle between vandV, and0is the angle between v0andV. Ifv=v0=cthen
cos=cos0+V
c
1 +V
ccos0; (1.6.112)
which is referred to as the aberration of a signal. Suppose an observer in frame Kmeasures a
periodic signal with period T, frequency =1
Tand wavelength =c
, propagating in the z
direction; the number of pulses in time dtisn=dt. A second observer in frame K0, moving in the
zdirection with velocity Vrelative to the rst one, travels a distance Vdtand measuresVdt
more
pulses:n0=(1 +V
c)dt. Because the time interval dtwith respect to K0isdt0=dt
, the frequency
of the signal in K0is0=
(1 +V
c) or
0=e: (1.6.113)
This dependence of the frequency of a signal on a frame of reference is referred to as the Doppler
e ect .
Whenc!1 (at which
!1) the above formulae, referring to relativistic kinematics , reduce to
their nonrelativistic limit. The Lorentz transformation (1.6.97) reduces to the Galileo transformation ,
t=t0;
x=x0+Vt0; (1.6.114)
so the time is an absolute (invariant) quantity in nonrelativistic ( Newtonian ) physics. Any two
Galileo transformations commute. The transformation law for velocities (1.6.105) reduces to the
simple addition of vectors,
v=v0+V: (1.6.115)
1.6.8 Four-acceleration
In a locally inertial frame of reference, the four-velocity is
ui=
;
v
c
; ui=
;
v
c
; (1.6.116)
where vis the velocity and
= (1 v2
c2) 1=2. We de ne the four-acceleration
wi=Dui
ds=D2xi
ds2=ukui
;k; (1.6.117)
which is orthogonal to uibecause of (1.4.19):
wiui= 0; (1.6.118)
thus having 3 independent components. In a locally inertial frame of reference, the four-acceleration
is
wi=dui
ds=d2xi
ds2=c 2
4va
c;
2a+
4(va)v
c2
; (1.6.119)
where ais the three-dimensional acceleration vector
a=dv
dt=d2x
dt2: (1.6.120)
The invariant square of the four-acceleration is thus
wiwi=
4
c4
a2+
2
c2(va)2
: (1.6.121)
Ifv= 0 at a given instant of time, the corresponding frame of reference is referred to as the
instantaneous rest frame . In this frame
wiwi= a2
c4; (1.6.122)
45
so
a0=c2p
wiwi (1.6.123)
is the absolute value of the acceleration in the instantaneous rest frame, called the proper acceleration .
References: [2, 3].
1.7 Spinors
1.7.1 Spinor representation of Lorentz group
Let
abe the coordinate-invariant 4 4Dirac matrices de ned as
a
b+
b
a= 2abI; (1.7.1)
whereIis the unit 44 matrix (4 is the lowest dimension for which (1.7.1) has solutions). Accord-
ingly, the spacetime-dependent Dirac matrices,
i=ei
a
a, satisfy
i
j+
j
i= 2gijI: (1.7.2)
Under a tetrad rotation, (1.5.8) gives
~
a= a
b
b: (1.7.3)
LetLbe a 44 matrix such that
a= a
bL
bL 1=L~
aL 1; (1.7.4)
whereL 1is the matrix inverse to L:LL 1=L 1L=I. The condition (1.7.4) represents the
constancy of the Dirac matrices
aunder the combined tetrad rotation and transformation
!
L
L 1. We refer to Las the spinor representation of the Lorentz group. The relation (1.7.4) gives
the matrix Las a function of the Lorentz matrix a
b. For an in nitesimal Lorentz transformation
(1.6.7), the solution for Lis
L=I+1
2abGab; L 1=I 1
2abGab; (1.7.5)
whereGabare the generators of the spinor representation of the Lorentz group:
Gab=1
4(
a
b
b
a): (1.7.6)
Aspinor is de ned as a quantity that, under tetrad rotations, transforms according to
~ =L : (1.7.7)
Anadjoint spinor is de ned as a quantity that transforms according to
~ = L 1; (1.7.8)
so the product is a scalar:
~ ~ = : (1.7.9)
The indices of the
aandLthat are implicit in the 4 4 matrix multiplication in (1.7.1), (1.7.2) and
(1.7.4) are spinor indices. The relation (1.7.4) implies that the Dirac matrices
acan be regarded as
quantities that have, in addition to the invariant index a, one spinor index and one adjoint-spinor
index. The product transforms like the Dirac matrices:
~ ~ =L L 1: (1.7.10)
The spinors and can be used to construct tensors. For example,
a transforms like a
contravariant Lorentz vector:
a ! L 1a
bL
bL 1L = a
b
b : (1.7.11)
46
1.7.2 Spinor connection
The derivative of a spinor does not transform like a spinor:
~ ;i=L ;i+L;i : (1.7.12)
If we introduce the spinor connection ithat transforms according to
~ i=L iL 1+L;iL 1; (1.7.13)
then a covariant derivative of a spinor,
;i= ;i i ; (1.7.14)
is a spinor:
~ ;i=~ ;i ~ i~ =L ;i+L;i (L iL 1+L;iL 1)L =L ;i: (1.7.15)
Because is a scalar,
( );i= ( );i; (1.7.16)
the chain rule for covariant di erentiation gives a covariant derivative of an adjoint spinor
;i= ;i+ i: (1.7.17)
Also
ji= ;i; ji= ;i: (1.7.18)
The Dirac matrices
atransform like , whose covariant derivative is
( );i= ;i + ;i= ( );i i + i= ( );i [ i; ]: (1.7.19)
Therefore a covariant derivative of the Dirac matrices is
a
;i=
a
;i [ i;
a] = [ i;
a]; (1.7.20)
so
j
;i=
j
ji=
j
;i+ j
k;i
k [ i;
j]: (1.7.21)
Accordingly
a
ji=!a
bi
b [ i;
a]: (1.7.22)
The quantity
i jitransforms under Lorentz rotations like a scalar:
i ji! L 1L
iL 1L ji=
i ji: (1.7.23)
The relation abji= 0 implies that
a
ji= 0; (1.7.24)
because the Dirac matrices
aonly depend on ab. Multiplying both sides of (1.7.22) by
afrom
the left gives
!abi
a
b
a i
a+ 4 i= 0: (1.7.25)
We seek the solution of (1.7.25) in the form
i= 1
4!abi
a
b Ai; (1.7.26)
whereAiis a spinor-tensor quantity with one vector index. Substituting (1.7.26) to (1.7.25), together
with the identity
c
a
b
c= 4ab, gives
aAi
a+ 4Ai= 0; (1.7.27)
soAiis an arbitrary vector multiple of I. Therefore the spinor connection iis given, up to the
addition of an arbitrary vector multiple of I, by the Fock-Ivanenko coecients :
i= 1
4!abi
a
b= 1
2!abiGab: (1.7.28)
Using the de nition (1.5.15), we can also write (1.7.28) as
i= 1
8ej
c;i[
j;
c] =1
8[
j
;i;
j]: (1.7.29)
47
1.7.3 Curvature spinor
The commutator of total covariant derivatives of a spinor is
jji jij= ( jj);i i jj k
ji jk ( ji);j+ j ji+ k
ij jk
= j;i + i j + i;j j i + 2Sk
ij jk=Kij + 2Sk
ij jk; (1.7.30)
whereKij= Kjiis de ned as
Kij= i;j j;i+ [ i; j]: (1.7.31)
Substituting (1.7.13) to (1.7.31) gives
~Kij=~ i;j ~ j;i+ [~ i;~ j] =L( i;j j;i+ [ i; j])L 1=LKijL 1; (1.7.32)
soKijtransforms under tetrad rotations like the Dirac matrices
a, that is,Kijis a spinor with one
spinor index and one adjoint-spinor index. We refer to Kijas the curvature spinor .
The relation (1.7.24) leads to
k
ji= 0: (1.7.33)
Thus the commutator of covariant derivatives of the spacetime-dependent Dirac matrices vanishes:
2
k
j[ji]=Rk
lij
l+ 2Sl
ij
k
jl+ [Kij;
k] =Rk
lij
l+ [Kij;
k] = 0: (1.7.34)
Multiplying both sides of (1.7.34) by
kfrom the left gives
Rklij
k
l+
kKij
k 4Kij= 0: (1.7.35)
We seek the solution of (1.7.35) in the form
Kij=1
4Rklij
k
l+Bij; (1.7.36)
whereBijis a spinor-tensor quantity with two vector indices. Substituting (1.7.36) to (1.7.35) gives
kBij
k 4Bij= 0; (1.7.37)
soBijis an antisymmetric-tensor multiple of I. The tensor Bijis related to the vector Aiin (1.7.26)
by
Bij=Aj;i Ai;j+ [Ai;Aj]: (1.7.38)
Because has no indices other than spinor indices, Aiis a vector and [ Ai;Aj] = 0. The invariance
of (1.7.35) under the addition of an antisymmetric-tensor multiple Bijof the unit matrix to the
curvature spinor is related to the invariance of (1.7.25) under the addition of a vector multiple Aiof
the unit matrix to the spinor connection. Setting Ai= 0, which corresponds to the Fock-Ivanenko
spinor connection, gives Bij= 0. Therefore the curvature spinor Kijis given, up to the addition of
an arbitrary antisymmetric-tensor multiple of I, by
Kij=1
4Rklij
k
l=1
2RklijGkl: (1.7.39)
References: [3, 4].
48
2 Fields
2.1 Principle of least action
The most general formulation of the law that governs the dynamics of classical systems is Hamilton's
principle of least action , according to which every classical system is characterized by a de nite
scalar-density function L, and the dynamics of the system is such that a certain condition is satis ed.
Let A(xi) be a set of physical elds , being di erentiable functions of the coordinates, and let Lbe
a Lorentz covariant quantity constructed from the Aand their derivatives. We consider a scalar
quantity
S=1
cZ
Ld
; (2.1.1)
where the integration is over some region in locally Minkowski spacetime. Let Abe arbitrary
small changes in A(regarded as a dynamical variable) over the region of integration, which vanish
on the boundary. Then the change in Scan be written as
S=1
cZ
FA Ad
: (2.1.2)
The principle of least action states that the dynamics of a physical system is given by the condition
the scalarSbe a local minimum. Therefore any in nitesimal change in the dynamics of the system
does not alter the value of S:
S= 0 (2.1.3)
(Sis a local extremum). If Lis covariant and Atransform covariantly under the Lorentz group,
the variational condition (2.1.3) gives the Lorentz covariant equations
A= 0: (2.1.4)
These equations are also invariant for any other transformations (internal symmetries) for which
Lis invariant. Lis referred to as the Lagrangian density ,Sis the action functional, S= 0 is
the principle of least action, and FA= 0 are the eld equations . The eld equations of a physical
system are the result of the action being a local extremum. The condition that the action be a local
minimum imposes additional restrictions on possible choices for S. The number of independent eld
equations for a given system is referred to as the number of the degrees of freedom representing this
system.
In most cases Lcontains only Aand their rst derivatives (the Lagrangian density for the
gravitational eld contains second derivatives). A Lagrangian density containing higher derivatives
can always be written in terms of rst derivatives by increasing the number of the components of
A. We consider a physical system in the galilean frame of reference. If Ldepends only on Aand
@i A,L=L( ; ;i), then
S=1
cZ@L
@ +@L
@( ;i)( ;i)
d
=1
cZ@L
@ +@L
@( ;i)( );i
d
=1
cZ@L
@ @i@L
@( ;i)
+@i@L
@( ;i)
d
: (2.1.5)
The last term in the second line of (2.1.5) is a divergence, which, after integration, can be transformed
into a hypersurface integral over the boundary of integration region, where = 0 on the boundary,
so this term does not contribute to the action variation:
S=1
cZ@L
@ @i@L
@( ;i)
d
+Z@L
@( ;i) dSi=1
cZ@L
@ @i@L
@( ;i)
d
:(2.1.6)
IfS= 0 for arbitrary variations that vanish on the boundary then
@L
@ @i@L
@( ;i)
= 0; (2.1.7)
49
orL
= 0; (2.1.8)
where
L
=@L
@ @i@L
@( ;i)
(2.1.9)
is avariational derivative ofLwith respect to . This set of equations, for each component A, is
referred to as the Lagrange equations . Generalizing the Lagrange equations to an arbitrary coordinate
frame gives
@L
@ rfg
i@L
@( ;i)
= 0: (2.1.10)
There is some arbitrariness in the choice of L; adding to it the divergence of an arbitrary vector
density or multiplying it by a constant produces the same eld equations. If a system consists
of two noninteracting parts AandB, with corresponding Lagrangian densitites LA( A;@ A) and
LB( B;@ B), then the Lagrangian for this system is the sum LA+LB. This additivity of the
Lagrangian density means that the eld equations for either of the two parts do not involve quantities
pertaining to the other part. If LAalso depends on Band/or@ B, and/or LBdepends on A
and/or@ A, then the subsystems AandBinteract.
References: [1, 2, 3].
2.2 Action for gravitational eld
We consider a Lagrangian density that depends on the ane (or spin) connection and its rst
derivatives. Such Lagrangian density can be decomposed into the covariant part that contains
derivatives of the ane/spin connection, which is referred to as the Lagrangian density for the
gravitational eld , and the covariant part that does not contain these derivatives, which is referred
to as the Lagrangian density for matter . The simplest covariant scalar that can be constructed
from the ane/spin connection and its rst derivatives is the Ricci scalar R. The corresponding
Lagrangian density for the gravitational eld is proportional to the product of Rand the scalar
densityp g:
Lg= 1
2p gR; (2.2.1)
whereisEinstein's gravitational constant . There exist two variational principles in the theory
of the gravitational eld. The metric variational principle regards the metric tensor or tetrad as a
dynamical variable and assumes the ane connection to be the Levi-Civita connection. The metric-
ane variational principle regards both the metric tensor (or tetrad) and the metric-compatible
ane connection (or spin connection) as dynamical variables.
In the metric variational formulation, the Lagrangian density for the gravitational eld is pro-
portional to the Riemann scalar P:
Lg= 1
2p gP: (2.2.2)
BecausePis linear in derivatives of fi
klg:
p gP=p ggik(fl
ikg;l fl
ilg;k+fm
ikgfl
mlg fm
ilgfl
mkg)
= (p ggikfl
ikg);l fl
ikg(p ggik);l (p ggikfl
ilg);k+fl
ilg(p ggik);k
+p ggik(fm
ikgfl
mlg fm
ilgfl
mkg); (2.2.3)
we can subtract fromp gPtotal derivatives without altering the eld equations, replacing Pby a
50
noncovariant quantity G:
p gG=fl
ilg(p ggik);k fl
ikg(p ggik);l+p ggik(fm
ikgfl
mlg fm
ilgfl
mkg)
=fl
ilg
(p ggik):k+fj
jkgp ggik p gfi
jkggjk p gfk
jkggij
fl
ikg
(p ggik):l+fj
jlgp ggik p gfi
jlggjk p gfk
jlggij
+p ggik(fm
ikgfl
mlg fm
ilgfl
mkg) =fl
ilg
fj
jkgp ggik p gfi
jkggjk
p gfk
jkggij
fl
ikg
fj
jlgp ggik p gfi
jlggjk p gfk
jlggij
+p ggik(fm
ikgfl
mlg fm
ilgfl
mkg) =p ggik(fm
ilgfl
mkg fm
ikgfl
mlg): (2.2.4)
Therefore
G=gik(fm
ilgfl
mkg fm
ikgfl
mlg); (2.2.5)
and the Lagrangian density for the gravitational eld is
Lg= 1
2p gG: (2.2.6)
.
Any coordinate transformation results in variations of gik, soSis not necessarily a minimum
with respect to these variations (only an extremum) because not all gikcorrespond to actual
variations of the gravitational eld. In order to exclude the variations gikresulting from changing
the coordinates, we must impose on the metric tensor 4 arbitrary constraints. If we choose
g0 = 0;jg j= const; (2.2.7)
then Gbecomes
G= 1
4g00g g
g
;0g ;0: (2.2.8)
In the locally galilean frame g = , so
G= 1
4g00(g ;0)2: (2.2.9)
For physical systems g00>0. Therefore in order for Sto have a minimum, must be positive,
otherwise an arbitrarily rapid change of g in time would result in an arbitrarily low value of S
and there would be no minimum.
References: [2, 3].
2.3 Matter
2.3.1 Metric dynamical energy-momentum density
The variation of the matter action Sm=R
Lmd
with respect to the metric tensor,
Sm=1
2cZ
Tijgijd
= 1
2cZ
Tijgijd
; (2.3.1)
de nes the metric dynamical energy-momentum density Tij, which is symmetric:
Tij=Tji: (2.3.2)
Equivalently
Tij= 2Lm
gij= 2@Lm
@gij @k(@Lm
@gij
: (2.3.3)
The metric dynamical energy-momentum tensor Tijis de ned as
Tij=Tijp g: (2.3.4)
51
2.3.2 Tetrad dynamical energy-momentum density
The variation of the matter action Smwith respect to the tetrad,
Sm=1
cZ
Ta
iei
ad
; (2.3.5)
de nes the tetrad dynamical energy-momentum density Ta
i. Equivalently
Lm=Ta
iei
a (2.3.6)
or
Ta
i=Lm
eia: (2.3.7)
IfLmdepends only on tensor matter elds expressed in terms of the coordinate indices and it
depends on neither derivatives of the metric tensor nor derivatives of the tetrad then the tetrad
enters Lmonly through the metric tensor, in a combination gij=abei
aej
b. Thus
ei
a=1
2eajgij: (2.3.8)
Substituting (2.3.8) to (2.3.5) gives
Sm=1
2cZ
Tijgijd
; (2.3.9)
where
Tij=eajTa
i: (2.3.10)
The tensor Tijis generally not symmetric. Comparing (2.3.9) with (2.3.1) gives the relation be-
tween the tetrad dynamical energy-momentum density and the metric dynamical energy-momentum
density for tensor matter elds:
T(ij)=Tij: (2.3.11)
2.3.3 Canonical energy-momentum density
If we express the matter Lagrangian density Lm, depending on matter elds and their rst deriva-
tives ;i, only in terms of Lorentz and spinor indices, then the tetrad appears in Lmonly through a
derivative of , in a covariant combination ei
a ji. Since Lm=eL, whereLis a scalar, we obtain
Lm=eL eea
iLei
a=e@L
@ ja jiei
a Lmea
iei
a=@Lm
@ ja ji Lmea
i
ei
a
=@Lm
@ ;a ji Lmea
i
ei
a: (2.3.12)
The last term in (2.3.12),
a
i=@Lm
@ ;a ji ea
iLm; (2.3.13)
is referred to as the canonical energy-momentum density . Accordingly
i
j=@Lm
@ ;i jj i
jLm: (2.3.14)
Comparing (2.3.12) with (2.3.6) shows that the canonical energy-momentum density is identical with
the dynamical tetrad energy-momentum density:
a
i=Ta
i: (2.3.15)
52
2.3.4 Spin density
The variation of the matter action Smwith respect to the spin connection,
Sm=1
2cZ
Si
ab!ab
id
; (2.3.16)
de nes the dynamical spin density Si
ab:
Si
ab= 2Lm
!ab
i; (2.3.17)
which is antisymmetric in the Lorentz indices:
Si
ab= Si
ba: (2.3.18)
In the metric variational formulation of gravity, the variations !ab
i=!fgab
iare functions of
the variations ei
aand their derivatives, so the spin density is a function of the energy-momentum
density. In the metric-ane variational formulation of gravity, the variations !ab
iare independent
ofei
aand their derivatives. The relation (1.5.29) indicates that the spin density is generated by the
contortion tensor:
Sk
ij= 2Lm
Cij
k: (2.3.19)
Accordingly, the variation of Lmwith respect to the torsion tensor,
jk
i= 2Lm
Si
jk; (2.3.20)
is a homogeneous linear function of the spin connection because of (1.4.29):
ijk= 2Lm
Sijk= 2Lm
Clmn@Clmn
@Sijk=Slmn(l
im
[jn
k]+m
in
[jl
k]+n
im
[jl
k])
=Sijk Sjki+Skij; (2.3.21)
Sijk= [ij]k; (2.3.22)
antisymmetric in the last two indices:
ijk= ikj: (2.3.23)
The variation of Lmwith respect to the metric-compatible ane connection in the metric-ane
variational formulation of gravity is equivalent to the variation with respect to the torsion (or con-
tortion) tensor. The spin connection !ab
ienters Lmonly through derivatives of , in a combination
@L
@ ;i i , where iis the covariant derivative acting on :
i= 1
2!abiGab: (2.3.24)
Consequently, the dynamical spin density Si
abis identical with
i
ab=@Lm
@ ;iGab ; (2.3.25)
referred to as the canonical spin density .Spin tensor is de ned as
sijk=Sijkp g: (2.3.26)
53
2.3.5 Belinfante-Rosenfeld relation
The total variation of the matter action with respect to geometrical variables is either
Sm=1
cZ
d
Ta
iei
a+1
2cZ
d
Si
ab!ab
i (2.3.27)
or
Sm=1
2cZ
d
Tikgik+1
2cZ
d
ik
jSj
ik: (2.3.28)
Equation (1.5.25) gives
1
2Z
d
ik
jSj
ik=1
2Z
d
ik
j
(ej
aeib!ab
k) +ea
i;kej
a+ea
i;kej
a
=1
2Z
d
li
j(ej
aelb)!ab
i+ i
ab!ab
i+ ( ik
jej
aea
i);k ( ik
jej
a);kea
i+ ik
jea
i;kej
a
=1
2Z
d
lk
j!cb
kelbej
c+ li
j!ab
iej
aelb+ i
ab!ab
i ( ik
jej
a);kea
i+ lm
jeb
l;mej
b
+1
2Z
dSk ik
jej
aea
i=1
2Z
d
lk
j!cb
kelbei
cej
aea
i+ il
j!b
alej
bea
i+ i
ab!ab
i
( ik
ajk Si
jk jk
a 2Sj ij
a+!b
ak ik
b)ea
i lm
jeb
l;mei
bej
aea
i
=1
2Z
d
i
ab!ab
i ik
j;kej
aea
i+ 2Sj ij
aea
i
; (2.3.29)
so comparing of (2.3.27) with (2.3.28) leads to
Z
d
Ta
iei
a+1
2Z
d
Si
ab!ab
i=1
2Z
d
Tikgik+1
2Z
d
i
ab!ab
i ik
j;kej
aea
i
+2Sj ij
aea
i
=Z
d
Tikekaei
a+1
2Z
d
i
ab!ab
i+1
2Z
d
jk
i;kea
jei
a
Z
d
Sj kj
bea
keb
iei
a: (2.3.30)
The terms with !ab
igive (2.3.22), while the terms with ei
agive
Ta
i=Tikeka+1
2 jk
i;kea
j Sj aj
i (2.3.31)
or
Tik=Tik 1
2rj(Sj
ik Sj
k i+Sj
ik) +Sj(Sj
ik Sj
k i+Sj
ik): (2.3.32)
Equation (2.3.32) is referred to as the Belinfante-Rosenfeld relation between the dynamical metric
and dynamical tetrad (canonical) energy-momentum densites. In the absence of torsion, (2.3.32) is
consistent with (2.3.11). The Belinfante-Rosenfeld relation can be written as
Tik=1p gTik 1
2r
j(sj
ik sj
k i+sj
ik); (2.3.33)
where
r
i=ri 2Si (2.3.34)
is the modi ed covariant derivative .
References: [2, 3, 4, 6, 8]
54
2.4 Symmetries and conservation laws
2.4.1 Noether theorem
We consider a physical system in the galilean frame of reference, described by the Lagrangian density
Lthat depends on matter elds A, their rst derivatives ;i, and the coordinates xi. The change
of the Lagrangian density Lunder an in nitesimal coordinate transformation (1.2.55) is thus
L=@L
@ +@L
@ ;i( ;i) +@L
@xii; (2.4.1)
where the changes and( ;i) are brought by the transformation (1.2.55) and @denotes partial
di erentiation with respect to xiat constant and ;i. The variation Lunder this transformation
is also given by (1.2.64):
L=i
;iL: (2.4.2)
Using the Lagrange equations (2.1.7) and the identities
L;i=@L
@xi+@L
@ ;i+@L
@ ;j ;ji; (2.4.3)
( ;i) = ( );i j
;i ;j; (2.4.4)
we bring (2.4.1) to
L=iL;i+@L
@ ;i( j ;j)
;i: (2.4.5)
Combining (2.4.2) and (2.4.5) gives the conservation law,
Ji
;i= 0; (2.4.6)
for the current
Ji=iL+@L
@ ;i( j ;j) =iL+@L
@ ;i =@L
@ ;i i
jj: (2.4.7)
Equations (2.4.6) and (2.4.7) represent the Noether theorem , which states that to each continuous
symmetry of a Lagrangian density there corresponds a conservation law. Generalizing (2.4.6) to an
arbitrary coordinate frame gives
Ji
:i= 0: (2.4.8)
2.4.2 Conservation of spin
The Lorentz group is the group of tetrad rotations. Since a physical matter Lagrangian density
Lm( ; ;i) is invariant under local, proper Lorentz transformations, it is invariant under tetrad
rotations:
Lm=@Lm
@ +@Lm
@ ;i( ;i) +Ta
iei
a+1
2Si
ab!ab
i= 0; (2.4.9)
where the changes correspond to a tetrad rotation. Under integration of (2.4.9) over spacetime,
the rst two terms vanish because of the Lagrange equations for (2.1.7):
Z
Ta
iei
a+1
2Si
ab!ab
i
d4x= 0: (2.4.10)
For an in nitesimal Lorentz transformation (1.6.7), the tetrad ea
ichanges by
ea
i= ~ea
i ea
i= a
beb
i ea
i=a
i; (2.4.11)
and the tetrad ei
a, because of the identity (ea
iej
a) = 0, according to
ei
a= i
a: (2.4.12)
55
The spin connection changes by
!ab
i=(ea
j!jb
i) =a
j!jb
i ea
jjb
;i=a
c!cb
i ea
jjb
ji+a
c!bc
i= ab
ji: (2.4.13)
Substituting (2.4.12) and (2.4.13) to (2.4.10), together with partial integration (1.2.34), gives
Z
Ta
ii
a+1
2Si
abab
ji
d4x= Z
Tijij+1
2Sk
ijij
jk
d4x
=Z
T[ij] SkSk
ij+1
2Sk
ij;k
ijd4x: (2.4.14)
Since the in nitesimal Lorentz rotation ijis arbitrary, we obtain the covariant conservation law for
the spin density :
Sk
ij;k=Tij Tji+ 2SkSk
ij (2.4.15)
or
r
ksk
ij=1p g(Tij Tji): (2.4.16)
The conservation law for the spin density (2.4.16) also results from antisymmetrizing the Belinfante-
Rosenfeld relation (2.3.33) with respect to the indices i;k. If we use the metric-compatible ane
connection k
ij, which is invariant under tetrad rotations, instead of the spin connection !ab
ias a
variable in Lmthen we must replace the term with !ab
iin (2.4.9) by a term with (ei
a;j).
2.4.3 Conservation of metric energy-momentum
We consider the metric variational formulation of gravity. Under an in nitesimal coordinate trans-
formation (1.2.55), the matter Lagrangian density Lm( ; ;i) changes according to
Lm=@Lm
@ +@Lm
@ ;i( ;i) +@Lm
@gikgik+@Lm
@gik
;l(gik
;l): (2.4.17)
The matter action Sm=1
cR
Lm( ; ;i)d
is a scalar, so it does not change under this transformation:
Sm=1
cZ@Lm
@ +@Lm
@ ;i( ;i) +@Lm
@gikgik+@Lm
@gik
;l(gik
;l)
d
= 0: (2.4.18)
The rst two terms in (2.4.18) vanish because of the Lagrange equations for (2.1.7), so
Sm=1
cZ@Lm
@gik @l@Lm
@gik
;l
gikd
=1
cZLm
gikgikd
=1
2cZ
Tijgijd
= 0: (2.4.19)
If the components of the metric tensor change because of an in nitesimal coordinate transformation
(1.2.55) then the corresponding variation of the metric tensor is given by (1.4.41):
gij=gij= 2(i:j); (2.4.20)
so
Sm=Sm= 1
2cZ
Tijgijd
= 1
cZ
Tij(i:j)d
= 1
cZ
Tiji:jd
= 1
cZ
(Tiji):jd
+1
cZ
Tij
:jid
= 1
cZ
(Tiji);jd
+1
cZ
Tij
:jid
= 1
cZ
TijidSj+1
cZ
Tij
:jid
= 0: (2.4.21)
If the variation of the coordinates ivanishes on the boundary of the region of integration then
Z
Tij
:jid
= 0; (2.4.22)
56
which, for arbitrary variations igives the covariant conservation of the metric energy-momentum
density (4 equations):
Tij
:j= 0: (2.4.23)
Equivalently
Tij
:j= 0: (2.4.24)
Note that vanishing ofR
Tijgijd
in (2.4.21) does not imply Tij= 0, because 10 variations gij
are functions of 4 variations iand thus not independent.
2.4.4 Conservation of tetrad energy-momentum
The matter Lagrangian density Lmis invariant under in nitesimal translations of the coordinate
system (1.2.55). The corresponding changes of the tetrad and spin connection are given by Lie
derivatives
ei
a= Lei
a=i
;jej
a jei
a;j; (2.4.25)
!ab
i= L!ab
i= j
;i!ab
j j!ab
i;j: (2.4.26)
Equation (2.4.10) becomes now
Z
Ta
iei
a+1
2Si
ab!ab
i
d4x= 0: (2.4.27)
Substituting (2.4.25) and (2.4.26) into (2.4.27) gives
Z
Ta
ii
;jej
a Ta
ijei
a;j 1
2Si
abj
;i!ab
j 1
2Si
abj!ab
i;j
d4x
=Z
Tj
i ;j Ta
jej
a;i+1
2(Sj
ab!ab
i);j 1
2Sj
ab!ab
j;i
id4x= 0: (2.4.28)
This equation holds for an arbitrary vector i, so we obtain
Sj
ab ;j!ab
i+Sj
ab(!ab
i;j !ab
j;i) 2Tj
i ;j 2Ta
jej
a;i
= (Sj
abjj 2SkSk
ab+Sj
cb!c
aj+Sj
ac!c
bj)!ab
i 2Tj
i ;j 2Ta
jej
a;i
+Sj
ab( Rab
ij+!a
ci!cb
j !a
cj!cb
i) = 0; (2.4.29)
which reduces to
(Sj
abjj 2SkSk
ab)!ab
i Rab
ijSj
ab 2Tj
i;j+ 4SjTj
i 2Tjk!jk
i+ 4Sjk
iTjk
= (Sk
jl;k 2SkSk
jl)!jl
i Rkl
ijSj
kl 2Tj
i;j+ 4SjTj
i 2Tjk!jk
i
+4Sjk
iTjk= 0: (2.4.30)
The conservation law for the spin density (2.4.15) brings (2.4.30) to the covariant conservation law
for the energy-momentum density :
Tj
i;j= 2SjTj
i+ 2Sj
kiTk
j+1
2Sj
klRkl
ji (2.4.31)
or
Tij
:j=Ci
jkTjk+1
2SkljRklji: (2.4.32)
57
2.4.5 Conservation laws for Lorentz group
We consider a matter Lagrangian Lmfor a physical system in the galilean and geodesic frame
of reference, that depends on the coordinates only through a eld and its rst derivatives ;i.
Therefore
@iLm=Lm
@ ;i+@Lm
@ ;j ;ji=@j@Lm
@ ;j
;i+@Lm
@ ;j ;ji=@jLm
@ ;j ;i
; (2.4.33)
where we use the Lagrange equations (2.1.7), from which we obtain the conservation law,
j
i ;j= 0; (2.4.34)
for
j
i=@Lm
@ ;j ;i j
iLm: (2.4.35)
The conservation law (2.4.34) is a special case of (2.4.31) in the absence of torsion and spin, expressed
in the Galilean and geodesic frame. The quantity (2.4.35) is a special case of the canonical energy-
momentum density (2.3.14) in the absence of torsion and spin, expressed in the galilean and geodesic
frame.
Ifxiare Cartesian coordinates then for translations, i=i=const and = 0, the current
(2.4.7) is
Ji=iLm @Lm
@ ;ij ;j: (2.4.36)
The conservation law (2.4.6) gives
ji
j ;i= 0; (2.4.37)
which gives (2.4.34) because iare arbitrary. For Lorentz rotations, i=i
jxjand =1
2ijGij ,
whereGijare the generators of the Lorentz group, the current (2.4.7) is
Ji=ijxjLm+@Lm
@ ;i1
2klGkl jkxkj ;j
=kl
xk@Lm
@ ;i ;l xki
lLm+1
2@Lm
@ ;iGkl
:(2.4.38)
The conservation law (2.4.6) gives
kl@Lm
@ ;i ;[lxk] i
[lxk]Lm+1
2@Lm
@ ;iGkl
;i; (2.4.39)
which, because klare arbitrary, gives
Mi
kl ;i= 0; (2.4.40)
where
Mi
kl=xki
l xli
k+@Lm
@ ;iGkl : (2.4.41)
The quantity Mi
klis referred to as the angular momentum density , and is the sum,
Mi
kl= i
kl+ i
kl; (2.4.42)
of two densities: the orbital angular momentum density ,
i
kl=xki
l xli
k; (2.4.43)
and the canonical spin density (2.3.25).
The conservation law (2.4.40) for the angular momentum density is equivalent to
kl lk i
kl ;i= 0; (2.4.44)
58
which is a special case of the conservation law for the spin density (2.4.15) in the absence of torsion,
expressed in the galilean and geodesic frame. The canonical energy-momentum density ikis not
symmetric. However, the quantity
ik=ik 1
2@j(j
ik j
k i+ j
ik); (2.4.45)
is symmetric, which follows from (2.4.44), and conserved:
ik= ki; (2.4.46)
i
k ;i= 0: (2.4.47)
The symmetric energy-momentum density ikcorresponds to the metric dynamical energy-momentum
density (2.3.3), expressed in the galilean and geodesic frame. Equation (2.4.45) is a special case of
the Belinfante-Rosenfeld relation (2.3.32) in the absence of torsion, expressed in the Galilean and
geodesic frame. The second term on the right-hand side of (2.4.45) has the form @j ikj, where
ikj= ijk. Adding such term to ikpreserves the conservation law (2.4.34) and brings ikto a
symmetric form.
2.4.6 Components of energy-momentum tensor
Integrating the conservation law (2.4.34), valid in the galilean and geodesic frame of reference, over
a hypersurface enclosing matter represented by ikand using the Gau-Stokes theorem gives
I
ikdSk= 0; (2.4.48)
which gives the conservation of the four-momentum vector
Pi=1
cZ
ikdSk= const: (2.4.49)
Choosing the volume hypersurface dV=dS0gives
Pi=1
cZ
i0dV; (2.4.50)
so the components1
c i0form the four-momentum density . The component 00, referred to as the
energy density ,
W= 00=_ @Lm
@_ Lm; (2.4.51)
integrated over the volume gives the time component of the four-momentum, the energy
E=cP0; cP0=Z
00dV=_ @L
@_ L; (2.4.52)
where
L=Z
LmdV (2.4.53)
is the Lagrange function or Lagrangian . Hereinafter, a dot above any quantity denotes the partial
derivative of with respect to time, _ =d
dt, and two dots above denote the second derivative of
with respect to time, =d2
dt2. Consequently, the action of a physical system is the time integral
of the Lagrangian,
S=Z
Ldt: (2.4.54)
The components1
c 0, referred to as the momentum density , integrated over the volume give the
spatial components of the four-momentum, the momentum vector
P:P =1
cZ
0dV: (2.4.55)
59
Adding a total divergence @j ikjto ikdoes not alter the de nition of the four-momentum vector
(2.4.49).
The symmetry of ikcan be written as
@l(xi kl xk il) = 0; (2.4.56)
which upon the integration over a hypersurface enclosing matter represented by ikand using the
Gau-Stokes theorem gives I
(xi kl xk il)dSl= 0; (2.4.57)
which gives the conservation of the angular momentum tensor
Mik=Z
(xidPk xkdPi) =1
cZ
(xi kl xk il)dSl= const: (2.4.58)
Choosing the volume hypersurface dV=dS0gives
Mik=1
cZ
(xi k0 xk i0)dV: (2.4.59)
The conservation of M0 ,
M 0=1
cZ
x 00dV x0Z
0dV
=1
cZ
x 00dV ctP = const; (2.4.60)
divided by the conservation of P0(2.4.50),P0= const, gives
X =V t+ const; (2.4.61)
where
V =cP
P0(2.4.62)
and
X =R
x 00dVR
00dV: (2.4.63)
The relation (2.4.61) describes a uniform motion of the center of inertia , whose coordinates are X ,
with velocity V . The coordinates of the center of inertia (2.4.63) are not the spatial components
of a four-dimensional vector.
The conservation law (2.4.34) can be written as
1
c@ 00
@t+@ 0
@x = 0; (2.4.64)
1
c@ 0
@t+@
@x = 0: (2.4.65)
Integrating these equations over the volume hypersurface and using the Gau-Stokes theorem gives
@
@tZ
00dV= cI
0 df ; (2.4.66)
@
@tZ1
c 0dV= I
df ; (2.4.67)
where
df =df?
0 (2.4.68)
is the spatial surface element (1.1.30). The integral of a three-dimensional vector V over the two-
dimensional surface element df ,H
V df , is referred to as the
uxof this vector. Therefore the
components
S =c 0 (2.4.69)
60
of the energy current Sform, upon integrating over df , the energy
ux . The components
represent the momentum current and give, upon integrating over df , the momentum
ux . The
stress tensor is de ned as
= : (2.4.70)
The components of the energy-momentum tensor form the matrix
ik=
WS
cS
c
: (2.4.71)
We de ne the spatial surface force vector,
F =I
df : (2.4.72)
The relations (2.4.55), (2.4.67), (2.4.70) and (2.4.72) equal the time derivative of the momentum P
to the surface force F ,
_P =F : (2.4.73)
In an arbitrary frame of reference, the metric dynamical energy-momentum density Tikdescribing
isotropic spinless matter (without a preferred direction in its rest frame) can be decomposed into
the part proportional to uiuk, as in a system of particles (2.4.194), the part proportional to the
projection tensor ,
hik=gik uiuk; (2.4.74)
which is orthogonal to ui,
hikuk= 0; (2.4.75)
and parts containing covariant derivatives of ui. The projection tensor satis es
hj
ihk
j=hk
i: (2.4.76)
We assume thatTikdoes not depend on derivatives of ui. Therefore
Tik=uiuk phik= (+p)uiuk pgik; (2.4.77)
where a scalar is equal to the energy density Win the locally Galilean rest frame and a scalar pis
thepressure . In this frame Tik= diag(;p;p;p ) and the stress tensor = p , so (2.4.72) gives
F = pI
df = pI
n df; (2.4.78)
which states that the force per unit surface dfacting on a surface is parallel, with the opposite sign,
to the outward normal vector of this surface n ,dF
df= pn , and which is referred to as Pascal's
law. The scalars andpcan also be written as
=Tikuiuk; (2.4.79)
p= 1
3Tikhik: (2.4.80)
Matter described by the tensor (2.4.77) represents an ideal
uid . The relation (2.4.77) can be written
as
Tik= (ci+pui)uk pgik; (2.4.81)
where
i=1
cTikuk=
cui (2.4.82)
is equal to the four-momentum density in the locally Galilean rest frame (cf. (2.4.50)):
i=1
cTi0: (2.4.83)
61
We also have
iui=
c: (2.4.84)
The relation between andpis referred to as the equation of state .
In the Galilean frame of reference, combining (1.6.116), (2.4.71) and (2.4.77) gives
W=+pv2=c2
1 v2=c2; (2.4.85)
S=(+p)v
1 v2=c2; (2.4.86)
= (+p)v v
c2 v2 p : (2.4.87)
The relation (2.4.77) gives
T=Ti
i= 3p: (2.4.88)
The component T00=u2
0+p(u2
0 g00) is, usingu0=g00dx0+g0 dx
ds, (1.4.97) and (1.4.98), equal to
T00=u2
0+pg00dl
ds2
; (2.4.89)
so it is positive under physical conditions >0,p>0 andg00>0. IfTikdepends also on derivatives
ofuithen matter described by the tensor (2.4.77) with the corresponding additional terms represents
aviscous
uid .
In an arbitrary frame of reference, the tetrad dynamical energy-momentum density Tikdescribing
isotropic matter with spin cannot be decomposed, because of its asymmetry, as in (2.4.77). However,
it can be decomposed as in (2.4.81):
Tik=p g
(cpi+pui)uk pgik
; (2.4.90)
where
pi=1
cp gTikuk(2.4.91)
is the corresponding four-momentum density in the locally Galilean rest frame. The conservation
law for the spin density (2.4.16) gives
c(piuj pjui) =r
ksk
ij: (2.4.92)
De ning
=cpiui(2.4.93)
and multiplying (2.4.92) by ujgives
pi=
cui+1
cr
ksk
ijuj: (2.4.94)
Thus we obtain
Tij=p g(uiuj phij+r
ksk
iluluj); (2.4.95)
which, using the Belinfante-Rosenfeld relation (2.3.33), gives the metric dynamical energy-momentum
tensor for isotropic matter with spin:
Tij=uiuj phij+r
ksk
iluluj 1
2r
k(sk
ij+ 2sk
(ij)): (2.4.96)
Substituting (2.4.94) into (2.4.92) gives the dynamical equation for the spin tensor:
r
ksk
ij r
ksk
iluluj+r
ksk
jlului= 0: (2.4.97)
62
Ifsijk= 0 then (2.4.94) gives pi/ui. Multiplying (2.4.97) by ujgives the identity, so any 3
components of (2.4.97) are linear combinations of the other components. Thus we can impose 3
constraints on sk
ij.
The spin density Sijkdescribing isotropic matter with spin can be decomposed as (cf. (2.4.180))
Sijk=p gsijuk; (2.4.98)
where
sij=1p gSijkuk: (2.4.99)
The 3 constraints on sijk, which are now the 3 constraints on sij, can be taken as sijuj= 0 (cf.
(2.4.182)). Substituting (2.4.90) and (2.4.98) into the conservation law for the energy-momentum
density (2.4.32) gives
(cpiuj):j+ (puiuj pgij):j=Ci
jkcpjuk+1
2sklujRklji: (2.4.100)
Ifp= 0 then integrating (2.4.100) over the volume gives
cDfgPi
ds=Ci
jkcPjuk+1
2SklujRklji; (2.4.101)
where
Pi=Z
pidV; Sij=Z
sijdV: (2.4.102)
2.4.7 Mass and Papapetrou equations of motion
We consider matter which is distributed over a small region in space and consists of points with the
coordinates xi, forming an extended body whose motion is represented by a world tube in spacetime.
The motion of the body as a whole is represented by an arbitrary timelike world line
inside the
world tube, which consists of points with the coordinates Xi( ), where is the proper time on
.
We de ne
xi=xi Xi; x0= 0; ui=dXi
ds: (2.4.103)
Also de ne the following integrals:
Mik=u0Z
TikdV; (2.4.104)
Mijk= u0Z
xiTjkdV; (2.4.105)
Nijk=u0Z
SijkdV; (2.4.106)
Jik=Z
(xiTk0 xkTi0+Sik0)dV=1
u0(Mki0 Mik0+Nik0): (2.4.107)
The quantity Jikis equal toR
(xiTkl xkTil+Sikl)dSltaken for the volume hypersurface, so it
is a tensor, which we call the total spin tensor . The quantity Nijkis also a tensor. The relation
x0= 0 gives
M0jk= 0: (2.4.108)
We assume that the dimensions of the body are small, so integrals with two or more factors xi
multiplying Tjkand integrals with one or more factors ximultiplying Sjklcan be neglected.
The conservation law for the tetrad energy-momentum density (2.4.32) is
Tji
;i+fj
ikgTik Cj
ikTik 1
2Rj
iklSikl= 0: (2.4.109)
63
Integrating (2.4.109) over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals gives
Z
Tj0
;0dV+Z
fj
ikgTikdV Z
Cj
ikTikdV 1
2Z
Rj
iklSikldV= 0: (2.4.110)
Expanding
i
jk= i(0)
jk+ i(0)
jk;lxl; (2.4.111)
where the superscripts 0 denote the values at Xi, and substituting these expressions into (2.4.110)
gives (omitting the superscripts)
Z
Tj0dV
;0+fj
ikgZ
TikdV+fj
ikg;lZ
xlTikdV Cj
ikZ
TikdV
Cj
ik ;lZ
xlTikdV 1
2Z
Rj
iklSikldV= 0 (2.4.112)
or, using the de nitions (2.4.104), (2.4.105) and (2.4.106),
d
dsMj0
u0
+fj
ikgM(ik) fj
ikg;lMl(ik) Cj
ikM[ik]+Cj
ik ;lMl[ik] 1
2Rj
iklNikl= 0:(2.4.113)
The conservation law (2.4.109) gives
(xlTji);i=Tjl xlfj
ikgTik+xlCj
ikTik+1
2xlRj
ikmSikm; (2.4.114)
(xlxmTji);i=xmTjl+xlTjm xlxmfj
ikgTik+xlxmCj
ikTik
+1
2xlxmRj
iknSikn: (2.4.115)
Integrating (2.4.114) over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals gives
Z
(xlTj0);0dV=Z
TjldV Z
xlfj
ikgTikdV+Z
xlCj
ikTikdV+1
2Z
xlRj
ikmSikmdV: (2.4.116)
Substituting (2.4.103) into (2.4.116) gives
ul
u0Z
Tj0dV+XlZ
Tj0
;0dV+Z
(xlTj0);0dV=Z
TjldV XlZ
fj
ikgTikdV
Z
xlfj
ikgTikdV+XlZ
Cj
ikTikdV+Z
xlCj
ikTikdV
+1
2XlZ
Rj
ikmSikmdV; (2.4.117)
which reduces, due to (2.4.110), to
ul
u0Z
Tj0dV+Z
(xlTj0dV
;0=Z
TjldV Z
xlfj
ikgTikdV+Z
xlCj
ikTikdV: (2.4.118)
Substituting (2.4.111) into (2.4.118), omitting the superscripts and using the de nitions (2.4.104),
(2.4.105) and (2.4.106), turns (2.4.118) into
ul
u0Mj0 d
dsMlj0
u0
=Mjl+fj
ikgMlik Cj
ikMlik: (2.4.119)
Puttingl= 0 in (2.4.119) gives the identity because of (2.4.108).
64
Integrating (2.4.115) over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals gives
Z
(xlxmTj0);0dV=Z
xmTjldV+Z
xlTjmdV Z
xlxmfj
ikgTikdV
+Z
xlxmCj
ikTikdV+1
2Z
xlxmRj
iknSikndV: (2.4.120)
Substituting (2.4.103) into (2.4.116) gives
XlXmZ
Tj0
;0dV+ul
u0XmZ
Tj0dV+ul
u0Z
xmTj0dV+um
u0XlZ
Tj0dV
+um
u0Z
xlTj0dV+XlZ
xmTj0
;0dV+XmZ
xlTj0
;0dV
= XlXmZ
fj
ikgTikdV Z
Cj
ikTikdV 1
2Z
Rj
iklSikldV
+XlZ
TjmdV Z
xmfj
ikgTikdV+Z
xmCj
ikTikdV
+XmZ
TjldV Z
xlfj
ikgTikdV+Z
xlCj
ikTikdV
+Z
xmTjldV+Z
xlTjmdV; (2.4.121)
which reduces, due to (2.4.110) and (2.4.118), to
ul
u0Z
xmTj0dV+um
u0Z
xlTj0dV=Z
xmTjldV+Z
xlTjmdV (2.4.122)
or
ul
u0Mmi0+um
u0Mli0=Mmil+Mlim: (2.4.123)
The expressions analogous to (2.4.114) and (2.4.115) with higher multiples of xido not introduce
new relations.
The conservation law for the angular momentum density (2.4.15) is
Sijk
;k i
lkSjlk+ j
lkSilk 2T[ij]= 0: (2.4.124)
Integrating (2.4.124) over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals gives
Z
Sij0
;0dV Z
i
lkSjlkdV+Z
j
lkSilkdV 2Z
T[ij]dV= 0: (2.4.125)
Substituting (2.4.111) into (2.4.125), omitting the superscripts and using the de nitions (2.4.104)
and (2.4.106), turns (2.4.125) into
d
dsNij0
u0
i
lkNjlk+ j
lkNilk 2M[ij]= 0: (2.4.126)
The conservation law (2.4.124) gives
(xlSijk);k=Sijl+xl i
lkSjlk xl j
lkSilk+ 2xlT[ij]: (2.4.127)
Integrating (2.4.127) over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals gives
Z
(xlSij0);0dV=Z
SijldV+Z
xl i
mkSjmkdV Z
xl j
mkSimkdV+ 2Z
xlT[ij]dV: (2.4.128)
65
Substituting (2.4.103) into (2.4.128) gives
ul
u0Z
Sij0dV+XlZ
Sij0
;0dV=Z
SijldV+XlZ
i
mkSjmkdV Z
j
mkSimkdV
+2Z
T[ij]dV
+ 2Z
xlT[ij]dV; (2.4.129)
which reduces, due to (2.4.125), to
ul
u0Z
Sij0dV=Z
SijldV+ 2Z
xlT[ij]dV (2.4.130)
or
Ml[ij]= 1
2ul
u0Nij0 Nijl
: (2.4.131)
Puttingl= 0 in (2.4.131) gives the identity because of (2.4.108). The expressions analogous to
(2.4.127) with higher multiples of xido not introduce new relations.
Taking the cyclic permutations of the indices i;l;m in (2.4.123), adding the rst and second of
these relations, and subtracting the third, gives
ul
u0M[mi]0+ui
u0M[ml]0+um
u0M(li)0=Ml[im]+Mi[lm]+Mm(il): (2.4.132)
Substituting (2.4.107) and (2.4.131) into (2.4.132) gives
Mm(il)=u(iJl)m+um
u0M(il)0+Nm(il): (2.4.133)
Puttingm= 0 in (2.4.133), substituting it into (2.4.133) and using (2.4.108) gives
Mm(il)=u(iJl)m um
u0(u(iJl)0+N0(il)) +Nm(il): (2.4.134)
Combining (2.4.131) and (2.4.134) gives
Mmil=u(iJl)m um
u0(u(iJl)0+N0(il)) +Nm(il) 1
2um
u0Nil0 Nilm
: (2.4.135)
Therefore
M(ik)0= (u(iJk)0+N0(ik)): (2.4.136)
Combining the antisymmetric part of (2.4.119) and (2.4.126) gives
Mjl Mlj=ul
u0Mj0 uj
u0Ml0 (fj
ikg Cj
ik)Mlik+ (fl
ikg Cl
ik)Mjik
d
dsMlj0 Mjl0
u0
: (2.4.137)
Using (2.4.107), (2.4.131) and (2.4.134) brings (2.4.137) to
j
ikNlik+ l
ikNjik=ul
u0Mj0 uj
u0Ml0 fj
ikg
uiJkl+Nlik ul
u0(uiJk0+N0ik)
1
2Cj
ikul
u0Nik0 Nikl
+fl
ikg
uiJkj+Njik uj
u0(uiJk0+N0ik)
+1
2Cl
ikuj
u0Nik0 Nikj
+d
dsJlj; (2.4.138)
which, usingDfg
dsJlj=d
dsJlj+ukfl
ikgJij+ukfj
ikgJli, turns into
Dfg
dsJlj=uj
u0Ml0+uj
u0fl
ikg(uiJk0+N0ik) 1
2Cl
ikuj
u0Nik0 Nikj
+Cl
ikNjik
l$j
(2.4.139)
66
or, using the four-momentum
Pl=1
cZ
Tl0dV; (2.4.140)
into
Dfg
dsJlj=
cujPl+uj
u0fl
ikg(uiJk0+N0ik) 1
2Cl
ikuj
u0Nik0 Nikj
+Cl
ikNjik
l$j
:(2.4.141)
Therefore
DfgJli
dsui=cPl+1
u0fl
ikg(uiJk0+N0ik) 1
2Cl
ik1
u0Nik0 Nikjuj
+Cl
ikNjikuj
culujPj uluj
u0fj
ikg(uiJk0+N0ik) +1
2Cj
ikul
u0Nik0 Nikl
uj
Cj
ikNlikuj; (2.4.142)
which gives with (2.4.141)
DfgJlj
ds+uluiDfgJji
ds ujuiDfgJli
ds=Cl
ikNjik+1
2Cl
ikNikj Cj
ikNlik
1
2Cj
ikNikl ujum
Cl
ikNmik+1
2Cl
ikNikm Cm
ikNlik 1
2Cm
ikNikl
+ulum
Cj
ikNmik+1
2Cj
ikNikm Cm
ikNjik 1
2Cm
ikNikj
= 2(l
[nj
m] uju[ml
n]+ulu[mj
n])
Cn
ikNmik+1
2Cn
ikNikm
: (2.4.143)
This equation for Jik, except for the terms with Nijk, resembles (2.4.97) for r
ksk
ij.
Multiplying (2.4.143) by ujgives the identity, so only 3 equations in (2.4.143) are independent.
Thus 3 components of Jikare arbitrary and we can impose 3 constraints on Jik. A simple and
natural choice is
Jikuk= 0; (2.4.144)
which means that in the local rest frame J 0= 0, so the three independent components of Jik
are the spatial J . Analogously to the Pauli-Luba nski pseudovector (1.6.71), de ne the four-spin
pseudovector
Ji=1
2eijklujJkl; (2.4.145)
which is orthogonal to ui,
Jiui= 0: (2.4.146)
The condition (2.4.144) gives the relation inverse to (2.4.145):
Jik= eikjlujJl: (2.4.147)
Di erentiating (2.4.145) covariantly with respect to fi
jkgand using (2.4.143) gives
DfgJi
ds= 1
2eijklDfguj
dseklmnumJn+eijklujk
[nl
m]
Cn
prNmpr+1
2Cn
prNprm
= uiDfguk
dsJk+eij
nmuj
Cn
prNmpr+1
2Cn
prNprm
= uiDfguk
dsJk+Dfgui
dsukJk+eij
nmuj
Cn
prNmpr+1
2Cn
prNprm
: (2.4.148)
Thus the covariant (with respect to the Levi-Civita connection) change of the spin pseudovector
along the world line
is the sum of the corresponding Fermi-Walker transport (with respect to the
Levi-Civita connection) and a term which depends on the torsion and spin density.
67
In (2.4.142), the four-momentum Pldepends on terms proportional to the four-velocity uland
terms in which the index lappears in other quantities. We de ne the mass of the system described
by the energy-momentum density Tikas the coecient mofulin the expansion forPl
c,
Pl
c=mul+ terms not proportional to ul; (2.4.149)
so
m=uj
cPj+uj
c2u0fj
ikg(uiJk0+N0ik) uj
2c2u0Cj
ikNik0=uj
cj; (2.4.150)
where
j=Pj+1
cu0fj
ikg(uiJk0+N0ik) 1
2cu0Cj
ikNik0(2.4.151)
is the modi ed four-momentum . Substituting (2.4.150) and (2.4.151) into (2.4.142) gives
Dfg
dsJliui=cl mc2ul+uj
Cl
ikNjik+1
2Cl
ikNikj
uj
Cj
ikNlik+1
2Cj
ikNikl
;(2.4.152)
so l mcul= j(l
j uluj) is a vector. Thus the modi ed four-momentum iis a vector and the
massmis a scalar. Substituting (2.4.151) into (2.4.141) gives the Papapetrou equation of motion
for the spin :
Dfg
dsJlj=cujl culj+Cl
ikNjik+1
2Cl
ikNikj Cj
ikNlik 1
2Cj
ikNikl: (2.4.153)
Putting (2.4.119), (2.4.126), (2.4.131), (2.4.134), (2.4.136) and (2.4.151) into (2.4.113) gives
d
ds
cj 1
u0fj
ikg(uiJk0+N0ik) +1
2u0Cj
ikNik0
+fj
ikguk
ci 1
u0fj
lmg(ulJm0
+N0lm) +1
2u0Cj
lmNlm0
fj
ikg(fi
lmg Ci
lm)Mklm fj
ikgd
dsM(ik)0
u0
fj
ikg;l
u(iJk)l ul
u0(u(iJk)0+N0(ik)) +Nl(ik)
1
2Cj
ikd
dsNik0
u0
1
2Cj
ik( i
lmNklm+ k
lmNilm) 1
2Cj
ik ;lul
u0Nik0 Nikl
1
2RikljNikl
=cDfgj
ds fj
ikgfi
lmg(ulJmk+Nklm) fj
ikg;m(uiJkm+Nmik)
+1
2fj
ikgCi
lmNlmk+Cj
ik i
lmNklm+1
2Cj
ik ;mNikm 1
2RikljNikl= 0: (2.4.154)
Using (1.4.51) turns (2.4.154) into the Papapetrou equation of motion for the momentum :
Dfgj
ds= 1
2cPj
imkuiJmk 1
2cNiklCikl:j: (2.4.155)
If the spin density vanishes then the Einstein-Cartan gravitational eld equations reduce to the
Einstein-Hilbert gravitational eld equations. The conservation law for the spin density (2.4.15)
with the condition Sijk= 0 gives the symmetry of the energy-momentum density, Tik=Tki. The
relations (2.4.104), (2.4.105), (2.4.106) and (2.4.107) give then
Mik=Mki; (2.4.156)
Mijk=Mikj; (2.4.157)
Nijk= 0; (2.4.158)
Jik=cLik=Z
(xiTk0 xkTi0)dV=1
u0( Mik0+Mki0); (2.4.159)
68
whereLikis the angular momentum tensor, analogous to (2.4.58). The modi ed four-momentum
(2.4.151) reduces to
j=Pj+1
u0fj
ikguiLk0(2.4.160)
and (2.4.152) gives
l=mcul+DfgLli
dsui: (2.4.161)
The relation (2.4.148) reduces to
DfgJi
ds= uiDfguk
dsJk+Dfgui
dsukJk; (2.4.162)
so the covariant (with respect to the Levi-Civita connection) change of the spin pseudovector along
the world line
is equal to the corresponding Fermi-Walker transport. Multiplying (2.4.162) by Ji
and using (2.4.146) gives
JiJi= const; (2.4.163)
so the change of the spin pseudovector along a world line is a rotation, called spin precession . The
Papapetrou equation of motion for the spin (2.4.153) reduces to
DfgLlj
ds=ujl ulj; (2.4.164)
while the Papapetrou equation of motion for the momentum (2.4.155) reduces to
Dfgj
ds= 1
2Pj
imkuiLmk: (2.4.165)
The change of the mass malong the world line
is, using (1.4.48), (2.4.144), (2.4.150), (2.4.161)
and (2.4.165),
dm
ds=Dfgm
ds=1
cujDfgj
ds+1
cjDfguj
ds=1
cjDfguj
ds=1
cDfgLji
dsuiDfguj
ds
= 1
cLjiDfgui
dsDfguj
ds= 0; (2.4.166)
so
m= const: (2.4.167)
In the absence of the external gravitational eld and neglecting the gravitational eld of the
body, the relation (2.4.160) gives
j=Pj; (2.4.168)
so (2.4.165) reduces to
dPj
ds= 0; (2.4.169)
whose integration gives the conservation of the four-momentum along a world line:
Pi= const: (2.4.170)
The equation of motion for the spin (2.4.164) becomes
dLlj
ds=ujPl ulPj; (2.4.171)
whose integration gives the conservation of the angular momentum along a world line:
Lik+XiPk XkPi= const: (2.4.172)
69
The tensor Likis the intrinsic angular momentum of the body, while the tensor (in the absence of
the gravitational eld) XiPk XkPiis the orbital angular momentum associated with the motion of
the body as a whole. If Lik= 0 then (2.4.172) gives Pi/ui, so (2.4.170) is equivalent to ui= const
and thusXiis a linear function of the proper time . IfLik6= 0 thenXican be given by 3 arbitrary
functions of (sinceuiui= 1). In the momentum rest frame , in which P = 0,u 6= 0, so the
body has an arbitrary internal motion. The 3 constraints (2.4.144) eliminate this arbitrariness, so
the equations of motion entirely determine the motion of the body.
2.4.8 Spin tensor for particles
If the body is not spatially extended then it is referred to as a particle . The corresponding condition
x = 0 gives
Mijk= 0; Lik= 0: (2.4.173)
Therefore (2.4.131) reduces toul
u0Nij0 Nijl= 0, which with (2.4.107) gives
Nijl=Jijul; (2.4.174)
so
Jij=Sij=Nijkuk; (2.4.175)
whereSijis the intrinsic spin tensor . If the body is spatially extended then the di erence
Rik=Jik Sik(2.4.176)
is the rotational spin tensor . The di erence between the rotational spin tensor and angular momen-
tum tensor is, due to (2.4.107) and (2.4.131),
Rik cLik=Nik0
u0 Niklul= 2Ml[ik]ul: (2.4.177)
This expression vanishes, because of (2.4.108), in the velocity rest frame , in whichu = 0, which is
also locally Galilean, so u = 0. The relation (2.4.152) becomes
cl=mc2ul+Dfg
dsSliui uj
Cl
ikSjiuk+1
2Cl
ikSikuj
+uj
Cj
ikSliuk+1
2Cj
ikSikul
;(2.4.178)
while the Papapetrou equation of motion for the spin (2.4.153) becomes
Dfg
dsSlj=cujl culj+Cl
ikSjiuk+1
2Cl
ikSikuj Cj
ikSliuk 1
2Cj
ikSikul: (2.4.179)
The relation (2.4.174) implies that the spin tensor for a system of particles satis es
sijl=sijul; (2.4.180)
where
sij=sijlul (2.4.181)
is the spin-density tensor , which is orthogonal to ui(cf. (2.4.97) and (2.4.144)):
sijuj= 0: (2.4.182)
Thussijhas 3 independent components. A system satisfying (2.4.180) is referred to as a spin
uid .
The spin tensor (2.4.180) is traceless, due to (2.4.182). The relation (2.4.92) gives
c(piuj pjui) =ukr
ksij+sijr
kuk=Dfg
dssij+ other terms ; (2.4.183)
70
which resembles (2.4.179) in which Sikis replaced by sikand ibypi. Moreover, (2.4.93) corre-
sponds to (2.4.150) in which mc2is replaced by . Similarly, (2.4.94) gives
pi=
cui+1
c(ukr
ksij+sijr
kuk)uj=
cui+uj
cDfg
dssij+ other terms ; (2.4.184)
which resembles (2.4.178). In a locally Galilean frame of reference which is also a rest frame, (2.4.182)
becomes
s0 = 0: (2.4.185)
In this frame, the 3 components of sijare spatial, s , and are equivalent to 3 components of the
spatial spin-density pseudovector :
s =1
2e
s
: (2.4.186)
2.4.9 Energy-momentum tensor for particles
If a particle is spinless then its four-momentum is proportional to its four-velocity due to (2.4.160)
and (2.4.161):
Pl=mcul; (2.4.187)
which gives
P2=m2c2; (2.4.188)
in agreement with (1.6.81). Equations (2.4.119), (2.4.140), (2.4.156), (2.4.158) and (2.4.187) give
Mik=ui
u0Mk0=uiuk
(u0)2M00=mc2uiuk; (2.4.189)
so Z
TikdV=mc2uiuk
u0(2.4.190)
or
Tik(x) =mc2(x x0)uiuk
u0; (2.4.191)
where(x x0) is the spatial Dirac delta representing a point mass located at x0. We de ne the
mass density such that
psdV=dm; (2.4.192)
where sis given by (1.4.110). The mass density for a particle located at xais
(x) =mps(x xa); (2.4.193)
so (2.4.191) turns into
Tik=c2psuiuk
u0: (2.4.194)
Thus the energy-momentum tensor for a spinless particle is given by
Tik(x) =(x)c2uiuk
pg00u0=(x)cpg00dxi
dsdxk
dt=mc2(x xa)uiuk
p gu0
=mc2Zuiuk
p g(x xa( ))d ; (2.4.195)
wherexa( ) is the particle's wordline as a function of its proper time . For a system of particles,
we have
Tik(x) =X
amac2(x xa)uiuk
p gu0: (2.4.196)
71
The Papapetrou equation of motion (2.4.165) for a spinless particle reduces to the metric geodesic
equation (1.4.80),
Dfgui
ds= 0: (2.4.197)
In the absence of torsion and in the locally Galilean frame of reference, the conservation law for
the energy-momentum tensor is given by (2.4.34), so
Ti
;i= 0: (2.4.198)
We consider a closed system of particles which carry out a nite motion, in which all quantities
vary over nite ranges. We de ne the average over a certain time interval of a function fof these
quantities as f=1
R
0fdt. The average of the derivative of a bounded quantity_f=1
f( )
f(0)
!0 as !1 . Thus averaging (2.4.198) over the time gives
T
; = 0: (2.4.199)
Multiplying (2.4.199) by x and integrating over the volume gives, omitting surface integrals,
Z
x T
; dV= Z
T
dV= 0: (2.4.200)
The average energy of the system (2.4.52) is thus
E=Z
T0
0dV=Z
Ti
idV: (2.4.201)
Substituting (1.6.116) into (2.4.196) gives
Ti
i(x) =X
amac2(x xa)
1 v2
c21=2
; (2.4.202)
soTi
i0. Putting (2.4.202) into (2.4.201) gives
E=X
amac2
1 v2
c21=2
; (2.4.203)
which is referred to as the virial theorem .
Comparing (2.4.88) with (2.4.202) gives
3p=X
amac2
1 v2
c21=2
; (2.4.204)
where the summation extends over all particles in unit volume, so p=3. In the nonrelativistic limit
p0, while in the ultrarelativistic limit ( vc)p=3. We consider a system of noninteracting
identical particles of mass m, which we call an ideal gas , with the number of particles in unit volume
(concentration )n, so
=nm: (2.4.205)
Comparing (2.4.77) in the locally Galilean rest frame with (2.4.195) gives the kinetic formulae for
ideal gases:
=nmc2
; (2.4.206)
p=nm
3
v2: (2.4.207)
In a locally inertial frame of reference, (1.6.116) and (2.4.187) give
Pi=mc
1;v
c
; (2.4.208)
72
so the energy and momentum of the particle are
E=mc2
; (2.4.209)
P=m
v: (2.4.210)
Thus (2.4.188) gives
E2= (Pc)2+ (mc2)2: (2.4.211)
In the rest frame of the particle, P= 0, (2.4.211) reduces to Einstein's formula for the rest energy ,
E=mc2: (2.4.212)
The formulae (2.4.209) and (2.4.210) give
v=Pc2
E: (2.4.213)
Taking the di erential of (2.4.211) gives EdE =c2PdP, from which we obtain, using (2.4.213),
dE=vdP: (2.4.214)
If a particle is massless, m= 0, then (2.4.211) and (2.4.213) give
E=Pc; v =c: (2.4.215)
References: [2, 3, 4, 5, 6, 8, 9].
2.5 Gravitational eld equations
2.5.1 Einstein-Hilbert action and Einstein equations
The Einstein-Hilbert action for the gravitational eld and matter is, due to (2.2.2),
S= 1
2cZ
Pp gd
+Sm; (2.5.1)
where the metric tensor is regarded as a variational variable and the ane connection is the Levi-
Civita connection. Varying (2.5.1) with respect to the metric tensor gives, using (2.3.1) and the
identityp g= 1
2p ggikgik(which results from g=ggikgik= ggikgik),
S= 1
2cZ
Pikgikp g+Pikgikp g 1
2Pp ggikgik
d
+1
2cZ
Tikp ggikd
:(2.5.2)
Partial integration of the rst term on the right-hand side of (2.5.2), using (1.4.57), brings this term
to zero:
Z
Pikgikd
=Z
(fl
ikg):l (fl
ilg):k
gikd
= Z
(gik
:lfl
ikg gik
:kfl
ilg)d
= 0;(2.5.3)
where
gik=p ggik(2.5.4)
is the contravariant metric density , whose covariant derivative with respect to the Christo el symbols
vanishes, gik
:l= 0. Equaling S= 0 in (2.5.2) gives the Einstein equations of the general theory of
relativity:
Gik=Tik (2.5.5)
or
Pik=
Tik 1
2Tgik
: (2.5.6)
73
BecauseR
Pp gd
=R
Gp gd
, where the noncovariant quantity Gis given by (2.2.5), the
left-hand side of the Einstein equations is
Gik=1p g@(p gG)
@gik: (2.5.7)
The covariant conservation of the Einstein tensor (1.4.67) imposes the conservation of the metric
dynamical energy-momentum tensor (2.4.23). Therefore the gravitational eld equations contain
the equations of motion of matter. In vacuum , whereTik= 0, the Ricci tensor in (2.5.6) vanishes:
Pik= 0: (2.5.8)
Thus vanishing of Pikat a given point in spacetime is a covariant criterion of whether matter is
present or absent at this point.
The Einstein equations (2.5.5) are 10 second-order partial di erential equations for: 10 4 = 6
independent components of the metric tensor gik(the factor 4 is the number of the coordinates which
can be chosen arbitrarily), 3 independent components of the four-velocity ui, and either orp(which
are related to each other by the equation of state). The contracted Bianchi identity (1.4.67) gives
the equations of motion of matter. In vacuum, the Einstein equations are 10 4 = 6 independent
equations (the factor 4 is the number of constraints from the contracted Bianchi identity) for 6
independent components of the metric tensor gik.
In the Einstein equations, the only second time-derivatives of gikare the derivatives of the spatial
components of the metric tensor, g , and they appear only in the components of the eld
equations (2.5.5). Therefore the initial values (at t= 0) forg and _g can be chosen arbitrarily.
The rst time-derivatives _ g0 and _g00appear only in the components of the eld equations
(2.5.5). The 0 and 00 components of the eld equations (2.5.5) give the initial values for g0 and
g00. The undetermined initial values for _ g0 and _g00correspond to 4 degrees of freedom for a free
gravitational eld. A general gravitational eld has 8 degrees of freedom: 4 degrees of freedom for
a free gravitational eld, 3 related to the four-velocity, and 1 related to (orp).
2.5.2 Einstein pseudotensor and principle of equivalence
We de ne
G=p gG; (2.5.9)
where Gis the noncovariant quantity (2.2.5). The action for the gravitational eld and matter,
S= 1
2cZ
Gd
+Sm=1
cZ
1
2G+Lm
d
; (2.5.10)
produces the Einstein eld equations by varying the metric tensor, because Gdi ers fromp gPby
a total divergence:
gik
1
2G+Lm
= 0: (2.5.11)
Construct a canonical energy-momentum density (2.4.35) corresponding to the gravitational eld,
treating 1
2G(which depends only on gijand its rst derivatives gij
;k) like Lmandgiklike a matter
eld :
ti
k= 1
2@G
@gjl
;igjl
;k i
kG
: (2.5.12)
This quantity is not a tensor density since Gis not a scalar density and its division byp g,ti
kp g,
is referred to as the Einstein energy-momentum pseudotensor for the gravitational eld. The four-
momentum corresponding to the total energy-momentum density for the gravitational eld and
matter (which is not a vector) is then
Pi=1
cZ
(tk
i+Tk
i)dSk; (2.5.13)
74
where the sum tk
i+Tk
iis called the Einstein energy-momentum complex . The de nition (2.5.12)
gives
2ti
k ;i=@i@G
@gjl
;igjl
;k @G
@gjl
;igjl
;ki+G;k=@i@G
@gjl
;igjl
;k @G
@gjl
;igjl
;ki+@G
@gjlgjl
;k
+@G
@gjl
;igjl
;ik=@G
@gjl @i@G
@gjl
;i
gjl
;k=G
gjlgjl
;k; (2.5.14)
which, using (2.5.11), gives
ti
k ;i=Lm
gjlgjl
;k=1
2Tjlgjl
;k: (2.5.15)
The covariant conservation (2.4.23) gives
Ti
k ;i=fl
kigTi
l=1
2glmgim;kTi
l= 1
2glm
;kTlm; (2.5.16)
so the total energy-momentum density for the gravitational eld and matter is ordinarily conserved:
(ti
k+Ti
k);i= 0: (2.5.17)
Integrating (2.5.17) over the four-dimensional volume and using the Gau-Stokes theorem gives
I
(ti
k+Ti
k)dSi= 0; (2.5.18)
so the four-momentum (2.5.13) is conserved, Pi= const. Because the quantity tikis not symmetric
in the indices i;k, the total angular momentum constructed from Pias in (2.4.58),
Mik=Z
(xidPk xkdPi) =1
cZ
xi(tkl+Tkl) xk(til+Til)
dSl; (2.5.19)
is not conserved. The conservation law (2.5.17) gives tl
k+Tl
k=li
k ;i, wherekli= kil, so
tl
k tl
k= (li
k l i
k);i. Analogously to (2.4.44) and (2.4.45), we could bring tl
k+Tl
kto a symmetric
form. However, using ( li
k l i
k) instead of li
kin (2.4.45), where k
iis replaced by tk
i+Tk
i, gives
k
i= 0, so this symmetrization procedure does not work for the Einstein pseudotensor.
The Einstein pseudotensor (2.5.12) can be explicitly written as
ti
k=1
2(fi
lmgglm
;k fl
mlggmi
;k+i
kG); (2.5.20)
so it is a homogeneous quadratic function of the Christo el symbols. Thus it vanishes in the local
Galilean frame of reference. It can also di er from zero in the Minkowski spacetime (in the absence
of the gravitational eld) if we choose the coordinates such that the Christo el symbols do not
vanish. Therefore the energy of the gravitational eld is not absolutely localized in spacetime; it
depends on the choice of the coordinates. The gravitational eld can be always eliminated locally
by transforming the coordinate system to the local Galilean frame of reference in which the Einstein
pseudotensor vanishes. This property of the gravitational eld is referred to as the principle of
equivalence .
The construction of a conserved four-momentum for the gravitational eld and matter is possible
because the Lagrangian density for the gravitational eld Lgis linear in the second derivatives of
the metric tensor. The Lagrangian density (2.2.2) can be generalized to
Lg= 1
2p g(P+ 2); (2.5.21)
where is referred to as the cosmological constant , without altering the Einstein energy-momentum
pseudotensor (2.5.12). Another scalar density which is linear in curvature is ijklPijkl, but this parity-
violating expression vanishes due to the cyclic identity (1.4.64). Therefore the simplest choice for
75
a gravitational Lagrangian density, linear in P, is the only one which admits ordinary conservation
laws for the gravitational eld and matter, and thus it is physical. Other possible gravitational
Lagrangians can contain terms which depend explicitly on the torsion tensor, but such modi cations
of general relativity or the Einstein-Cartan gravity would involve free parameters and thus they would
be less fundamental.
2.5.3 Landau-Lifshitz energy-momentum pseudotensor
The covariant conservation (2.4.23) in the local Galilean frame of reference is
Ti
k ;i= 0; (2.5.22)
soTikcan be expressed as Tik=ikl
;l, whereikl= ilk. The Einstein equations (2.5.5) in the
Galilean frame are
( g)Tik=hikl
;l; (2.5.23)
where
hikl=iklm
;m= hilk; (2.5.24)
iklm=1
2( g)(gikglm gilgkm): (2.5.25)
In an arbitrary frame of reference, (2.5.23) is not valid. We de ne tiksuch that
( g)(tik+Tik) =hikl
;l: (2.5.26)
Therefore
( g)(tik+Tik)
;k= 0; (2.5.27)
so there is a conservation of the four-momentum of the gravitational eld and matter,
Pi=1
cZ
( g)(tik+Tik)dSk: (2.5.28)
The quantity tikis not a tensor density, so the conserved four-momentum Pi(2.5.28) is not a vector.
The four-momentum Piis not a vector even for Lorentz transformations, because of the factor g
instead of the correct (weight 1) densityp gin (2.5.28). Dividing Pibyp gat some xed point
(a natural choice is in nity) turns it into a vector under Lorentz transformations. Using (2.5.26)
turns (2.5.28), for the hypersurface dS0=dV, into
Pi=1
cZ
hikl
;ldSk=1
2cI
hikldf
kl=1
cI
hi0 df : (2.5.29)
The quantity tikis referred to as the Landau-Lifshitz energy-momentum pseudotensor for the grav-
itational eld, and the sum ( g)(tik+Tik) is called the Landau-Lifshitz complex .
The explicit expression for the Landau-Lifshitz pseudotensor is
tik=1
2
(gilgkm gikglm)(2fn
lmgfp
npg fn
lpgfp
mng fn
lngfp
mpg)
+gilgmn(fk
lpgfp
mng+fk
mngfp
lpg fk
npgfp
lmg fk
lmgfp
npg)
+gklgmn(fi
lpgfp
mng+fi
mngfp
lpg fi
npgfp
lmg fi
lmgfp
npg)
+glmgnp(fi
lngfk
mpg fi
lmgfk
npg)
(2.5.30)
or
( g)tik=1
2
gik
;lglm
;m gil
;lgkm
;m+1
2gikglmgln
;pgpm
;n
(gilgmngkn
;pgmp
;l+gklgmngin
;pgmp
;l) +glmgnpgil
;ngkm
;p
+1
8(2gilgkm gikglm)(2gnpgqr gpqgnr)gnr
;lgpq
;m
: (2.5.31)
76
This pseudotensor is symmetric in the indices i;k, so there is a conservation of the total angular
momentum constructed from Pias in (2.4.58),
Mik=Z
(xidPk xkdPi) =1
cZ
xi(tkl+Tkl) xk(til+Til)
( g)dSl: (2.5.32)
DividingMikbyp gat in nity turns it into an antisymmetric tensor under Lorentz transforma-
tions. Using (2.5.24) and (2.5.26) turns (2.5.28), for the hypersurface dS0=dV, into
Mik=1
cZ
(xiklmn
;nm xkilmn
;nm)dSl=1
2cI
(xiklmn
;n xkilmn
;n)df
lm
1
cI
(klin ilkn);ndSl=1
cI
(xihk0 xkhi0 +i0 k)df : (2.5.33)
Choosing the volume hypersurface dV=dS0gives
Mik=1
cZ
xi(tk0+Tk0) xk(ti0+Ti0)
( g)dV: (2.5.34)
The conservation of M0 in (2.5.34) divided by the conservation of P0in (2.5.28) gives a uniform
motion (2.4.61) with velocity (2.4.62) of the center of inertia for the gravitational eld and matter,
with the coordinates
X =R
x (t00+T00)( g)dVR
(t00+T00)( g)dV: (2.5.35)
The coordinates of the center of inertia (2.5.35), like (2.4.63), are not the spatial components of a
four-dimensional vector.
2.5.4 Utiyama action
The Utiyama action for the gravitational eld and matter is equal to (2.5.1), where the tetrad is
regarded as a variational variable and the spin connection is the Levi-Civita spin connection (1.5.31).
Thus (2.3.5) gives
S= 1
2cZ
(eP)d
+1
cZ
Ta
iei
ad
: (2.5.36)
The Lagrangian density for the gravitational eld is given by (2.2.2), with the Riemann scalar P
given by (1.5.39) and (1.5.41):
eP=eei
aejb($a
bj;i $a
bi;j+$a
ci$c
bj $a
cj$c
bi) = 2eij
ab($ab
j;i+$a
ci$cb
j); (2.5.37)
where
eij
ab=ee[i
aej]
b: (2.5.38)
Varying ePand omitting total derivatives gives in the absence of torsion, using e=eei
aea
iand
eij
abjj=eij
ab;j $c
ajeij
cb $c
bjeij
ac= 0 (which results from (1.5.22)),
(eP) = (2Pa
i Pea
i)eei
a+ 2eij
ab($ab
j;i+$a
ci$cb
j) = (2Pa
i Pea
i)eei
a
+2(eij
ab;j $c
ajeij
cb $c
bjeij
ac)$ab
i= (2Pa
i Pea
i)eei
a: (2.5.39)
EqualingS= 0 gives the tetrad Einstein equations:
Pa
i 1
2Pea
i=
eTa
i; (2.5.40)
equivalent to the metric Einstein equations (2.5.5) because of (2.3.4) and (2.3.32) (in the absence of
torsion).
77
2.5.5 M ller pseudotensor
The Riemann scalar Pis linear in derivatives of $a
bi:
eP= (eei
aej
b$ab
j);i (eei
aej
b);i$ab
j (eei
aej
b$ab
i);j+ (eei
aej
b);j$ab
i+eei
aej
b$ac
i$b
c j
eei
aej
b$ac
j$b
c i= 2(eei
aej
b$ab
j);i 2(eei
aej
b);i$ab
j+eei
aej
b$ac
i$b
c j
eei
aej
b$ac
j$b
c i: (2.5.41)
Thus we can subtract from ePtotal derivatives without altering the eld equations, replacing Pby
a noncovariant quantity M:
eM= 2(eei
aej
b);i$ab
j+eei
aej
b$ac
i$b
c j eei
aej
b$ac
j$b
c i
= 2e(fk
kig$ij
j+$i
ai$aj
j fi
kig$kj
j+$j
bi$ib
j fj
kig$ik
j)
+e(!ic
i!j
c j !ic
j!j
c i) =e(!ia
i!j
aj !ia
j!j
ai); (2.5.42)
using (1.4.35) and (1.5.31). Therefore
M=!ia
i!j
aj !ia
j!j
ai: (2.5.43)
Because of (1.5.15), the quantity (2.5.43) depends on the tetrad ei
aand its rst derivatives ei
a;j.
Therefore, analogously to (2.5.12), we can construct a canonical energy-momentum density corre-
sponding to the gravitational eld, treating 1
2eM:
mi
k= 1
2@(eM)
@ej
a;iej
a;k i
keM
: (2.5.44)
This quantity is not a tensor density since eMis not a scalar density and its division by e,mi
k
e,
is referred to as the M ller energy-momentum pseudotensor for the gravitational eld. One can
show, analogously to the steps leading to (2.5.17, that the total energy-momentum density for the
gravitational eld and matter is ordinarily conserved:
(mi
k+Ti
k);i= 0: (2.5.45)
Thus the corresponding total four-momentum is conserved:
Pi=1
cZ
(mk
i+Tk
i)dSk= const; (2.5.46)
where the sum mk
i+Tk
iis called the M ller energy-momentum complex . The M ller pseudotensor
depends on the choice of both the coordinates and the tetrad. To x the tetrad, one can impose on
it 6 constraints which are covariant under constant Lorentz transformations but not under general
Lorentz transformations (otherwise these constraints would not x the tetrad since Lorentz trans-
formations are tetrad rotations). A natural choice is to constrain the 6 components of the spin
connection !ijkin which the last index is contracted with a covariant derivative or the trace of the
spin connection.
2.5.6 Einstein-Cartan action
If we regard the torsion tensor as a variational variable (in addition to the metric tensor) then the
action for the gravitational eld and matter is, due to (2.2.1),
S= 1
2cZ
Rp gd
+Sm; (2.5.47)
and it is referred to as the Einstein-Cartan action . Using (1.4.54) gives
S= 1
2cZ
P gik(2Cl
il:k+Cj
ijCl
kl Cl
imCm
kl)p gd
+Sm: (2.5.48)
78
Partial integration of the terms with covariant derivatives : and omitting total derivatives (which do
not contribute to the eld equations) reduces (2.5.48) to
S= 1
2cZ
P gik(Cj
ijCl
kl Cl
imCm
kl)p gd
+Sm: (2.5.49)
Varying (2.5.49) with respect to the metric tensor and contortion tensor (which is equivalent to
varying with respect to the torsion tensor) gives, using (2.3.19), (2.3.26) and (2.5.3),
S= 1
2cZ
Pik 1
2Pgik Cj
ijCl
kl+Cl
imCm
kl+1
2gik(Cjm
jCl
ml Clj
mCm
jl)
p ggikd
1
cZ
(Ckj
i Clj
lk
i)p gCi
jkd
+1
2cZ
Tikp ggikd
+1
2cZ
sik
jp gCj
ikd
: (2.5.50)
For variations gik,S= 0 gives the rst Einstein-Cartan equation
Gik=(Tik+Uik); (2.5.51)
where
Uik=1
Cj
ijCl
kl Cl
ijCj
kl 1
2gik(Cjm
jCl
ml CmjlCljm)
(2.5.52)
or
Uik=1
(Sl
ij+ 2Sl
(ij))(Sj
kl+ 2Sj
(kl)) + 4SiSk+1
2gik(Smjl+ 2S(jl)m)
(Sljm+ 2S(jm)l) 2gikSjSj
: (2.5.53)
For variations Cj
ik,S= 0 gives the second Einstein-Cartan equation
Ck
[ji] k
[iCl
j]l=
2sk
ij (2.5.54)
or
Tj
ik=
2sj
ik; (2.5.55)
where
Tj
ik=Sj
ik Sij
k+Skj
i (2.5.56)
is the modi ed torsion tensor . The relation (2.5.55) is equivalent to
Sk
ij=
2(sk
ij+k
[isl
j]l): (2.5.57)
This relation between the torsion and spin tensors is algebraic: torsion at a given point in spacetime
does not vanish only if there is matter at this point, represented in the Lagrangian density by
a function which depends on torsion. Unlike the metric, which is related to matter through a
di erential eld equation, torsion does not propagate. Combining (2.5.52) and (2.5.57) gives
Uik=
sij
[lskl
j] 1
2sijlsk
jl+1
4sjlisk
jl+1
8gik( 4sl
j[msjm
l]+sjlmsjlm)
: (2.5.58)
The tensor (2.5.58) represents a correction to the dynamical energy-momentum tensor from the
spin contributions to the geometry of spacetime, quadratic in the spin density (so the sign of the
spin density does not a ect this correction) and corresponding to a spin-spin contact interaction. If
matter elds do not depend on torsion then Uik= 0 and the rst Einstein-Cartan equation (2.5.51)
reduces to the Einstein equations (2.5.5).
79
2.5.7 Sciama-Kibble action
The Sciama-Kibble action for the gravitational eld and matter is equal to (2.5.47), where both the
tetrad and spin connection are regarded as variational variables. Thus (2.3.27) gives
S= 1
2cZ
(eR)d
+1
cZ
Ta
iei
ad
+1
2cZ
Si
ab!ab
id
: (2.5.59)
The Lagrangian density for the gravitational eld is given by (2.2.1), with the curvature scalar R
given by (1.5.36) and (1.5.38):
eR=eei
aejb(!a
bj;i !a
bi;j+!a
ci!c
bj !a
cj!c
bi) = 2eij
ab(!ab
j;i+!a
ci!cb
j): (2.5.60)
Varying eRand omitting total derivatives gives, using eij
abjj=eij
ab;j !c
ajeij
cb !c
bjeij
ac+ i
kjekj
ab+
j
kjeik
ab k
kjeij
ab= 0 (which results from (1.5.22)),
(eR) = (2Ra
i Rea
i)eei
a+ 2eij
ab(!ab
j;i+!a
ci!cb
j)
= (2Ra
i Rea
i)eei
a+ 2(eij
ab;j !c
ajeij
cb !c
bjeij
ac)!ab
i
= (2Ra
i Pea
i)eei
a 2(Si
kjekj
ab+ 2Sjeij
ab)!ab
i: (2.5.61)
For variations !ab
i,S= 0 gives
Si
ab Saei
b+Sbei
a=
2eSi
ab; (2.5.62)
equivalent to the second Einstein-Cartan equation (2.5.55). For variations ei
a,S= 0 gives
Ra
i 1
2Rea
i=
eTa
i (2.5.63)
or
Rki 1
2Rgik=p gTik: (2.5.64)
Substituting (2.5.55) and (2.5.64) into the conservation law for the spin density (2.4.16) gives
2(Sk
ij;k Si;j+Sj;i) =Rji Rij 4Sk(Sk
ij Sik
j+Sjk
i); (2.5.65)
which is equivalent to the contracted cyclic identity (1.4.61). Thus the contracted cyclic identity
imposes the conservation law for the spin density in the Einstein-Cartan gravity. Substituting
(2.5.55) and (2.5.64) into the conservation law for the energy-momentum density (2.4.31) gives
Rj
i;j 1
2R;i= 2Sj
Rj
i 1
2Rj
i
+ 2Sj
ki
Rk
j 1
2Rk
j
(Sj
kl Skj
l+Slj
k)Rkl
ji;(2.5.66)
which is equivalent to the contracted Bianchi identity (1.4.62). Thus the contracted Bianchi identity
imposes the conservation law for the energy-momentum density in the Einstein-Cartan gravity.
Substituting (2.5.55) and (2.5.64) into the Belinfante-Rosenfeld relation (2.3.33) gives
Tik=Rki 1
2Rgik+r
j(Sj
ik+ 2Skj
i 2Sj
(ik) 2Sjgik) =Rki 1
2Rgik
+r
j( Cj
ki+Cl
klj
i Clj
lgik): (2.5.67)
Combining (1.4.52), (1.4.54) and (2.5.67) gives
Tik=Pik 1
2Pgik+Cl
ki:l Cl
kl:i+Cj
kiCl
jl Cj
klCl
ji 1
2gik( 2Clj
l:j
Clj
lCm
jm+CmjlCljm) Cj
ki:j Cj
ljCl
ki+Cl
kjCj
li+Cl
ijCj
kl+Cj
kj:i
Cl
kiCj
lj gik(Clj
l:j+Cj
ljCml
m) Cj( Cj
ki+Cl
klj
i Clj
lgik); (2.5.68)
which is equivalent to the rst Einstein-Cartan equation (2.5.51). Thus the relation between the
Ricci tensor and the Riemannian Ricci tensor is equivalent to the Belinfante-Rosenfeld relation in
the Einstein-Cartan gravity, and (2.5.64) is another form of the rst Einstein-Cartan equation.
80
2.5.8 Einstein-Cartan pseudotensor
Replacing the action for the gravitational eld and matter (2.5.49) by
S= 1
2cZ
G gik(Cj
ijCl
kl Cl
imCm
kl)p gd
+Sm (2.5.69)
produces the rst Einstein-Cartan equation by varying the metric tensor, becausep gGdi ers
fromp gPby a total divergence:
gik
1
2
G gnp(Cj
njCl
pl Cl
nmCm
pl)
+Lm
= 0: (2.5.70)
The canonical energy-momentum density for the gravitational eld is also given by (2.5.12). The
relations (2.5.14) and (2.5.70) give
ti
k ;i=
Lm+ 2gnp(Cj
njCl
pl Cl
nmCm
pl)
gjlgjl
;k=1
2(Tjl+p gUjl)gjl
;k: (2.5.71)
The covariant conservation (2.4.23) gives
(Ti
k+p gUi
k);i=fl
kig(Ti
l+p gUi
l) =1
2glmgim;k(Ti
l+p gUi
l)
= 1
2glm
;k(Tlm+p gUlm); (2.5.72)
so the total energy-momentum density for the gravitational eld and matter is ordinarily conserved:
(ti
k+Ti
k+p gUi
k);i= 0: (2.5.73)
Thus the corresponding four-momentum is conserved:
Pi=1
cZ
(tk
i+Tk
i+p gUk
i)dSk= const: (2.5.74)
The quantitytk
ip g+Uk
iis referred to as the Einstein-Cartan energy-momentum pseudotensor for the
gravitational eld, and the sum tk
i+Tk
i+p gUk
iis called the Einstein-Cartan energy-momentum
complex .
2.5.9 Palatini variation
If the matter action Smdoes not depend on the ane connection, its variation with respect to the
metric and connection ( j
ikis a tensor) is referred to as the Palatini variation . Varying (2.5.47)
with respect to k
ijgives, due to (1.3.39),
S= 1
2cZ
Rikgikd
= 1
2cZ
( l
ik);l ( l
il);k 2Sj
lk l
ij
gikp gd
: (2.5.75)
Partial integration and omitting total derivatives in (2.5.75) gives, using (1.2.34),
S=1
2cZ
( l
ikgik
;l 2Sl l
ikgik l
ilgik
;k+ 2Sk l
ilgik+ 2Sj
lk l
ijgik)d
: (2.5.76)
Since the ane connection is metric-compatible, gij;k= 0,S= 0 turns the torsion tensor into
zero, so the connection is formed by the Christo el symbols and the eld equations are the Einstein
equations (2.5.5). Thus varying the action for matter elds, which do not depend on the ane
81
connection, with respect to the connection is equivalent to varying it with respect to the torsion
tensor. However, if the matter action Smdepends on the ane connection then (2.5.76) becomes
S=1
2cZ
( l
ikgik
;l 2Sl l
ikgik l
ilgik
;k+ 2Sk l
ilgik+ 2Sj
lk l
ijgik)d
+1
2cZ
i k
j j
ikd
; (2.5.77)
where the hypermomentum density is de ned as
i k
j= 2Lm
j
ik: (2.5.78)
Since the connection is metric-compatible, S= 0 gives
gikSj k
jSi Ski
j=
2p gi k
j: (2.5.79)
Contracting the indices i;jgives
i k
i= 0; (2.5.80)
which also results from the invariance of the Lagrangian density under a projective transformation
(1.2.50) (the symmetric part of the Ricci tensor is invariant under this transformation):
L=Lm=1
2i k
j j
ik=1
2i k
jj
iAk= 0: (2.5.81)
The relation (2.5.80) constrains possible forms of matter Lagrangians algebraically, so it is not a
conservation law. Therefore varying the action with respect to the ane connection, unlike that
with respect to the torsion (or spin connection), does not constitute a physical variational principle.
Only the antisymmetric part of the connection (torsion) can be regarded as a dynamical variable; its
symmetric part can always be brought locally to zero by a suitable transformation of the coordinates.
2.5.10 Gravitational potential
If the metric tensor gijis approximately equal to the Minkowski metric tensor ijthen the corre-
sponding gravitational eld is weak. We can write
g001 +2
c2; (2.5.82)
where is referred to as the gravitational potential . Thus nonrelativistic gravitational elds, cor-
responding to the limit c!1 , are weak. Also u01 andu 0. In this limit, the leading
component of the Levi-Civita connection is
f
0 0g 1
2g @g00
@x =1
c2@
@x ; (2.5.83)
so the metric geodesic equation (1.4.80) reduces to
dv
dt=g= r ; (2.5.84)
where gis the acceleration due to gravity. The quantity Gin (2.2.5) reduces to
G=2
c4(r )2: (2.5.85)
The leading component of the Riemannian Ricci tensor is
P00@f
0 0g
@x =1
c2@2
@x 2=1
c24 : (2.5.86)
82
The leading component of the energy-momentum tensor (2.4.195) is
T00=c2: (2.5.87)
Therefore the Einstein equations in the nonrelativistic limit reduce to the Poisson equation :
4 = 4G; (2.5.88)
where
G=c4
8(2.5.89)
isNewton's gravitational constant . In vacuum, where = 0, the Poisson equation reduces to the
Laplace equation:
4 = 0: (2.5.90)
2.5.11 Relativistic ideal
uids
The covariant conservation (2.4.24) of the metric energy-momentum tensor (2.4.77) gives
(+p)uk
:kui+ (+p)ukui
:k=p;kgik: (2.5.91)
Multiplying (2.5.91) by uigives
(+p)uk
:k=p;kuk; (2.5.92)
which, upon substituting into (2.5.91) yields the Euler equation :
(+p)Dfgui
ds=p;khik: (2.5.93)
Ifp;i/ui(which includes the case p= const) then (2.5.93) reduces to the metric geodesic equation
(1.4.80). De ning a quantity nsuch that
dn
n=d
+p(2.5.94)
brings (2.5.92) to the conservation law
(nui):i= 0: (2.5.95)
The quantity nmay thus represent the proper (in the rest frame) number density of particles com-
posing the
uid.
In the nonrelativistic limit, c!1 ,u01,u v
c,c2andp , so (2.5.92) reduces to
theequation of continuity :
@
@t+ div s= 0; (2.5.96)
where
s=v (2.5.97)
is referred to as the mass current . Integrating (2.5.96) over the volume gives
@
@tZ
dV +I
sdf= 0; (2.5.98)
which means that the change in time of the total mass inside a volume, m=R
dV, is balanced by
themass
ux through the surface bounding this volume, representing the conservation of the total
mass of a
uid. The Euler equation (2.5.93) reduces in this limit to
@v
@t+v
; v
= ; +p; (2.5.99)
83
or
dv
dt=@v
@t+ (vr)v
= r rp: (2.5.100)
Integrating (2.5.100) over the volume gives, using P=R
vdV, the change in time of the total
momentum of a
uid:dP
dt= Z
r dm I
pdf: (2.5.101)
Without pressure gradients, (2.5.100) reduces to (2.5.84).
The number of particles dNin a volume element dVin the rest frame of reference is equal to
dN=ndV; (2.5.102)
wherenis the proper number density. In a frame of reference moving relative to the rest frame with
velocity v, the same volume element is given by dV0=dVp
1 v2=c2(1.6.107), and the number
density isn0. SincedN0=n0dV0anddN0=dNis an invariant, we have
n0=np
1 v2=c2: (2.5.103)
2.5.12 Relativistic spin
uids
For a spin
uid, substituting (2.4.180) into the second Einstein-Cartan equation (2.5.57) and using
(2.4.182) gives
Si= 0;r
i=ri: (2.5.104)
Thus the corresponding metric dynamical energy-momentum tensor (2.4.96) reduces to
Tij=uiuj phij rksk
(ij)+rksk
iluluj 1
2rksk
ij: (2.5.105)
Accordingly, the torsion tensor is
Sj
ik= 1
2sikuj: (2.5.106)
The last two terms on the right of the second line of (2.5.105) can be written, using (2.4.92), (2.4.94)
and (2.4.182), as
rksk
iluluj 1
2rksk
ij=c(piul plui)uluj 1
2c(piuj pjui)
=1
2c(piuj+pjui) uiuj=1
2(rksk
iluluj+rksk
jlului) = rlsl
k(iukuj)
= rl(sk(iul)ukuj)= rlsk(iulukuj)= rl(sk(iuj))uluk
= rl(sk
(iuj))ukul: (2.5.107)
The metric dynamical energy-momentum tensor (2.5.105) is then
Tij=uiuj phij (l
k+ukul)rl(sk(iuj)): (2.5.108)
The last term on the right of (2.5.108) can be decomposed according to (1.4.28) into
(l
k+ukul)rl(sk(iuj)) = (l
k+ukul)rfg
l(sk(iuj)) (l
k+ukul)(Ck
mlsm(iuj)
+Ci
mlsk(muj)+Cj
mlsk(ium)): (2.5.109)
84
This term reduces, using (2.4.182) and (2.5.54), to
(l
k+ukul)rfg
l(sk(iuj)) l
k(Ci
mlsk(muj)+Cj
mlsk(ium)) ukulCk
mlsm(iuj)
= (l
k+ukul)rfg
l(sk(iuj)) Ci
mksk(muj) Cj
mksk(ium)
= (l
k+ukul)rfg
l(sk(iuj)) +1
2(smkui+si
kum+si
muk)sk(muj)
+1
2(smkuj+sj
kum+sj
muk)sk(mui)
= (l
k+ukul)rfg
l(sk(iuj)) 1
2(sklskluiuj siksj
k): (2.5.110)
Thus (2.5.108) becomes
Tij=uiuj phij (l
k+ukul)rfg
l(sk(iuj)) s2uiuj+1
2siksj
k; (2.5.111)
where
s2=1
2sijsij>0: (2.5.112)
Substituting (2.4.180) into (2.5.58) and using (2.4.182 gives
Uij=1
2s2uiuj+1
4s2gij 1
2siksj
k: (2.5.113)
Adding (2.5.111) and (2.5.113) brings the combined energy-momentum tensor Tij+Uijin the rst
Einstein-Cartan equation (2.5.51) to
Tij+Uij=
1
4s2
uiuj
p 1
4s2
hij
(l
k+ukul)rfg
l(sk(iuj)): (2.5.114)
If the spin orientation of particles (cf. (2.6.60)) in a spin
uid is random then the macroscopic
spacetime average of sijand of its gradients, such as of the last term on the right of (2.5.114), vanish.
On the contrary, the terms that are quadratic in the spin tensor do not vanish after averaging. Thus
the combined energy-momentum tensor of a macroscopic spin
uid describes a perfect
uid with the
e ective energy density
~= (Tij+Uij)uiuj= 1
4s2(2.5.115)
and the e ective pressure
~p= 1
3(Tij+Uij)hij=p 1
4s2: (2.5.116)
If the spin orientation of particles (cf. (2.6.60)) in a spin
uid is not random then the combined
energy density of a macroscopic spin
uid is
~= 1
4s2 (l
k+ukul)uirfg
lsk
i= 1
4s2 rfg
ksk
iui
= 1
4s2+skirfg
[kui]= 1
4s2+ski@[kui]: (2.5.117)
In a locally Galilean frame of reference which is also a rest frame, (2.5.117) becomes
~= 1
4s2+1
2scurlv: (2.5.118)
where sis the spatial spin-density pseudovector (2.4.186). The quantityp
s2is proportional to the
square of the proper particle number density, n2.
85
2.5.13 Raychaudhuri equation
We consider a congruence of particles with four-velocity ui. We de ne the expansion/contraction
scalar, the traceless shear tensor ik, and the antisymmetric rotation/vorticity tensor !ikaccording
to
=ui
:i; (2.5.119)
ik=u(i:k) 1
3hik w(iuk); (2.5.120)
!ik=u[i:k] w[iuk]; (2.5.121)
wherewiis the four-acceleration (1.6.117). Expansion has >0 and contraction has <0. These
de nitions give
ui:jhj
k=ik+!ik+1
3hik: (2.5.122)
Contracting ui
:jk ui
:kj=Ri
lkjulwith respect to the indices i;jgives:kuk uj
:kjuk= Rklukul
or
Rklukul=:kuk wi
:i+ui:kuk:i: (2.5.123)
De ning
2=1
2ikik; !2=1
2!ik!ik(2.5.124)
gives
ui:kuk:i= 2(2 !2) +1
32; (2.5.125)
which brings (2.5.123) to the Raychaudhuri equation :
d
ds= 2(2 !2) 1
32+wi
:i Rikuiuk: (2.5.126)
For a perfect
uid, the Einstein equations give
Rikuiuk=
2(+ 3p): (2.5.127)
We de ne four energy conditions. The null energy condition is satis ed if
Tijkikj0 (2.5.128)
for any null, future-pointing vector ki. For a perfect
uid, this condition gives
+p0: (2.5.129)
The weak energy condition is satis ed if
Tijuiuj0 (2.5.130)
for any causal (null or timelike), future-pointing vector ui. For a perfect
uid, this condition gives
+p0; 0: (2.5.131)
The strong energy condition is satis ed if
Tij 1
2Tgij
uiuj0 (2.5.132)
for any causal, future-pointing vector ui. For a perfect
uid, this condition gives
+p0; + 3p0: (2.5.133)
86
The dominant energy condition is satis ed if the weak condition is satis ed and Tijujis a causal,
future-pointing vector. For a perfect
uid, this condition gives
+p0; jpj: (2.5.134)
This condition guarantees that particles in a congruence do not move faster than light. The dominant
condition implies the weak condition. The weak condition implies the null condition. The strong
condition implies the null condition.
If the strong condition is satis ed and particles in a congruence move without rotation and
acceleration ( !ik=wi= 0) then (2.5.126) and the Einstein equations give
d
d c
32: (2.5.135)
If0isat = 0 then
1 1
0+c
3: (2.5.136)
Therefore, if 0<0 (initial contraction) then diverges to a curvature singularity (a point in
spacetime where the density of matter and curvature are in nite) as increases:! 1 .
2.5.14 Event horizon
If a hypersurface Sin spacetime is given by an equation of constraint,
f(xi) = 0; (2.5.137)
then the vector normal to this hypersurface is
ni=@f
@xi: (2.5.138)
If such a vector is a null vector,
nini= 0; (2.5.139)
thenSis a null hypersurface. All in nitesimal displacements dxialong such a hypersurface satisfy,
according to (2.5.138),
df=nidxi= 0: (2.5.140)
Equations (2.5.139) and (2.5.140) indicate that nilies itself on the null hypersurface to which it is
normal,
dxi/ni; (2.5.141)
which also gives
ds2=dxidxi/nini= 0: (2.5.142)
Thus all world lines on a null hypersurface are null. The light cones at the points of such a hy-
persurface are tangent to this hypersurface. Since all physical world lines must lie within the local
light cones, the forward-time motion through a null hypersurface can occur in only one direction.
To avoid any discontinuities, this direction is the same for all points on such a hypersurface. A null
hypersurface is therefore an event horizon : a boundary in spacetime beyond which events cannot
a ect events on the other side. All laws of classical physics are known to be time-symmetric , that
is, symmetric under the transformation t! t. However, the existence of event horizons, which are
solutions to these laws and provide boundary conditions for spacetime, violates this symmetry. The
unidirectional character of the motion of matter through an event horizon can be used to de ne the
past and the future: the arrow of time .
References: [1, 2, 3, 4, 5, 6, 8].
87
2.6 Spinor elds
2.6.1 Dirac matrices
The Dirac matrices de ned by (1.7.1) are complex. A particular solution of (1.7.1) is given by the
Dirac representation :
0=I0
0 I
;
=0
0
; (2.6.1)
whereIis the unit 22 matrix and
1=0 1
1 0
; 2=0 i
i0
; 3=1 0
0 1
(2.6.2)
are the Pauli matrices (all indices are coordinate invariant). The Pauli matrices are traceless tr( ) =
0 and Hermitian y= (the Hermitian conjugation of a matrix Ais the combination of the
complex conjugation and transposition, Ay=AT), satisfy
= +i
; (2.6.3)
and their square is I. The identity (2.6.3) gives the anticommutation relation
h
2;
2i
=i
2; (2.6.4)
so
2form the lowest, two-dimensional representation of the angular momentum operator M
(1.6.77). The properties of imply that the Dirac matrices are traceless tr(
i) = 0 and satisfy
0y=
0;
y=
;
iy=
0
i
0;
i=
2
i
2: (2.6.5)
The relation (1.7.1) yields the total antisymmetry of
0
1
2
3:
0
1
2
3=
[0
1
2
3]: (2.6.6)
We de ne
5= i
24eijkl
i
j
k
l=i
0
1
2
3; (2.6.7)
which is traceless tr(
5) = 0 and Hermitian
5y=
5, and satis es
f
i;
5g= 0;(
5)2= 1: (2.6.8)
In the Dirac representation,
5=
5=0I
I0
: (2.6.9)
The anticommutation relation (1.7.1) gives
i
i= 4; (2.6.10)
i
j
i= 2
j; (2.6.11)
i
j
k
i= 4jkI; (2.6.12)
i
j
k
l
i= 2
l
k
j; (2.6.13)
i
j
k=ij
k+jk
i ik
j+ieijkl
l
5: (2.6.14)
The Dirac representation is not unique; the relation (1.7.1) is invariant under a similarity trans-
formation
i!S
iS 1, whereSis a nondegenerate (det S6= 0) matrix. Accordingly, !S
and ! S 1. TakingS=1p
2I I
I I
turns the Dirac representation into the chiral orWeyl
representation , in which
0=0I
I0
;
=0
0
;
5= I0
0I
: (2.6.15)
88
For an in nitesimal Lorentz transformation (1.6.7), the relations (1.7.5) and (1.7.6) give L=
I+1
8ab(
a
b
b
a), so
Ly=I+1
8ab(
by
ay
ay
by) (2.6.16)
is equal to L 1(soLis unitary) for rotations and equal to Lfor boosts. The relation (2.6.5) gives
then
Ly
0=
0+1
8ab(
by
ay
ay
by)
0=
0 1
8ab
0(
a
b
b
a) =
0L 1: (2.6.17)
Thus the quantity y
0transforms under (1.7.7) like an adjoint spinor:
y
0! yLy
0= y
0L 1: (2.6.18)
The spinors and y
0can be used to construct tensors, as in (1.7.11): y
0 transforms like
a scalar, y
0
i is a vector, y
0
[i
j] is an antisymmetric tensor, y
0
5 is a pseudoscalar
(dual scalar density), and y
0
i
5 is a pseudovector (dual vector density). Higher-rank tensors
constructed from and y
0reduce to the above 5 kinds of tensors because of (2.6.14). Hereinafter,
we will use to denote y
0. To show that
5 transforms like a pseudoscalar, we substitute
(1.7.4) into (2.6.7) and use (2.6.6), which gives
5=i0
a1
b2
c3
dL
a
b
c
dL 1=i0
a1
b2
c3
dL
[a
b
c
d]L 1: (2.6.19)
Using (1.1.22) and
[a
b
c
d]= ieabcd
5; (2.6.20)
which results from (2.6.7), gives
5=eabcd0
a1
b2
c3
dL
5L 1= det(a
b)L
5L 1: (2.6.21)
Therefore we have
5 ! L 1det(a
b)L
5L 1L = det(a
b)
5 ; (2.6.22)
which is the transformation law for a Lorentz scalar density. Similarly,
c
5 ! L 1c
ddet(a
b)L
d
5L 1L = det(a
b)c
d
d
5 ; (2.6.23)
which is the transformation law for a Lorentz vector density.
We de ne the chirality projection operators
P=I
5
2; P++P =I; P2
=I; P +P =P P+= 0: (2.6.24)
They project a spinor into the right-handed spinor Rand left-handed spinor L,
R=P+ ; L=P ; = R+ L: (2.6.25)
If we split a spinor into two two-component parts,
=u
v
; (2.6.26)
then, in the chiral representation:
L=
u
0
; R=
0
v
: (2.6.27)
An in nitesimal rotation is described by a Lorentz matrix (1.6.7) with 0 = 0. The correspond-
ing spinor transformation matrix (1.7.5) in the Dirac representation is, using (1.6.25),
L=I+1
4
=I+1
4e
#
=I i
2# 0
0
: (2.6.28)
89
Thus the (unitary) spinor transformation matrix for a nite rotation by an angle #about an axis
parallel to a unit vector n,#=#n, is
L= exp"
i
2#
0
0#
= cos#
2I isin#
2n
0
0
: (2.6.29)
If we split a spinor into two parts uandv(2.6.26), then both uandvtransform under rotations
according to
u!Su; v!Sv; (2.6.30)
where
S= cos#
2I isin#
2n: (2.6.31)
A rotation by a full angle 2 changes the sign of a spinor. A rotation by 4 brings a spinor into its
original position.
An in nitesimal boost is described by a Lorentz matrix (1.6.7) with = 0. The corresponding
spinor transformation matrix (1.7.5) in the Dirac representation is, using (1.6.26),
L=I+1
20
0
=I+1
2
0
=I+1
2
0
0
: (2.6.32)
Thus the spinor transformation matrix for a nite boost with a rapidity along an axis parallel to
a unit vector n,=n, is
L= exp"
1
2
0
0#
= cosh
2I+ sinh
2n
0
0
: (2.6.33)
Using (1.6.90) gives
L=1p
2(1 +
)1 +
1 +
: (2.6.34)
If this boost transforms a particle of mass mfrom rest to a motion with momentum pand energy
Ethen (2.6.34) is equivalent, due to (2.4.209) and (2.4.210), to
L=1p
2mc2(E+mc2)E+mc2cp
cpE+mc2
: (2.6.35)
2.6.2 Dirac equation
A Lagrangian density for dynamical spinor elds must contain rst derivatives of spinors. The
simplest scalar containing derivatives of spinors is quadratic in ,
i ;i, where ;iis the covariant
derivative of (1.7.14). This quantity is complex. In the locally inertial frame of reference, its
complex conjugate is
(
i ;i)= (
i ;i)y= y
;i
iy y= ;i
0
iy
0 = ;i
i ; (2.6.36)
so both
i ;i+ ;i
i andi(
i ;i ;i
i ) are real. The former is, however, equal to a total
divergence (
i );i, so a Lagrangian density proportional to such term does not contribute to eld
equations. Thus the simplest dynamical part of a spinor Lagrangian density is proportional to
i(
i ;i ;i
i ). Another scalar that can be used in a spinor Lagrangian is proportional to .
Therefore the simplest Lagrangian density for spinor elds, in the locally Galilean frame of reference,
has the form
L =i
2(
i ;i ;i
i ) m ; (2.6.37)
wheremis a real scalar constant called the spinor mass , and it is referred to as the Dirac Lagrangian
density . For any frame of reference,
L =ie
2(
i ;i ;i
i ) me =ie
2ei
a(
a ;i ;i
a ) me : (2.6.38)
90
We consider the metric formulation of gravity with the Einstein-Hilbert action (2.5.1). Therefore
spacetime has the Riemannian geometry, so ;i= :i. Varying (2.6.38) with respect to and
omitting total derivatives gives
L = (i
i :i m ); (2.6.39)
so the stationarity of the action S= 0 under gives the Dirac equation :
i
i :i=m : (2.6.40)
Varying (2.6.38) with respect to and omitting total derivatives gives the adjoint conjugate of
(2.6.40):
i :i
i=m : (2.6.41)
The Dirac equation is linear in , so can be multiplied by an arbitrary constant without altering
(2.6.40). Varying (2.6.38) with respect to ei
agives the tetrad energy-momentum density for the
spinor eld,
Ta
i=ie
2(
a :i :i
a ea
i
j :j+ea
i :j
j ) +meea
i ; (2.6.42)
so
Tik=i
2(
(k :i) :(i
k) gik
j :j+gik :j
j ) +mgik : (2.6.43)
The conservation law (2.4.24) applied to the energy-momentum tensor (2.6.43) gives the Dirac
equations (2.6.40) and (2.6.41).
Subtracting (2.6.41) multiplied by from (2.6.40) multiplied by gives, using (1.7.33) and
ji= :i,
(
i )ji= (
i ):i= 0; (2.6.44)
so the vector density
ji
V=e
i ; (2.6.45)
called the vector Dirac current , is conserved: ji
V;i= 0 or
@
@t+rj= 0; (2.6.46)
where
c=e y ;j=e
: (2.6.47)
The vector Dirac current is real because
(
i )= ( y
0
i )y= y
iy
0 = y
0
i =
i : (2.6.48)
The spinor density is real and positive. The conservation law (2.6.46) is referred to as the equation
of continuity, like (2.5.96).
The Dirac equation (2.6.40) gives
j(
i :i)jj=im
j :j (2.6.49)
or, due to (1.7.33),
j
i :ij=m2 : (2.6.50)
Using (1.7.30) (in the absence of torsion) and (1.7.39) turns (2.6.50) into the Klein-Gordon-Fock
equation :
i
:i+m2 =1
8Pklij
k
l
i
j ; (2.6.51)
wherePijklis the Riemann curvature tensor. If a spinor is equal to its either left- or right-handed
projection, = Lor = R, then it is called a Weyl spinor . Multiplying (2.6.40) by Pgives
iP
i :i=i
iP :i=mP (2.6.52)
or
i
i L(R)
:i=m R(L): (2.6.53)
Thus if is a Weyl spinor then m= 0.
91
2.6.3 Spinors in Einstein-Cartan-Sciama-Kibble gravity
We consider the metric-ane formulation of gravity with the Einstein-Cartan action (2.5.47), in
which spacetime has the Riemann-Cartan geometry. Varying (2.6.38) with respect to ei
agives the
tetrad energy-momentum density for the spinor eld,
Ta
i=ie
2(
a ;i ;i
a ea
i
j ;j+ea
i ;j
j ) +meea
i : (2.6.54)
Putting the de nition of the covariant derivative of a spinor (1.7.14) into (2.6.38) gives
L =ie
2(
i ;i ;i
i ) ie
2 f
i; ig me : (2.6.55)
Using the Fock-Ivanenko coecients (1.7.28) as the spinor connection iturns (2.6.55) into
L =ie
2(
i ;i ;i
i ) +ie
8!abi f
i;
a
bg me : (2.6.56)
The spin density (2.3.17) corresponding to the Lagrangian density (2.6.56) is, due to the identity
f
i;
j
kg= 2
[i
j
k],
Sijk=ie
2
[i
j
k] : (2.6.57)
The spin density (2.6.57) is totally antisymmetric,
Sijk=S[ijk]; (2.6.58)
and independent of the spinor mass m. The corresponding spin tensor is also totally antisymmetric,
sijk=i
2
[i
j
k] =s[ijk]= eijklsl; (2.6.59)
where
si=1
2
i
5 : (2.6.60)
The second Einstein-Cartan equation (2.5.57) for the spin tensor (2.6.59) gives a totally antisym-
metric torsion tensor,
Sijk= i
4
[i
j
k] ; (2.6.61)
soSi= 0. Thus the contortion tensor is, using (2.6.14),
Cijk=
4ijkl
l
5 : (2.6.62)
The pseudovector density
ji
A=e
i
5 = 2esi(2.6.63)
is called the axial Dirac current . The axial Dirac current and the pseudovector (2.6.60) are real
because
(
i
5 )= ( y
0
i
5 )y= y
5
iy
0 = y
5
0
i = y
0
i
5 =
i
5 : (2.6.64)
Thus the spin tensor (2.6.59) is also real.
Varying (2.6.56) with respect to gives, after omitting total divergences,
i
2
e
k ;k+ (e
k );k ef k;
kg
em = 0: (2.6.65)
Substituting
(e
k );k=e
k ;k+e
k
;k 2eSk
k =e
k ;k+e[ k;
k] (2.6.66)
92
into (2.6.65) gives
i
k ;k i
k k m =i
k ;k m = 0: (2.6.67)
The relation (1.5.33) gives
;k= :k+1
4Cijk
i
j ; (2.6.68)
from which we obtain, upon substituting (2.6.62),
k ;k=
k :k+
16ijkl(
l
5 )
k
i
j =
k :k+i
16ijkl(
l
5 )ijkm
m
5
=
k :k 3i
8(
l
5 )
l
5 : (2.6.69)
Therefore (2.6.67) becomes the Hehl-Datta equation :
i
k :k+3
8(
k
5 )
k
5 =m : (2.6.70)
Varying (2.6.56) with respect to gives the adjoint conjugate of (2.6.70),
i :k
k+3
8(
k
5 )
k
5=m : (2.6.71)
The Hehl-Datta equation (2.6.70) di ers from the Dirac equation (2.6.40) by a nonlinear term,
cubic in the spinor eld and representing a spinor self-interaction, corresponding to a spin-spin
interaction in the tensor (2.5.58). The conservation law (2.4.32) applied to the energy-momentum
density (2.6.54) gives the Hehl-Datta equations (2.6.70) and (2.6.71). Subtracting (2.6.71) multiplied
by from (2.6.70) multiplied by gives the conservation of the vector Dirac current (2.6.45). In
the metric formulation of gravity, (2.6.70) and (2.6.71) are the eld equations corresponding to the
Hehl-Datta Lagrangian density :
L =ie
2(
i :i :i
i ) me +3e
16(
k
5 )(
k
5 ): (2.6.72)
The total antisymmetry of the spin density (2.6.58) implies
Nijk=N[ijk]; (2.6.73)
whereNijkis given by (2.4.106). We also have
Nijk= 3S[ijuk]; (2.6.74)
whereSijis the intrinsic spin tensor (2.4.175). The covariant (with respect to the Levi-Civita
connection) change (2.4.148) of the spin pseudovector along a world line becomes
DfgJi
ds= uiDfguk
dsJk+Dfgui
dsukJk+3
2eij
nmujSn
ikNikm= 3Si
jkujNk; (2.6.75)
where
Ni=1
6eijklNjkl: (2.6.76)
IfNi/Jithen (2.6.75) gives JiJi= const. For a point or a system of points, Mi[jk]given by
(2.4.105) vanishes. Thus (2.4.131) reduces to
Nijl=ul
u0Nij0; (2.6.77)
which for a spinor particle gives Nil0= ul
u0Ni00and thusNijk= 0 or
= 0: (2.6.78)
93
Therefore a spinor eld in the Einstein-Cartan-Sciama-Kibble gravity cannot be approximated as a
point particle or a system of point particles.
For a totally antisymmetric spin tensor, the correction to the dynamical energy-momentum tensor
from the spin contributions to the geometry of spacetime (2.5.58) reduces to
Uik=
4
sijlsk
jl 1
2giksjlmsjlm
: (2.6.79)
The relation (2.6.74) suggests that
sijk= 3s[ijuk]; (2.6.80)
wheresijis the spin-density tensor (2.4.181). Analogously to (2.4.145), we can de ne the spin-density
pseudovector
si=1
2eijklujskl; (2.6.81)
which is orthogonal to ui,
siui= 0: (2.6.82)
The inverse relation is
sik= eikjlujsl: (2.6.83)
The spin-density pseudovector (2.6.81) coincides with the pseudovector (2.6.60).
2.6.4 Discrete symmetries of spinors
The spinor representation of the parity transformation (1.6.3) is given by
LP=C
0; (2.6.84)
whereC= const. Indeed, substituting (3.1.170) into (1.7.4) gives, using (2.6.5),
a= a
b
0
b
0= a
b
by; (2.6.85)
which holds if a
b= a
b(P). Since the double parity transformation is equivalent to the iden-
tity transformation, L2
P=I, we haveC=1. The spinor representation of the time-reversal
transformation (1.6.4) is given by
LT=C
0
5; (2.6.86)
whereC= const. Indeed, substituting (3.1.172) into (1.7.4) gives, using (2.6.5) and (2.6.8),
a= a
b
0
5
b
5
0= a
b
by; (2.6.87)
which holds if a
b= a
b(T). Since the double time-reversal transformation is equivalent to the
identity transformation, L2
T=I, we haveC=i.
The charge conjugation of a spinor is de ned as
c= i
2 ; = i
2 c: (2.6.88)
The double charge-conjugation transformation is equivalent to the identity transformation:
( c)c= i
2( c)= i
2( i
2 )=
2
2 = : (2.6.89)
The charge conjugation of the left-handed projection of a spinor is the right-handed projection of
the charge conjugation of the spinor and vice versa (cf. (2.6.25)):
(I
5) c
= i
2
(I
5)
= i
2(I
5) = i(I
5)
2
= (I
5)
c: (2.6.90)
References: [3, 4, 11].
94
2.7 Electromagnetic eld
2.7.1 Gauge invariance and electromagnetic potential
The Lagrangian density (2.6.38) is a real combination of the complex Dirac matrices
iand spinors
, . It is invariant under a gauge transformation of the rst type of the spinor elds,
! 0=eie ; ! 0=e ie ; (2.7.1)
ife is a real scalar constant, but it is not invariant for e (xi), because
0
;=eie ( ;+ie ; ): (2.7.2)
Introduce a compensating vector eld A, called the electromagnetic potential , such that the Weyl
orelectromagnetic covariant derivative
D=r ieA (2.7.3)
of a spinor ,
D = ; ieA ; (2.7.4)
transforms under (2.7.1) like :
D 0=eie D : (2.7.5)
This requirement gives
0
; ieA0
0=eie ( ; ieA ); (2.7.6)
which, with (2.7.1) and (2.7.2), yields the transformation law for the electromagnetic potential,
A0
=A+ ;; (2.7.7)
called a gauge transformation of the second type . The real scalar constant eis called the spinor
electric charge . The adjoint conjugation of (2.7.4) is
D = ;+ieA
: (2.7.8)
The scalar is invariant under (2.7.1), so
D( ) =@( ); (2.7.9)
which constraints the electromagnetic potential to be real:
A
=A: (2.7.10)
The time component of A, =A0, is called the electric potential and the spatial components
A form the magnetic potential A:
A= ( ;A): (2.7.11)
The gauge transformation (2.7.7) reads
0= +@
c@t;A0=A r : (2.7.12)
In the local Minkowski spacetime, Atransforms according to (1.6.99),
A
=
1 +(
1)
2 !
0
A0
: (2.7.13)
The gauge-invariant modi cation of the Dirac Lagrangian density (2.6.38) is
L =ie
2ei
a(
aDi Di
a ) me : (2.7.14)
95
The spin density corresponding to the Lagrangian density (2.7.14) remains equal to (2.6.57); it is
independent of the electric charge e. The electromagnetic potential corresponds, up to the multipli-
cation by an arbitrary constant, to the vector multiple of Iin the formula for the spinor connection
(1.7.26). The electromagnetic potential is analogous to the ane connection: it modi es a derivative
of a spinor so such derivative transforms like a spinor under unitary gauge transformations of the
rst type, while the connection modi es a derivative of a tensor so such derivative transforms like a
tensor under coordinate transformations.
The gauge-invariant modi cation of the Dirac equation (2.6.40) is
i
k :k+eAk
k =m ; (2.7.15)
whose adjoint conjugate is
i :k
k+eAk
k=m : (2.7.16)
The gauge-invariant modi cation of the Hehl-Datta equation (2.6.70) is
i
k :k+eAk
k =m 3
8(
5
k )
5
k ; (2.7.17)
whose adjoint conjugate is
i :k
k+eAk
k=m 3
8(
5
k )
5
k: (2.7.18)
Taking the complex conjugate of (2.7.17) gives, using (2.6.64),
i
k
:k+eAk
k =m 3
8(
5
k )
5
k : (2.7.19)
Using (2.6.5), (2.6.8), (2.6.9) and (2.6.88) turns (2.7.19) into
i
k c
:k eAk
k c=m c 3
8(
5
k )
5
k c: (2.7.20)
Finally, one can show that
5
k = (
5
k )= c
5
k c; (2.7.21)
so (2.7.20) reads
i
k c
:k eAk
k c=m c+3
8( c
5
k c)
5
k c: (2.7.22)
Comparing (2.7.22) with (2.6.70) shows that and csatisfy, apart from the cubic term, the same
equation for eand e, respectively. Thus the charge-conjugation transformation changes sign of the
spinor electric charge. In the Einstein gravity, the cubic term vanishes; ( ;Ai) and ( c; Ai) are
symmetric under the charge-conjugation transformation. In the Einstein-Cartan gravity, the cubic
term does not vanish and it changes sign under the charge-conjugation transformation. In this case,
( ;Ai) and ( c; Ai) are not symmetric under the charge-conjugation transformation: the torsion
generates an asymmetry between a spinor and its charge conjugate.
2.7.2 Electromagnetic eld tensor
The commutator of total covariant derivatives of a spinor is given by (1.7.30) with the curvature
spinorKijgiven by (1.7.36), where the tensor Bijis related to the vector Aiin (1.7.26) by (1.7.38).
Therefore the commutator of the electromagnetic covariant derivatives of a spinor, [ Di;Dj] , is
given by (1.7.30) with the curvature spinor
Kij=1
4Rklij
k
l+ieFijI; (2.7.23)
where the antisymmetric tensor
Fij=Aj;i Ai;j=Aj:i Ai:j (2.7.24)
96
is referred to as the electromagnetic eld tensor . The electromagnetic eld tensor is analogous to
the curvature tensor: it appears in the expression for the commutator of electromagnetic covariant
derivatives of a spinor, while the curvature tensor appears in the expression for the commutator of
coordinate-covariant derivatives of a tensor. Substituting (2.7.7) into (2.7.24) gives
F0
ij=Fij; (2.7.25)
so the electromagnetic eld tensor is gauge invariant. The de nition (2.7.24) is equivalent to the
rst Maxwell-Minkowski equation
Fij;k+Fjk;i+Fki;j=Fij:k+Fjk:i+Fki:j= 0 (2.7.26)
or
ijklFjk;l=ijklFjk:l= 0: (2.7.27)
We de ne a spatial vector Ewhose covariant components are related to the 0 components of
the electromagnetic eld tensor (2.7.24):
E =F0 ; (2.7.28)
and a spatial tensor B equal to the spatial part of (2.7.24):
B =F ; B = 1
2ps
B
; B = ps
B
; (2.7.29)
where sis given by (1.4.110). The component of (2.7.26) with all spatial indices, B ;
+B
; +
B
; = 0, gives, using (1.4.126),
divB= 0: (2.7.30)
The components of (2.7.26) with one temporal index, B ;0+E ; E ; = 0, gives, using (1.4.127),
curlE= 1
cps@(psB)
@t: (2.7.31)
The spatial vector Eis called the electric eld and the spatial pseudovector Bis the magnetic eld .
In the locally geodesic and Galilean frame of reference, these elds depend on the components
of the electromagnetic potential (2.7.11) according to (2.7.24):
E= @A
c@t r ; (2.7.32)
B=rA; (2.7.33)
and they are invariant under (2.7.12). The tensor Fijis given by
Fij=0
BB@0ExEyEz
Ex 0 BzBy
EyBz 0 Bx
Ez ByBx 01
CCA;E= (F01;F02;F03);B= (F32;F13;F21); (2.7.34)
and transforms according to (1.6.100). Thus the electric and magnetic elds transform according to
E=
(E0 B0) +1
2( E0) ; (2.7.35)
B=
(B0+ E0) +1
2( B0) : (2.7.36)
In this frame, (2.7.30) and (2.7.31) become the rst pair of the Maxwell equations :
divB= 0; (2.7.37)
curlE= @B
c@t: (2.7.38)
97
Applying the div operator to (2.7.33) gives (2.7.37) and applying the curl operator to (2.7.32) gives
(2.7.38). Applying the div operator to (2.7.38) gives (2.7.37). Integrating the rst pair of the
Maxwell equations over the volume and surface area, respectively, gives
I
Bdf= 0; (2.7.39)
I
Edl= @
c@tZ
Bdf
: (2.7.40)
The integralH
Adfis the
ux of a vector Athrough the surface fand the integralH
Adlis called
thecirculation ofAalong the contour l. Thus the
ux of the magnetic eld through a closed surface
vanishes and the circulation of the electric eld along a contour, which is called the electromotive
force, is equal to the minus time derivative of the
ux of the magnetic eld through the surface
enclosed by this contour ( Faraday's law ).
In a locally Galilean frame of reference, the simplest invariants (under proper Lorentz transfor-
mations) of the electromagnetic eld are quadratic in Fij:
FijFij= 2(B2 E2) = const; eijklFijFkl= 8EB= const: (2.7.41)
If the vectors EandBare mutually perpendicular in frame K,EB= 0, then they are mutually
perpendicular in other inertial frames. If EandBare equal in magnitude in K,B2 E2= 0, then
they are equal in magnitude in other inertial frames. The transformation laws (2.7.35) and (2.7.36)
imply that if E0= 0 in the frame K0then in the frame K
E=
B0= B; (2.7.42)
and if B0= 0 in the frame K0then in the frame K
B=
E0= E: (2.7.43)
If the vectors EandBare mutually perpendicular in K, but not equal in magnitude, then there
exists a frame K0in which the eld is either electric, B0= 0 (ifE > B ), or magnetic, E0= 0 (if
E <B ). The velocity of K0relative toKis perpendicular to EandB, and it is equal in magnitude
to respectively either cB
EorcE
B. Equivalently, if one of the vectors E;Bvanishes in one frame of
reference then these vectors are mutually perpendicular in other inertial frames. Except for the case
where the vectors EandBare mutually perpendicular and equal in magnitude, there exist frames
in which these vectors are parallel to each other at a given point. These frames move relative to one
another with velocities parallel to both vectors. One of such frames, K0(in which E0kB0), has a
velocity Vrelative toKwhich is perpendicular to both vectors EandB. Substituting the formulae
E0=
(E+ B) +1
2( E) ; (2.7.44)
B0=
(B E) +1
2( B) ; (2.7.45)
which are inverse to (2.7.35) and (2.7.36), into the condition E0B0= 0 and using =kEB,
wherekis a constant of proportionality, gives
E+k(EB)B
B k(EB)E
= 0 or
k=1+ 2
E2+B2, so
V
c
1 +V2
c2=EB
E2+B2: (2.7.46)
2.7.3 Lagrangian density for electromagnetic eld
The simplest gauge-invariant Lagrangian density representing the electromagnetic eld is a linear
combination of terms quadratic in Fij:p gFijFijandijklFijFkl, which a locally Galilean frame
of reference reduce to (2.7.41). The second term is a total divergence because of (2.7.27):
ijklFijFkl= 2(ijklFijAl);k; (2.7.47)
98
so it does not contribute to the eld equations. Thus the Lagrangian density for the electromagnetic
eld is given by
LEM= 1
16p gFijFij; (2.7.48)
where the Gauian factor1
16sets the units of Ai. In the locally geodesic and Galilean frame of
reference, (2.7.48) becomes
LEM=1
8(E2 B2): (2.7.49)
Therefore in order for the action Sto have a minimum, there must be the minus sign in front of
the right-hand side of (2.7.48). Otherwise an arbitrarily rapid change of Ain time would result in
an arbitrarily large value of E, according to (2.7.32), and thus an arbitrarily low value of S, so the
action would have no minimum. A generalization of the tensor (2.7.24) to a covariant derivative
with respect to the ane connection k
ij,Aj;i Ai;j=Fij+ 2Sk
ijAk, is not gauge invariant, so
the torsion tensor cannot appear in a gauge-invariant Lagrangian density which is quadratic in Fij.
Thus the electromagnetic eld, unlike spinor elds, does not couple to torsion.
2.7.4 Electromagnetic current
We de ne the electromagnetic current density
ji= cLm
Ai; (2.7.50)
and the electromagnetic current vector
ji=ji
p g: (2.7.51)
The invariance of the action under an arbitrary in nitesimal gauge transformation Ai=A0
i Ai=
;igives, upon partial integration and omitting a total divergence,
S= 1
c2Z
jiAjd
= 1
c2Z
ji ;id
=1
c2Z
ji
;i d
= 0; (2.7.52)
so the electromagnetic current is conserved,
ji
;i= 0; ji
:i= 0: (2.7.53)
The gauge-invariant Lagrangian density (2.7.14) for spinor matter is
L =ie
2ei
a(
a ;i ;i
a ) me eAie
i ; (2.7.54)
so the electromagnetic current for the spinor eld is
ji=ece
i ; (2.7.55)
which is proportional to the conserved vector Dirac current (2.6.45). The electromagnetic current
density (2.7.50) corresponds to the current (2.4.7) with i=0 and =ie (which is equal to the
in nitesimal 0 due to (2.7.1)).
We consider matter which is distributed over a small region in space, as in section (2.4.7). Inte-
grating (2.7.53) over the volume hypersurface and using Gau-Stokes theorem to eliminate surface
integrals gives Z
j0
;0dV= 0: (2.7.56)
The conservation law (2.7.53) also gives
(xkji);i=xk
;iji+xkji
;i=k
iji=jk; (2.7.57)
99
which, upon integrating over the volume hypersurface and using Gau-Stokes theorem to eliminate
surface integrals, givesZ
xkj0dV
;0=Z
jkdV: (2.7.58)
Using (2.4.103) turns (2.7.58) into
uk
u0Z
j0dV+Z
xkj0dV
;0=Z
jkdV: (2.7.59)
For a particle located at xa,R
xkj0dV= 0 and ji(x) is thus proportional to (x xa), so
jk=uk
u0j0: (2.7.60)
We de ne the electric charge density such that
j0=cpg00: (2.7.61)
The electric charge density is not a tensor density. We de ne the electric charge esuch that
psdV=de: (2.7.62)
The electric charge density for particles with charges ealocated at xais
(x) =X
aeaps(x xa); (2.7.63)
andR
j0dV(which is equal toR
jidSifor a volume hypersurface, so it is a scalar) is
Z
j0dV=X
aZp gceapg00ps(x xa)dV=cX
aea; (2.7.64)
so the electric charge is a scalar. Thus the electromagnetic current vector for a system of charged
particles is
jk(x) =X
acuk
u0eap g(x xa); (2.7.65)
analogously to (2.4.201). The relation (2.7.56) represents the conservation of the total electric charge
of a physical system. For a particle moving along a wordline xa( ), we have
jk(x) =cuk
u0ep g(x xa) =ecZuk
p g(x xa( ))d : (2.7.66)
In the locally geodesic and Galilean frame of reference,ui
u0= (1;v=c), so
ji= (c;j); (2.7.67)
where jis the spatial current vector ,
j=v: (2.7.68)
The conservation law (2.7.53) in this frame, ji
;i= 0, has the form of the equation of continuity
(2.6.46). For one particle located at x0(t),(x) =e(x x0), (2.6.46) is explicitly satis ed since
@
@t=e@
@t(x x0) =ev@
@x0(x x0) = ev@
@x(x x0)
= @
@x
ev(x x0)
= rj; (2.7.69)
100
where v=dx0
dt. For a system of charged particles, we also have
Z
jdV=X
aeava: (2.7.70)
The equation of continuity (2.6.46) represents, upon integrating over the volume, the conservation
of the total electric charge:
@
@t Z
dV!
+I
jdf= 0: (2.7.71)
2.7.5 Maxwell equations
The total Lagrangian density for the electromagnetic eld and matter is the sum of (2.7.48) and the
term p gAijidue to (2.7.50):
LEM= 1
16p gFikFik 1
cp gAkjk; (2.7.72)
where we omit the terms corresponding to the gravitational eld and matter which does not depend
onAk. Varying (2.7.72) with respect to Ak, integrating partially and omitting total divergences
gives
LEM= 1
8p gFikFik jk
cAk= 1
8p gFik(Ak;i Ai;k) jk
cAk
=1
4p gFikAk;i jk
cAk=1
4(p gFik);iAk 1
cp gjkAk; (2.7.73)
so the principle of least action S= 0 for arbitrary variations Akyields the second Maxwell-
Minkowski equation
(p gFik);i=4
cjk(2.7.74)
or
Fik
:i=4
cjk: (2.7.75)
The electromagnetic eld equation (2.7.74) implies that jiis conserved, ji
;i= 0, which corresponds
to the conservation of the total electric charge, but does not constrain the motion of particles.
Therefore a con guration of charged particles producing the electromagnetic eld can be arbitrary,
subject only to the condition that the total charge be conserved, unlike a con guration of particles
producing the gravitational eld which is not arbitrary but constrained by the gravitational eld
equations.
We de ne
D = pg00F0 ; (2.7.76)
H =pg00F ; H = 1
2ps
H
; H = 1ps
H
: (2.7.77)
The relations F0 =g0ig jFijandF =g ig jFijgive then
D =E pg00+g H ; (2.7.78)
B =H
pg00 g E +g E ; (2.7.79)
or, in the spatial-vector notation,
D=Epg00 gH; (2.7.80)
B=Hpg00+gE: (2.7.81)
101
Using (1.4.107) brings the temporal component of (2.7.74) to
1ps(psD ); = 4 (2.7.82)
or
divD= 4: (2.7.83)
The spatial components of (2.7.74) read
1ps(psH ); +1ps(psD );0= 4dx
dx0(2.7.84)
or
curlH=1
cps@(psD)
@t+4
cj: (2.7.85)
The conservation law (2.7.53) reads
1ps@(ps)
@t+ div j= 0: (2.7.86)
In the locally geodesic and Galilean frame of reference, (2.7.80) and (2.7.81) reduce to
D=E; (2.7.87)
B=H: (2.7.88)
In this frame, (2.7.83) and (2.7.85) become the second pair of the Maxwell equations :
divE= 4;) (2.7.89)
curlB=@E
c@t+4
cj: (2.7.90)
Applying the div operator to (2.7.90) and using (2.7.89) gives (2.6.46). Integrating the second pair
of the Maxwell equations over the volume and surface area, respectively, gives
I
Edf= 4q; (2.7.91)
I
Bdl=@
c@tZ
Edf
+4
cZ
jdf: (2.7.92)
Thus the
ux of the electric eld through a closed surface is proportional to the total charge inside
the volume enclosed by the surface f(Gau' law ) and the circulation of the magnetic eld along a
contour is equal to the time derivative of the
ux of the electric eld through the surface enclosed
by this contour, called the displacement current, plus the surface integral of the current vector (the
Amp ere- rsted law ).
The two pairs of the Maxwell equations are linear in the elds EandB. The sum of any two
solutions of the Maxwell equations is also a solution of these equations. Thus the electromagnetic
eld of a system of sources (particles) is the sum of the elds from each source. The additivity of
the electromagnetic eld is referred to as the principle of superposition .
2.7.6 Energy-momentum tensor for electromagnetic eld
The metric energy-momentum tensor (2.3.3) for the electromagnetic eld TEM
ik is given by the
Lagrangian density (2.7.48):
LEM=1
32p gglmFikFikglm 1
8p gFikFlmgilgkm
=1
8p g1
4gikFlmFlm Fj
iFkj
gik; (2.7.93)
102
so
TEM
ik=1
41
4gikFlmFlm Fj
iFkj
: (2.7.94)
The corresponding energy density W, energy current Scalled the Poynting vector , and stress tensor
called the Maxwell stress tensor , are given in the locally geodesic and Galilean frame of reference,
due to (2.4.71), by
W=1
8(E2+B2); (2.7.95)
S=c
4EB; (2.7.96)
=1
4
E E +B B 1
2 (E2+B2)
: (2.7.97)
Multiplying (2.7.38) by Band (2.7.90) by Eand adding these scalar products gives
1
cE@E
@t+1
cB@B
@t= 4
cjE (BcurlE EcurlB); (2.7.98)
from which we obtain1
2c@
@t(E2+B2) = 4
cjE div(EB) (2.7.99)
or@W
@t+jE+ div S= 0: (2.7.100)
Under Lorentz transformations, W,Sand transform like the corresponding components of a
tensor of rank (0,2) (2.4.71), according to (1.6.100).
The energy-momentum tensor for the electromagnetic eld is traceless,
TEM
ikgik= 0; (2.7.101)
so (2.4.202) and the virial theorem (2.4.203) remain unchanged if the particles interact electromag-
netically. The condition (2.7.101) also gives, using (2.4.88),
EM= 3pEM; (2.7.102)
so (2.4.204) implies that the free electromagnetic eld is ultrarelativistic. In a frame of reference, in
which the vectors EandBare parallel to one another or one of them vanishes (and the x-axis is
along the direction of these vectors), the nonzero components of the tensor Tikare
T00= T11=T22=T33=W: (2.7.103)
If the vectors E(along thex-axis) and B(along they-axis) are mutually perpendicular and equal
in magnitude then
T00=T03=T33=W: (2.7.104)
2.7.7 Lorentz force
We consider a charge particle interacting with the electromagnetic eld. The total energy-momentum
tensor for the particle and electromagnetic eld is covariantly conserved, which gives the motion of
the particle. The electromagnetic part yields, using (2.7.26) and (2.7.75),
Tk
i:k=1
41
2Flm:iFlm Fil:kFkl FilFkl
:k
=1
4
1
2Fmi:lFlm 1
2Fil:mFlm
Fil:kFkl FilFkl
:k
=1
4FilFkl
:k= 1
cFiljl: (2.7.105)
103
The particle part gives, using (2.4.195),
Tk
i:k=
c2uiuk
pg00u0!
:k; (2.7.106)
so we obtain
c2uiuk
pg00u0!
:k 1
cFiljl= 0: (2.7.107)
Multiplying (2.7.107) by uiand using (2.7.60) gives
c2uk
pg00u0!
:k; (2.7.108)
which turns (2.7.107) into
c2uk
pg00u0ui:k=1
cFilul
pg00u0(2.7.109)
or
mcDfgui
ds=e
cFijuj; (2.7.110)
which is the equation of motion of a particle of mass mand charge ein the electromagnetic eld
Fij. Multiplying (2.7.110) by uigives the identity, so (2.7.110) has 3 independent components. The
right-hand side of (2.7.110) is referred to as the Lorentz force .
In the locally geodesic and Galilean frame of reference,Dfg
ds=d
ds=u0
cd
dtandui= (
;
v=c), so
(2.7.110) reads (we choose the spatial components as the 3 independent ones)
mcdu
dt=eF 0+e
cF v (2.7.111)
or, using (2.4.187),
dP
dt=eE+e
cvB: (2.7.112)
The temporal component of (2.7.110) is
mcdu0
dt=e
cF0 v (2.7.113)
ordE
dt=evE; (2.7.114)
which also results from multiplying (2.7.112) by vand using (2.4.214). Integrating (2.7.100) over
the volume gives
@
@tZ
WdV +Z
jEdV+I
Sdf= 0; (2.7.115)
which, with (2.7.70) and (2.7.114), yields the conservation of the total energy (2.4.66) of the elec-
tromagnetic eld and particles:
@
@t Z
WdV +X
aEa!
+I
Sdf= 0: (2.7.116)
References: [2, 3].
104
3 Particles
3.1 Lagrangian mechanics
3.1.1 Coordinates, velocities and accelerations
For a system of particles, which is localized at a nite number of points in space, spatial coordinates qs
of particles can be taken for physical elds . Any quantities qswhich completely de ne the position
of a system of particles are called generalized coordinates of the system. They are di erentiable
functions of the time coordinate t, the rst derivatives _ qsare called generalized velocities of the
system, and the second derivatives qsare called generalized accelerations .
We consider the frame of reference which is locally geodesic and Galilean. The position of
the particle in space is de ned by its three-dimensional radius vector r, whose components are its
Cartesian coordinates x;y;z . The velocity of the particle is v=_rand its acceleration is a=r. We
consider a particle moving along a curve r(t). We de ne a tangent vector,
t=dr
ds; (3.1.1)
where
ds=jdrj=p
dx2+dy2+dz2 (3.1.2)
is the length of an in nitesimal arc of the curve. The tangent vector is a unit vector: jtj= 1. The
velocity is thus tangent to the path of the particle:
v=dr
dt=dr
ds_s= _st=vt: (3.1.3)
We de ne a normal vector,
n=dt
ds
(3.1.4)
where
= dt
ds (3.1.5)
is the curvature of the curve. The normal vector is also a unit vector: jnj= 1, and it is orthogonal
to the tangent vector: nt= 0, which follows from di erentiating tt= 1 with respect to s. The
inverse of the curvature is called the radius of curvature of the curve,
= 1: (3.1.6)
The acceleration of the particle is then, using (3.1.3),
a= st+ _s_t= st+ _sdt
dsds
dt= st+_s2
n; (3.1.7)
so it is the sum of the tangent acceleration st= _vtand the normal orcentripetal acceleration
_s2
n=v2
n.
Higher time derivatives of the radius vector, such as the jerkvector
j=_r; (3.1.8)
involve another unit vector,
b=tn; (3.1.9)
called the binormal vector. The three mutually perpendicular unit vectors t;n;bconstitute a Frenet
trihedron , which can be used as a local basis of a spatial frame of reference associated with a given
particle. We de ne the torsion of a curve,
=dn
dsb; (3.1.10)
105
whose inverse is called the radius of torsion of the curve,
= 1: (3.1.11)
The derivatives of the three vectors t;n;bwith respect to sdepend on these vectors according to
theFrenet-Serret formulae :
0
@dt=ds
dn=ds
db=ds1
A=0
@00
0
0 01
A0
@t
n
b1
A: (3.1.12)
3.1.2 Hamilton principle and Lagrange equations
According to the principle of least action or Hamilton's principle , the dynamics of every classical
system is governed by minimizing the action of the system (2.1.1). We consider a locally Galilean
frame of reference, in which generalized coordinates coincide with Cartesian radius vectors of parti-
cles. The energy-momentum tensor Tik(x) as a function of xfor one particle located at the position
x0is proportional to (x x0), and so is the Lagrangian density L. Thus we can use, instead of L,
the Lagrangian L=R
LdV(2.4.53). The action (2.1.1) becomes
S=Zt2
t1L(q;_q;t)dt; (3.1.13)
where the integration over tis between the instants t1andt2, andqdenotes the generalized coordi-
natesqs.
The principle of least action states that the dynamics of a mechanical system is given by the
condition (2.1.3): the system moves between the positions q(t1) andq(t2) at these instants in such
a way that the integral (3.1.13) takes the least possible value. In most cases Ldepends only qand
_q, but not higher derivatives with respect to t, so the mechanical state of the system is completely
de ned by the coordinates and velocities (a Lagrangian containing higher derivatives can always be
written in terms of rst derivatives by increasing the number of the coordinates qs). The condition
S= 0 gives, using _q=dq
dtand integrating by parts,
Zt2
t1L(q;_q;t)dt=Zt2
t1@L
@qq+@L
@_q_q
dt=Zt2
t1@L
@q d
dt@L
@_q
qdt
+@L
@_qqt2
t1= 0: (3.1.14)
Since the positions of the system are xed at t1andt2, the second term vanishes, so S= 0 for
arbitrary variations qgives the Euler-Lagrange or Lagrange equations for each degree of freedom
s= 1::n:d
dt@L
@_qs @L
@qs= 0: (3.1.15)
Thesensecond-order di erential equations give the relations between accelerations, velocities and
coordinates, i.e. they are the equations of motion of the system. The general solution of the equations
of motion contains thus 2 narbitrary constants, related to the initial velocities and coordinates. If
we do not x q= 0 att1andt2then the variation of the action (3.1.14) is, using the Lagrange
equations,
S=@L
@_qsqst2
t1: (3.1.16)
Adding the total derivative with respect to time of some function f(q;t) to a Lagrangian Lgives
a new Lagrangian L0,
L0=L+df
dt; (3.1.17)
106
the new action is thus
S0=S+f
q(t1);t1
f
q(t2);t2
; (3.1.18)
so the conditions S= 0 andS0= 0 are equivalent, and the Lagrange equations do not change.
Thus the Lagrangian of a system is de ned up to the additive total time derivative of an arbitrary
function of the coordinates and time. If a system consists of two noninteracting parts AandB,
with corresponding Lagrangian LA(qA) andLB(qB), then the Lagrangian for this system is the sum
LA+LB. This additivity of the Lagrangian means that the eld equations for either of the two parts
do not involve quantities pertaining to the other part. If LAalso depends on qBand/or _qB, and/or
LBdepends on qAand/or _qA, then the subsystems AandBinteract. Multiplying the Lagrangian L
by an arbitrary constant does not change the Lagrange equations for the corresponding system. The
additive property of Lagrangians admits only the simultaneous multiplication of the Lagrangians of
all the systems by the same constant, corresponding to choosing the unit of the Lagrangian.
If we vary the time, in addition to varying q, then the variation fof any function f(q;t) can be
decomposed into the part _ftresulting from varying tand the part fwhich equals fift= 0:
f=_ft+f: (3.1.19)
Thus the variation of the action is, usingd
dtf=_fand (3.1.19) for f=q,
S=_St+S=Ltjt2
t1+Zt2
t1Ldt =Ltjt2
t1+Zt2
t1@L
@qq+@L
@_q_q
dt
=Ltjt2
t1+Zt2
t1@L
@qq+@L
@_qd
dtq
dt=Ltjt2
t1+Zt2
t1@L
@q d
dt@L
@_q
qdt
+@L
@_qq t2
t1=Zt2
t1@L
@q d
dt@L
@_q
qdt +@L
@_qq @L
@_q_q L
tt2
t1: (3.1.20)
Ifq= 0 andt= 0 att1andt2then (3.1.20) gives for arbitrary variations qthe Lagrange
equations (3.1.15). If we do not x the endpoints t1andt2and the positions qat these instants
then the variation of the action (3.1.20) is, using the Lagrange equations,
S=@L
@_qq @L
@_q_q L
tt2
t1: (3.1.21)
3.1.3 Action for particles
The Lagrange equations (3.1.15) in the Cartesian coordinates are
d
dt@L
@v @L
@r= 0: (3.1.22)
The energy-momentum tensor for a point particle of mass mlocated at the radius vector r0is given
by (2.4.196):
Tik(r) =mc2(r r0)uiuk
p gu0; (3.1.23)
so the variation of the action with respect to the metric tensor (2.3.1) gives
S= 1
2cZ
Tikgikp gd
= mc
2Zuiuk
u0gikdx0= mc
2Z
uiukgikds
= mc
2Zgikdxidxk
ds= mcZ
p
gikdxidxk= mcZ
ds: (3.1.24)
Thus the action for a free(interacting only with the gravitational eld) particle is
S= mcZ2
1ds; (3.1.25)
107
where 1 and 2 denote the world points corresponding to the arrival of the particle at the initial and
nal position. The equation of motion of a free particle is thus the metric geodesic equation (1.4.80).
The action for a system of noninteracting particles is the sum of the actions corresponding to each
particle:
S= X
amacZ
dsa: (3.1.26)
If the particles interact with each other then we must add to (3.1.26) the action describing elds
which carry this interaction.
Multiplying the action (3.1.26) by an arbitrary constant does not change the Lagrange equations
for the corresponding system. The additive property of actions admits only the simultaneous multi-
plication of the masses of all the particles by the same constant, corresponding to choosing the unit
of the mass. The integralR2
1dshas its maximum value along a straight world line (cf. text after
(1.6.110)). By integrating dsalong a curved world line we can makeR
dsarbitrarily small. Thus if
m< 0 then the action (3.1.25) cannot have a minimum, so the mass of a particle must be a positive
quantity.
The form of the action for a free particle (3.1.25) can also be obtained from general considerations.
The action is a scalar integral whose integrand is a di erential of rst order. The only scalar of this
kind that one can construct for a free particle is proportional to the integral of the interval ds
along the world line of the particle between two given world points. The constant of proportionality
characterizes the particle and we denote it mc, calling the scalar mthe mass of the particle.
In a frame of reference which is inertial (locally geodesic and Galilean), space is homogeneous and
isotropic and time is homogeneous. The homogeneity of space and time implies that the Lagrangian
of a particle cannot depend explicitly on either rort, so it must be a function of vonly. The
isotropy of space implies that this Lagrangian cannot depend on the direction of v, so it must be a
function of v2. In an inertial frame, the action (3.1.25) is
S= mc2Z
1dt= mc2Zr
1 v2
c2dt: (3.1.27)
Thus the corresponding Lagrangian is
L= mc2r
1 v2
c2: (3.1.28)
For a free particle, the Lagrange equations in the Cartesian coordinates (3.1.22) give
@L
@v= const: (3.1.29)
SinceLis a function of the velocity only, we obtain
v= const; (3.1.30)
so a free particle moves with a velocity which is constant in both magnitude and direction (uniform
in a straight line). This is referred to as the law of inertia orNewton's rst law of dynamics . This
law is invariant under Lorentz transformations which preserve the Minkowski metric tensor. Thus
there exists an in nity of inertial frames, moving relative to one another uniformly in a straight
line, in which the properties of space and time and the laws of mechanics are the same (Einstein's
principle of relativity).
The electromagnetic current vector for a point particle of charge elocated at the radius vector
r0is given by (2.7.65):
jk(r) =cuk
u0ep g(r r0): (3.1.31)
Substituting (3.1.31) into the second term of (2.7.72) gives
S= 1
c2Zp gAkjkdV dx0= e
cZ
Akuk
u0dx0= e
cZ
Akdxk; (3.1.32)
108
so the total action for a particle of mass mand chargeeinteracting with the electromagnetic potential
Aiis
S= mcZ
ds e
cZ
Aidxi: (3.1.33)
For a system of particles, the total action is the sum of the actions (3.1.33) for each particle. Under
the gauge transformation (2.7.7), the action (3.1.33) changes by the integral of a total di erential:
S0= mcZ
ds e
cZ
Aidxi e
cZ
d ; (3.1.34)
so the conditions S= 0 andS0= 0 are equivalent, and the corresponding Lagrange equations are
gauge invariant.
The form of the term e
cR
Aidxiin (3.1.33) can also be obtained from general considerations.
The action is a scalar integral whose integrand is a di erential of rst order, and the only gauge-
invariant scalar of this kind that one can construct from the electromagnetic potential is proportional
to the integral of the product Aidxialong the world line of the particle between two given world
points. The constant of proportionality characterizes the interaction of the particle with the elec-
tromagnetic eld and we denote it e
c, calling the scalar ethe electric charge of the particle.
The variation of (3.1.25) with respect to the coordinates xigives, using (1.4.78), (1.4.79) and
(1.4.80),
S=mcZDfgui
dsxids Z
d(uixi)
: (3.1.35)
The variation ofR
Aidxiis
Z
Aidxi=Z
Aidxi+Z
Aidxi=Z
Ai;jxjdxi+Z
Aidxi
=Z
Ai;jxjdxi+Z
d(Aixi) Z
dAixi=Z
Ai;jxjdxi+Z
d(Aixi)
Z
Ai;jdxjxi=Z
Fijdxjxi+Z
d(Aixi)
=Z
Fijujxids+Z
d(Aixi): (3.1.36)
Thus the variation of (3.1.33) is, using the expression for the four-momentum pl=mcul(2.4.187),
S=mcZDfgui
dsxids e
cZ
Fijujxids mcZ
d(uixi) e
cZ
d(Aixi)
=ZDfgpi
ds e
cZ
Fijuj
xids Z
d
pixi+e
cAixi
=Z2
1Dfgpi
ds e
cZ
Fijuj
xids
pi+e
cAi
xi 2
1; (3.1.37)
where the limits 1 and 2 denote the endpoints of the particle's world line. The principle of least
actionS= 0 for arbitrary xivanishing at the endpoints gives the Lorentz equation of motion of
a particle of mass mand charge ein the electromagnetic eld Fij(2.7.110).
In a frame of reference which is locally geodesic and Galilean, the action (3.1.33) for a particle
in the presence of the electromagnetic eld gives
S=Z
mcds +e
cAdr e dt
(3.1.38)
or
S=Z
mc2
+e
cAv e
dt: (3.1.39)
109
Thus the Lagrangian for this particle is
L= mc2r
1 v2
c2+e
cAv e : (3.1.40)
The Lagrange equations (3.1.22) for the Lagrangian (3.1.40) give
d
dt
p+e
cA
=e
cr(Av) er =e
c(vr)A+e
cv(rA) er
=e
cdA
dt @A
@t
+e
cvB er ; (3.1.41)
which is equivalent to (2.7.112). The Lagrangian for a system of particles is the sum of the La-
grangians (3.1.40) for each particle:
L= X
mc2r
1 v2
c2+e
cAv e
: (3.1.42)
3.1.4 Conservation laws and integrals of motion
During the motion of a mechanical system with ndegrees of freedom, the 2 nquantitiesqsand _qs
vary with time according to the Lagrange equations. Since the general solution of the equations of
motion contains 2 narbitrary constants, we can express these constants as functions of qsand _qs
that remain constant during the motion. Such functions are called integrals of motion . A system of
particles which interact with one another but with no other bodies is said to be closed . For a closed
system of particles, the Lagrangian does not depend explicitly on time, so the origin of time t0may
be chosen arbitrarily as an additive constant. The remaining 2 n 1 constants are functions of qs
and _qs, so a closed mechanical system with ndegrees of freedom has 2 n 1 independent integrals
of motion.
According to the Noether theorem, each continuous symmetry of a Lagrangian leads to some
conservation law. For mechanical systems, the homogeneity and isotropy of space and time yields
the constancy of certain quantities, which are said to be conserved . These quantities are additive,
their values for a system composed of noninteracting parts are equal to the sums of their values for
each part. Therefore the additivity of conserved integrals of motion relates the states of interacting
bodies after the interaction to their states before the interaction, regardless of the nature of this
interaction.
The variation of the action as a function of the coordinates (3.1.35) for a system of particles of
massesma, using the geodesic equation of motion,
S= X
a[piaxi
a]2
1; (3.1.43)
where
pia=macuia (3.1.44)
is the four-momentum for the ath particle. The variation (3.1.43) is equivalent to
pia= @S
@xia; (3.1.45)
so the four-momentum operator (1.6.75) (which is proportional to the di erential generator of trans-
lation (1.6.66)) acting on the action Sgives the four-momentum pi(with the minus sign). If the
spacetime is homogeneous then the mechanical properties of a closed system do not change un-
der any translation of the system in space or time. Thus the variation S(3.1.43) vanishes if
xi
a=i= const:
S= X
a[pia]2
1i= 0 (3.1.46)
110
or, using the arbitrariness of i,X
apia2
1= 0; (3.1.47)
so the total four-momentum of a closed system is conserved:
X
api
a= const: (3.1.48)
This conservation, using the decomposition of piinto the energy Eand momentum p:
pi=E
c;p
; (3.1.49)
contains the conservation of the total energy of the system,
X
aEa= const; (3.1.50)
and the conservation of the total momentum of the system,
X
apa= const: (3.1.51)
For a system composed of one particle (3.1.43) becomes
S= [px Et]2
1; (3.1.52)
Comparing (3.1.52) with (3.1.21) gives
p=@L
@v; (3.1.53)
E=@L
@vv L=pv L: (3.1.54)
In generalized coordinates, we de ne the generalized momenta psconjugate to the coordinates qs:
ps=@L
@_qs; (3.1.55)
while the energy is
E=@L
@_qs_qs L=ps_qs L; (3.1.56)
in agreement with (2.4.52). The variation of the action (3.1.21) in this notation is
S= [psqs Et]t2
t1; (3.1.57)
which is equivalent to
ps=@S
@qs; E= @S
@t: (3.1.58)
Substituting the Lagrangian (3.1.28) into (3.1.53) gives
p=mvq
1 v2
c2; (3.1.59)
as in (2.4.210), while substituting it into (3.1.54) gives
E=mc2
q
1 v2
c2; (3.1.60)
111
as in (2.4.209).
Under Lorentz transformations, the four-momentum pi(3.1.49) transforms like the Lorentzian
coordinate vector xi, according to (1.6.97). The energy and momentum of a particle, which form
such a four-vector, transform according to (1.6.95):
E=
(E0+Vp0);
p?=p?0;pk=
pk0+VE0
c2
; (3.1.61)
or equivalently
E0=
(E Vp);
p?0=p?;pk0=
pk VE
c2
; (3.1.62)
where pis the momentum of the particle in a frame of reference K,p0is the momentum of the
particle in a frame of reference K0,Vis the velocity of the reference frame K0relative toK, and
=1p
1 V2=c2. Accordingly, E=p
p2c2+m2c4is the kinetic energy of the particle in K, and
E0=p
p2
0c2+m2c4is the kinetic energy of the particle in K0.
If the spacetime is isotropic then the mechanical properties of a closed system do not change under
any rotation of the system in spacetime (spatial rotation or boost). Thus the variation Svanishes
ifxi
a=ikxka, whereik= ki= const represents an in nitesimal Lorentz transformation:
S= X
a[pi
axk
a]2
1ik= 1
2X
a[pi
axk
a pk
axi
a]2
1ik= 0 (3.1.63)
or, using the arbitrariness of ik,X
aMik
a2
1= 0; (3.1.64)
where
Mik
a=xi
apk
a xk
api
a (3.1.65)
is the tensor of angular momentum (2.4.58) for the ath particle. Therefore the total angular mo-
mentum of a closed system is conserved:
X
aMik
a= const: (3.1.66)
We de ne the spatial angular momentum vector :
M:M =1
2e
M
(3.1.67)
or, due to (3.1.65),
M=rp: (3.1.68)
The isotropy of space thus implies the conservation of the total angular momentum vector.
The conservation of the total spatial vector M0 ,
M0 =X
aM0
a=X
a(ctpa Eara) = const; (3.1.69)
divided by the conservation of energy (3.1.50) gives
R=Vt+ const; (3.1.70)
where
V=c2P
apaP
aEa(3.1.71)
112
and
R=P
aEaraP
aEa: (3.1.72)
The relation (3.1.70) describes a uniform rectilinear motion (2.4.61) of the center of inertia of the
system, that is, of the system as a whole. The constant (due to (3.1.51)) velocity of this motion is
equal to V(2.4.62) and the coordinates of the center of inertia are given by the radius vector R.
This is a generalization of the law of inertia for one particle (3.1.30) to a system of particles. so
a free particle moves with a velocity which is constant in both magnitude and direction (uniform
in a straight line). This is referred to as the law of inertia The velocity of the system of particles
as a whole (3.1.71) has the same form as the velocity for one particle (2.4.213). We can always
choose a frame of reference in which the total momentumP
apavanishes, so the system as a whole
is at rest, V= 0. If the particles interact then we must add to Eathe energy of the elds which
carry the interaction. Since the components of the center of inertia (3.1.72) do not form the spatial
components of any four-vector, they do not transform like the coordinates of a point under Lorentz
transformations. Thus the location of the center of inertia relative to the particles depends on the
frame of reference.
The angular momentum (3.1.68) contains the radius vector, so it depends on the choice of the
point with respect to which it is de ned. If the radius vectors of the ath particle in two di erent
frames are raandr0
a, and the origins of these frames are separated by a constant vector a, then
ra=r0
a+a: (3.1.73)
IfMais the angular momentum of the particle in the unprimed frame and
M0
a=r0
apa (3.1.74)
is the angular momentum of this particle in the primed frame then
M=X
aMa=X
arapa=X
ar0
apa+aX
apa=M0+aP; (3.1.75)
where Pis the total momentum of the system. Thus, in the rest frame of reference, in which P= 0,
the angular momentum of the system does not depend on the choice of the origin of the Cartesian
coordinates. If the system is closed, the conservation of its total angular momentum is independent
of the choice of the origin because the total momentum of such system is also conserved.
The antisymmetric tensor of angular momentum (3.1.65) transforms under Lorentz transforma-
tions according to (1.6.100). If a system of particles is at rest as a whole in the frame K, in which
its angular momentum is M, and moves with velocity V=V^zin the frame K0, in which its angular
momentum M0is de ned with respect to the center of inertia RinK(M0inK0depends on the
choice of the rbecause P6= 0), thenKmoves relative to K0withV, so (1.6.100) gives
M012=M12; M013=
M13+V
cM10
; M023=
M23+V
cM20
; (3.1.76)
where
=q
1 V2
c2. InK,P
apa= 0 (system at rest) andP
aEara= 0 (origin at the center of
inertia), so (3.1.69) gives M0 = 0. Thus (3.1.76) reduces to
M0
x=
Mx; M0
y=
My; M0
z=Mz: (3.1.77)
The Lagrangian for a charged particle in the presence of the electromagnetic eld (3.1.40) gives,
upon substitution into (3.1.53),
p=mvq
1 v2
c2+e
cA; (3.1.78)
while substituting it into (3.1.54) gives
E=mc2
q
1 v2
c2+e : (3.1.79)
113
The relation (2.4.211) between the momentum and energy turns into
(E e )2= (pc eA)2+ (mc2)2: (3.1.80)
The momentum (3.1.78) and energy (3.1.79) of a charged particle are the sums of their values in
the absence of the electromagnetic eld, (3.1.59) and (3.1.60), called respectively the kinematic
momentum and kinematic energy , and terms linear in the electromagnetic potential.
3.1.5 Nonrelativistic mechanics and Galileo principle of relativity
In the nonrelativistic limit, the Lagrangian for a free particle (3.1.28) becomes
L= mc2+mv2
2: (3.1.81)
The rst term in (3.1.81) is constant and thus it does not contribute to the Lagrange equations.
The form of the second term can also be obtained from the invariance of the variation of the action
under Galileo transformations (1.6.114), which is referred to as Galileo's principle of relativity . If
an inertial frame Kis moving with an in nitesimal velocity relative to another inertial frame K0
then the addition law for velocities (1.6.115) gives v0=v+. The Lagrangian in K0is related to
the Lagrangian in Kby
L(v02) =L(v2+ 2v+2)L(v2) +@L
@v22v: (3.1.82)
Since the Lagrangian L0inK0can di er from the Lagrangian LinKonly by the total time derivative
of a function of the coordinates and time, the second term on the right-hand side of (3.1.82) must
be linear in v, so@L
@v2is independent of v, and thus Lis proportional to v2. The constant of
proportionality characterizes the particle and we can denote itm
2, calling the scalar mthe mass of
the particle. If Kis moving with a nite velocity Vrelative toK0then the addition law for velocities
(1.6.115) gives v0=v+V. The Lagrangian in K0is thus related to the Lagrangian in Kby
L0=1
2mv02=1
2m(v+V)2=1
2mv2+mvV+1
2mV2=L+d
dt
mrV+1
2mV2t
;(3.1.83)
in which the last term is a total time derivative and may be omitted. The integral S=m
2R2
1v2dt
has a minimum if mis not negative, otherwise Swould take arbitrarily large negative values for
an arbitrarily rapid motion of the particle from point 1 to point 2. In the nonrelativistic limit, the
Lagrangian (3.1.40) for a particle in the electromagnetic eld is
L= mc2+mv2
2+e
cAv e : (3.1.84)
We consider a closed system of particles. If the particles are nonrelativistic ( c!1 ) then the
interactions between them propagate instantaneously. Otherwise the nonrelativistic addition law
for velocities (1.6.115) would imply that the velocity of propagation of interaction and thus the
interaction vanishes in some frame of reference, contradicting Galileo's principle of relativity. Thus
the elds which carry these interactions depend on the positions of the particles at a given instant,
so the interaction can be described by adding to the total Lagrangian of noninteracting particlesP
a1
2mav2
aa certain function of the coordinates U(ra), called the potential energy :
L=X
a1
2mav2
a U(ra) =T(va) U(ra): (3.1.85)
The termTin (3.1.85) is referred to as the kinetic energy . The Lagrangian (3.1.85) is invariant
under the time reversal t! t, so all motions which obey the laws of mechanics are reversible.
114
For arbitrary generalized coordinates qs, the transformations
ra=ra(qs);va=@ra
@qs_qs (3.1.86)
turn the Lagrangian for a system of interacting nonrelativistic particles (3.1.85) into
L=1
2mrs(q) _qr_qs U(q) =T(q;_q) U(q); (3.1.87)
where
mrs=X
ama@ra
@qr@ra
@qs(3.1.88)
is the symmetric mass tensor ,mrs=msr. The Lagrangian L, due to the kinetic energy T, is a
quadratic function of the velocities.
We consider a system Awhich is not closed and interacts with another system Bexecuting a
given motion; the system Amoves in an external eld due to the system B. If the system A+Bis
closed then
L=TA(qA;_qA) +TB(qB;_qB) U(qA;qB): (3.1.89)
Substituting for qBthe given functions of time and omitting the term T(qB(t);_qB(t)), which depends
on time only so it is the total time derivative of some function of time, gives
L=TA(qA;_qA) U(qA;qB(t)): (3.1.90)
Thus the motion of a system in an external eld is described by a Lagrangian in which the potential
energy depends explicitly on time.
3.1.6 Momentum, force and Newton equations of motion
If the space is homogeneous then the mechanical properties of a closed system do not change under
any translation of the system in space. In the Cartesian coordinates, a translation by a constant
vectorresults in r!r+, so the corresponding change in Lfor a closed system is
L=X
a@L
@rara=X
a@L
@ra= 0: (3.1.91)
Sinceis arbitrary, this condition is equivalent to
X
a@L
@ra= 0; (3.1.92)
so the Lagrange equations (3.1.22) give
P=X
apa=X
a@L
@va= const; (3.1.93)
where Pis the additive momentum of the system, equal to the sum of its values pafor each particle,
regardless of the nature of the interaction between them. The additivity of the momentum follows
from the additivity of the Lagrangian. If a system is not closed but its Lagrangian does not depend
explicitly on all the Cartesian coordinates, then the components of Pwhich are parallel to the
Cartesian coordinates which do not appear in Lare conserved.
The Lagrange equations in the Cartesian coordinates (3.1.22) for the ath particle are equivalent
to
_pa=fa; (3.1.94)
where
fa=@L
@ra(3.1.95)
115
is referred to as the force acting on the ath particle. This form of the equations of motion is
called Newton's equations and constitutes Newton's second law of dynamics . The relation (3.1.94) is
consistent with (2.4.73). The covariant form of (3.1.94) is
dpi
ds=gi; (3.1.96)
wheregiis the force four-vector, parallel to the four-acceleration and thus perpendicular to the
four-velocity, giui= 0. The components of giare
gi=fv
c2;f
c
: (3.1.97)
Combining (3.1.59) and (3.1.94) gives the equation of motion of a particle of mass mmoving under
the in
uence of the force f:d
dt(m
v) =f: (3.1.98)
Asv!c, the momentum tends to in nity, so accelerating a massive particle to the velocity of
propagation of interaction crequires an in nite force and thus is impossible. In the nonrelativistic
limit, the momentum (3.1.59) reduces to
p=mv: (3.1.99)
The equation of motion (3.1.98) reduces to
ma=f= @U
@r; (3.1.100)
which also results from substituting (3.1.85) into (3.1.22). Adding a constant to the potential energy
does not a ect Newton's equations (such a constant is a total time derivative of some function), so
Uis de ned up to an additive constant, which usually is chosen such that U!0 asra!1 . A
eld in which the same force facts on a particle at any point is uniform . The potential energy in
such eld is
U= fr: (3.1.101)
In a weak gravitational eld, the Lagrangian of a particle is
L= mc2+mv2
2 m g; (3.1.102)
where gis the gravitational potential (2.5.82), because the corresponding Lagrange equations
(3.1.22) coincide with (2.5.84). The term m gis thus the potential energy of the particle in a
nonrelativistic gravitational eld,
Ug=m g: (3.1.103)
The action for (3.1.102) is
S=Z
Ldt= mcZ
c v2
2c+ g
c
dt; (3.1.104)
which, compared with (3.1.25), gives
ds=
c v2
2c+ g
c
dt: (3.1.105)
Omitting terms vanishing in the limit c!1 gives
ds2= (c2+ 2 g)dt2 dr2; (3.1.106)
which is consistent with (2.5.82). In the presence of a weak gravitational eld, the Lagrangian
(3.1.84) turns into
L= mc2+mv2
2 m g+e
cAv e : (3.1.107)
116
The relation (3.1.92) is equivalent to
X
afa= 0; (3.1.108)
so the sum of the forces in a closed system vanishes. For a system of two particles, f1+f2= 0, so the
force on the rst particle exerted by the second is equal in magnitude and opposite in direction to
the force on the second particle exerted by the rst (this statement is valid only for nonrelativistic
particles, for which we can neglect elds propagating the interaction). This is the action-reaction
laworNewton's third law of dynamics . For a system of particles, summing (3.1.94) over agives
_P=X
a_pa=X
afa: (3.1.109)
Since the forces of interaction between particles cancel out due to the action-reaction law, (3.1.109)
becomes
_P=F; (3.1.110)
where Fis the sum of external forces acting on the particles. If no external forces act on a mechanical
system then the system is closed, _P= 0.
The Lagrangian Lof a closed system does not depend explicitly on qs,@L
@qs= 0, so the Lagrange
equations (3.1.15) with (3.1.55) give,
ps= const: (3.1.111)
IfLdepends on qsthen
Fs=@L
@qs(3.1.112)
are called generalized forces . The Lagrange equations (3.1.15) are in this notation
_ps=Fs: (3.1.113)
The form of the Lagrangian (3.1.87) gives the relation between generalized momenta and generalized
velocities:
pr=mrs(q) _qs: (3.1.114)
Substituting (3.1.87) into the Lagrange equations (3.1.15) gives
mrsqr+ _mrs_qr=1
2mrt;s_qr_qt @U
@qs; (3.1.115)
where
mrs;t=@mrs
@qt; (3.1.116)
or
mrsqr+mrs;t_qt_qr 1
2mrt;s_qr_qt= @U
@qs: (3.1.117)
De ning the inverse of the mass tensor m 1
rs,
m 1
rsmst=rt; (3.1.118)
and multiplying (3.1.117) by m 1
usgives
qu+ [rt;u] _qr_qt= m 1
su@U
@qs; (3.1.119)
where the quantities [ rt;u] are analogous to the Christo el symbols (1.4.26):
[rt;u] =1
2m 1
us(msr;t+mst;r mrt;s): (3.1.120)
117
3.1.7 Energy
If the time is homogeneous then the mechanical properties of a closed system do not change under
any translation of the system in time, so the Lagrangian of a closed system does not depend explicitly
on time. The total time derivative of the Lagrangian L(q;_q) is thus, using the Lagrange equations
(3.1.15),
dL
dt=@L
@qs_qs+@L
@_qsqs=d
dt@L
@_qs_qs+@L
@_qsqs=d
dt@L
@_qs_qs
(3.1.121)
or
d
dt@L
@_qs_qs L
=dE
dt= 0: (3.1.122)
Hence the energy of such system is conserved. The additivity of the energy follows from the additivity
of the Lagrangian. Mechanical systems whose energy is conserved are said to be conservative . In
the nonrelativistic limit, the energy of a particle of mass mmoving with velocity v(3.1.60) reduces
to
E=mc2+1
2mv2; (3.1.123)
where it is the sum of its rest energy mc2and its kinetic energy1
2mv2. The energy is a positive
quantity because m > 0 (ifm= 0 thenE=Pc > 0). In the nonrelativistic mechanics the energy
can be of either sign, since adding a constant to the potential energy does not a ect the Newton
equations. The equivalence of mass and energy (2.4.212) is valid not only for a single particle, but
also for a system of particles which is at rest as a whole, if mis the total mass of such system
andEis its total energy. The energy of a composite body at rest contains, in addition to the rest
energies of its constituent particles, their kinetic energies and the energies of their interactions. Thus
mc26=P
amac2, so the mass, unlike energy, is not additive; the mass of a composite body is not
equal to the sum of the masses of its parts:
m6=X
ama: (3.1.124)
Since the Lagrangian L(q;_q) =T(q;_q) U(q) for a system of nonrelativistic particles is a quadratic
function of _ q, Euler's theorem on homogeneous functions gives
@L
@_qs_qs=@T
@_qs_qs= 2T: (3.1.125)
Combining (3.1.56) and (3.1.125) gives
E=T(q;_q) +U(q): (3.1.126)
In the Cartesian coordinates the energy of a system of nonrelativistic particles is the sum of the
kinetic energy, which depends on the veloctites of the particles, and the potential energy, which
depends on their coordinates:
E=T(va) +U(ra) =X
a1
2mav2
a+U(ra): (3.1.127)
The change of kinetic energy in time is, due to (2.4.214) and (3.1.94),
dE
dt=vdp
dt=vF=dW
dt; (3.1.128)
where
dW=Fdr (3.1.129)
is an in nitesimal work done by a force Fover a displacement dr. Thus a work over a nite
displacement is
W=Z
Fdr: (3.1.130)
118
The quantity P=dW
dtis called a power . If we regard a system of particles as a
uid and the force
on this system is a surface force due to pressure (2.4.78) then (3.1.129) gives
dW= pdfndr= pdV: (3.1.131)
Thus the pressure is given by
p= @E
@V: (3.1.132)
3.1.8 Center of mass
We consider a system of nonrelativistic particles. If a frame of reference K0moves with velocity V
relative to a frame Kthen the velocities v0
aandvaof the particles in these two frames are related
by
va=v0
a+V: (3.1.133)
The momenta P0andPof the system are related by
P=X
amava=X
amav0
a+VX
ama=P0+VX
ama: (3.1.134)
Thus we can choose a frame K0in which the system as a whole is at rest, P0= 0. The velocity of
this rest frame in Kis
V=PP
ama=P
amavaP
ama: (3.1.135)
This velocity is the velocity of the system of particles as a whole. The relation (3.1.135) between the
momentum Pand velocity Vof the system of particles is the same as that for one particle (3.1.99),
P=MV; (3.1.136)
of mass
M=X
ama: (3.1.137)
Thus the mass in nonrelativistic mechanics is additive. The relation (3.1.135) is the time derivative
of
R=P
amara
M: (3.1.138)
The radius vector Rof the system as a whole is the nonrelativistic limit of the center of inertia
(3.1.72) of such system (since Eamac2), and it is referred to as its center of mass . Since the
components of the center of mass (3.1.138) transform like the coordinates of a point under Galileo
transformations, the location of the center of mass relative to the particles does not depend on the
frame of reference. The center of mass (3.1.138) contains the radius vectors of the particles, so it
depends on the choice of the point with respect to which it is de ned, like the angular momentum
(3.1.68). If the origins of two frames are separated by a constant vector athen the radius vectors of
theath particle in these frames, raandr0
a, satisfy (3.1.73). If R0is the center of mass in the primed
frame,
R0=P
amar0
a
M; (3.1.139)
then
R0=R a: (3.1.140)
If we choose a=Rthen R0becomes zero, corresponding to setting the origin of the frame of
reference at the center of mass of the system.
119
The energy of a mechanical system which is at rest as a whole is referred to as its internal energy
". The energies E0andEof the system in the two frames K0andKare related by
E=1
2X
amav2
a+U=1
2X
ama(v0
a+V)2+U=1
2X
amav02
a+U+VX
amav0
a
+1
2MV2=E0+VP0+1
2MV2: (3.1.141)
The corresponding Lagrangians are related by
L=L0+VP0+1
2MV2; (3.1.142)
so the corresponding actions are related by
S=S0+MVR0+1
2MV2t: (3.1.143)
If the center of mass is at rest in K0thenP0= 0, soE0=", and (3.1.141) reduces to
E="+1
2MV2: (3.1.144)
Thus the energy of a system moving as a whole is the sum of its internal energy and the kinetic
energy of its center of mass, which is called K onig's theorem .
3.1.9 Angular momentum and torque
If the space is isotropic then the mechanical properties of a closed system do not change under
any rotation of the system in space. A Lorentz matrix (1.6.7) with 0 = 0, which represents an
in nitesimal rotation, changes the spatial coordinates according to
x = x = e
x
(3.1.145)
or
r= r; (3.1.146)
where we donote the angle of this rotation (1.6.25) . The corresponding change in the velocity
relative to an inertial frame of reference is
v= v: (3.1.147)
Thus the corresponding change in Lfor a closed system of particles is
L=X
a@L
@rara+@L
@vava
=X
a(_pa ra+pa va)
= X
a(ra_pa+vapa) = d
dtX
arapa
= X
a_Ma= 0: (3.1.148)
Since is arbitrary, this condition gives the conservation of the total angular momentum of the
system (3.1.68):X
aMa= 0: (3.1.149)
The additivity of the angular momentum follows from the additivity of the Lagrangian. There
are no other additive integrals of motion; every closed system has 7 such integrals: energy, the
3 components of momentum and the 3 components of angular momentum. The component of the
120
angular momentum of a particle along any axis icoincides with its generalized momentum conjugate
to the angle of rotation iabout this axis, regarded as a generalized coordinate qs:
Mi=@L
@_ i: (3.1.150)
The Lagrange equations (3.1.15) give then
_Mi=@L
@ i: (3.1.151)
If a system is not closed but interacts with an external eld which is symmetrical about some axis
then the component of Malong this axis relative to a point lying on the axis is conserved.
Di erentiating the angular momentum for the ath particle with respect to time gives, using
(3.1.59) and (3.1.94),
_Ma=d
dt(rapa) =vapa+raFa= a; (3.1.152)
where
a=raFa (3.1.153)
is referred to as the torque acting on the ath particle. For a system of particles, summing (3.1.152)
overagives
_M=X
a_Ma=X
a a: (3.1.154)
Since the total angular momentum Mfor a closed system is conserved, the sum of internal (within
the system) torques exerted on the particles vanishes. Thus (3.1.154) becomes
_M=K; (3.1.155)
where Kis the sum of external torques acting on the particles. If no external torques act on a
mechanical system then _M= 0. The relation (3.1.151) implies that the component of the torque
acting on a particle along any axis icoincides with its generalized force related to the angle of
rotation iabout this axis, regarded as a generalized coordinate qs:
i=@L
@ i: (3.1.156)
Equation (3.1.155) is a Lagrange equation for the rotational coordinate .
If a frame of reference K0moves with velocity Vrelative to a frame Kthen the velocities v0
a
andvaof the particles in these two frames are related by (3.1.133). Thus the angular momenta M0
andMof the system are related by
M=X
amarava=X
amarav0
a+X
amaraV=M0+MRV: (3.1.157)
If the system as a whole is at rest in K0thenVis the velocity of its center of mass and MVis its
total momentum P(3.1.136) relative to K, so
M=M0+RP: (3.1.158)
Thus the angular momentum of a moving system is the sum of its intrinsic angular momentum in a
frame in which it is at rest and the (orbital) angular momentum of its motion as a whole.
The torque contains the radius vector, so it depends on the choice of the point with respect
to which it is de ned, like the angular momentum. If the origins of two frames are separated by
a constant vector athen the radius vectors of the ath particle in these frames, raandr0
a, satisfy
(3.1.73). If ais the torque acting on the particle in the unprimed frame and
0
a=r0
afa (3.1.159)
121
is the torque acting on this particle in the primed frame then
K=X
a a=X
arafa=X
ar0
afa+aX
afa=K0+aF: (3.1.160)
Thus, if the total external force acting on the system of particles F= 0 then the total torque acting
on this system does not depend on the choice of the origin of the Cartesian coordinates. In this case,
the system is said to be acted on by a couple . IfFandKare perpendicular then we can choose
the vector afor which K0= 0, so K=aF. In this case, the e ect of all the forces acting on the
system reduces to that of a single force F. This choice is not unique: adding to aany vector parallel
toFdoes not change K0= 0.
3.1.10 Mechanical similarity
We consider a similarity transformation in which the coordinates riof all the particles (indexed by
i) in a nonrelativistic mechanical system are multiplied by a constant factor aand the time by a
constant factor b:
ri!r0
i=ari; t!t0=bt: (3.1.161)
This transformation changes the velocities viby a factora
b, so the kinetic energy Tchanges by
a factora2
b2. If the potential energy of the system is a homogeneous function of the Cartesian
coordinates,
U(ari) =akU(ri); (3.1.162)
wherekis the degree of homogeneity of U, then it changes by a factor ak. The equations of motion
remain unchanged if the Lagrangian T Uof the system changes by the same factor:
a2
b2=ak; (3.1.163)
which transforms a path of each particle into a geometrically similar path of di erent size. The
relations between the ratios of mechanical quantities corresponding to the two paths and the ratio
l0
lof linear dimensions of these paths are
t0
t=l0
l1 k
2
;v0
v=l0
lk
2
;p0
p=l0
lk
2
;E0
E=l0
lk
;M0
M=l0
l1+k
2
: (3.1.164)
The kinetic energy Tof a nonrelativistic system of particles is a quadratic function of the veloc-
ities, so Euler's theorem on homogeneous functions gives
2T=X
ivi@T
@vi=X
ivipi=d
dtX
iripi
X
iri_pi: (3.1.165)
If the motion of the system is nite in space then averaging (3.1.165) over the time and using
vanishing of the average of the time derivative of any bounded quantity gives
2T= X
iri_pi= X
irifi=X
iri@U
@ri: (3.1.166)
If the potential energy of a system of particles is a homogeneous function of the coordinates (3.1.162)
then Euler's theorem turns (3.1.166) into the nonrelativistic virial theorem :
2T=kU: (3.1.167)
If the system is closed then its energy E=T+Uis constant, so (3.1.167) gives
T=kE
k+ 2;U=2E
k+ 2: (3.1.168)
122
The nonrelativistic limit of (2.4.208) gives
E=X
amac2 X
a1
2mav2=X
amac2 T; (3.1.169)
which is consistent with (3.1.168), if we omit the rest energies of the particles, for k= 1. Thus
nonrelativistic charged particles interact with each other through a force which is proportional to
the inverse of the distance between them.
3.1.11 Dissipation
If the mechanical energy Eof a system is transformed into other, nonmechanical forms of energy,
we refer to a process of such a transformation as the dissipation of energy . We de ne the dissipative
function ,
F=1
2 st_qs_qt; (3.1.170)
which modi es the Lagrange equations (3.1.15) to include dissipation:
d
dt@L
@_qs=@L
@qs @F
@_qs: (3.1.171)
Thus the generalized force fs=@L
@qsis modi ed by an additional part,
f(fr)
s= st_qt; (3.1.172)
called the resistive force . The change of energy during dissipation is
dE
dt=d
dtX
s_qs@L
@_qs L
=X
s_qsd
dt@L
@_qs @L
@qs
= X
s_qs@F
@_qs= 2F: (3.1.173)
ThereforeF > 0, so the quadratic form (3.1.170) is positive de nite. If Fwere a linear function of
_qsthen the sign ofdE
dtcould not be xed negative. If Fdid not depend on _ qsthendE
dt= 0. Thus
(3.1.170) is the simplest function which can represent dissipation of energy in (3.1.171). For one
particle, stis a 33 symmetric matrix. Since a particle is mechanically isotropic, stis diagonal,
st= st; (3.1.174)
where >0 is a scalar. The resistive force (3.1.172) acting on a particle is then
f(fr)= v: (3.1.175)
References: [2, 12].
3.2 Rigid bodies
3.2.1 Angular velocity
A system of particles in which the distances between the particles do not change is referred to as
arigid body . The interaction between the particles which imposes the constancy of their relative
distances must propagate instantaneously. Thus the concept of a rigid body is valid only in non-
relativistic mechanics. To describe the motion of a rigid body, we use two frames of reference: an
inertial frame Kand a moving frame K0rigidly xed in the body. The origin of K0is usually taken
at the center of mass of the body. The position of a rigid body with respect to Kis determined
by the position of K0: the 3 components of the radius vector Rof the origin of K0inKand 3
angles representing the orientation of the axes of K0relative to the axes of K. Thus a rigid body is
a mechanical system with 6 degrees of freedom. If the distances between the particles in a system
123
change under the action of external forces or torques, and they return to their original values when
this action ends, then such a system is referred to as an elastic body .
An arbitrary displacement of a rigid body is the sum of an in nitesimal translation of the body
as a whole dRand an in nitesimal rotation of the body about its center of mass. If ris the radius
vector of the particle in K,d is an in nitesimal angle of rotation of K0with respect to K, and r0
is the radius vector of the particle in K0, then (3.1.146) gives
dr=dR+d r0: (3.2.1)
Dividing (3.2.1) by dtgives
v=V+!r0; (3.2.2)
where the vector
V=dR
dt(3.2.3)
is the velocity of the center of mass of the body, called the translational velocity of the body, and
the pseudovector
!=d
dt(3.2.4)
is the angular velocity of the rotation of the body. The term !ris the rotational velocity of the
body. Under rotations, boosts and translations, the angular velocity behaves like a spatial vector.
The time derivative of the angular velocity is called the angular acceleration :
=_!: (3.2.5)
If we choose a di erent frame K00rigidly xed in the body, with the origin at a distance afrom
the origin of K0, then the radius vector r00of the particle in K00satis es
r0=r00+a: (3.2.6)
Thus (3.2.2) becomes
v=V+!a+!r00: (3.2.7)
If the translational velocity of the body in K00isV0and its angular velocity is !0then (3.2.2) gives
(vdoes not change because it is related to K)
v=V0+!0r00: (3.2.8)
Comparing (3.2.8) with (3.2.2) gives
V0=V+!a; (3.2.9)
!0=!: (3.2.10)
Thus the angular velocity of rotation of a rigid body at any instant is, unlike the translational
velocity, independent of a system of coordinates chosen for a moving frame rigidly xed in the body.
Thus!is the angular velocity of the body. If Vand!at any instant are perpendicular for some
choice of the origin of K0then (3.2.9) implies that V0and!0are also perpendicular. In this case,
(3.2.2) implies that the velocities vof all points of the body are perpendicular to !, so we can choose
the origin of K0whose velocity V0= 0 at this instant. The corresponding motion is a pure rotation
about an axis which passes through the origin of K0, called the instantaneous axis of rotation . IfV
and!are not perpendicular then we can choose the origin of K0for which Vand!are parallel at
a given instant.
124
3.2.2 Inertia tensor
The kinetic energy of a rigid body is given by substituting (3.2.2) (where we omit the prime) into
(3.1.123) and summing over all the particles:
T=X
a1
2mav2
a=X
a1
2ma(V+!ra)2
=X
a1
2maV2+X
amaV(!ra) +X
a1
2ma(!ra)2
=1
2MV2+X
amara(V!) +X
a1
2ma
!2r2
a (!ra)2
: (3.2.11)
If the origin of the moving frame rigidly xed to the body is at its center of mass thenP
amara= 0
and (3.2.11) reduces to
T=1
2MV2+X
a1
2ma(x2
a x ax a)! ! =1
2MV2+1
2I ! ! ; (3.2.12)
where the symmetric tensor
I =X
ama(x2
a x ax a) (3.2.13)
is referred to as the inertia tensor . Thus the kinetic energy of a rigid body is the sum of the
kinetic energy of the translational motion of its center of mass,1
2mV2, and the kinetic energy of the
rotational motion of the body ( rotational kinetic energy ) about an axis passing through the center
of mass,
Trot=1
2I ! ! : (3.2.14)
The inertia tensor is additive, which follows from the additivity of the kinetic energy. If a rigid body
is regarded as continuous then the mass of each particle must be replaced by the mass dV with
densitycontained in a volume element dV, and the summation over all particles by the integration
over the volume of the body. The inertia tensor (3.2.13) becomes then
I =Z
(x2 x x )dV: (3.2.15)
The inertia tensor (3.2.13) contains the radius vector, so it depends on the choice of the origin
of a frame rigidly xed in a rigid body. If the radius vectors of the ath particle composing the body
in two di erent frames are raandr0
a, and the origins of these frames are separated by a constant
vector a, then substituting (3.1.73) into (3.2.13) gives
I0
=X
ama
(ra a)2 (x a a )(x a a )
=X
ama
(x2
a 2xaa+a2) x ax a+x aa +x aa a a
=X
ama(x2
a x ax a) M
(2Ra a2) + (X a +X a a a )
=I M
(2Ra a2) + (X a +X a a a )
; (3.2.16)
where
I0
=X
ama(x02
a x0
ax0
a) (3.2.17)
andR= (X ) is the radius vector of the center of mass of the body (3.1.138). If the origin of the
unprimed frame is at the center of mass of the body then R= 0 and (3.2.16) reduces to
I0
=I +M(a2 a a ); (3.2.18)
125
which is called the parallel-axis or Steiner's theorem .
The diagonal components of the inertia-tensor matrix are called the moments of inertia about
the corresponding axes. The inertia tensor can be reduced to a diagonal form by an appropriate
choice of the axes of a frame rigidly xed in a rigid body, called the principal axes of inertia . The
corresponding values of the diagonal components of I are called the principal moments of inertia ,
I1;I2;I3. The rotational kinetic energy of the body (3.2.14) is then
Trot=1
2(I1!2
1+I2!2
2+I3!2
3); (3.2.19)
where!1;!2;!3are the components of the angular velocity along the principal axes of inertia. None
of the three principal moments of inertia can be larger than the sum of the other two. If the body
has a plane of symmetry then its center of mass and two of the principal axes of inertia must lie in
that plane. If the body has an axis of symmetry then this axis is one of the principal axes of inertia
and it passes through its center of mass.
The angular momentum of a rigid body, de ned with respect to its center of mass, is given by
substituting (3.2.2) (where we omit the prime) into (3.1.68) and summing over all the particles:
M=X
amara(V+!ra) =X
amara(!ra) =X
ama
r2
a! (ra!)ra
(3.2.20)
or
M =I ! : (3.2.21)
If the axes of a frame rigidly xed in the body coincide with the principal axes of inertia then
M1=I1!1; M 2=I2!2; M 3=I3!3: (3.2.22)
3.2.3 Eulerian angles
The orientation of the axes x0=x1;y0=x2;z0=x3of a frameK0moving with a rigid body relative
to the axes x;y;z of an inertial Kcan be represented by 3 Eulerian angles ; ; , as shown in Fig.
4. The origins of KandK0can be taken to coincide (at a point O) because the orientation of K0
relative toKdoes not change under translations. The x0y0plane intersects the xyplane along the
line of nodes ON, in the direction of zz0, perpendicular to both zandz0. The angle 2[0;]
is between the zandz0axes, 2[0;2] betweenONand thexaxis, and 2[0;2] betweenON
and thex0axis. The polar angle and azimuth of the direction z0with respect to Kareand
2.
The polar angle and azimuth of the direction zwith respect to K0areand
2 .
Figure 4: Eulerian angles.
The angular velocity _is alongON, so its nonvanishing components along the axes of K0are:
_1=_cos ,_2= _sin . The angular velocity _ is along the z-axis, so _ 3=_ cos,_ 1=
126
_ sinsin ,_ 2=_ sincos . The angular velocity _ is along the z0-axis, so its only nonvanishing
component is _ 3=_ . Thus the components of the angular velocity !along the axes of K0are
0
@!1
!2
!31
A=0
@cos sinsin 0
sin sincos 0
0 cos 11
A0
@_
_
_ 1
A: (3.2.23)
Substituting (3.2.23) into (3.2.21) gives the angular momentum of a rigid body in terms of the
Eulerian angles and its derivatives. If the axes of K0are the principal axes of inertia of the body
then (3.2.22) gives
M1=I1(cos _+ sinsin _ ); M 2=I2( sin _+ sincos _ ); M 3=I3(cos_ +_ ):(3.2.24)
Substituting (3.2.23) into (3.2.14) gives the rotational kinetic energy of a rigid body in terms of the
Eulerian angles and its derivatives. If the axes of K0are the principal axes of inertia of the body
then (3.2.19) gives
Trot=1
2
I1(cos _+ sinsin _ )2+I2( sin _+ sincos _ )2+I3(cos_ +_ )2
: (3.2.25)
A product of 3 rotations (which is a rotation) brings the frame KtoK0: a rotation about the z-
axis by an angle transforms the x-axis into a new x00-axis (which coincides with the line of nodes),
then a rotation about the x00-axis by an angle transforms the z-axis into the z0-axis, and then a
rotation about the z0-axis by an angle transforms the x00-axis into the x0-axis. The corresponding
orthogonal matrix of this rotation is
R( ;; ) =Rz0( )Rx0()Rz( ); (3.2.26)
whereR are given by (1.6.33) and describe rotations with respect to moving axes. If a rotation
Rtransforms an axis nto a new axis n0then a rotation R0about n0is related to the rotation R0
about nby
R0
n0=RR0
nR 1: (3.2.27)
Thus we can rewrite (3.2.26) in terms of rotations with respect to the xed axes of K:
R( ;; ) =
Rx0()Rz( )
Rz( )
Rx0()Rz( ) 1
Rx0()Rz( )
=Rx0()Rz( )Rz( ) =Rz( )Rx()R 1
z( )Rz( )Rz( )
=Rz( )Rx()Rz( ): (3.2.28)
3.2.4 Newton and Euler equations
We consider a rigid body in an inertial frame of reference. The Lagrangian of a rigid body is given
by
L=1
2MV2+1
2Iik!i!k U; (3.2.29)
where the potential energy Uis a function of the 6 degrees of freedom of the body. The Lagrange
equations (3.1.15) for the 3 components of the radius vector Rof the center of mass of the body are
d
dt@L
@V @L
@R= 0: (3.2.30)
The change of the potential energy under a translation of the body through a distance Ris
U=X
a@U
@rara=X
a@U
@raR= X
afaR= FR; (3.2.31)
so
F= @U
@R: (3.2.32)
127
Substituting (3.2.29) into (3.2.30) gives, using (3.1.136) and (3.2.32), the Newton equation of trans-
lational motion of a rigid body (3.1.110).
The Lagrange equations (3.1.15) for the 3 angles representing the orientation of the principal
axes of inertia of the body relative to the axes of the inertial frame are
d
dt@L
@! @L
@ = 0: (3.2.33)
The change of the potential energy under a rotation of the body by an angle is
U= X
afara= X
afa( r) = X
arafa= X
a a= K ;(3.2.34)
so
K= @U
@ : (3.2.35)
Substituting (3.2.29) into (3.2.33) and using (3.2.21) gives
_M = @U
@ (3.2.36)
or, using (3.2.35), the Newton equation of rotational motion of a rigid body (3.1.155).
If a rigid body interacts with a uniform eld Ethen the force on each particle composing the
body is f=eE, whereecharacterizes the properties of the particle with respect to the eld. The
corresponding total force F=EP
aeaand the total torque K=P
aearaE. IfP
aea6= 0 then
K=r0F; (3.2.37)
where
r0=P
aearaP
aea: (3.2.38)
Thus the e ect of a uniform eld on a moving rigid body reduces to the action of a single force F
applied at the point with the radius vector (3.2.38). In a uniform electric eld, Eis the electric
eld vector and eis the electric charge of the particle. In a uniform gravitational eld, Eis the
gravitational acceleration due to gravity, eis the mass of the particle and r0is the radius vector of
the center of mass of the body.
We consider a frame of reference K0moving with a rigid body, whose axes are the principal axes
of inertia, relative to an inertial frame K. If the change of any vector AinK0isd0A
dtthen its change
inKis, according to (3.2.2),
dA
dt=d0A
dt+!A: (3.2.39)
Thus the Newton equations of motion of a rigid body (3.1.110) and (3.1.155) are equivalent to
d0P
dt+!P=F; (3.2.40)
d0M
dt+!M=K: (3.2.41)
Projecting (3.2.40) onto the axes of K0and using (3.1.136) gives
M(_V1+!2V3 !3V2) =F1;
M(_V2+!3V1 !1V3) =F2;
M(_V3+!1V2 !2V1) =F3: (3.2.42)
Projecting (3.2.41) onto the axes of K0and using (3.2.22) (which is valid for the principal axes of
inertia) gives Euler's equations :
I1_!1+ (I3 I2)!2!3=K1;
I2_!2+ (I1 I3)!3!1=K2;
I3_!3+ (I2 I1)!1!2=K3: (3.2.43)
128
3.2.5 Noninertial frames of reference
In an inertial frame of reference K0, the Lagrangian for a nonrelativistic particle of mass mmoving
with velocity v0in an external eld represented by the potential energy Uis
L0=1
2mv2
0 U; (3.2.44)
which gives the Newton equation of motion mdv0
dt= @U
@r. We consider a frame of reference K0,
which moves relative to K0with translational velocity V(t). Substituting the relation v0=v0+V,
where v0=dr0
dtis the velocity of the particle in K0, into (3.2.44) gives
L0=1
2mv02+mv0V+1
2mV2 U=1
2mv02+d
dt(mr0V) mr0dV
dt+d
dt1
2mV2t
U;(3.2.45)
which, after omitting total time derivatives, reduces to
L0=1
2mv02 mr0A U; (3.2.46)
where A=dV
dtis the translational acceleration of K0. The Lagrange equations (3.1.22) in K0are
d
dt@L0
@v0 @L0
@r0= 0, giving the Newton equation of motion:
mdv0
dt= @U
@r0 mA: (3.2.47)
The e ect of a translational motion of a frame of reference with acceleration A(t) is equivalent to
the e ect of a uniform eld of force mA. This force acts on the particle in the direction opposite
to the acceleration.
We consider a frame of reference K, which rotates relative to K0with angular velocity !(t) and
whose origin coincides with the origin of K0, sor=r0. Substituting the relation v0=v+!r,
where v=dr
dtis the velocity of the particle in K, into (3.2.46) gives
L=1
2mv2+mv(!r) +1
2m(!r)2 mr0A U; (3.2.48)
so@L
@v=mv+m!r;@L
@r=mv!+m(!r)! mA @U
@r: (3.2.49)
Thus the Lagrange equations (3.1.22) in Kgive the Newton equation of motion:
mdv
dt= @U
@r0 mA m r 2m!v m!(!r); (3.2.50)
where (t) = _!is the angular acceleration of K. The e ect of a rotational motion of a frame of
reference with angular velocity !is equivalent to the e ect of three forces: the force m rarising
from the nonuniformity of the rotation, the Coriolis force
fC= 2m!v; (3.2.51)
and the centrifugal force
fc= m!(!r): (3.2.52)
The Coriolis force depends on the velocity v, but it is perpendicular to vso it does not produce any
work. The centrifugal force acts in the direction opposite to r(away from the center of K) and its
magnitude is m!2, whereis the distance of the particle from the axis of rotation.
129
3.2.6 Constraints and d'Alembert principle
A rigid body is in equilibrium if the total force and torque acting on it vanish:
F=X
f= 0;K=X
rf= 0: (3.2.53)
According to (3.1.110) and (3.1.155), such body is at rest ( static ). The equations of equilibrium
(3.2.53) do not depend on the choice of the origin with respect to which the torque is de ned
because of (3.1.160). For a system of rigid bodies, (3.2.53) must be satis ed for each body. The
consistency of these equations with Newton's third law for each pair of bodies implies the appearance
of additional forces exerted on each body by the bodies with which it is in contact and acting at
the points of contact. These forces are referred to reactions and must be included in (3.2.53). They
satisify Newton's third law: the mutual reactions of two bodies in contact are equal in magnitude
and opposite in direction. Solving the equations of equilibrium (3.2.53) gives the reactions. If the
system of bodies is not in equilibrium then reactions must be included in (3.1.110) and (3.1.155).
Solving these equations give the motion of the bodies and the reactions. In addition to reaction,
dissipative forces of friction opposing the motion appear if two bodies in contact are in relative
motion.
The motion of bodies in contact is a combination of sliding and rolling . Sliding is characterized
by a reaction Rperpendicular to the surfaces in contact and a friction Ttangential to these surfaces
and proportional in magnitude to the reaction,
T=R: (3.2.54)
The constant of proportionality is called the coecient of kinetic friction . Two bodies in contact
have a smaller number of degrees of freedom relative to these bodies in free motion. In sliding, the
constraints impose relations between the coordinates of the bodies only; such constraints are called
holonomic . The number of generalized coordinates ndescribing rigid bodies in sliding is n=n0 k,
wheren0is the number of degrees of freedom of the bodies in free motion and kis the number of
(holonomic) constraints.
Rolling is characterized by no relative motion of the bodies at their instantaneous point of contact,
with an arbitrarily oriented force of reaction and a friction torque opposing the rotation. In rolling,
the velocities of the points in contact are thus equal. This condition is expressed by the equations
of constraint on the generalized velocities _ qs:
X
scjs_qs= 0; (3.2.55)
where the index j= 1::knumbers these equations and the functions cjsdepend only on the gen-
eralized coordinates qs. IfP
scjs_qsis not the time derivative of some function of the coordinates
then (3.2.55) cannot be integrated to relations between the coordinates of the bodies only; such
constraints are called nonholonomic . For a rigid body rolling on another body at rest, (3.2.55) in
the Cartesian coordinates gives
V+!r= 0; (3.2.56)
where ris the radius vector of the instantaneous point of contact with respect to the center of mass
of the rolling body. For sliding, the reaction is perpendicular to the surface of contact of the bodies.
Since the motion in sliding is tangential to this surface, the work of forces of reaction along a virtual
displacement (displacement at a xed time) r, which is consistent with the constraints, vanishes:
X
Rr= 0; (3.2.57)
where the summation extends over all the bodies. This relation is equivalent to d'Alembert's prin-
ciple:X
(f _p)r= 0; (3.2.58)
where pis the momentum of the body and fis the total force acting on it. The static ( p= 0) case
of d'Alembert's principle is called the principle of virtual work .
130
Multiplying (3.2.55) by tgivesX
scjsqs= 0; (3.2.59)
so the variations qsare not independent. The dynamics of a mechanical system is given by the
stationarity of the action S= 0 withkadditional constraints (3.2.59). Lagrange's method of nding
conditional extrema gives
S+ZX
j;sjcjsqsdt= 0; (3.2.60)
wherejare some functions of the coordinates, called Lagrange multipliers . The variations qsin
(3.2.60) are now independent, so using (3.1.14) yields
Zt2
t1@L
@qs d
dt@L
@_qs+X
jjcjs
qsdt= 0: (3.2.61)
Since the variations qsare arbitrary, (3.2.61) gives the Lagrange equations with constraints for each
degree of freedom s= 1::n:
d
dt@L
@_qs @L
@qs=X
jjcjs: (3.2.62)
Thesensecond-order di erential equations, together with kequations of constraint (3.2.55), are the
equations of motion of the system for ncoordinates qsandkmultipliers j. The right-hand sides
of (3.2.62) are the generalized forces of reaction :
Rs=X
jjcjs: (3.2.63)
The condition (3.2.59) is also valid for holonomic constraints. In this case,P
scjs_qsin (3.2.55)
is the time derivative of some function of the coordinates fj, so (3.2.59) gives
fj=X
scjsqs; (3.2.64)
which then gives
cjs=@fj
@qs: (3.2.65)
The Lagrange equations with constraints (3.2.62) become thus
d
dt@L
@_qs @L
@qs=X
jj@fj
@qs: (3.2.66)
3.2.7 Maggi and Appell equations
References: [12].
3.3 Ideal
uids
3.3.1 Lagrange formulation
3.3.2 Euler formulation
References: [13, 14].
131
3.4 Hamiltonian mechanics
3.4.1 Legendre transformation, Hamilton and Routh equations
For a particle, the Hamiltonian is
H=p
p2c2+m2c4: (3.4.1)
In the nonrelativistic limit, (3.4.1) reduces to
H=p2
2m+mc2: (3.4.2)
3.4.2 Poisson brackets
fM ;M g=e
M
; (3.4.3)
which resembles (1.6.37). This relation gives
fM2;M g=fM M ;M g= 0;whichresembles (1:6:41): (3.4.4)
The square of the angular momentum vector is therefore the Casimir invariant of the rotation group,
that is, a quantity whose Poisson brackets with Mvanish.
3.4.3 Maupertuis principle
3.4.4 Canonical transformations
3.4.5 Liouville theorem and distribution functions
3.4.6 Adiabatic motion
References: [12, 14].
3.5 Hamilton-Jacobi mechanics
3.5.1 Hamilton-Jacobi equation
pi= @S
@xiandpipi=m2c2give the Hamilton-Jacobi equation
@S
@xi@S
@xkgik=m2c2: (3.5.1)
Substituting
S0=S+mc2t (3.5.2)
into (3.5.1) in the Galilean frame of reference gives
1
2m(rS0)2 1
2mc2@S0
@t2
+@S0
@t= 0: (3.5.3)
In the nonrelativistic limit, (3.5.3) reduces to
1
2m(rS0)2+@S0
@t= 0: (3.5.4)
3.5.2 Canonical variables
References: [12, 14].
132
4 Applications
4.1 Mechanics
4.1.1 Orthonormal systems of coordinates
The position of a particle in the Cartesian coordinates is given by the radius vector
r= (x;y;z ) =x e ; (4.1.1)
where the Cartesian unit vectors ( versors ) are constant vectors:
e1= (1;0;0);e2= (0;1;0);e3= (0;0;1): (4.1.2)
The velocity of the particle is
v=v e = _x e ; (4.1.3)
so
v = _x : (4.1.4)
The acceleration of the particle is
a=a e = _v e ; (4.1.5)
so
a = x : (4.1.6)
The Cartesian versors are orthonormal (unit and orthogonal):
e e = : (4.1.7)
We consider a transformation from the Cartesian coordinates x to generalized (curvilinear)
coordinates q :
x =x (q ): (4.1.8)
The di erential dris thus
dr=h dq ; (4.1.9)
where
h =@r
@q : (4.1.10)
The scalar product
h h =@x
@q @x
@q e
e=
; (4.1.11)
where
is the spatial metric tensor (1.4.98). If
is diagonal then one can de ne a vector
n =h p
(4.1.12)
for each . These vectors are orthonormal,
n n = ; (4.1.13)
and form a basis of an orthonormal system of coordinates in which the upper and lower indices
are identical.
The vectors n generally are not constant. The time derivative _n can be expressed as linear
combinations of n :
_n =! n ; (4.1.14)
where the coecients ! depend on the chosen coordinates q . Multiplying (4.1.14) by n
gives
_n n
=! n n
=!
=!
; (4.1.15)
133
so
! +! = (n n )= 0: (4.1.16)
Thus the spatial tensor ! is antisymmetric. The velocity is given by
v=@r
@q _q = _q h =X
p
_q n : (4.1.17)
The acceleration is given by
a=X
(p
_q )n +X
p
_q _n =X
;
! p
_q + (p
_q )
n : (4.1.18)
The passage from the Cartesian coordinates x;y;z to the cylindrical coordinates r; ;z is given
by
x=rcos ; y =rsin ; z =z: (4.1.19)
The radius vector is r= (rcos ;rsin ;z) and the metric tensor is
=0
@1 0 0
0r20
0 0 11
A; (4.1.20)
which gives
nr= (cos ;sin ;0);n = ( sin ;cos ;0);nz= (0;0;1): (4.1.21)
Thus _nr=_ n ,_n = _ nrand _nz= 0, so
! =0
@0 _ 0
_ 0 0
0 0 01
A: (4.1.22)
The velocity is given by (4.1.17):
v= _rnr+r_ n + _znz; (4.1.23)
so
vr= _r; v =r_ ; vz= _z: (4.1.24)
The acceleration is given by
a= ( _rnr+r_ n + _znz)= (r r_ 2)nr+ (r + 2 _r_ )n + znz; (4.1.25)
so
ar= r r_ 2; a =r + 2 _r_ ; az= z: (4.1.26)
The gradient (1.4.124), divergence (1.4.126) and Laplacian (1.4.128) in the cylindrical coordinates
are
r =@
@r;@
r@ ;@
@z
; (4.1.27)
divv=@vr
@r+vr
r+@v
r@ +@vz
@z; (4.1.28)
4=1
r@
@r
r@
@r
+@2
r2@ 2+@2
@z2: (4.1.29)
134
4.1.2 Uniformly accelerated motion
If the four-velocity uiof a particle is constant then its motion is uniform . If the four-acceleration
wiof a particle is constant then its motion is uniformly accelerated . We consider wi= const and
vka. Ifvis along the xdirection, v=vexanda=aex, then (1.6.119) gives
wx=
2a
c2+
4v2a
c4=
4a
c2; (4.1.30)
where
= (1 v2=c2) 1=2. We also have
wx=dux
ds=u0
c2d(
v)
dt=
c2d(
v)
dt; (4.1.31)
which gives
d
dt(
v) =
3a: (4.1.32)
The relations (1.6.121) and (1.6.123) give the proper acceleration (1.6.123)
a2
0=
4
a2+
2v2
c2a2
=
6a2; (4.1.33)
which brings (4.1.32) to
d
dt(
v) =a0: (4.1.34)
If the time is chosen such that v= 0 att= 0 then (4.1.34) integrates to
v=ca0t
(c2+ (a0t)2)1=2: (4.1.35)
Ast!1 ,v!c. If the coordinates are chosen such that x=c2
a0; y=z= 0 att= 0 then (4.1.35)
integrates, using v=dx
dt, to
x2 (ct)2=c4
a2
0: (4.1.36)
The motion is thus hyperbolic, with a parametric solution
ct=c2
a0sinh; x =c2
a0cosh: (4.1.37)
The proper time of the particle is given by
=Zt
0dt
u0=Zt
0(1 v2=c2)1=2dt=c
a0sinh 1a0t
c=c
a0: (4.1.38)
The world line (4.1.36) is represented by a hyperbola ACin Fig. 5. As t!1 , this curve
asymptotically tends to the line ct=x, which is represented by a line OB. No signal emitted in the
region on the left of this line can therefore reach a point moving along the world line (4.1.36). Such
a point has an event horizon: no signals from world points which are behind the given point (in the
instantaneous rest frame) at distances larger thanc2
a0att= 0 (jOAjin Fig. 5) can reach this point.
We consider the following transformation from the Cartesian coordinates ct;x;y;z to new coor-
dinates (cT;X;y;z ):
ct=XsinhcT
; x=XcoshcT
; (4.1.39)
where = const. The inverse transformation is
cT= tanh 1ct
x; X =p
x2 (ct)2: (4.1.40)
135
Figure 5: Event horizon in uniformly accelerated motion.
Substituting (4.1.39) into the Minkowski interval (1.4.85) gives
ds2=X2
2c2dT2 dX2 dy2 dz2: (4.1.41)
The interval (4.1.41) is called the Rindler metric . If a point is at rest in the Rindler metric,
X=c2
a0; y= const; z= const; (4.1.42)
wherea0= const, and we de ne
=cT
; (4.1.43)
then (4.1.39) gives (4.1.37). Thus a point at rest in the Rindler coordinates cT;X is moving with a
constant proper acceleration a0in the Galilean coordinates.
In the nonrelativistic limit, a0t c, (4.1.35) reduces to
v=a0t; (4.1.44)
which integrates to
x=1
2a0t2+x0: (4.1.45)
Therefore a nonrelativistic uniformly accelerated motion is parabolic. As t!1 ,v!1 , so the
applicability of this limit is restricted to velocities much smaller than c. Formulae (4.1.44) and
(4.1.45) are special cases of the nonrelativistic kinematics of a particle moving in a uniform eld of
force fwith a constant acceleration a=f
m, which integrates to
v=at+v0;r=1
2at2+v0t+r0; (4.1.46)
where v0andr0are the velocity and position of the particle at t= 0.
4.1.3 Uniformly rotating frame of reference
The Galilean frame of reference in the cylindrical coordinates is described, due to (4.1.20), by the
interval
ds2=c2dt2 dr2 r2d 2 dz2: (4.1.47)
We consider a transformation to a frame of reference which rotates uniformly about the z-axis with
angular velocity !:
= 0+!t: (4.1.48)
The interval in this frame is
ds2= (c2 !2r2)dt2 2!r2d 0dt dr2 r2d 02 dz2: (4.1.49)
136
The rotating frame can be realized with physical bodies if g00>0 which gives
r<c
!; (4.1.50)
representing the impossibility of exceeding the speed of propagation of interaction by physical bodies
rotating with this frame. The square of an in nitesimal distance (1.4.97) corresponding to the
interval (4.1.49) is
dl2=dr2+dz2+r2d 02
1 !2r2
c2; (4.1.51)
so the ratio of the circumference of a circle at z= const to its radius is 2 =q
1 !2r2
c2. The time
di erence between two synchronized in nitesimally separated points is, due to (1.4.101),
t=1
c2Z!r2d 0
1 !2r2
c2: (4.1.52)
Thus the time di erence along an in nitesimal closed curve connecting synchronized points is
t=1
c2I!r2d 0
1 !2r2
c2; (4.1.53)
which, in the limit!r
c 1, gives
t=!
c2I
r2d 0=2!
c2S; (4.1.54)
whereSis the surface area of the projection of the contour on the plane perpendicular to the z-axis
and the signdepends on whether the synchronization takes place in the direction of the rotation
or the opposite. If the length of the contour is Lthen the time of the propagation of a signal along
the contour is thus
t=L
c2!
c2S; (4.1.55)
so the speed of propagation of interaction in the rotating frame measured in the inertial frame is
c0=c2!S
L: (4.1.56)
The metric tensor corresponding to the interval (4.1.49) is
gik=0
BB@1 !2r2
c2 0 !r2
c0
0 1 0 0
!r2
c0 r20
0 0 0 11
CCA; gik=0
BB@1 0 !
c0
0 1 0 0
!
c0!2
c2 1
r20
0 0 0 11
CCA: (4.1.57)
Substituting the corresponding Christo el symbols (1.4.26) into the metric geodesic equation (1.4.80)
gives the equations of motion of a particle in a uniformly rotating frame of reference:
du0
ds= 0;duz
ds= 0; (4.1.58)
dur
ds r!
cu0+u 2
= 0; (4.1.59)
du
ds+2
ruru +2!
cru0ur= 0: (4.1.60)
Due to (4.1.58), we can take s=ct. Thus (4.1.59) and (4.1.60) give, using (4.1.26),
ar=@vr
@t rv 2=!2r+ 2!rv ; (4.1.61)
a =r@v
@t+ 2vrv = 2!vr: (4.1.62)
137
We consider a nonrelativistic particle in a frame of reference Kwhich rotates uniformly relative
to an inertial frame K0with angular velocity !and whose origin coincides with the origin of K0,
sor=r0. Substituting _!= 0 and A= 0 into the Lagrangian (3.2.48) gives
L=1
2mv2+mv(!r) +1
2m(!r)2 U; (4.1.63)
so the equation of motion (3.2.50) is
mdv
dt= @U
@r0 2m!v m!(!r): (4.1.64)
This nonrelativistic equation of motion in a uniformly rotating frame of reference is consistent with
(4.1.61) and (4.1.62) for U= 0 (no external forces). Thus the relativistic Coriolis and centrifugal
inertial forces have the same forms as their nonrelativistic expressions. The momentum p=@L
@vfor
the Lagrangian (4.1.63) is
p=mv+m!r=mv0=p0; (4.1.65)
where v0is the velocity of the particle in K0. The momentum of the particle pinKcoincides with
the momentum p0inK0. Thus the angular momentum of the particle M=rpinKcoincides
with the angular momentum M0=r0p0inK0. The energy E=pv Lfor the Lagrangian
(4.1.63) is
E=1
2mv2 1
2m(!r)2+U; (4.1.66)
where the term 1
2m(!r)2, which is independent of the velocity, is referred to as the centrifugal
potential energy . Substituting v0=v+!rin (4.1.66) gives, using E0=1
2mv2
0+U, the relation
between the energies EandE0in the two frames:
E=1
2mv2
0 mv0(!r) +U=E0 m!(rv0) =E0 M!: (4.1.67)
The relation E=E0 M!is also valid for a system of particles due to the additivity of the
Lagrangian.
4.1.4 Unidimensional nonrelativistic motion
In 1 dimension, the Lagrangian for a particle in an external eld Uis given by
L=1
2a(q) _q2 U(q); (4.1.68)
whereqis a generalized coordinate. In the Cartesian coordinates, the Lagrangian becomes
L=1
2m_x2 U(x): (4.1.69)
The conservation of energy,
1
2m_x2+U(x) =E; (4.1.70)
gives
t=rm
2Zdxp
E U(x)+ const: (4.1.71)
The motion of the particle is possible in the regions where U(x)<E. The points where U(x) =E
are called turning points . At these points, the particle is at rest: _ x= 0. If the region of motion is
bounded by 2 turning points then such a motion is nite , that is, it takes place in a nite region of
space. A nite motion is oscillatory with a period (the doubled time of the motion of the particle
from one turning point to the other)
T(E) =p
2mZx2(E)
x1(E)dxp
E U(x); (4.1.72)
138
wherex1(E) andx2(E) are the coordinates of the turning points. The inverse function x(U) is
two-valued. If the region of motion is bounded by 1 or no turning points then such a motion is
in nite .
If the potential energy U(x) has a local minimum, whose coordinate can be taken at x= 0 and
whose value can be set to 0, then (4.1.72) gives
T(E) =p
2mZ0
Edx1(U)
dUdUp
E U+p
2mZE
0dx2(U)
dUdUp
E U; (4.1.73)
wherex1(U) andx2(U) are the two branches of the function x(U), as shown in Fig. (6). Integrating
ZK
0T(E)dEp
K E=p
2mZK
0ZE
0dx2
dU dx1
dUdUdEp
(K E)(E U)
=p
2mZK
0dx2
dU dx1
dU
dUZK
UdEp
(K E)(E U): (4.1.74)
UsingRK
UdE=p
(K E)(E U) =and taking K=Ugives
x2 x1=1
p
2mZU
0T(E)dEp
U E: (4.1.75)
If the function U(x) is symmetric with respect to its minimum then x2= x1=x, yielding the
functionx(U):
x(U) =1
2p
2mZU
0T(E)dEp
U E: (4.1.76)
Inverting this function gives the potential energy U(x) as a functional of the period T(E).
Figure 6: Potential energy in a nite motion.
4.1.5 Central eld
We consider a nonrelativistic particle in an external eld. If the potential energy Udepends only
on the distance rof the particle from the a xed point, called the center of the eld, then the eld
is referred to as central . The force acting on the particle is given by
F= @U
@r= dU
drr
r: (4.1.77)
Since the angular momentum Mof the particle (or any system) in such a eld relative to the center
is conserved (cf. (3.1.151)),
_M=rF= 1
rdU
drrr= 0; (4.1.78)
andMis perpendicular to r, the radius vector rand the path of the particle lie in a plane per-
pendicular to the constant vector M. The motion of the particle in a central eld is therefore
139
two-dimensional. The Lagrangian of a particle in a central eld U(r) in the cylindrical coordinates
(r; ;z ) is given by
L=1
2m( _r2+r2_ 2) U(r): (4.1.79)
The relation@L
@ = 0 shows that is a cyclic coordinate. The Lagrange equations for ,d
dt@L
@_ =@L
@ ,
give the conservation of the corresponding generalized momentum,
p =@L
@_ =mr2_ =Mz; (4.1.80)
which coincides with the conserved angular momentum M=Mz^z.
The area of an in nitesimal, triangular region bounded by the radius vectors r,r+dr, and the
element of the path dr=rd is
df=1
2r2d ; (4.1.81)
as shown in Fig. (7). The absolute value of angular momentum M=mr2_ is therefore equal to
M= 2m_f; (4.1.82)
where _fis called the sectorial velocity of the particle. The conservation of angular momentum in a
central eld gives thus Kepler's second law : the radius vector of a particle in such a eld sweeps out
equal areas in equal times.
Figure 7: Radius vector.
The energy of the particle in a constant central eld U(r) is conserved,
E=1
2m( _r2+r2_ 2) +U(r): (4.1.83)
Using (4.1.80) in (4.1.83) gives
E=1
2m_r2+Ue ; (4.1.84)
where
Ue =U(r) +M2
2mr2(4.1.85)
is the e ective potential andM=Mz. The termM2
2mr2is the centrifugal potential energy (cf.
(4.1.66)). The centrifugal energy diverges as rtends to 0. Therefore, the particle cannot reach the
center of the eld unless Utends to 1 asr!0 suciently enough so that Ue tends there to a
nite value. The motion of the particle is possible in the regions where Ue <E. The points where
Ue =Eare the turning points for the rcoordinate, _ r= 0. At these points, however, the particle
is not at rest: _ 6= 0. If the region of motion is bounded by 2 turning points for r,r1rr2,
then such a motion is nite. If the region of motion is bounded from below by 1 turning point for
r,rr1, then such a motion is in nite.
Integrating (4.1.84) gives
t(r) =Zdrq
2
m(E U) M2
m2r2+ const; (4.1.86)
140
which upon inversion gives the motion of the particle, r(t). Substituting _ r=dr
d _ into (4.1.84) and
integrating gives
(r) =ZM
r2drq
2m(E U) M2
r2+ const; (4.1.87)
which upon inversion gives the path of the particle, r( ). Since _ does not change sign, (t) is a
monotonic function. Substituting
dr
dt=dr
d d
dt=M
mr2dr
d = M
md
d 1
r
(4.1.88)
into (4.1.84), and de ning
u=1
r; (4.1.89)
gives
E=M2
2mdu
d 2
+u2
+U(1=u): (4.1.90)
Di erentiating (4.1.90) with respect to and dividing byM2
mdu
d gives
d2u
d 2+u= m
M2d
duU(1=u): (4.1.91)
The path of a particle in a central eld is a closed curve if the change of during the period of the
oscillation of the rcoordinate,
= 2Zr2
r1M
r2drq
2m(E U) M2
r2; (4.1.92)
is a rational fraction of 2 . The only central potentials that lead to closed paths are Ur2( =)
andU1=r( = 2).
The concept of the potential energy is not valid in relativistic mechanics. For a charged particle
in an electromagnetic eld, however, the energy of the particle has an additive contribution from
the eld. We consider a relativistic particle of charge ein a central electrostatic eld produced by
a xed charge e0. The motion of the particle takes place in a plane perpendicular to the constant
angular momentum vector. The conserved energy of the particle is given by (3.1.80):
E=p
p2c2+m2c4+e ; (4.1.93)
where =e0
r(4.2.7) and A= 0. The momentum of the particle can be decomposed into the angular
and radial parts:
p2=M2
r2+p2
r: (4.1.94)
The corresponding Hamilton-Jacobi equation is
1
c2@S
@t+ee0
r2
+@S
@r2
+1
r2@S
@ 2
+m2c2= 0: (4.1.95)
PuttingS= Et+M +Srgives
S= Et+M +Z
drr
1
c2
E ee0
r2
M2
r2 m2c2: (4.1.96)
The motion of the particle is given by@S
@E= const, whereas its trajectory is given by@S
@M= const. If
ee0<0 then the motion is nite (and the path is not closed) for E <mc2and in nite for Emc2.
Ifee0>0 then the motion is in nite.
141
The motion of two interacting particles regarded as a closed system is referred to as the two-body
problem . For nonrelativistic particles, such a motion can be separated into the motion of the center
of mass of the system and the motion of the particles relative to the center of mass. The Lagrangian
of the system of two particles with masses m1andm2is given by
L=1
2m1_r2
1+1
2m2_r2
2 U(jr1 r2j): (4.1.97)
De ning the separation vector,
r=r1 r2; (4.1.98)
and using the center of mass of the system (3.1.138),
R=m1r1+m2r2
m; (4.1.99)
wheremis the total mass of the system (3.1.137),
m=m1+m2; (4.1.100)
gives
r1=m2r
m+R;r2= m1r
m+R: (4.1.101)
The rst terms on the right-hand sides in (4.1.101) are the coordinates of the two particles relative
to the center of mass. Accordingly, the Lagrangian (4.1.97) becomes
L=1
2m_R2+1
2_r2 U(jrj); (4.1.102)
where
=m1m2
m1+m2(4.1.103)
is referred to as the reduced mass . The Lagrangian (4.1.102) describes the free motion of a particle
of massmand the motion of a particle of mass in an external eld U(r). The motion of the
two particles is then given by substituting R(t) and r(t) into (4.1.101). The generalized momenta
corresponding to the coordinates Randrare
P=@L
@_R=m_R=m1_r1+m2_r2;p=@L
@_r=_r: (4.1.104)
The quantity Pis the total momentum of the system.
4.1.6 Kepler motion
We consider the motion of a particle with mass min a nonrelativistic (Newtonian) gravitational
eld of a body with mass . Such a eld is central, with the potential energy given by (3.1.103) and
(4.3.59):
U(r) = Gm
r: (4.1.105)
The motion in such a eld is referred to as a Keplerian motion . The e ective potential is thus
Ue =
r+M2
2mr2; (4.1.106)
where
=Gm> 0: (4.1.107)
The particle therefore cannot reach the center of the eld unless it moves radially, M= 0. The
e ective potential Ue (r) is shown in Fig. 8. Such a potential has a minimum at
Umin= m
2
2M2: (4.1.108)
142
Figure 8: E ective potential in an
attractive eld 1 =r.
Figure 9: E ective potential in a
repulsive eld 1 =r.
The motion of the particle is nite for E < 0 and in nite for E0. If
< 0 then the e ective
potentialUe (r) is shown in Fig. 9. The motion in such a eld is in nite.
The Newton equation of motion (3.1.100) in a Newtonian eld gives
mr=
r
r3: (4.1.109)
Since such a eld is central, the angular momentum of the particle, M=mr_r, is conserved (cf.
(4.1.78)). The eld is constant in time, so that the energy of the particle is conserved,
E=1
2m_r2+
r; (4.1.110)
which can also be obtained by multiplying (4.1.109) by _rand integrating. We consider a quantity
L=1
mpM
r
r: (4.1.111)
Di erentiating Lwith respect to time gives, using (4.1.109),
dL
dt=rM
d
dtrprr=
r3r(r_r) +
(r_r)r
r3 _r
r
= 0: (4.1.112)
The vector (4.1.111) is thus conserved and it is referred to as the Laplace-Runge-Lenz vector . Unlike
the energy and angular momentum, this conserved quantity is not an additive constant of motion.
The existence of such a one-valued function of randvis related to a speci c character of the motion
in the1
rpotential, called degeneracy (cf. ??). The Laplace-Runge-Lenz vector is perpendicular to
the angular momentum vector:
LM=1
m(pM)M
r
r(rp) = 0; (4.1.113)
so that it lies in a plane of the motion and has 2 independent components. The number of the
conserved quantities in the Keplerian motion is thus 6: 3 components of M, 2 components of L, and
E, related to 2 narbitrary constants of the equations of motion for n= 3 degrees of freedom of a
particle.
The square of the Laplace-Runge-Lenz vector is, using (4.1.110), equal to
L2=LL= (vM)2 2
rr(vM) +
2=v2M2 (vM)2
2
rM(rv) +
2=v2M2 2
mrM2+
2=2EM2
m+
2: (4.1.114)
We also have
Lr=1
mr(pM)
r=1
mM(rp)
r=M2
m
r: (4.1.115)
143
De ning as the angle between Land the radius vector rgives Lr=Lrcos , so that (4.1.115)
turns into
r=p
1 +ecos ; (4.1.116)
where
p=M2
m
; e=L
: (4.1.117)
The relation r( ) (4.1.116) is the equation of a conic-section curve with latus rectum 2 pand eccen-
tricitye, giving a mathematical formulation of Kepler's rst law . It is a circle or radius pfore= 0,
an ellipse for 0 <e< 1, a parabola for e= 1, and a hyperbola for e>1. Substituting (4.1.117) into
(4.1.114) gives
E=
(e2 1)
2p; e=s
1 +2EM2
m
2: (4.1.118)
The path of the particle (4.1.116) can also be obtained from (4.1.87) or (4.1.91). The latter gives
d2u
d 2+u=m
M2=1
p.
The path of a particle in a Newtonian eld for e<1 (E < 0) is shown in Fig. 10. The origin of
the eld is in one of the two foci of the ellipse, F. The center of the ellipse is at point O. The semi-
major axis of the ellipse is jOPj=jFBj=a, the semi-minor axis is jOBj=b=ap
1 e2=Mp 2mE
(due to (4.1.118)), jOFj=ae, andjFDj=p. The Laplace-Runge-Lenz vector is directed along
the vectorFP. When = 0, the particle is located at the point Pwhich is closest to the center,
called a periastron , for which r=rmin=p
1+e. When =, the particle is located at the point
Awhich is furthest from the center, called a apoastron , for which r=rmax=p
1 e. The relation
rmin+rmax= 2agives
p=a(1 e2); (4.1.119)
so thatrmin=a(1 e),rmax=a(1 +e), and (4.1.118) yields
a=
2E: (4.1.120)
Accordingly, the equation of path (4.1.116) is
r=a(1 e2)
1 +ecos : (4.1.121)
The periastron and apoastron are turning points: rminandrmaxare the two roots of the equation
Ue (r) =E. IfE=Uminthene= 0.
Some properties of the Keplerian motion can be obtained using mechanical similarity. The
potential energy in such a eld is a homogeneous function of the Cartesian coordinates (3.1.162)
with degree of homogeneity k= 1. The rst relation in (3.1.164) gives thus
t0
t=l0
l3=2
: (4.1.122)
Accordingly, the period Tof revolution of a particle along an elliptical orbit is related to the semi-
major axis aof the ellipse by
T2a3; (4.1.123)
which constitutes Kepler's third law . The value of Tis given by the constancy of the sectorial velocity
in (4.1.82), which constitutes Kepler's second law. This law gives MT= 2mab , from which
T= 2rm
a3=2: (4.1.124)
Substituting (4.1.118) and (4.1.120) into (4.1.86) gives
t=rma
Zrdrp
a2e2 (r a)2: (4.1.125)
144
Integrating (4.1.125) over the whole ellipse gives the period (4.1.124),
T= 2rma
Za(1+e)
a(1 e)rdrp
a2e2 (r a)2: (4.1.126)
The expression (4.1.126) can also be derived using (4.1.88) and the equation of path (4.1.116).
Substituting
r=a(1 ecos); (4.1.127)
whereis a parameter, into (4.1.125) and putting t= 0 when the particle is at the periastron gives
Kepler's equation
!t= esin; (4.1.128)
where
!=2
T: (4.1.129)
Combining the equation of path (4.1.121) with (4.1.127) leads to
ae+rcos =acos; (4.1.130)
tan
2=r
1 +e
1 etan
2: (4.1.131)
The relation (4.1.130) is represented in Fig. 11 which shows the auxiliary circle with radius a,
corresponding to the Keplerian ellipse in Fig. 10. The angle between the major axis and the
radius vector ris, in astronomy, denoted vand called the true anomaly . The angle is between the
major axis and a vector, which is directed from the center of the circle (and of the ellipse) to the
point on the circle with the same horizontal coordinate as the particle. This angle is, in astronomy,
denotedEand called the eccentric anomaly . The quantity !tin (4.1.128) is, in astronomy, denoted
Mand called the mean anomaly . Kepler's equation (4.1.128) is then
M=E esinE: (4.1.132)
Figure 10: Keplerian motion on
an ellipse.
Figure 11: Auxiliary circle.
ForE= 0, (4.1.118) indicates the motion on a parabola, e= 1. The only turning point is located
at the periastron, rmin=p
2, due to (4.1.116). Such a motion is given by (4.1.86):
t=Zrdrq
2
mr M2
m2: (4.1.133)
Substituting
r=1
2p(1 +2); (4.1.134)
145
whereis a parameter, into (4.1.133) and putting t= 0 when the particle is at the periastron gives
r
mp3t=1
2
+3
3
: (4.1.135)
ForE > 0, (4.1.118) indicates the motion on a hyperbola, e > 1, as shown in Fig. 12. The
equation of path is
r=a(e2 1)
1 +ecos ; (4.1.136)
where the semi-major axis of the hyperbola is given by
a=
2E=p
e2 1: (4.1.137)
The periastron is located at jFPj=rmin=a(e 1), andjFBj=p. The parametric equations
(4.1.127) and (4.1.128) are replaced by
r=a(ecosh 1); (4.1.138)
!t=esinh : (4.1.139)
De ning
tan
2=r
1 +e
e 1tanhH
2(4.1.140)
leads to Kepler's equation
M=esinhH H: (4.1.141)
If the eld is repulsive,
<0, then the motion is shown in Fig. 13. The equation of path is
r=p
ecos 1; (4.1.142)
where
p= M2
m
: (4.1.143)
The periastron is located at jFPj=rmin=a(e+ 1). The parametric equations (4.1.127) and
(4.1.128) are replaced by
r=a(ecosh+ 1); (4.1.144)
!t=esinh+: (4.1.145)
Figure 12: Keplerian motion on a
hyperbola.
Figure 13: Motion on a hyperbola
in a repulsive eld.
We consider the motion of a particle in a perturbed Keplerian eld, described by the Lagrangian
(4.1.79) with
U(r) =
r+U(r); (4.1.146)
146
whereU(r) is a small correction to the Newtonian potential
r. The change of during the period
of the oscillation of the rcoordinate (4.1.92) can be written as
= 2@
@MZrmax
rminr
2m(E U) M2
r2dr: (4.1.147)
Substituting (4.1.146) into (4.1.147) and expanding in Ugives
= 2+ ; (4.1.148)
where
=@
@MZrmax
rmin2mUdrq
2m(E+
r) M2
r2=@
@M2m
MZ
0r2Ud
: (4.1.149)
IfE6= 0 then we normalize the Laplace-Runge-Lenz vector by de ning
D=Lp
2mjEj: (4.1.150)
The Poisson brackets of the components of such a vector and the components of the angular mo-
mentum vector are
[D ;D ] = sgn(E)e
M
; (4.1.151)
[D ;M ] =e
D
: (4.1.152)
IfE < 0 then the relations (3.4.3), (4.1.151) and (4.1.152) form the Lie algebra of the rotation
group in the four-dimensional Euclidean space. The Casimir invariants Ciof such a group are the
quantities whose Poisson brackets with MandDvanish:
fCi;M g= 0;fCi;D g= 0: (4.1.153)
These invariants are
C1=MD= 0; C2=M2+D2=m
2
2jEj: (4.1.154)
4.1.7 Decays and collisions of particles
We consider a free body with mass Mat rest, decaying into two particles with masses m1;m2
and momenta p10;p20. The energies of the resulting particles are E10=p
p2
10c2+m2
1c4,E20=p
p2
20c2+m2
2c4. The conservation of momentum in this frame of reference gives p10+p20= 0, so that
E2
10 m2
1c4=E2
20 m2
2c4. Combining this relation with the conservation of energy, E10+E20=Mc2,
gives the values of E10andE20. The rest frame of the body is the center-of-inertia frame ( C-frame ) of
the two resulting particles. If the internal state of particles does not change during the collision then
such a collision is called elastic . The decay is possible if the binding energy Eb= (M m1 m2)c2
is positive, otherwise the body is unstable with respect to this decay.
In a laboratory frame ( L-frame ), the body moves with velocity Vand decays into the same two
particles with momenta p1;p2and energies E1=p
p2
1c2+m2
1c4,E2=p
p2
2c2+m2
2c4. We consider
one of the emerging particles, omitting the subscripts 1,2. We take the plane in which the vectors
V,pandp0lie as thexy-plane (the x-axis along the vector V). We choose the L-frame as K
and the C-frame as K0in (3.1.61) and (3.1.62), so that Vis the velocity of the C-frame relative to
the L-frame. If mis the mass of the particle, is the angle between pandV, and0is the angle
between p0andV, then (3.1.61) and (3.1.62) give
E0=E (V=c)p
E2 m2c4cosp
1 V2=c2; E=E0+ (V=c)p
E2
0 m2c4cos0p
1 V2=c2; (4.1.155)
pcos=px=p0cos0+E0V=c2
p
1 V2=c2; psin=py=p0sin0: (4.1.156)
147
The value of 0depends on the law of interaction of the particles and their relative positions. If we
regard the two resulting particles as a single composite system then the velocity of its motion as a
whole is given by (3.1.71),(p1+p2)c2
E1+E2=V. The invariant mass of such a composite system,
minv=1
c2sX
aEa2
X
apa2
c2; (4.1.157)
is equal to1
c2p
(E1+E2)2 (p1+p2)2c2=M.
The relations (4.1.156) give ( pxp
1 V2=c2 E0V=c2)2+p2
y=p2
0, which is represented by a
kinematic ellipse in Fig. 14. The major semiaxis of the ellipse is along the x-axis and it is equal to
p0p
1 V2=c2. The minor semiaxis is along the y-axis and it is equal to p0. The vector AO=VE0=c2p
1 V2=c2
and the vector AB=p. IfV < v 0=p0c2
E0thenAlies inside the ellipse. In this case, for each
there is 1 value of Eand thus 1 value of 0. IfV >v 0thenAlies outside the ellipse. In this case,
for eachthere are 2 values of Eand thus 2 values of 0. Also,cannot exceed a value max
given by sin max=p0p
1 V2=c2
mV. The angles of emergence of the two particles in the C-frame satisfy
10+20=. The separation angle in the L-frame, which is the angle between the directions of the
particles after decay, is equal to = 1+2.
In the nonrelativistic limit, the relations (4.1.155) and (4.1.156) reduce to v=v0+V, where
vis the velocity of a decay particle in the L-frame, and v0is the velocity of this particle in the
C-frame. This relation gives v2=v2
0+V2+ 2v0Vcos0andv2
0=v2+V2 2vVcos, from which
tan=v0sin0
v0cos0+V. Accordingly, the ellipse in Fig. 14 reduces to a circle of radius p0, where
AO=mVandAB=mv, so thatOB=mv0. An equivalent circle of radius v0=p0
m, where
AO=V,AB=vandOB=v0, is shown in Fig. 15. If V >v 0then sinmax=v0
V.
Figure 14: Kinematic ellipse of a
decay.
Figure 15: Nonrelativistic limit.
For a system of decaying particles that are randomly oriented in space, every decay particle of
a given kind has the same energy in the C-frame and the angular distribution of their directions
of motion is isotropic. The fraction of particles in a solid angle do0is thus equal to dN=do0
4,
wheredo0= 2sin0d0and02[0;]. The angular distributions dNand the ranges of possible
values of other quantities can be obtained from the relations between 0and those quantities, such
as (4.1.155). If a particle decays into 3 or more particles then the laws of conservation of energy
and momentum allow more freedom in directions of motion and velocities of the emerging particles.
The kinetic energy of each decay particle cannot exceed the maximum value which the particle has
if the system of the other particles has the least possible mass. Such a mass is equal to the sum of
the masses of those particles and describes the case where the particles move with the same velocity.
We consider two colliding particles with masses m1;m2and four-momenta pi
1;pi
2. We consider
anelastic collision , which is a collision during which the internal states of particles do not change.
The conservation law of four-momentum is
pi
1+pi
2=p0i
1+p0i
2; (4.1.158)
148
where the primed quantities refer to the particles after the collision. This law gives, using p1ipi
1=
p0
1ip0i
1=m2
1c2andp2ipi
2=p0
2ip0i
2=m2
2c2,
p1ipi
2 p1ip0i
1 p2ip0i
1+m2
1c2= 0;
p2ipi
1 p2ip0i
2 p1ip0i
2+m2
2c2= 0: (4.1.159)
Inelastic collisions, where the states of colliding particles change, are possible if the invariant mass
(4.1.157) of the resulting particles does not exceed the invariant mass of the original particles.
The L-frame for collisions is de ned as a frame of reference in which one of the colliding particles,
for example m2, is at rest, so that p2= 0 andE2=m2c2. The velocity of the motion of the system
of such particles as a whole (3.1.71),
V=p1c2
E1+m2c2; (4.1.160)
is the velocity of the C-frame relative to the L-frame. Therefore, before the collision, the particle
with massm2moves in the C-frame with velocity V, so that its momentum is
p20= m2V
1 V2=c2: (4.1.161)
If1is the angle between p1andp0
1, and2is the angle between p1andp0
2, then in the L-frame
we obtainp1ipi
2=E1m2,p1ip0i
1=E1E0
1=c2 p1p0
1=E1E0
1=c2 p1p0
1cos1,p2ip0i
1=m2E0
1,
p2ip0i
2=m2E0
2andp1ip0i
2=E1E0
2=c2 p1p0
2=E1E0
2=c2 p1p0
2cos2. These quantities are scalars
so that they have the same values in all frames of reference. Substituting the above expressions into
(4.1.159) gives the relations between the energies of the particles and the directions of their motion.
The time component of (4.1.158) in the L-frame is the conservation of energy, E1+m2c2=E0
1+E0
2.
The energy in the L-frame transferred from the incident particle m1to the target particle m2is
equal toEt=E0
2 m2c2. Ifm1>m 2then the scattering angle 1has a maximum value given by
sinmax=m2
m1.
In the C-frame, the momenta of the two particles before the collision, p10=p0andp20, satisfy
p10+p20= 0. The conservation of momentum causes that the momenta of the particles rotate
during the collision, remaining opposite in direction and equal in magnitude. The conservation
of energy causes that the values of the momenta (and of the energies) do not change during the
collision. If is the angle of such a rotation ( angle of scattering ), that is, the angle between p10
andp0
10or between p20andp0
20, then in the C-frame we obtain p1ipi
2=E10E20=c2 p10p20=p
(p2
0+m2
1c2)(p2
0+m2
2c2)+p2
0andp1ip0i
1=E10E0
10=c2 p10p0
10=E2
10 p2
0cos =p2
0(1 cos )+m2
1.
Substituting the above expressions into (4.1.159) gives the relations between the energies of the
particles and . A head-on collision corresponds to =.
An elastic collision of two particles with masses m1;m2and momenta p1;p2= 0 is equiva-
lent (with respect to the resulting particles) to a decay of a particle with mass equal to minv=
1
c2p
(E1+m2c2)2 p2
1c2(4.1.157), moving with velocity (4.1.160). Therefore, the vector AB=p0
1
lies on an ellipse in Fig. 16, which is analogous to that in Fig. 14. The major semiaxis of this ellipse
is equal top0p
1 V2=c2and its minor semiaxis is equal to p0, wherep0=p0
10=p0
20=p20=m2Vp
1 V2=c2
(due to (4.1.161)) and V=p1c2
E1+m2c2(due to (4.1.160)). The length of the vector ACis equal
toVE10=c2p
1 V2=c2+p10p
1 V2=c2, whereE10=p
p2
0c2+m2
1c4, leading tojACj=p1. Thus we obtain
AC=p1. The spatial components of (4.1.158) in the L-frame are the conservation of momentum,
p1=p0
1+p0
2, so that the vector BC=p0
2. Ifm1<m 2thenAlies inside the ellipse. If m1>m 2
thenAlies outside the ellipse.
In the nonrelativistic limit, the ellipse in Fig. 16 reduces to a circle of radius v1in Fig. 17,
whereis the reduced mass (4.1.103), and the angle between the vectors OBandOCis . Thus
we nd
tan1=m2sin
m1+m2cos ; 2=
2; v0
2=2m1v
m1+m2sin
2: (4.1.162)
149
This circle can also be constructed from the two-body problem. Di erentiating (4.1.101) with respect
to time gives
v1=m2v
m1+m2+V;v2= m1v
m1+m2+V; (4.1.163)
where v1andv2are the velocities of the particles before the collision, v=v1 v2is the relative
velocity (the time derivative of the the separation vector (4.1.98)), and V=m1v1+m2v2
m1+m2is the
velocity of the center of mass of the particles (3.1.135) (the time derivative of R(4.1.99)). During
the collision, the vector of relative velocity rotates without changing its magnitude, by an angle .
Ifnis the unit vector along the direction of the velocity of the particle m1after the collision, then
vin (4.1.163) turns into vn, so that v0
1=m2vn
m1+m2+V;v0
2= m1vn
m1+m2+Vor
p0
1=vn+m1
m1+m2(p1+p2);p0
2= vn+m2
m1+m2(p1+p2): (4.1.164)
In the L-frame, p2= 0, so that the formulae (4.1.164) correspond to Fig. 17 with OB=vn. A
head-on collision corresponds to n= v
v.
Figure 16: Kinematic ellipse of a
collision.
Figure 17: Nonrelativistic limit.
For two particles with masses m1;m2, velocities v1;v2and four-momenta pi
1;pi
2, we have from
(3.1.59) and (3.1.60):
p1ipi
2=m1m2c2 1 v1v2=c2
p
(1 v2
1=c2)(1 v2
2=c2): (4.1.165)
In the rest frame of the particle m2, (4.1.165) becomes
p1ipi
2=m1p
1 v2
rel=c2m2c2; (4.1.166)
wherevrelis the velocity of the particle m1in this frame,
vrel=cs
1 m2
1m2
2c4
(p1ipi
2)2: (4.1.167)
Because (4.1.167) is symmetric under interchanging m1andm2,vrelis the relative velocity of the
two particles. Substituting (4.1.165) into (4.1.167) gives
vrel=p
(v1 v2)2 (v1v2)2=c2
1 v1v2=c2: (4.1.168)
In the nonrelativistic limit, (4.1.168) reduces to vrel=jv1 v2j, in agreement with vin (4.1.163).
The relative velocity dlvbetween particles with velocities vandv+dvgives thus the element of
length in velocity space,
dl2
v=dv2 (vdv)2=c2
(1 v2=c2)2=dv2
(1 v2=c2)2+v2
1 v2=c2(d2+ sin2d 2); (4.1.169)
whereand are the polar angle and azimuth of the direction of v. In terms of the rapidity
(1.6.90), (4.1.169 is a line element in a three-dimensional Lobachevskii space (space of constant
negative curvature),
dl2
v=d2+ sinh2(d2+ sin2d 2): (4.1.170)
150
4.1.8 Scattering of particles
The trajectory of a particle with mass mmoving in an external central eld with potential U(r)
(in a plane) is de
ected from a straight line, as shown in Fig. 18. The impact parameter is the
distance at which the particle would pass the center of the eld if the eld were absent. The angle
of de
ection is the angle between the asymptotes representing the trajectory of the particle far
from the eld. It is given by
=j 2 0j; (4.1.171)
where 0is the angle in the polar coordinates of the point of the trajectory which is closest to the
center. If the motion is nonrelativistic then 0results from (4.1.87):
0=Z1
rminM
r2drq
2m(E U) M2
r2; (4.1.172)
whererminis the distance of this point from the center. Far from the eld, the conserved energy
and angular momentum of the particle are given by
E=1
2mv2
1; M =mv1; (4.1.173)
wherev1is the corresponding velocity of the particle. Accordingly, (4.1.172) becomes
0=Z1
rmin
r2drq
1 2=r2 2U
mv21: (4.1.174)
Figure 18: De
ection in a central eld.
The de
ection of a uniform (in cross section) beam of particles with the same velocities but with
di erent impact parameters results in di erent angles of de
ection. If dNis the number of particles
scattered per unit time into the angles of de
ection between and +d , andnis the number of
particles passing per unit time through unit area of the cross section of the beam, then the scattering
cross section is de ned as
d=dN
n: (4.1.175)
Particles which are scattered into the angles of de
ection between and +d have impact param-
eters between and+d, so thatdN=ndS, wheredS= 2d is the area between the circles
of radiiand+d. Therefore, we obtain
d= 2d = 2 d
d d =
sin d
d do; (4.1.176)
wheredo= 2sin d is the element of the solid angle between the cones with vertical angles
and +d . Particles that reach the center of the eld satisfy E1>U e ,max , whereUe ,max is the
151
maximum value of the e ective potential (4.1.85), which gives < max, wheremaxis a solution of
Ue ,max =E1. The cross section for a particle to reach the center is thus given by =2
max.
Scattering of two particles with masses m1;m2interacting through the potential energy U(r),
where ris the separation vector (4.1.98), is equivalent to scattering of a particle with reduced mass
(4.1.103) by the eld U(r) whose origin is located at the center of mass of the two particles
(4.1.99). Accordingly, scattering of a beam of particles by a central eld is equivalent to scattering
of two beams of particles. The cross section (4.1.176) is then a function of the angle of scattering
in the C-frame. In the L-frame, in which one beam is at rest, the scattering cross section for
incident particles is given by (4.1.176) upon substituting the relation (1) from (4.1.162), whereas
the scattering cross section for particles initially at rest is given by (4.1.176) upon substituting the
relation (2) from (4.1.162).
We consider two colliding beams of particles with particle number densities n1;n2. In the rest
frame of one beam (L-frame), the invariant number of collisions in a volume element dVand in time
intervaldtis equal to
d=vreln1n2dV dt; (4.1.177)
where=R
dis the total scattering cross section given by (4.1.176) and vrelis the relative velocity
of the two beams. In the frame of reference in which the beams have velocities v1;v2, an invariant
quantity which reduces in the L-frame to n1n2, is equal ton1n2
E1E2p1ipi
2, due ton=n0
(2.5.103) and
(3.1.60). The number of collisions in this frame in thus
d=vreln1n2
E1E2p1ipi
2dV dt: (4.1.178)
Substituting (4.1.165) and (4.1.168) into (4.1.178) gives the Pauli formula,
d=p
(v1 v2)2 (v1v2)2=c2n1n2dV dt: (4.1.179)
For a system of particles, the distribution function for the momenta of the particles, f(p), is
de ned according to
dN=f(p)dp; (4.1.180)
wheredNis the number of particles whose components of momenta lie in the intervals [ px;px+dpx],
[py;py+dpy] and [pz;pz+dpz], and
dp=dpxdpydpz (4.1.181)
is a volume element in momentum space . An element of a hypersurface is a four-vector along the
normal to the hypersurface. For the hypersurface pipi=m2c2, such a four-vector is parallel to
pi. The element (4.1.181), which is the 0th component of an element of this hypersurface, is thus
proportional to p0=E
c. Accordingly, a quantitydp
dEis an invariant, which also results from
2
c(pipi m2c2)d4p=dpp
p2c2+m2c4=dp
dE; (4.1.182)
where
d4p=dp0dp (4.1.183)
is an invariant. The invariance of dNin (4.1.180) yields the invariance of a quantity f(p)E:
f(p0)E0=f(p)E; (4.1.184)
where the energies and momenta transform according to (3.1.61) or (3.1.62). In the spherical coor-
dinates,dp=p2dpdo , wheredois the element of a solid angle around the direction of p, so that
pdp=EdE
c2leads to the invariance of pdEdo .
The distribution function for the momenta and positions of particles, f(p;r), is de ned according
to
dN=f(p;r)dpdV; (4.1.185)
152
wheredNis the number of particles whose components of momenta lie in a momentum-space volume
element (4.1.181) and in a volume element dV. The product
dq=dpdV (4.1.186)
is an element of phase space (space of the momenta and positions of particles). Since dV=
dV0p
1 v2=c2(due to (1.6.107)) and E=E0=p
1 v2=c2, where the subscripts 0 refer to the
rest frame of a particle in the volume element dV, a quantity EdV is an invariant. The invariance
ofdp
dEyields then the invariance of (4.1.186):
dq0=dq: (4.1.187)
Accordingly, the invariance of dNin (4.1.185) yields the invariance of f(p;r):
f(p0;r0) =f(p;r): (4.1.188)
4.1.9 Small oscillations
4.1.10 Motion of rigid bodies
We consider the motion of a free asymmetrical top with the principal moments of inertia I1<I2<I3.
The conserved energy Eof the top is equal to (3.2.19). Substituting (3.2.22) into (3.2.19) gives
M2
1
I1+M2
2
I2+M2
3
I3= 2E: (4.1.189)
The terminus of the angular momentum vector Mthus lies on an ellipsoid with semiaxes 2 EI1, 2EI2
and 2EI3. The angular momentum Mof the top is also conserved,
I2
1!2
1+I2
2!2
2+I2
3!2
3=M2
1+M2
2+M2
3=M2: (4.1.190)
The terminus of Mthus also lies on a sphere of radius M. When the vector Mmoves relative
to the axes of inertia, its terminus moves along the line of intersection of the two surfaces. Such
an intersection exists because 2 EI1< M2<2EI3. IfM2is near 2EI1then the sphere intersects
the ellipsoid along 2 small, closed curves around the poles of the ellipsoid on the x1axis. AsM2
increases, the curves become larger. If M2= 2EI2then the sphere intersects the ellipsoid along 2
ellipses intersecting at the poles of the ellipsoid on the x2axis. AsM2increases further, the curves
become smaller and separate. If M2is near 2EI3then the sphere intersects the ellipsoid along 2
closed curves around the poles of the ellipsoid on the x3axis. The curves of intersection are closed,
so that the motion of Mrelative to the axes of inertia is periodic. Paths passing near the poles on
thex1andx3axes lie entirely near these poles. However, paths passing near the poles on the x2
axis go far from these poles. Therefore, the rotation of a top about the x1orx3axis is stable, while
the rotation about the x2axis is unstable.
The relations (3.2.19) and (4.1.190) give !1and!3as functions of !2: Substituting these compo-
nents into Euler's equation (3.2.43) for !2(t) in the absence of external torques, I2_!2+(I1 I3)!3!1=
0, and de ning
=p
(I3 I2)(M2 2EI1)pI1I2I3t;
w=p
I2(I3 I2)p2EI3 M2!2;
k2=(I2 I1)(2EI3 M2)
(I3 I2)(M2 2EI1)<1; (4.1.191)
gives
dw
d =p
(1 s2)(1 k2s2): (4.1.192)
153
Settingt= 0 when!2= 0 gives (w) =Rw
0dxp
(1 x2)(1 k2x2), which upon inverting gives w( ) as a
Jacobian elliptic function w( ) = sn . The components of !are thus
!1=p
2EI3 M2p
I1(I3 I1)cn ;
!2=p
2EI3 M2p
I2(I3 I2)sn ;
!3=p
M2 2EI1p
I3(I3 I1)dn ; (4.1.193)
where cn =p
1 sn2 and dn =p
1 k2sn2 . These functions are periodic in (with period
4 (w= 1)). Within a nite time, the vector !returns to the same position relative to the principal
axes of inertia.
The polar angle and azimuth of the direction of the vector M(along the xed zaxis) with respect
to thex0=x1;y0=x2;z0=x3axes xed with the top are and
2 . The absolute motion of the
top relative to the xed x;y;z axes is therefore given by
Msinsin =M1=I1!1;
Msincos =M2=I2!2;
Mcos=M3=I3!3; (4.1.194)
in which we used (3.2.22). These relations give and as functions of !(4.1.193), cos =I3!3
M
and tan =I1!1
I2!2. Accordingly, (t) and (t) are periodic. The Eulerian angle does not appear in
(4.1.194). The function (t) can be instead found by integrating
_ =M(I1!2
1+I2!2
2)
I2
1!2
1+I2
2!2
2; (4.1.195)
derived from (3.2.23). It turns out that (t) is not periodic. A free asymmetrical top thus never
returns to the same position within a nite time.
We now consider the motion of a free symmetrical top with the principal moments of inertia
I1=I2< I3. The parameter k2in (4.1.191) vanishes, so that cn , sn and dn reduce to cos ,
sin and 1. Accordingly, the quantities (4.1.193) reduce to !1=Acos(
t),!2=Asin(
t) and
!3=p
M2 2EI1p
I3(I3 I1), whereA=p
2EI3 M2p
I1(I3 I1)and
=p
(I3 I1)(M2 2EI1)
I1pI3. Equivalently, Euler's equations
(3.2.43) with K= 0 give _!1=
!2, _!2=
!1and!3= const, where
=!3(I3 I1)
I1, in agreement
with (4.1.193). The vectors !andMtherefore rotate uniformly with a constant angular velocity
about the axis of the top (the zaxis). The relations (4.1.194) for the Eulerian angles give
cos=I3!3
M= const ( _= 0) and tan = cot(
t), so that =
2
t(_ =
). The relation
(4.1.195) reduces to _ =M
I1, which, using (3.2.24), gives _ =Mcos(1
I3 1
I1). We also nd
A=_ sin, in agreement with (3.2.23).
The motion of a free symmetrical top can also be directly obtained from (3.2.23) and (4.1.194).
Since the axes x0andy0can be chosen arbitrarily, because of the symmetry of the top, we can choose
thex0-axis along the line of nodes, so that = 0. Equations (3.2.23) therefore reduce to !1=_,
!2=_ sinand!3=_ cos+_ . The relations (4.1.194) reduce to M1= 0,M2=Msinand
M3=Mcos. The relations (3.2.24) give thus _= 0,I1_ =MandI3(_ cos+_ ) =Mcos, in
agreement with the formulae obtained from Euler's equations.
Finally, we derive the motion of a free symmetrical top solely from (3.2.22). Since the axes x0
andy0can be chosen arbitrarily, we can choose the y0-axis perpendicular to the plane containing
the constant angular momentum vector Mand thez0-axis. We have thus M2= 0 and then !2= 0,
so that the vectors M,!and the axis of the top (the z0-axis) are at every instant in the same
plane, as shown in Fig. 19. The velocity of all points on the axis of the top, v=!r, is therefore
perpendicular to that plane. Accordingly, the axis of the top rotates uniformly about the vector
M, along the surface of a circular cone. Such a rotation is referred to as precession . Ifis the
(constant) angle between the axis of the top and the vector Mthen the other components of the
154
angular momentum are M1=MsinandM3=Mcos. The angular velocity of the top about
its axis is thus !3=Mcos
I3. We also have !1=Msin
I1. Since M,!and thez0-axis lie in the
same plane, the vector !can be projected into the axis of the top and the direction of the angular
momentum. The rst component is equal to !3, whereas the second one causes the motion of the
axis of the top, so that it is equal to the angular velocity of precession !pr. The relation !1=!prsin
gives then!pr=M
I1. Therefore !pr=_ , in agreement with _ being the angular velocity of rotation
about thez-axis.
Figure 19: Symmetrical top.
4.1.11 Motion of ideal
uids
4.1.12 Motion in ideal
uids
4.2 Electromagnetism
4.2.1 Electrostatic eld
A static (constant in time) electric eld is called an electrostatic eld . In such eld, (2.7.38) reduces
to
curlE= 0; (4.2.1)
which implies
E= r ; (4.2.2)
in agreement with (2.7.32). Substituting (4.2.2) into (2.7.89) gives the Poisson equation
4 = 4: (4.2.3)
The general solution of (4.2.3) is
=Z
RdV; (4.2.4)
where Ris the vector from the element of volume dVto the point at which is calculated. If the
source of an electric eld is a system of static point charges then (4.2.4) becomes
=X
aea
Ra; (4.2.5)
where Rais the vector from a charge eato the point at which is calculated. Thus the electric eld
is
E=X
aeaRa
R3a; (4.2.6)
155
in agreement with the principle of superposition for the electric eld. For one charge ewe have
=e
R;E=eR
R3; (4.2.7)
so the Lorentz force (2.7.112) exerted on a static charge eafrom a static charge ebgives the Cavendish-
Coulomb law :
fab=eaebRab
R3
ab; (4.2.8)
where
Rab=Ra Rb: (4.2.9)
The Coulomb force (4.2.8) is repulsive for charges of the same sign and attractive for charges with
opposite signs. According to Newton's third law, the force fbaexerted on a static charge ebfrom a
static charge eais equal in magnitude and opposite in direction to fab:
fba= fab: (4.2.10)
The energy of an electrostatic eld is, due to (2.7.95),
U=1
8Z
E2dV= 1
8Z
Er dV = 1
8Z
r(E )dV+1
8Z
rEdV: (4.2.11)
The rst term on the right of (4.2.11) can be transformed into a surface integral using the Gau'
theorem. This integral vanishes if the surface of integration is taken at in nity, so (2.7.89) and
(4.2.11) give
U=1
2Z
dV =1
2X
aea a; (4.2.12)
where ais the electric potential at the point occupied by a charge ea. In calculating a, we must not
include the electric eld produced by eabecause the corresponding contribution to the electrostatic
energy, which is a self-energy (energy of an interaction of a point with itself), is in nite. Subtracting
this self-energy, which process is a renormalization of the energy, from Ugives
U=1
2X
aea 0
a; (4.2.13)
where
0
a=X
b6=aeb
Rab: (4.2.14)
For two static charges e1ande2separated by a distance r, the energy of their interaction is thus
U=e1e2
r, in agreement with rU=f, where fis given by (4.2.8).
We consider a system of static point charges ealocated at raand take the origin of the Cartesian
system of coordinates anywhere within the system. The electric potential at a point with the radius
vector Ris thus
=X
aea
jR raj: (4.2.15)
IfR rathen the electric potential (4.2.15) can be expanded in a Taylor series with respect to the
small quantityra
R:
0+ 1=1
RX
aea r1
RX
aeara=1
RX
aea r1
Rd; (4.2.16)
where the spatial vector
d=X
aeara (4.2.17)
156
is referred to as the electric dipole moment of the system. It has 3 independent components. If we
shift the origin by a constant vector athen
r0
a=ra+a; (4.2.18)
so
d0=d+aX
aea: (4.2.19)
Therefore if the total electric charge of the system is zero then its dipole moment does not depend
on the choice of the origin. In this case, (4.2.16) becomes
=dR
R3; (4.2.20)
which gives
E=3(dn)n d
R3; (4.2.21)
where
n=R
R(4.2.22)
is the unit vector in the direction of R. If the charge-to-mass ratioea
ma=e
mis equal for all the
charges in the system then the dipole moment of the system is proportional to the radius vector of
its center of mass X:
d=e
mX: (4.2.23)
The next term in a Taylor expansion of the electric potential (4.2.15) is (omitting the subscripts
a)
2=1
2X
ex x @2
@X @X 1
R=1
6D @2
@X @X 1
R=1
2R3D n n ; (4.2.24)
wherex are the components of r,X are the components of R, and the symmetric spatial tensor
D =X
e(3x x r2 ) =D (4.2.25)
is the electric quadrupole moment of the system. This tensor is traceless,
D = 0; (4.2.26)
so it has 5 independent components. Similarly, the l-th term lin a Taylor expansion of the electric
potential (4.2.15) depends on the 2l-pole moment of a system of charges which is a totally symmetric
tensor of rank lwith 2l+ 1 independent components. The expansion of1
jR rjcan be written as
1
jR rj=1X
l=0rl
Rl+1Pl(cos ) =1X
l=0lX
m= l4
2l+ 1rl
Rl+1Y
lm(; )Ylm(#;'); (4.2.27)
where is the angle between Randr,and are the polar angle and azimuth of the direction
along R, and#and'are the polar angle and azimuth of the direction along r. Therefore lis given
by
l=1
Rl+1lX
m= lr
4
2l+ 1QlmY
lm(; ); (4.2.28)
where
Qlm=r
4
2l+ 1X
aearl
aYlm(#a;'a): (4.2.29)
157
We consider a system of static charges in an external eld with potential , whose spatial vari-
ations are small within the system,Rjr j
1. The electrostatic energy (4.2.12) of such a system
is
U=X
aea (ra)U0+U1+U2; (4.2.30)
where
U0= 0X
aea; (4.2.31)
U1=r 0X
aeara= dE0; (4.2.32)
U2=1
2X
ex x @2 0
@x @x =1
6D @2 0
@x @x ; (4.2.33)
where the subscripts 0 refer to the eld at the origin. For two static dipoles d1andd2separated by
a vector R, the energy of their interaction is given by (4.2.21) and (4.2.32):
U=(d1d2) 3(d1n)(d2n)
R3: (4.2.34)
The total force on a system in an electrostatic eld in the dipole approximation is
F=E0X
aea+ (dr)E0: (4.2.35)
The total torque on such a system is
K=X
aearaE0=dE0: (4.2.36)
For a nonrelativistic scattering in a Coulomb eld, U(r) =
r, where
= const, (4.1.171) and
(4.1.174) give the impact parameter as a function of the scattering angle,
2=
2
m2v41cot2
2: (4.2.37)
Substituting (4.2.37) into (4.1.176) gives the Rutherford formula for the scattering cross section in
such a eld:
d=
2mv212do
sin4
2: (4.2.38)
4.2.2 Magnetostatic eld
A static magnetic eld is called a magnetostatic eld . We consider a system of charges which move
with nite velocities within a nite region of space. Such a motion is stationary ; in this case we can
average the magnetic eld over the time and regard it as magnetostatic. Thus (2.7.33) gives
divB= 0; (4.2.39)
which implies
B=rA; (4.2.40)
as in (2.7.33). After averaging, (2.7.90) simpli es to
curlB=4
cj: (4.2.41)
Substituting (4.2.40) into (4.2.41) gives
r(rA) 4A=4
cj: (4.2.42)
158
Gauge invariance of the electromagnetic potential allows to impose one condition on Ai. If we impose
theCoulomb gauge ,
rA= 0; (4.2.43)
then (4.2.42) reduces to the Poisson equation
4A= 4
cj: (4.2.44)
The general solution of (4.2.44) is
A=1
cZj
RdV; (4.2.45)
where Ris the vector from the element of volume dVto the point at which Ais calculated. Thus
the magnetic eld is given by the Biot-Savart-Laplace law :
B=1
cZjR
R3dV: (4.2.46)
If the source of a magnetic eld is a system of stationary point charges then (4.2.45) becomes
A=1
cX
aeava
RadV; (4.2.47)
where Rais the vector from a charge eato the point at which Ais calculated.
We consider a system of stationary point charges ealocated raand take the origin of the Cartesian
system of coordinates anywhere within the system. The magnetic potential at a point with the radius
vector Ris thus
A=1
cX
aeava
jR raj: (4.2.48)
IfR rathen the magnetic potential (4.2.48) can be expanded in a Taylor series with respect to
the small quantityra
R(omitting the subscripts a):
A1
cRX
ev 1
cX
ev
rr1
R
=1
cRd
dtX
er+1
cR3X
ev(rR)
=1
cR31
2d
dtX
er(rR) +1
2X
v(rR) r(vR)
=1
2cR3X
e(rv)R
=mR
R3; (4.2.49)
where the spatial pseudovector
m=1
2cX
aearava (4.2.50)
is referred to as the magnetic moment of the system. Thus the magnetic eld is
B=3(mn)n m
R3: (4.2.51)
The total force on a system of charges in a magnetostatic eld vanishes:
F=Xe
cvB=d
dtXe
crB= 0: (4.2.52)
The total torque on such a system is, analogously to (4.2.49),
K=Xe
cr(vB) =Xe
c
v(rB) 1
2Bd
dtr2
=mB: (4.2.53)
159
The expression for the torque on a magnetic dipole (4.2.53) is similar to that for the torque on an
electric dipole (4.2.36). If the charges are nonrelativistic, va c, and their charge-to-mass ratio
ea
ma=e
mis equal for all the charges in the system then the magnetic moment of the system is
proportional to its angular momentum:
m=e
2mcM: (4.2.54)
A general system with a total mass mand electric charge ehas
m=ge
2mcM; (4.2.55)
where the constant of proportionality gis called the gyromagnetic ratio . Thus a system of nonrel-
ativistic particles with the same charge-to-mass ratio has g= 1. Combining (4.2.53) and (4.2.54)
gives the Larmor precession ,
_M=!M; (4.2.56)
with angular velocity
!= e
2mcB: (4.2.57)
If a magnetostatic eld is uniform (in space) then
A=1
2Br (4.2.58)
is the solution of (2.7.33). Substituting (4.2.58) into the termPe
cAvin the Lagrangian (3.1.42)
turns this term into mB. Thus the magnetostatic (potential) energy of such a system is
U= mB: (4.2.59)
The expression for the energy of a magnetic dipole (4.2.59) is similar to that for the energy of an
electric dipole (4.2.32).
4.2.3 Motion in constant uniform electromagnetic eld
The equation of motion (2.7.110) of a charged particle in an electromagnetic eld integrates to
ui( ) = (ee
mcR
0F( 0)d 0)i
juj(0); (4.2.60)
where the exponent of a matrix Ais de ned as a Taylor expansion:
(eA)i
j=i
j+Ai
j+1
2Ai
kAk
j+1
6Ai
kAk
lAl
j+:::: (4.2.61)
If the eld is constant and uniform then (4.2.60) becomes
ui( ) = (ee
mcF)i
juj(0): (4.2.62)
Equations (2.7.112) and (2.7.114) integrate to
p p0=e(Et+rB);
E E0=eEr: (4.2.63)
The mixed components of the electromagnetic eld tensor are
Fi
j=0
BB@0ExEyEz
Ex 0Bz By
Ey Bz 0Bx
EzBy Bx 01
CCA; (4.2.64)
160
which gives
(F2)i
j=Fi
kFk
j=
E2 EB
EBE E +B B B2
: (4.2.65)
IfEB= 0 then (F3)i
j=Fi
kFk
lFl
j= (E2 B2)Fi
j. This case includes a constant uniform electric
eld and a constant uniform magnetic eld.
We consider the motion of a particle with mass mand charge ein a constant uniform electric
eldE, which we choose to be along the zaxis. The particle moves in such a eld in a plane, which
we choose as the xz-plane. If the initial velocity of the particle is v(0) =v(0)^exthen its initial
four-velocity is
u0(0) =
(0); ux(0) =
(0)v(0)
c;
(0) =1q
1 v2(0)
c2: (4.2.66)
The equation of motion (4.2.62) gives
u0=
(0) cosh(w ); ux=ux(0); uy= 0; uz=
(0) sinh(w ); (4.2.67)
where
w=ejEj
mc: (4.2.68)
Ift(0) =x(0) =y(0) =z(0) = 0 then the trajectory of the particle is a catenary curve:
t=
(0)
wsinh(w ); x=
(0)v(0) ; y = const; z=
(0)c
w
cosh(w ) 1
: (4.2.69)
In the nonrelativistic limit,
(0)1 andw 1, this curve reduces to a parabola:
t= ; x =v(0)t; y = const; z=1
2cwt2=1
2a0t2; (4.2.70)
where
a0=ejEj
m: (4.2.71)
Ifv(0) = 0 then the motion of a particle in a constant uniform electric eld Ecoincides with the
relativistic motion (4.1.37) with constant proper acceleration a0(the electric eld does not change
under boosts along the direction of the eld). This motion can also be obtained by integrating
(2.7.112) and using (2.4.213).
We consider the motion of a particle with mass mand charge ein a constant uniform magnetic
eldB, which we choose to be along the zaxis. If the initial velocity of the particle is v(0) =
(vx(0);0;vz(0)) then its initial four-velocity is
u0(0) =
(0); ux(0) =
(0)vx(0)
c;uz(0) =
(0)vz(0)
c;
(0) =1q
1 v2x(0)+v2z(0)
c2: (4.2.72)
The equation of motion (4.2.62) gives
u0=
(0); ux=ux(0) cos(! ); uy= ux(0) sin(! ); uz=uz(0); (4.2.73)
where
!=ejBj
mc: (4.2.74)
The rst equation in (4.2.73) represents the constancy of the energy in the magnetic eld. If
t(0) =x(0) =z(0) = 0 and y(0) =ux(0)
!then the trajectory of the particle is a helix:
t=
(0) ; x =sin(! )
!ux(0); y=cos(! )
!ux(0); z=uz(0) : (4.2.75)
161
The axis of this helix is parallel to the direction of the eld and its radius is
r=ux(0)
!: (4.2.76)
This motion can also be obtained by integrating (2.7.112) and using (2.4.213) and the constancy of
the energy in the magnetic eld.
We consider the motion of a particle with mass mand charge ein a superposition of a constant
uniform electric eld and a constant uniform magnetic eld if both elds are mutually perpendicular
and equal in magnitude. Thus ( F3)i
j= 0, so if the particle is at rest at = 0 then the equation of
motion (4.2.62) gives
u0= 1 +1
2(! )2;u=E
E! +EB
2E2(! )2: (4.2.77)
As !1 , the direction of the velocity of the particle tends to that of EB.
4.2.4 Electromagnetic waves
In the absence of electric charges, the Maxwell equations (2.7.37), (2.7.38), (2.7.89) and (2.7.90)
reduce to
rB= 0; (4.2.78)
rE= 1
c_B; (4.2.79)
rE= 0; (4.2.80)
rB=1
c_E: (4.2.81)
Gauge invariance of the electromagnetic potential allows to impose on it one condition. If we impose
= 0 (4.2.82)
then (2.7.32) simpli es to
E= 1
c_A: (4.2.83)
Thus (4.2.81) gives
r(rA) = 4A+r(rA) = 1
c2A: (4.2.84)
This equation is invariant under a transformation
A!A+rf(r); (4.2.85)
wheref(r) is any function of the spatial coordinates. Therefore we can impose another constraint on
A. If we choose the Coulomb gauge (4.2.43) then (4.2.84) reduces to the d'Alembert wave equation :
A=1
c2A 4A= 0: (4.2.86)
Its solutions are called electromagnetic waves . They have 2 degrees of freedom because the 2 con-
ditions (4.2.82) and (4.2.43) constrain the 4 components of Ai. The elds EandBalso satisfy the
wave equation (4.2.86). The second Maxwell-Minkowski equation in the absence of charges (2.7.75)
becomes
Fik
;k= 0: (4.2.87)
Substituting (2.7.24) into (4.2.87) gives
@i@iAj @j(@iAi) = 0: (4.2.88)
162
If we impose on the electromagnetic potential, due to gauge invariance, the Lorentz gauge
@iAi= 0 (4.2.89)
then (4.2.88) simpli es to
@i@iAj= 0; (4.2.90)
which is identical with (4.2.86). This equation is invariant under a transformation
Ai!Ai+@if; (4.2.91)
wherefis any function satisfying the wave equation @i@if= 0. This invariance allows imposing on
Aianother condition, such as A0= 0 (4.2.82).
A solution of (4.2.86) which depends on time and only one spatial coordinate describes a plane
wave . If we choose the Cartesian frame of reference such that this coordinate is xthen (4.2.86) gives
@2Ai(t;x)
c2@t2=@2Ai(t;x)
@2x; (4.2.92)
whose solution is
Ai=f(1)
i(t x=c) +f(2)
i(t+x=c); (4.2.93)
wheref(1)
iandf(2)
iare vector functions of their arguments. If f(2)= 0 then the eld has the same
values for events ( t;x) which satisfy the relation
x=ct+ const: (4.2.94)
Thus this eld propagates along the x-axis with velocity c, representing a plane wave moving in
the positive direction along this axis. Therefore electromagnetic waves move with the velocity of
propagation of interaction. If f(1)= 0 then the eld represents a plane wave moving in the negative
direction along this axis. If we impose the Coulomb gauge then@Ax
@x= 0, which gives, due to
(4.2.92), cEx=@Ax
@t= const. A constant electric eld does not represent a wave, so
Ax= 0: (4.2.95)
The electric and magnetic elds corresponding to a plane wave propagating along the + x-axis are
E= 1
c@A
@(t x=c); (4.2.96)
B=r(t x=c)@A
@(t x=c)=nE; (4.2.97)
where n=rxis the unit vector in the direction of propagation of the wave. A plane wave is
transverse (perpendicular to the wave vector):
E?n;B?n; (4.2.98)
which gives
E2=B2; (4.2.99)
W=E2
4= xx; (4.2.100)
S=c
4E2n=cWn: (4.2.101)
The relation between the energy density Wand the momentum densityS
c2is the same as for a
particle moving with the velocity of propagation of interaction c. Under a Lorentz transformation
withV=c ^z,Wtransforms like the 00 component of the tensor (2.4.71), according to (1.6.100):
W=1
1 V2
c2
W0+ 2V
c2S0
z V2
c20
zz
: (4.2.102)
163
If 0is the angle between the z0-axis (which coincides with the z-axis) and nthenS0
z=cW0cos 0
and0
zz= W0cos2 0, so
W=(1 +V
ccos 0)2
1 V2
c2W0: (4.2.103)
A plane wave re
ecting from a plane whose normal vector is Nexerts on the plane the force f =
( +0
)N , where and0
are the stress tensors of the wave before and after the re
ection.
This force is thus f=Wn(Nn) +W0n0(Nn0).
A solution of (4.2.86) which is a periodic function of time can be represented as a superposition
ofmonochromatic waves . A monochromatic plane wave along the x-axis is a sinusoidal solution of
(4.2.86),
A= Re( A0e i!(t x=c)); (4.2.104)
where!is the angular frequency of the wave. If this wave propagates in the direction of rthen
A= Re( A0ei(kr !t)); (4.2.105)
where
k=!
cn (4.2.106)
is the wave vector . A monochromatic plane wave satis es, due to the wave equation,
4A+!2
c2A= 0: (4.2.107)
For linear operations on Awe can omit the real-part operator Re. Thus A=A0ei(kr !t), from
which we obtain
E=ikA;B=ikA; (4.2.108)
where
k=jkj: (4.2.109)
The electric eld is a real vector,
E= Re( E0ei(kr !t)); (4.2.110)
where E0is a complex vector. Its square can be written as
E2
0=jE2
0je 2i ; (4.2.111)
where is the phase of the wave, so
E0=be i ;b2=jE2
0j; (4.2.112)
where bis a complex vector whose square is a real scalar. Writing
b=b1+ib2; (4.2.113)
where b1andb2are real vectors, gives then
b1b2= 0; (4.2.114)
so these vectors are perpendicular to each other. Equation (4.2.110) gives
E= Re( bei(kr !t )) (4.2.115)
or, if b1is along the y-axis and b2along thez-axis,
Ey=b1cos(!t kr+ ); Ez=b2sin(!t kr+ ); (4.2.116)
164
which describes an elliptically polarized wave:
E2
y
b2
1+E2
z
b2
2= 1: (4.2.117)
Ifb1=b2then a wave is circularly polarized . Ifb1= 0 or b2= 0 then a wave is plane polarized or
linearly polarized . Similar expressions can be obtained for the vector B.
The frequency and wave vector form a four-vector:
ki=!
c;k
; (4.2.118)
which is a null vector,
kiki= 0: (4.2.119)
The potential (4.2.105) is then
A= Re( A0e ikixi): (4.2.120)
The energy-momentum tensor (2.7.94) for a monochromatic plane wave is
Tij=Wc2
!2kikj: (4.2.121)
If a source at rest in a frame Kemits a monochromatic wave with frequency !then its frequency !0
in a frameK0in which the source moves with velocity valong thex-axis is given by the corresponding
Lorentz transformation of the vector ki:
k0=k00 v
ck01
p
1 v2=c2: (4.2.122)
If 0is the angle between vandk0thenk01=k0cos 0, so
!0=!p
1 v2=c2
1 v
ccos 0; (4.2.123)
as in formula (1.6.113) for the Doppler e ect.
A wave can be represented as a superposition of monochromatic waves with di erent frequencies,
which is referred to as spectral resolution . A periodic eld f(t) can be represented as a discrete super-
position of monochromatic waves with frequencies which are integral multiples of the fundamental
frequency!0(Fourier series):
f(t) =1X
n= 1fne in!0t: (4.2.124)
The inverse relation is
fn=1
TZT=2
T=2f(t)ein!0tdt; (4.2.125)
where
T=2
!0(4.2.126)
is the period of the eld. Since fis real,f n=f
n. The time average of f2satis es, using f= 0,
f2= 21X
n=1jfnj2: (4.2.127)
A nonperiodic eld f(t) which vanishes for t!1 can also be represented as a superposition of
monochromatic waves, but the distribution of the frequencies is continuous (Fourier transform):
f(t) =1
2Z1
1f!e i!td!: (4.2.128)
165
The inverse relation is
f!=Z1
1f(t)ei!tdt: (4.2.129)
Sincefis real,f !=f
!. The time integral of f2satis es
Z1
1f2dt=1
Z1
1jf!j2d!: (4.2.130)
We consider an electromagnetic eld in the absence of charges enclosed inside a nite volume
taken as a parallelepiped with sides a;b;c . We can expand the eld Ainside this parallelepiped into
plane waves through a Fourier series:
A(r) =X
kAkeikr; (4.2.131)
where the summation extends over all values of k= (kx;ky;kz) with
kx=2nx
a; ky=2ny
b; kz=2nz
c; (4.2.132)
wherenx;ny;nzare integers. The eld Ais real, so
A k=A
k: (4.2.133)
The Coulomb gauge implies
kAk= 0; (4.2.134)
while the wave equation gives
Ak+!2
kAk= 0; (4.2.135)
where
!k=ck: (4.2.136)
The vectors Akare thus harmonic functions of time. We can choose these functions to represent
running waves:
A=X
k(akeikr+a
ke ikr); (4.2.137)
or
Ak=ak+a
k: (4.2.138)
Thus akei!ktanda
ke i!kt, so
_Ak= ick(ak a
k): (4.2.139)
The electric and magnetic elds are given by
E= 1
cX
k_Akeikr; (4.2.140)
B=iX
k(kAk)eikr; (4.2.141)
so the total energy and momentum of the electromagnetic eld inside the parallelepiped are
E=1
8Z
(E2+B2)dV=X
kEk; (4.2.142)
p=1
4cZ
EBdV=X
kk
kEk
c; (4.2.143)
166
where
Ek=k2V
2aka
k (4.2.144)
andV=abcis the volume of the parallelepiped.
De ning
Qk=r
V
4c2(ak+a
k);) (4.2.145)
Pk=_Qk= i!kr
V
4c2(ak a
k); (4.2.146)
allows to write the energy of the eld (4.2.142) as a Hamiltonian:
H=X
k1
2(P2
k+!2
kQ2
k): (4.2.147)
The Hamilton equations for this Hamiltonian are
_Qk=@H
@Pk!Pk=_Qk; (4.2.148)
_Pk= @H
@Qk!Qk+!2
kQk= 0; (4.2.149)
in accordance with (4.2.135) and (4.2.145). The vectors QkandPkare perpendicular to the wave
vector k, so they have 2 independent components = 1;2 which determine the polarization of the
wave. Thus we can put in (4.2.147)
Q2
k=X
Q2
k ;P2
k=X
P2
k : (4.2.150)
The Hamiltonian (4.2.147) is thus the sum of the Hamiltonians corresponding to one-dimensional
harmonic oscillators:
H=X
k Hk ; Hk =1
2(P2
k +!2
kQ2
k ): (4.2.151)
The eld of charges can also be expanded into plane waves,
f(r) =Z1
1fkeikrdk
(2)3; fk=Z1
1f(r)e ikrdV: (4.2.152)
We consider a charge qin uniform rectilinear motion with velocity v, so that=q(r vt). The
scalar potential satis es then = 4q(r vt), so that it Fourier transform satis es ( )k=
4qe ikvt. The relation ( )k= k
c2+k2 kgives then
k= 4qe ikvt
k2 (kv=c)2: (4.2.153)
The vector potential is given by Ak=v
c k. The dependence of the Fourier-transformed elds on
the Fourier-transformed potentials results from (2.7.32) and (2.7.33) by substituting rbyik:
Ek= ik k _Ak
c; (4.2.154)
Bk=ikAk: (4.2.155)
For a static charge, k=4q
k2andEk= 4iq
k2k, so that its electric eld is resolved into waves
directed along the wave vector ( longitudinal waves).
167
4.2.5 Retarded potentials
In the presence of electric charges, the Maxwell-Minkowski equation (2.7.75) and the Lorentz gauge
(4.2.89) give
Ai=1
c2@2Ai
@t2 4Ai=4
cji: (4.2.156)
The time component of (4.2.156) is
= 4; (4.2.157)
whose physical solution vanishing at in nity is
(t;r) =Z(t R=c;r0)
RdV0; (4.2.158)
where
R=jr r0j (4.2.159)
is the distance from the source at r0to the eld point r. The spatial components of (4.2.156) are
A=4
cj; (4.2.160)
whose physical solution vanishing at in nity is
A(t;r) =Zj(t R=c;r0)
cRdV0: (4.2.161)
The potentials (4.2.158) and (4.2.161) are referred to as the retarded potentials . If we replace the
minus sign in (4.2.158) and (4.2.161) with the plus sign then the corresponding potentials, which
also are solutions of (4.2.157) and (4.2.160), are called the advanced potentials . Only the retarded
potentials are physical, in accordance with the nite velocity of propagation of interaction cand the
principle of causality. Their values at tdepend on the distribution of electric charges at the earlier,
retarded time ,
tR=t R=c; (4.2.162)
which is the time at which a signal emitted from the source located at r0reaches the eld point
located at rat timet. Adding to the right-hand sides of (4.2.158) and (4.2.161) the general solutions
0andA0of the corresponding homogeneous equations@2
c2@t2 4 = 0 and (4.2.86) does not alter
(4.2.156). The quantities 0andA0correspond to the external eld acting on the system. For a
static eld, (4.2.157) and (4.2.160) reduce to (4.2.3) and (4.2.44). In the absence of charges, (4.2.157)
and (4.2.160) reduce to the wave equations @i@i = 0 and (4.2.86).
We consider an electric charge emoving along a path given by a radius vector r0(t). The vector
from the charge to the eld point located at ris thus R(t) =r r0. The electromagnetic potential
Ai(t;r) depends on the motion of the charge at the retarded time t0satisfying
t0+R(t0)
c=t: (4.2.163)
In the frame of reference in which the charge is at rest at time t0,v0(t0) =_r0(t0) = 0, the potentials
(at timet) are
=e
R(t0);A= 0: (4.2.164)
Therefore, in the frame of reference in which the charge is moving at time t0and its four-velocity is
ui, the electromagnetic potential is
Ai=eui
ukRk; (4.2.165)
whereRiis the four-vector with components
Ri=
c(t t0);R
: (4.2.166)
168
This vector is a null vector, RiRi= 0. The temporal and spatial components of (4.2.165) are referred
to as the Lienard-Wiechert potentials :
=e
R vR
c; (4.2.167)
A=ev
c(R vR
c): (4.2.168)
The left-hand sides of (4.2.167) and (4.2.168) are evaluated at t, while their right-hand sides must
be evaluated at t0. The Lienard-Wiechert potentials can also be obtained by substituting (t;r) =
e(r r0(t)) and j(t;r) =ev(r r0(t)) into the retarded potentials (4.2.158) and (4.2.161).
Di erentiating R(t0) =c(t t0) with respect to tand using@R(t0)
@t0= v(t0) gives@R
@t=@R
@t0@t0
@t=
Rv
R@t0
@t=c(1 @t0
@t), so
@t0
@t=R
R vR
c: (4.2.169)
Di erentiating R(t0) =c(t t0) with respect to rgives@t0
@r= 1
c@R(t0)
@r= 1
c(@R
@t0@t0
@r+R
R), so
@t0
@r= R
c(R vR
c): (4.2.170)
Substituting these formulae into (2.7.32) and (2.7.33), written as
E= @A
c@t0@t0
@t r @
@t0@t0
@r; (4.2.171)
B=rA+@t0
@r@A
@t0; (4.2.172)
gives the electric and magnetic eld produced by a moving charge:
E=e(1 v2
c2)(R v
cR)
(R Rv
c)3+eR
(R v
cR)_v
c2(R Rv
c)3; (4.2.173)
B=1
RRE; (4.2.174)
where
_v=@v
@t0: (4.2.175)
The left-hand side of (4.2.173) is evaluated at t, while its right-hand side must be evaluated at t0.
For largeR, the rst term on the right of (4.2.173) varies like1
R2, while the second term varies like
1
R. The rst term on the right of (4.2.173) does not depend on the acceleration _v, so it correponds
to the electric eld produced by a uniformly moving charge. If the velocity vis constant then
R(t0) v
cR(t0) =R(t0) v(t t0) =R(t) (4.2.176)
and
R(t0) R(t0)v
c=r
R2(t) 1
c2(vR(t))2=R(t)r
1 v2
c2sin2(t); (4.2.177)
where(t) is the angle between R(t) and v. Thus (4.2.173) reduces to
E=eR(t)
R3(t)1 v2
c2
(1 v2
c2sin2(t))3=2; (4.2.178)
while (4.2.174) is equivalent to
B=1
cvE: (4.2.179)
169
These elds can also be obtained by applying a Lorentz transformation from the frame in which the
charge is at rest and where =e
RandA= 0 to the frame in which the charge moves with velocity
v.
Spectral resolution (4.2.128) of and into!e i!tand !e i!tturns the wave equation
(4.2.157) into4 !+k2 != 4 !, and the formula for the retarded potential (4.2.158) into
!=Z
!eikR
RdV: (4.2.180)
Analogously, (4.2.161) gives
A!=Z
j!eikR
cRdV: (4.2.181)
The inverse relation (4.2.129) applied to !and substituted into (4.2.180) gives !=R
Rei(!t+kR)dV dt .
Substituting to this relation the charge density of an electric charge emoving along a path given
by a radius vector r0(t),(t;r) =e(r r0(t)), gives the spectral representation of the scalar
Lienard-Wiechert potential:
!=eZ1
Rei!(t+R=c)dt: (4.2.182)
Analogously, the vector potential gives
A!=eZ_r0
cRei!(t+R=c)dt: (4.2.183)
For a discrete spectral resolution of the eld, (4.2.182) becomes
n=e
TZT
01
Rein!0(t+R=c)dt: (4.2.184)
4.2.6 Electromagnetic eld in second approximation
The Lagrangian for a particle of mass mand charge emoving with velocity vin a eld represented
by the potentials andAis given by (3.1.40). Because the velocity of propagation of interaction
is nite, the eld must be considered as a system with degrees of freedom independent of those of
the particles. If the velocities of the particles in a system of charges are small compared to cthen
the degrees of freedom of the eld can be represented in terms of the coordinates and velocities of
the particles. The Lagrangian for the system can then be described in terms of the coordinates and
velocities of the particles only up to terms of second order inv
c, because moving charges radiate
electromagnetic waves (which have their own degrees of freedom) at the third order. Expanding the
retarded scalar potential in series of powers of R=cand using the conservation of the total charge
yields
=Z
RdV @
c@tZ
dV+1
2c2@2
@t2Z
RdV =Z
RdV+1
2c2@2
@t2Z
RdV; (4.2.185)
where all the quantities are evaluated at the same time t. Expanding the retarded vector potential
in series of powers of R=cyields
A=1
cZv
RdV: (4.2.186)
The retarded potentials due to one charge are
=e
R+e
2c2@2R
@t2; (4.2.187)
A=ev
cR: (4.2.188)
170
Applying a gauge transformation (2.7.12) with
= e
2c@R
@t(4.2.189)
brings the potentials (4.2.187) and (4.2.188) to
=e
R; (4.2.190)
A=ev
cR+e
2cr@R
@t=e
2cR
v+ (vn)n
: (4.2.191)
The Lagrangian for a charge eais given by (3.1.40) in which m=ma,e=ea, and andAare the
potentials of the eld produced by all the other charges at the position of ea, given by the sums of
(4.2.190) and (4.2.191). This Lagrangian is thus, omitting the constant term mac2,
La=mav2
a
2+mav4
a
8 eaX
b6=aeb
Rab+ea
2c2X
b6=aeb
Rab
vavb+ (vanab)(vbnab)
; (4.2.192)
where
Rab=ra rb;nab=Rab
Rab: (4.2.193)
Consequently, the Lagrangian for a system of charges is
L=X
amav2
a
2+X
amav4
a
8c2 X
a;b6=aeaeb
2Rab+X
a;b6=aeaeb
4c2Rab
vavb+ (vanab)(vbnab)
;(4.2.194)
and is called the Darwin Lagrangian .
The corresponding Darwin Hamiltonian is obtained from (4.2.194) by substituting va=pa
maand
changing the signs of all the terms except the nonrelativistic kinetic termP
amav2
a
2(these other
terms are regarded as the potential energy):
H=X
ap2
a
2ma X
ap4
a
8c2m3a+X
a;b6=aeaeb
2Rab X
a;b6=aeaeb
4c2mambRab
papb+(panab)(pbnab)
:(4.2.195)
The center of inertia of the system Rin this approximation is given by (3.1.72) in which we must
add toEathe energy of the eld:
R=P
aEara+R
WrdVP
aEa+R
WdV: (4.2.196)
In the numerator, we have
Z
WrdV=1
8Z
E2rdV=1
8Z
(r )2rdV=1
8Z
r
r 2
2df
1
8Z
r 2
2dV 1
8Z
4 rdV: (4.2.197)
At in nity, the rst two terms on the right of (4.2.197) vanish, which gives
Z
WrdV= 1
8Z
4 rdV=1
2Z
rdV=1
2X
aea ara: (4.2.198)
Therefore the radius vector of the center of inertia is
R=1
EX
ara
mac2+p2
a
2ma+eaX
b6=aeb
2Rab
; (4.2.199)
where the energy of the system is
E=X
a
mac2+p2
a
2ma
+X
a;b6=aeaeb
2Rab: (4.2.200)
171
4.2.7 Electromagnetic radiation
We consider a system of charges dV located rand take the origin of the Cartesian system of
coordinates anywhere within the system. If Ris the radius vector of a point at which we measure
the eld and this point is located at a distance which is large compared to the size of the system,
R r, thenjR rjR rn, where n=R
R. The retarded potentials in this approximation are
t=1
RZ
t R=c+rn=cdV; (4.2.201)
At=1
cRZ
jt R=c+rn=cdV: (4.2.202)
The vector Lienard-Wiechert potential (4.2.168) from a single charge is
At=evt0
cR(1 vt0n
c); (4.2.203)
where the retarded time t0satis es
t0+R
c 1
crt0n=t: (4.2.204)
If the eld point is also located at a distance which is large compared to the wavelength of the
radiated electromagnetic waves (in a wave zone ),R , then these waves can be approximated as
plane waves. The magnetic eld of this wave is given by (4.2.97):
B=1
c_An; (4.2.205)
while the electric eld is
E=Bn=1
c(_An)n: (4.2.206)
The energy of the radiation per unit time and per unit solid angle dois called the intensity of the
radiation. It is equal to the
ux of the Poynting vector through a surface element R2doof a sphere
of radiusR:
dI=Sdf; (4.2.207)
which gives, due to (4.2.101),
dI=c
4B2R2do: (4.2.208)
Since, for large R, (4.2.173) varies like1
R, the intensity (4.2.208) is independent of R. Continuous
spectral resolution of the vector potential,
A!=eikR
cRZ
j!e ikrdV; (4.2.209)
gives, using (4.2.130),
B=ikA!; (4.2.210)
E=ic
!k(A!k); (4.2.211)
dIn!=c
2jB!j2R2dod!
2: (4.2.212)
Analogously, discrete spectral resolution gives
dInn=c
2jBnj2R2do: (4.2.213)
172
If a radiating system of charges is nonrelativistic then the wavelength of the radiated electro-
magnetic waves is large compared to the size of the system abecauseac=v. Expanding the
retarded vector potential (4.2.202) in a Taylor series with respect to the small quantity rn=Rgives
At=1
cRZ
jt R=cdV+1
c2R@
@(t R=c)Z
(rn)jt R=cdV; (4.2.214)
which for the system of charges becomes
A=1
cRX
ev+1
c2R@
@tX
ev(rn: (4.2.215)
The quantity rn=cis on the order of a=c, where Using v(rn=1
2@
@t
r(rn)
+1
2(rv)nand
the de nitions of the electric and magnetic dipole momenta turns (4.2.215) into
A=1
cR_d+1
2c2R@2
@2tX
er(rn) +1
cR_mn: (4.2.216)
Since adding to Aany vector proportional to ndoes not alter the elds (4.2.205) and (4.2.206), we
can write (4.2.216) as
A=1
cR_d+1
6c2R@2
@2tX
e
3r(rn) nr2
+1
cR_mn=1
cR_d+1
6c2RD+1
cR_mn;(4.2.217)
where the components of the electric quadrupole vector Dare
D =D n : (4.2.218)
Thus the magnetic eld is given by
B=1
c2R
dn+1
6c_Dn+ (mn)n
: (4.2.219)
Substituting (4.2.219) into (4.2.208) and integrating over the solid angle gives, usingR
n n do=
4
3 andR
n n n
ndo=4
15(
+
+
),
I=2
3c3d2+1
180c5_D2
+2
3c3m2: (4.2.220)
The rst term on the right of (4.2.220) represents the electric dipole radiation , the second one the
electric quadrupole radiation , and the third one the magnetic dipole radiation . These terms must be
evaluated at the time t R=c. For the electric dipole radiation, (4.2.212) and (4.2.213) integrated
over the solid angle give
dI!=4!4
3c3jd!j2d!
2; (4.2.221)
In=4!4
0n4
3c3jdnj2; (4.2.222)
where d!anddnare the spectral representations of d. The recoil force acting on a closed radiating
system is
F =Z
n R2do; (4.2.223)
which, using the Maxwell stress tensor (2.7.97) and the radiation elds for the vector potential
(4.2.217), gives
F = 1
c41
15c_D d +2
3(dm)
: (4.2.224)
If all the charges in a closed system move with velocities v cand have the same charge-to-mass
ratio then they do not produce the electric dipole radiation because the dipole moment of the system
173
is proportional to the conserved radius vector of its center of mass (4.2.23). They also do not produce
the magnetic dipole radiation because the magnetic moment of such a system is proportional to its
conserved angular momentum (4.2.54).
If the eld point is located at a distance which is not large compared to the wavelength of the
radiated electromagnetic waves then these waves cannot be approximated as plane waves. In the
electric-dipole approximation, (4.2.217) gives A=_d
cR, so the magnetic eld is
B=1
cr_d
R: (4.2.225)
After spectral resolution, in which dd!e i!(t R=c)andBB!e i!t, (4.2.225) gives
B!= ikr
d!eikR
R
: (4.2.226)
To calculate the scalar potential, we use the Lorentz gauge, obtaining
= rd
R; (4.2.227)
so the electric eld is
E=r
rd
R
d
c2R=r
rd
R
: (4.2.228)
The angular momentum radiated away by electromagnetic waves is
_M =Z
e
x
nR2do; (4.2.229)
which gives, using the Maxwell stress tensor (2.7.97),
_M=2
3c3_dd: (4.2.230)
In the next approximation, the potentials due to the electric quadrupole moment are
=1
6@2
@X @X D
R; A = 1
6c@
@X _D
R; (4.2.231)
and the potentials due to the magnetic dipole moment are
= 0;A=rm
R: (4.2.232)
If the intensity of the radiation in the rest frame K0of the radiating source is
dI0=f(cos0; 0)d(cos0)d 0; (4.2.233)
then in the frame Kmoving relative to K0with velocity Vwe havedE0=dE VdPp
1 V2=c2=dE1 Vcos=cp
1 V2=c2,
= 0anddt=dt0=p
1 V2=c2. ThusdI= (dE=dt )d(cos)d gives, using (1.6.112),
dI=(1 V2=c2)2
(1 Vcos=c)3fcos V=c
1 Vcos=c;
d(cos)d : (4.2.234)
For the dipole radiation, f= constsin20, (4.2.234) gives
dI= const(1 V2=c2)3
(1 Vcos=c)5sin2d(cos)d : (4.2.235)
174
4.2.8 Radiation reaction
Expanding the retarded scalar potential in series of powers of R=cup to the cubic term yields
=Z
RdV+1
2c2@2
@t2Z
RdV 1
6c3@3
@t3Z
R2dV; (4.2.236)
while expanding the retarded vector potential yields
A=1
cZv
RdV 1
c2@
@tZ
vdV: (4.2.237)
The last terms on the right-hand sides of (4.2.236) and (4.2.237) correspond to the radiation of
electromagnetic waves. Applying a gauge transformation (2.7.12) with
=1
6c2@2
@t2Z
R2dV (4.2.238)
brings these terms to
rad= 0;Arad= 2
3c3X
e_v; (4.2.239)
so the corresponding electric eld is
Erad=2
3c3_d: (4.2.240)
The force acting on a charge ein this eld is then
f=2e
3c3_d (4.2.241)
The sum of the forces (4.2.241) acting on charges in a closed, electrically neutral system vanishes,
so it does not appear in the total recoil force (4.2.224). If the charge is point-like then expanding
Ato terms of higher power of R=cdoes not change (4.2.241) because these terms vanish as R!0.
The energy radiated per unit time equals to the total power:
X
fv=2
3c3_dX
ev=2
3c3_d_d=2
3c3d
dt(d_d) 2
3c3d2; (4.2.242)
so the average loss of the energy ( radiation damping ) through the electric dipole radiation is
dE
dt=X
fv= 2
3c3d2; (4.2.243)
in agreement with the rst term on the right of (4.2.220). The angular momentum radiated per unit
time equals to the total torque:
X
rf=2
3c3X
er_d=2
3c3d_d=2
3c3d
dt(dd) 2
3c3_dd; (4.2.244)
so the average loss of the angular momentum through the electric dipole radiation is
dM
dt=X
rf= 2
3c3_dd; (4.2.245)
in agreement with (4.2.230).
For a nonrelativistic particle with mass mand electric charge e, its equation of motion must also
include the recoil force (4.2.241) due to the interaction with its own electromagnetic eld, which is
called the Abraham-Lorentz force :
f=2e2
3c3v: (4.2.246)
175
This force is a recoil force on an accelerating charged particle caused by emitting an electromagnetic
radiation, and it is also called the radiation-reaction force . It is proportional to the particle's jerk.
Thus the equation of motion is
m_v=eE+e
cvB+2e2
3c3v; (4.2.247)
or
m_v=Fext+mt0v; (4.2.248)
where
t0=2e2
3mc3(4.2.249)
is on the order of the time of propagation of interaction through a distance equal to the classical
radius of the particle,
r0=e2
mc2: (4.2.250)
The classical radius of a particle is on the order of the radius of a uniformly charged sphere whose
electrostatic energy and rest energy are on the same order. Integrating (4.2.247) once gives
m_v=1
t0Z1
te t0 t
t0Fext(t0)dt0; (4.2.251)
which means that future values of the external force unphysically (violating causality) a ect the
acceleration of the particle in the present, indicating limits of the classical (based on the principle
of least action) description of electrodynamics. The weight factor e t0 t
t0falls of rapidly for times
greater than t0in the future.
To eliminate these noncausal pre-acceleration e ects, caused by the third-order character of
(4.2.247), we can apply to this equation a reduction of order due to Landau and Lifshitz. Dif-
ferentiating (4.2.247) with respect to the time, in the frame of reference in which the particle is
instantaneously at rest, gives
v=e
m_E+e
mc_vB: (4.2.252)
Substituting (4.2.247) with v= 0 and without (4.2.246) into (4.2.252) gives
v=e
m_E+e2
m2cEB; (4.2.253)
so the Abraham-Lorentz force (4.2.246) becomes in this approximation
f=2e3
3mc3_E+2e4
3m2c4EB: (4.2.254)
Thus (4.2.247) turns into a second-order equation, which is free of noncausal solutions,
m_v=eE+e
cvB+2e3
3mc3_E+2e4
3m2c4EB: (4.2.255)
The above method of reduction of order is valid if the force f eE. If the frequency of the
electromagnetic waves radiated by the particle is !then _E!E. The rst term in (4.2.254) is
much smaller than eEif the wavelength of these waves is much larger than the classical radius of
the particle:
=c
! r0: (4.2.256)
The second term in (4.2.254) is much smaller than eEif the magnetic eld is not too strong:
jBj m2c4
e3: (4.2.257)
176
The above conditions limit the classical description of the interaction of charged particles with
electromagnetic waves and thus the classical theory of electromagnetic elds.
In the frame of reference in which the particle is instantaneously at rest, the radiated energy and
momentum are
dE=2e2
3c3a2dt; dp= 0: (4.2.258)
Thus the relativistic expression for the radiated momentum four-vector in any frame of reference is,
using (1.6.122),
dpi= 2e2
3cduk
dsduk
dsdxi: (4.2.259)
Substituting here (2.7.110) in the locally Galilean system of coordinates gives
dpi= 2e4
3m2c5FklulFkmumdxi: (4.2.260)
The time components of (4.2.259) and (4.2.260) gives the intensity of the radiation:
I=2e2
3c3a2 (va)2=c2
(1 v2=c2)3=2e4
3m2c3(E+v
cB)2 (v
cE)2
1 v2=c2: (4.2.261)
In the case of a uniform motion on a circle in a constant uniform magnetic eld, E= 0 and B?v,
leading to the synchrotron radiation :
I=2e4
3m2c5B2v2
1 v2=c2: (4.2.262)
Substituting the Lienard-Wiechert electric (4.2.173) and magnetic (4.2.174) elds in the wave zone
(for largeR),
E=e
c2Rn
(n v
c)a
(1 nv
c)3;B=nE; (4.2.263)
into (4.2.208) gives
dI=e2
4c32(na)(va)
c(1 nv=c)5+a2
(1 nv=c)4 (1 v2=c2)(na)2
(1 nv=c)6
do; (4.2.264)
whose right-hand side must be evaluated at the retarded time t0. Integrating (4.2.264) over the solid
angle gives (4.2.261), while the angular distribution of the total radiated energy is
dEn=Z
dIdt =Z
dI(1 nv=c)dt0: (4.2.265)
The relativistic equation of motion of a charged particle in an external electromagnetic eld in
the locally Galilean system of coordinates, including radiation damping, is
mcdui
ds=e
cFikuk+gi; (4.2.266)
where the relativistic Abraham-Lorentz force four-vector gimust be orthogonal to uiand its spatial
components must reduce in the nonrelativistic limit to (4.2.246). These requirements yield
gi=2e2
3cd2ui
ds2 uiukd2uk
ds2
: (4.2.267)
Substituting here (4.2.266) without gigives the approximated expression for the relativistic Abraham-
Lorentz force:
gi=2e3
3mc3@Fik
@xlukul 2e4
3m2c5FilFkluk+2e4
3m2c5FklulFkmumui: (4.2.268)
In the nonrelativistic limit, the spatial components of (4.2.268) reduce to (4.2.255).
177
4.2.9 Scattering of electromagnetic waves
We consider a plane electromagnetic wave impinging on a system of charges. The resulting motion
of the charges produces an electromagnetic radiation. This process can be regarded as a scattering
of the incident wave on the system of charges. The intensity of the radiation dIis proportional to
the energy
ux of the incident wave S=jSj. Thus the scattering of an electromagnetic wave on the
system is characterized by their ratio, called the scattering cross section for waves:
d=dI
S: (4.2.269)
For a free nonrelativistic charge at rest, we can neglect the magnetic eld in the Lorentz force,
so the equation of motion of the charge is mr=eEand thus d=e2
mE, from which (4.2.208) and
(4.2.219) give
dI=e4
4m2c3(En)2do: (4.2.270)
This expression also results from (4.2.264) for v= 0. For a linearly polarized wave, E=E0cos(k
r !t). If the wave is weak then the charge executes a small motion, so r0. Since the Poynting
vector is S=c
4E2, (4.2.269) gives
d=r2
0sin2do; (4.2.271)
wherer0is the classical radius of the charge (4.2.250) and is the angle between nandE. Integrating
(4.2.271) over the solid angle gives the Thomson formula :
=8
3r2
0: (4.2.272)
The time-averaged force (4.2.254) exerted on a scattered charge is
f=2e4
3m2c4E2n=Wn: (4.2.273)
In the (locally) rest frame of the charge, its acceleration is thus a0=W0=m. In any frame, using
(4.1.34) and (4.2.103), this equation gives the speed of the charge v:
d
dtvp
1 v2=c2=W
m1 v=c
1 +v=c: (4.2.274)
In the rest frame of the charge, the frequency of the scattering wave is equal to the frequency of the
incident wave, !0=!. In any frame, in which the charge moves with velocity V, this condition is
k0
iu0i=kiuior
!0
1 V
ccos0
=!
1 V
ccos
: (4.2.275)
For an elliptically polarized wave (with r0),E=E1cos!t+E2sin!t, which gives for a free charge
at rest:
d=r2
0(E1n)2+ (E2n)2
E2
1+E2
2do: (4.2.276)
For a linearly polarized wave impinging on a free electric dipole which is a rotator with moment of
inertiaIand frequency !0,_d=!0d, the equation of motion of the dipole is K=I_!0=dE.
Ifj!0j !then d=I 1(dE)d, which leads to dupon using (4.2.208) and (4.2.219). If an
incident wave is unpolarized then averaging (4.2.271) over all the directions perpendicular to nand
usingE E =1
2E2( k k
k2) gives
d=1
2r2
0(1 + cos2#)do; (4.2.277)
where#is the angle between nandk.
178
4.2.10 Light
The wave vector
ki=dxi
d; (4.2.278)
whereis the ane parameter. The wave vector satis es
dki
d+ i
jlkjkl= 0 (4.2.279)
and
kiki= 0: (4.2.280)
It also satis es
ki=@
@xi; (4.2.281)
where is the eikonal. Thus (4.2.280) gives
gik@
@xi@
@xk= 0: (4.2.282)
The Fermat principle is
Z
k dx = 0: (4.2.283)
4.3 Gravity
4.3.1 Constant gravitational eld
A gravitational eld is constant if the metric and torsion tensors do not depend on the time coordinate
x0. In this case, the time coordinate is called the world time . The constancy of the metric tensor
gik=gik(x ) is invariant under transformations
x0!x0+f(x ); (4.3.1)
wherefis an arbitrary function of the spatial coordinates. The three-dimensional scalar g00, three-
dimensional tensor
(1.4.98) and three-dimensional curl of g (1.4.100),
f =g ; g ; ; (4.3.2)
are invariant under (4.3.1). The spatial metric tensor
is used to lower contravariant spatial
indices. The tensor
, which is inverse to
, is used to raise covariant spatial indices. If the
source of a eld is at rest, both directions of time, tand t, are equivalent. Such a eld is referred
to as static and the corresponding metric tensor has
g0 = 0: (4.3.3)
Ifg0 6= 0 thentand tare not equivalent and the eld is referred to as stationary . The proper
time is given by
=1
cpg00x0: (4.3.4)
For a weak eld,
=x0
c
1 +
c2
: (4.3.5)
In a stationary gravitational eld, the interval is
ds2=g00(dx0 g dx )2 dl2; (4.3.6)
179
wheredl2is given by (1.4.97). The three-dimensional velocity of a particle,
v =dx
d ; (4.3.7)
is de ned in terms of the synchronized proper time (corresponding to the di erence in x0between
two synchronized in nitesimally separated points (1.4.101)):
d =1
cpg00(dx0 x0) =1
cpg00(dx0 g dx ): (4.3.8)
Thus the interval (4.3.6) becomes
ds2=g00(dx0 g dx )2
1 v2
c2
; (4.3.9)
where
v2=
v v : (4.3.10)
Using (4.3.6) in the de nition of the four-velocity gives
u =v
cp
1 v2=c2; u0=1
pg00p
1 v2=c2+g u ; (4.3.11)
from which we nd
u0=pg00p
1 v2=c2: (4.3.12)
Thus a particle in a stationary gravitational eld has a constant energy:
E0= c@S
@x0=mc2u0=mc2pg00p
1 v2=c2= const; (4.3.13)
whereSis the principal function. Since a static eld is the special case of a stationary eld, (4.3.13)
is also the energy of a particle in a static gravitational eld. In a weak eld, (4.3.13) reduces to
E0=mc2+mv2
2+m : (4.3.14)
The force acting on a particle in a constant gravitational eld is equal to
f =rp
dt=c
1 v2
c2rp
ds=c
1 v2
c2
u p
: ; (4.3.15)
whereris the three-dimensional Riemannian covariant di erential and colon denotes the three-
dimensional Riemannian covariant derivative, both de ned with respect to the three-dimensional
Christo el symbols (1.4.26) constructed from the spatial metric tensor
(1.4.98) instead of gik,
f
g=1
2
(
; +
;
;): (4.3.16)
Using the metric geodesic equation (1.4.80) valid for a particle in the gravitational eld turns (4.3.15)
into
f =mc2
p
1 v2=c2
(lnpg00) +pg00(g ; g ; )v
c
: (4.3.17)
In the spatial-vector notation, this equation is
f=mc2
p
1 v2=c2
rlnpg00+pg00v
c(rg)
: (4.3.18)
180
If a particle is at rest, v= 0, then the force (4.3.18) has a potential:
f= r(mc2lnpg00): (4.3.19)
Ifv cthen the second term on the right of (4.3.18) has the form of the Coriolis force (3.2.51)
with the angular velocity
!= c
2pg00rg: (4.3.20)
For a ray of light in a stationary gravitational eld, its frequency, measured in terms of the world
time, is constant:
!0= c@
@x0= const; (4.3.21)
where is the eikonal. If the frequency is measured in terms of the proper time then
!= @
@ = @
@x0cpg00=!0pg00: (4.3.22)
For a weak eld, (4.3.22) reduces to
!=!0
1
c2
: (4.3.23)
This decrease of the frequency of light in a gravitational eld is referred to as the redshift .
The null condition for the wave vector, kiki= 0, can be written, analogously to (4.3.6), as
g00(k0 g k )2
k k = 0: (4.3.24)
The constant frequency satis es
!0
c=k0=g00(k0 g k ); (4.3.25)
so (4.3.24) gives
k =!0
cpg00dx
dl: (4.3.26)
The Fermat principle (4.2.283) reads
Zdlpg00+g dx
= 0: (4.3.27)
In a constant gravitational eld, g appears in the Einstein eld equations only in the tensor
f which is invariant under (4.3.1). The terms containing g org( : ), which are not invariant
under (4.3.1), vanish. The component R0
0of the Riemannian Ricci tensor Rikcan be written, using
(1.4.32) and (1.4.35), as
R0
0=gi0Ri0=gi0(fj
i0g;j fj
ijg;0+fk
i0gfj
kjg fk
ijgfj
k0g) =1p g(p ggi0fj
i0g);j
1p g(p g);jgi0fj
i0g gi0
;jfj
i0g+gi0(fk
i0gfj
kjg fk
ijgfj
k0g)
=1p g(p ggi0fj
i0g);j fk
kjggi0fj
i0g (gi0
:j fi
kjggk0 f0
kjggik)fj
i0g
+gi0(fk
i0gfj
kjg fk
ijgfj
k0g) =1p g(p ggi0f
i0g); +gikf0
kjgfj
i0g
=1p g(p ggi0f
i0g); +1
2gikf0
kjggjl(gli;0+gl0;i gi0;l)
=1p g(p ggi0f
i0g); +gk(igl)jf0
kjgg0[l;i]=1p g(p ggi0f
i0g); : (4.3.28)
181
Therefore, we have Z
R0
0p gdV=Ip ggi0f
i0gdf : (4.3.29)
We de ne
h=g00: (4.3.30)
If the metric tensor satis es g0 = 0 theng0 = 0 andg00= 1=g00. In this case, R0
0in (4.3.28)
reduces, using (1.4.106), (1.4.107) and (1.4.128), to the Levi-Civita identity :
R0
0=1p
hs(p
hsg00f
0 0g); =1
2p
hsp
hsh 1g (2g 0;0 g00; )
;
=1p
hs(ps
p
h; ); =1p
h4p
h: (4.3.31)
If the torsion tensor vanishes then the Einstein equations for a constant eld are
1
hR00=1p
h(p
h)
: +h
4f f =+p
1 v2=c2 p
2
;
1p
hR
0= p
h
2f
: 3
2f (p
h): =+p
1 v2=c2v
c;
R =P +h
2f
f
1p
h(p
h): =+p
1 v2=c2v v
c2
+ p
2
; (4.3.32)
whereP is the three-dimensional Ricci tensor constructed from the spatial metric tensor
instead ofgik.
4.3.2 Synchronous system of reference
A system of coordinates in which
g00= 1; g0 = 0 (4.3.33)
is referred to as a synchronous system of reference . In such a system, we also have
g = 0;
= g ;g= s; g00= 1; g0 = 0; (4.3.34)
and the interval reduces to
ds2=c2dt2+g dx dx : (4.3.35)
In a synchronous system of reference, the di erence in x0between two synchronized in nitesimally
separated points (1.4.101) vanishes. For a given metric gik, there exists an in nite number of coor-
dinate transformations leading to synchronous systems of reference. An in nitesimal transformation
from one synchronous system to another is given by
ct!ct+ (x ); x !x + (xi); (4.3.36)
where andare in nitesimal functions. Such a transformation does not change g00= 1, whereas
the condition g0 = 0 yields the relation between and :
=c ; Z
dt+f (x ); (4.3.37)
wheref are arbitrary in nitesimal functions. The corresponding change in
is given by
!
: : : (4.3.38)
182
In a synchronous system of reference, time lines u0= 1; u = 0 are geodesic, that is, they
identically satisfy the metric geodesic equation (1.4.80). They are also normal to the hypersurface
t= const:ni= (ct);igives the normalized normal vector n0= 1; n = 0, which then gives
n0= 1; n = 0, leading to
ni/ui: (4.3.39)
A synchronous system of reference for a particle in a gravitational eld can be constructed from
the Hamilton-Jacobi equation (3.5.1) for the principal function S. A complete integral of this
equation depends on four parameters, out of which one is an additive function of the other three:
S=f(xi; ) +A( ). The world trajectory, which is a metric geodesic, is given by
@S
@x = 0: (4.3.40)
Choosing
xi0=
S
mc;
(4.3.41)
as the new coordinates gives, using (3.5.1),
g0000=gik@x00
@xi@x00
@xk=gik@( S=mc )
@xi@( S=mc )
@xk= 1: (4.3.42)
The four-velocity of the particle in the new coordinates is, using mcui=pi= @S
@xi, (4.3.40) and
(4.3.41),
ui0= @(S=mc )
@xi0= (1;0): (4.3.43)
The new four-velocity ui0satis es the metric geodesic equation (1.4.80) in the new coordinates. The
hypersurface of constant time in the new coordinates is S=mc = const. The corresponding normal
vector is, using (4.3.40),
ni0=@( S=mc )
@xi0 = ( 1;0); (4.3.44)
which is proportional to ui0. Thus the primed coordinate system of reference (4.3.41) for this particle
is synchronous:
g00 0= 0: (4.3.45)
We de ne
=
;0; (4.3.46)
which gives
=
= (ln s);0: (4.3.47)
The components of the Riemann tensor are
R0 0 =1
2 ;0 1
4
;
R0
=1
2(
: :
);
R
= P
+1
4(
); (4.3.48)
where colon denotes the three-dimensional Riemannian covariant derivative with respect to the
three-dimensional Christo el symbols (4.3.16) and P
is the three-dimensional Riemann tensor
constructed from the spatial metric tensor
instead ofgik. The components of the corresponding
Ricci tensor are
R0
0= 1
2
;0 1
4
;
R0
=1
2(
:
: );
R
= P
1
2ps(ps
);0; (4.3.49)
183
whereP is the three-dimensional Riemannian Ricci tensor constructed from the spatial metric
tensor
instead ofgik. For a perfect
uid, the Einstein equations for R0
0give
R0
0=
T0
0 1
2T
=+ 3p
2++p
1 v2=c2v2
c2
: (4.3.50)
If = 0 then the rst equation in (4.3.49) and the Einstein equations require vanishing of the
energy-momentum tensor Tik. In this case, the third equation in (4.3.49) leads to P = 0, which
then givesP
= 0 because of (1.4.74). However, the condition P
= 0 means that the space
is Euclidean: the gravitational eld vanishes. Therefore a synchronous system of reference in the
presence of a gravitational eld is nonstationary.
For a perfect
uid, if p6= 0 then the motion is nongeodesic (cf. (2.5.93)). Therefore in a
synchronous system of reference, the wordlines of particles are not time lines, so matter is not at
rest. In such a coordinate system one cannot construct a comoving frame of reference. If p= 0 then
a synchronous and comoving frame of reference yields u0= 1; u = 0, which then gives
u : u : = 0: (4.3.51)
This equation is equivalent to
curlv= 0: (4.3.52)
Thus a synchronous and comoving frame of reference can be constructed only if the motion of dust
(p= 0) is irrotational.
For a perfect
uid, the rst equation in (4.3.49) and (4.3.50) give
1
2
;0+1
4
0: (4.3.53)
The algebraic inequality
1
3(
)2(4.3.54)
leads then to the Landau inequality ,
(
) 1
;01
6: (4.3.55)
This inequality, together with (4.3.47), means that the determinant of the spatial metric tensor, s,
becomes zero in a nite time. The determinant of the metric tensor, g, becomes zero in a nite time
as well. Thus physical synchronous systems of reference cannot be constructed for all values of the
time coordinate t.
4.3.3 Nonrelativistic gravity
The Poisson equation for a nonrelativistic gravitational eld (2.5.88) has the general solution
= GZ
RdV; (4.3.56)
whereR=jRjandRis the vector from the element of volume dVto the point at which is
calculated. If the source of a gravitational eld is a system of static point masses then (4.3.56)
becomes
= GX
ama
Ra; (4.3.57)
where Rais the vector from a mass mato the point at which is calculated. Thus the gravitational
acceleration in (2.5.84) is
g= GX
amaRa
R3a; (4.3.58)
184
representing the principle of superposition for the gravitational eld. For one mass mwe have
= Gm
R;g= GmR
R3; (4.3.59)
so the force f=mgexerted on a nonrelativistic mass mafrom a nonrelativistic mass mbgives
Newton's law of universal gravitation :
fab= GmambRab
R3
ab; (4.3.60)
where
Rab=Ra Rb: (4.3.61)
The gravitational potential (4.3.59) is negative. Thus the force of gravitation (4.3.60) is attrac-
tive. According to Newton's third law, the force fbaexerted on a nonrelativistic mass mbfrom a
nonrelativistic mass mais equal in magnitude and opposite in direction to fab:
fba= fab: (4.3.62)
The action for the gravitational eld corresponding to G(2.5.85) is
Sg= 1
8GZ Z
(r )2dV dt; (4.3.63)
which gives the total action for the nonrelativistic gravitational eld and matter:
S=Z Zv2
2 1
8G(r )2
dV dt: (4.3.64)
Varying the action (4.3.64) with respect to the gravitational potential gives, for vanishing at
the surface embracing the volume of integration,
S=Z Z
+1
4G4
dV dt: (4.3.65)
The stationarity of this action, S= 0, under gives the Poisson equation (2.5.85).
The integrand in (4.3.64) is the nonrelativistic Lagrangian density for the gravitational eld
and matter. The corresponding Hamiltonian density is given by changing the signs of the terms in
(4.3.64) containing (corresponding to the potential energy in nonrelativistic mechanics). Thus the
energy for the gravitational eld and matter is
E=Zv2
2+ +1
8G(r )2
dV: (4.3.66)
UsingR
(r )2dV= R
4 dV +H
r dSand that the surface integral vanishes at in nity (since
depends on Raccording to (4.3.59)), gives, together with (2.5.85),
E=Zv2
2 1
8G(r )2
dV: (4.3.67)
Therefore the energy of the nonrelativistic gravitational eld is given by
U= 1
8GZ
(r )2dV= 1
8GZ
(g)2dV: (4.3.68)
This energy is not equal to the ( g)t00component of the Landau-Lifshitz energy-momentum com-
plex, where tikis the Landau-Lifshitz energy-momentum pseudotensor (2.5.30); ( g)Tikalso con-
tributes to U. Analogously to (4.2.11), we have
U=1
8GZ
gr dV =1
8GZ
r(g )dV 1
8GZ
rgdV: (4.3.69)
185
The rst term on the right of (4.3.69) can be transformed into a surface integral using the Gau'
theorem. This integral vanishes if the surface of integration is taken at in nity, so (2.5.85) and
(4.3.69) give
U=1
2Z
dV =1
2X
ama a; (4.3.70)
where ais the gravitational potential at the point occupied by a mass ma. In calculating a, we must
not include the gravitational eld produced by mabecause the corresponding contribution to the
gravitational energy (a self-energy), is in nite. Subtracting this self-energy from U(renormalizing
the masses of the particles) gives
U=1
2X
ama 0
a; (4.3.71)
where
0
a= GX
b6=amb
Rab: (4.3.72)
We consider a system of static point masses malocated at raand take the origin of the Cartesian
system of coordinates anywhere within the system. The gravitational potential at a point with the
radius vector Ris thus
= GX
ama
jR raj: (4.3.73)
IfR rathen the gravitational potential (4.3.73) can be expanded in a Taylor series with respect
to the small quantityra
R:
0+ 1+ 2= G
RX
ama+GX
amarar1
R G
2X
max x @2
@X @X 1
R
a
= GM
R+GMr1
R G
6D @2
@X @X 1
R
a= GM
R GMn
R2
G
2R3D n n ; (4.3.74)
whereMis the total mass of the system (3.1.137), denotes the radius vector of its center of mass
(3.1.138),x are the components of r,X are the components of R,nis the unit vector in the
direction of R, and the symmetric spatial tensor
D =X
ama(3x x r2 )ja=D (4.3.75)
is the tensor of mass quadrupole moment of the system. This tensor can be written as
D =Z
(3x x r2 )dV; (4.3.76)
whereis the mass density, M=R
dV. The tensor of mass quadrupole moment is traceless,
D = 0; (4.3.77)
so it has 5 independent components. It is related to the inertia tensor (3.2.13) by
D =I
3I : (4.3.78)
If we choose the origin of the frame of reference at the center of mass of the system then = 0 and
(4.3.74) reduces to
= GM
R G
2R3D n n : (4.3.79)
This reduction is possible because the masses of the particles are of the same (positive) sign. The
electric charges of the particles can have di erent signs. In the analogous expression for the electric
potential (4.2.16), we can eliminate the electric-dipole term 1, which corresponds to the center-of-
mass term 1in (4.3.74), only if the total electric charge of the system is di erent from zero.
186
4.3.4 Schwarzschild metric
Acentrally orspherically symmetric gravitational eld is the eld in which the generators of rotations
(1.6.58) are also generators of isometries (transformations that do not change the metric tensor).
The generators (1.6.58) in the Cartesian coordinates are
Jx=y@
@z z@
@y; Jy=z@
@x x@
@z; Jz=x@
@y y@
@x: (4.3.80)
In the spherical coordinates ( r;; ), they are
Jx= sin @
@ cotcos @
@ ;
Jy= cos @
@ cotsin @
@ ;
Jz=@
@ : (4.3.81)
We can write them as
J =i
@i; (4.3.82)
where
0
x=0
y=0
z= 0; r
x=r
y=r
z= 0;
x= sin ;
x= cotcos ;
y= cos ;
x= cotsin ;
z= 0;
z= 1: (4.3.83)
They are generators of isometries if i
are Killing vectors. For the Levi-Civita connection, (1.4.42)
gives
gikk
;j+gjkk
;i+gij;kk
= 0: (4.3.84)
For =z, (4.3.84) gives
gij; = 0; (4.3.85)
so the metric tensor gijis independent of . For =x;y, we consider di erent components gij. If
i;j2f0;rgthen (4.3.84) gives
=x: gij;sin gij; cotcos = 0;
=y:gij;cos gij; cotsin = 0; (4.3.86)
which, using (4.3.85), yields
gij;= 0: (4.3.87)
Thus
g00=g00(t;r); g0r=g0r(t;r); grr=grr(t;r): (4.3.88)
Ifi2f0;rgandj=then (4.3.84) gives
=x:gi cos
sin2 gi;sin gi; cotcos = 0;
=y:gi sin
sin2+gi;cos gi; cotsin = 0; (4.3.89)
which, using (4.3.85), yields
gi = 0; gi;= 0: (4.3.90)
187
Ifi2f0;rgandj= then (4.3.84) gives
=x: gicos +gi cotsin gi ;sin gi ; cotcos = 0;
=y: gisin gi cotcos +gi ;cos gi ; cotsin = 0; (4.3.91)
which, using (4.3.85) and (4.3.90), yields
gi= 0: (4.3.92)
Thus
g0=g0 =gr=gr = 0: (4.3.93)
Ifi=andj=then (4.3.84) gives
=x: 2g cos
sin2cos g;sin = 0;
=y: 2g sin
sin2cos +g;cos = 0; (4.3.94)
which yields
g = 0; g;= 0: (4.3.95)
Ifi=andj= then (4.3.84) gives
=x: gcos +g cotsin +g cos
sin2 g ;sin = 0;
=y: gsin g cotcos +g sin
sin2+g ;cos = 0; (4.3.96)
which, using (4.3.95), yields
g = sin2g: (4.3.97)
Ifi= andj= then (4.3.84) gives
=x: 2g cotsin 2g cos g ;sin = 0;
=y: 2g cotcos 2g sin +g ;cos = 0; (4.3.98)
which, using (4.3.95) and (4.3.97), leads to the identity. Thus
g=g(t;r); g = 0; g = sin2g: (4.3.99)
Combining (4.3.88), (4.3.93) and (4.3.99) gives the general form of the metric tensor in a spher-
ically symmetric gravitational eld ( r= 1,= 2, = 3):
gij=0
BB@g00(t;r)g01(t;r) 0 0
g01(t;r)g11(t;r) 0 0
0 0 g22(t;r) 0
0 0 0 sin2g22(t;r)1
CCA(4.3.100)
or
ds2=g00(t;r)c2dt2+ 2g01(t;r)cdtdr +g11(t;r)dr2+g22(t;r)(d2+ sin2d 2): (4.3.101)
This form can also be obtained from general considerations. We can apply a coordinate transforma-
tion fromt;r;; tot0(t;r);r0(t;r);; such that
g0010= 0; g22=g22(r0): (4.3.102)
Accordingly, the interval (4.3.101) becomes (omitting the primes)
ds2=g00(t;r)c2dt2+g11(t;r)dr2+g22(r)(d2+ sin2d 2): (4.3.103)
188
In the locally Galilean frame of reference, (4.3.103) should coincide with the Minkowski metric tensor
(1.4.84) in the spherical coordinates,
ds2=c2dt2 dr2 r2(d2+ sin2d 2); (4.3.104)
which implies that g00(t;r)>0; g11<0; g22<0. Thus we can write
g00(t;r) =e(t;r); g11= e(t;r); g22= f2(r); (4.3.105)
where,andfare real functions. We can still apply coordinate transformations t!t0(t) and
r!r0(r) without changing the form of the interval (4.3.104). Speci cally, we can use r!r0(r) to
de ne the radial coordinate rsuch that
f(r) =r; (4.3.106)
for which a circle composed of points with r=Rhas the circumference 2 Rand the sphere composed
of points with r=Rhas the surface area 4 R2. The corresponding radius, de ned as the distance
from the origin to a point where r=R, isRR
0e=2dr, which is equal to Ronly if= 0. Thus the
interval for a spherically symmetric gravitational eld is given by
ds2=e(t;r)c2dt2 e(t;r)dr2 r2(d2+ sin2d 2); (4.3.107)
where we can still use t!t0(t) to rede ne the time coordinate t. The gravitational eld given by
the interval (4.3.107) is symmetric about the origin r= 0.
The Christo el symbols for the interval (4.3.107) that do not vanish are
f0
0 0g=_
2;f0
0 1g=0
2;f0
1 1g=_
2e ;
f1
0 0g=0
2e ;f1
0 1g=_
2;f1
1 1g=0
2;f1
2 2g= re ;f1
3 3g= rsin2e ;
f2
1 2g=f3
1 3g=1
r;f2
3;3g= sincos;f3
2 3g= cot; (4.3.108)
where dot denotes a derivative with respect to ctand prime denotes a derivative with respect to r.
The corresponding components of the Einstein tensor that do not vanish identically are
G0
0=1
r2 e 1
r2 0
r
;
G1
1=1
r2 e 1
r2+0
r
;
G2
2=G3
3= 1
2e
00+02
2 00
2+0 0
r
+1
2e
+_2
2 __
2
;
G1
0= e _
r: (4.3.109)
In vacuum, the Einstein equations (as well as the Einstein-Cartan equations) give Gi
j= 0. The last
equation in (4.3.109) gives
_= 0!=(r): (4.3.110)
Adding the rst and second equation in (4.3.109) gives
(+)0= 0!+=f(t): (4.3.111)
However, we can use t!t0(t) to rede ne the time coordinate tsuch thatf(t) = 0. Thus
= ; (4.3.112)
which also leads to
_= 0!=(r): (4.3.113)
189
Therefore the metric tensor of a spherically symmetric gravitational eld does not depend on time.
The existence of the three Killing vectors associated with the generators of rotations (1.6.58) implies
the existence of the fourth Killing vector associated with time translation:
i
0=i
0: (4.3.114)
A spherically symmetric gravitational eld in vacuum is static, which constitutes the Birkho theo-
rem.
We consider a spherically symmetric gravitational eld in vacuum around a massive sphere. Such
a eld vanishes at in nity. Integrating the rst equation in (4.3.109), with the condition e!1 at
r!1 , gives
e =e= 1 +C
r; (4.3.115)
whereCis a constant. Comparing (4.3.115) with (2.5.82) and (4.3.59) yields
C= rg; (4.3.116)
where
rg=2Gm
c2(4.3.117)
is called the gravitational or Schwarzschild radius corresponding to the mass m. The corresponding
interval,
ds2=
1 rg
r
c2dt2
1 rg
r 1
dr2 r2(d2+ sin2d 2); (4.3.118)
describes the Schwarzschild metric for a spherically symmetric gravitational eld. For a spherically
symmetric cavity inside the sphere, the regularity of the metric at the center, r= 0, requires C= 0.
Thus the gravitational eld there vanishes. For a spherically symmetric gravitational eld inside
a massive sphere of radius R, the regularity of the metric at the center yields = 0 atr= 0.
Integrating the rst equation in (4.3.109) then gives
= ln
1
rZr
0T0
0r2dr
; (4.3.119)
yieldinge1. The rst and second equation in (4.3.109) give also 0+00, yielding+0.
Outside the sphere, (4.3.119) gives
= ln
1
rZR
0T0
0r2dr
= ln
1
rZR
0~r2dr
; (4.3.120)
where ~is the e ective energy density (2.5.115) of a spin
uid. Without spin, ~ =is the ordinary
energy density of an ideal
uid. Comparing (4.3.120) with (4.3.115) and (4.3.117) gives
m=4
c2ZR
0~r2dr<4
c2ZR
0~e=2r2dr=1
c2Z
~dV: (4.3.121)
This inequality indicates the gravitational mass defect .
The spatial interval (1.4.97) for the Schwarzschild metric is given by
dl2=
1 rg
r 1
dr2+r2(d2+ sin2d 2): (4.3.122)
The proper time is related to the coordinate time tbyd =pg00dt, yieldingd dt. Thus
the time is slower in the presence of a gravitational eld, in accordance with the time dilation in
noninertial frames of reference and the principle of equivalence. The distance between two points
with the same angular components of the spherical coordinates, located at r1andr2, is bigger than
190
the di erence between their radial components:Rr2
r1drp
1 rg=rr2 r1. For a weak gravitational
eld, the metric (4.3.118) can be approximated as
ds2=ds2
0 rg
r(c2dt2+dr2); (4.3.123)
whereds2
0is the interval corresponding to the Minkowski tensor of a
at spacetime (1.4.84). For
a plane passing through the origin of the Schwarzschild eld, we can take ==2. The spatial
interval (4.3.122) on this plane is
dl2=
1 rg
r 1
dr2+r2d 2: (4.3.124)
The form of the surface of rotation on which the geometry is the same as on this plane is
dl2=dr2+dz2+r2d 2=dr2(1 +z02) +r2d 2; (4.3.125)
wherez0=dz
drand 1 +z02= (1 rg=r) 1, yielding Flamm's paraboloid :
z= 2q
rg(r rg): (4.3.126)
The surface of rotation (4.3.126) represents embedding of the non-Euclidean, two-dimensional equa-
torial plane of the Schwarzschild geometry into the three-dimensional Euclidean space. Such a space
has no physical meaning and merely serves to visualize Flamm's paraboloid, as shown in Fig. 20.
Figure 20: Flamm paraboloid.
The nonzero components of the curvature tensor which for the Schwarzschild eld is equal to the
Riemann tensor (1.4.44) are
R01
01=R23
23= rg
r3; R02
02=R03
03=R12
12=R13
13=rg
2r3: (4.3.127)
The Kretschmann scalar ,
I=RijklRijkl; (4.3.128)
is singular at the origin r= 0:
I=12r2
g
r6: (4.3.129)
Thus the Schwarzschild eld has a spacetime (curvature) singularity at the origin, indicating its
unphysical character. The nonzero components of the spatial Ricci tensor (1.4.74) (the Ricci tensor
constructed from the spatial metric tensor (1.4.98)) are
P1
1= rg
r3; P2
2=P3
3=rg
2r3: (4.3.130)
Thus the spatial Ricci scalar P= 0. For a plane perpendicular to the radius, the Gau curvature
(1.4.76) is
K=P2323
22
33= P1
1>0: (4.3.131)
191
Thus the sum of the angles of a small triangle on such a plane embracing its intersection with the
above radius is >. For a plane passing through the origin, K < 0. Thus the sum of the angles
of a small triangle on such a plane is < , except for triangles embracing the origin, for which
>.
The choice (4.3.106) of the radial coordinate ris not unique. Using the Birkho theorem, we
can write the interval for a spherically symmetric gravitational eld as
ds2=e(r)c2dt2 e(r)dr2 e(r)r2(d2+ sin2d 2); (4.3.132)
and use the freedom of applying a transformation r!r0(r) to impose a constraint
f(;; ) = 0: (4.3.133)
The Einstein equations in vacuum for the metric (4.3.132) give the de Sitter formulae :
e= 1 rg
R; e=R02
1 rg
R; e=R2
r2; (4.3.134)
whereR=R(r) is a function xed by (4.3.133) and R0=dR
dr. The Schwarzschild-Hilbert constraint
= 0 givesR=r(4.3.106) and the Schwarzschild metric (4.3.118). Droste's constraint= 0 yields
r=R
1 rg
R1=2
+rg
2lnp
R+p
R rgp
R p
R rg+ const: (4.3.135)
Schwarzschild's constraint+ 2+= 0 gives
R= (r3+r3
g)1=3; (4.3.136)
which is equivalent to (4.3.118) with the origin at r= rg.Weyl's constraint = 0 gives
R=r
1 +rg
4r2
; (4.3.137)
which yields the metric describing a spherically symmetric gravitational eld in the isotropic spherical
coordinates :
ds2=(1 rg=(4r))2
(1 +rg=(4r))2c2dt2 (1 +rg=(4r))4
dr2+r2(d2+ sin2d 2)
: (4.3.138)
The Einstein-Rosen constraint ++ ln4 = 0 yields
R=rg+r2
4rg; (4.3.139)
which is equivalent to (4.3.138).
A hypersurface of constant radius, f(r) =r const = 0, has the normalized normal vector
(2.5.138) given in the spherical coordinates ( ct;r;; ) by
ni= (0;1;0;0): (4.3.140)
In a Schwarzschild eld,
nini=rg
r 1: (4.3.141)
Thus the Schwarzschild sphere r=rgis an event horizon.
192
4.3.5 Motion in Schwarzschild eld
de ne black hole and white hole
d
2=d2+ sin2d 2(4.3.142)
tortoise radial coordinate
dr=dr
1 rg
r(4.3.143)
r=r+rglnr
rg 1
(4.3.144)
free-fall velocity
V
c=rrg
r(4.3.145)
proper coordinates
dT2=
1 rg
r
dt2;
dR2=
1 rg
r 1
dr2: (4.3.146)
We consider a massive particle moving radially in the gravitational eld described by the metric
(??). For brevity, we use
h=g00=(1 rg=(4r))2
(1 +rg=(4r))2; f= grr= (1 +rg=(4r))4: (4.3.147)
The motion of the particle is given by the radial geodesic equations. If the particle is at rest at
r=r0, then these equations are
dt
d =u0=p
h0=h; (4.3.148)
dr
cd =ur=(h0h 1f 1 f 1)1=2; (4.3.149)
where is the proper time of the particle, h0=hjr=r0, and= 1 (+1) for an infalling (outgoing)
motion. We consider a particle falling into a black hole, = 1. Asr!rg=4,hgoes to zero and
f!16, so both u0andurbecome in nite. Even if the initial motion were not purely radial, the
components u0; urwould still become in nite at r=rg=4, withu; u remaining nite. Therefore,
each motion of a massive particle becomes e ectively radial at the surface r=rg=4.
A distant observer situated in a nearly Galilean spacetime measures the velocity of the infalling
particle as
vd=dr
dt=cur
u0= c(h0f 1h f 1h2)1=2
ph0: (4.3.150)
Asr!rg=4,vdgoes to zero. Writing r=rg=4 +, where 0< rg, givesvd c=(2rg) and
thusr rg=4exp( ct=(2rg)), so the particle reaches the surface of a black hole r=rg=4 after
an in nite time t. This surface is an event horizon for a distant observer, as it is for the standard
Schwarzschild metric [ ?]. The proper time of the particle for moving radially from r=r0to
r=rg=4 is nite, which can be shown by considering r0=rg=4 +:
c =Zrg=4+
rg=4dr=urrg: (4.3.151)
After reaching the surface r=rg=4, the particle continues moving; its radial coordinate rdecreases
tor1=r2
g=(16r0)rg=4 (at whichur= 0) in a proper time =rg=c. The radial motion of
193
a massive particle (in terms of the proper time) in the spacetime ( ??) forrrg=4 is the image (in
the sense of the method of image charges for spheres in electrostatics) of the particle's motion for
rrg=4. Applying the transformation ( ??) to Eqs. (4.3.148) and (4.3.149) in the region rrg=4
gives
dt
d =p
h0=h; (4.3.152)
dr0
cd = (h0h 1f 1 f 1)1=2; (4.3.153)
whereh= (1 rg=(4r0))2=(1 +rg=(4r0))2andf= (1 +rg=(4r0))4. An infalling radial geodesic
motion inside a black hole appears, in terms of the new radial coordinate r0, as an outgoing motion
from a white hole (the time reversal of a black hole).
The local velocity of the particle vl, measured in terms of the proper time, as determined by static
clocks synchronized along the trajectory of the particle, is related to u0byu0= (h(1 v2
l=c2)) 1=2[2].
As the particle moves from r=r0tor=rg=4,vlincreases from zero to c, and as the particle moves
fromr=rg=4 tor=r1,vldecreases to zero. In a Schwarzschild eld, vlexceedscinside a black
hole, which does not violate Einstein's theory of relativity because the interior of a Schwarzschild
black hole is not static and neither can be clocks synchronized along the trajectory of the particle.
4.3.6 Other coordinates for Schwarzschild eld
The Schwarzschild metric (4.3.118) has a coordinate singularity at the surface r=rg, whereg00= 0
andg11is in nite. The components of the curvature tensor (4.3.127) and the Kretschmann scalar
(4.3.129) are nite at this surface, though. Therefore an appropriate coordinate transformation,
that is singular at r=rg, can remove this apparent singularity. De ning new coordinates ;R
such that
cd =cdt+f(r)dr;
dR=cdt+dr
f(r); (4.3.154)
wheredris the tortoise coordinate (4.3.143), turns (4.3.118) into
ds2=1 rg=r
1 f2(c2d 2
f2dR2) r2d
2: (4.3.155)
We also have
Rc =Z1 f2
fdr: (4.3.156)
The physical case of gravitational attraction (black hole) corresponds to plus signs in (4.3.154). The
metric (4.3.155) is regular at r=rgiff(rg) = 1. Taking
f(r) =rrg
r(4.3.157)
turns (4.3.155) into the Lema^ tre metric :
ds2=c2d 2 rg
r( ;R)dR2 r2( ;R)d
2; (4.3.158)
where = +andr( ;R) is given by integrating (4.3.156):
r=3
2(R c )2=3
r1=3
g: (4.3.159)
The Lema^ tre metric (4.3.158) is regular at the gravitational radius. It is singular at r= 0, where
c =R. The Lema^ tre metric is synchronous, g00= 1 andg0 = 0. Thus the bodies at rest in the
194
Lema^ tre coordinates ( ;R) are freely falling in the gravitational eld. The time coordinate is the
proper time. Radial null geodesics, ds=d
= 0, are given by
cd
dR=rrg
r; (4.3.160)
where plus sign corresponds to a radially outgoing signal and minus sign to a radially infalling signal.
Since all physical world lines must lie within the local light cones, no signal can escape from inside
the Schwarzschild sphere, wheredr
d <0, as shown in Fig. 21. Thus signals emitted there radially
inwards and outwards both reach the origin. The radially falling bodies reach the gravitational
radius and then the origin within nite proper time. The unphysical case of gravitational repulsion
(white hole) corresponds to minus signs in (4.3.154). In this case, no signal can enter inside the
Schwarzschild sphere. In Fig. 21, this case corresponds to = .
Figure 21: Lema^ tre coordinates.
We can de ne new time coordinates v;uaccording to
v=ct+r;
u=ct r: (4.3.161)
Thevcoordinate is the ingoing null coordinate or advanced Eddington-Finkelstein time anduis the
outgoing null coordinate or retarded Eddington-Finkelstein time , analogously to the advanced and
retarded time (4.2.162). Replacing tbyvturns (4.3.118) into the ingoing Eddington-Finkelstein
metric :
ds2=
1 rg
r
dv2 2dvdr r2d
2: (4.3.162)
The metric (4.3.162) is regular at r=rgand describes a black hole. Radial null geodesics are given
by
dv= 0; dv = 2dr; (4.3.163)
where the rst case corresponds to a radially infalling signal and the second one to a radially outgoing
signal. Since all physical world lines must lie within the local light cones, no signal can escape from
inside the Schwarzschild sphere, wheredr
dv<0, as shown in Fig. 22. Replacing tbyuturns (4.3.118)
into the outgoing Eddington-Finkelstein metric :
ds2=
1 rg
r
du2+ 2dudr r2d
2: (4.3.164)
The metric (4.3.164) is regular at r=rgand describes a white hole. Radial null geodesics are given
by
du= 0; du = 2dr; (4.3.165)
where the rst case corresponds to a radially outgoing signal and the second one to a radially infalling
signal. No signal can enter inside the Schwarzschild sphere. This case is described by Fig. 22, in
whichvis replaced by u.
195
Figure 22: Eddington-Finkelstein coordinates.
We can de ne new time coordinates ~ v;~uaccording to
d~v=cdt+rrg
rdr;
d~u=cdt rrg
rdr: (4.3.166)
The ~vcoordinate is the advanced Painlev e-Gullstrand time and ~uis the retarded Painlev e-Gullstrand
time. Integrating (4.3.166) gives
~v=ct+ 2prgr+rglnpr prgpr+prg;
~u=ct 2prgr rglnpr prgpr+prg(4.3.167)
Replacingtby ~vturns (4.3.118) into the ingoing Painlev e-Gullstrand metric :
ds2=
1 rg
r
d~v2 2rrg
rd~vdr dr2 r2d
2: (4.3.168)
The metric (4.3.168) is regular at r=rgand describes a black hole. Radial null geodesics are given
by
d~v=rrg
r1
dr; (4.3.169)
where plus sign corresponds to a radially outgoing signal and minus sign to a radially infalling signal.
Replacingtby ~uturns (4.3.118) into the outgoing Painlev e-Gullstrand metric :
ds2=
1 rg
r
d~u2+ 2rrg
rd~udr dr2 r2d
2: (4.3.170)
The metric (4.3.170) is regular at r=rgand describes a white hole. Radial null geodesics are given
by
d~u=
rrg
r1
dr; (4.3.171)
where plus sign corresponds to a radially outgoing signal and minus sign to a radially infalling signal.
The Painlev e-Gullstrand coordinates can also be obtained through a special Lorentz transfor-
mation (1.6.93) with the free-fall velocity V(4.3.145) in the R-direction applied (locally) to the
stationary proper time T(4.3.146):
d~v=
1 V2
c2 1=2
cdT+V
cdR
;
d~u=
1 V2
c2 1=2
cdT V
cdR
: (4.3.172)
196
Thus the Painlev e-Gullstrand time is the proper time measured by clocks that fall freely in the
spherically symmetric gravitational eld. This result agrees with the metric geodesic equations
(1.4.81) corresponding to (4.3.168) and (4.3.170), for which, respectively,d~v
ds= 1 andd~u
ds= 1.
Replacing both tandrby the null coordinates vandu(4.3.161) turns (4.3.118) into
ds2=
1 rg
r
dudv r2d
2: (4.3.173)
De ning the Kruskal coordinates ,
U=e u=(2rg); V=ev=(2rg);
T+R=V; T R= U; (4.3.174)
turns (4.3.173) into the Kruskal-Szekeres metric :
ds2=4r3
g
re r=rg(dT2 dR2) r2d
2; (4.3.175)
wherer=r(T;R) is given by a transcendental equation,
er=rgr
rg 1
=R2 T2: (4.3.176)
The metric (4.3.175) is regular at r=rgand singular at r= 0. It is also singular (det gij= 0) for
rg= 0.Tis the time coordinate and Ris the radial coordinate. Region I in Fig. 23 ( V > 0; U > 0)
represents the exterior of a black hole ( r > rg), where the transformation from ( t;r) to (T;R) is
given by
T=r
rg 11=2
er=(2rg)sinhct
2rg;
R=r
rg 11=2
er=(2rg)coshct
2rg: (4.3.177)
Region II in Fig. 23 ( V > 0; U < 0) represents the interior of a black hole ( r < rg), where the
transformation from ( t;r) to (T;R) is given by
T=
1 r
rg1=2
er=(2rg)coshct
2rg;
R=
1 r
rg1=2
er=(2rg)sinhct
2rg: (4.3.178)
The linesr= const are given by UV= const and the lines t= const by V=U = const. The curvature
singularity r= 0 is given by UV= 1 and the event horizon by UV= 0. Radial null geodesics,
which are the boundaries of light cones, are given by dUdV = 0:V= const describes an ingoing
null geodesic and U= const describes an outgoing null geodesic.
A Schwarzschild black hole is represented by V > 0. However, the coordinates ( V;U) can be
extended to V < 0. Region III in Fig. 23 ( V < 0; U < 0) represents the exterior of a white hole
(r>rg), where the transformation from ( t;r) to (T;R) is given by
T= r
rg 11=2
er=(2rg)sinhct
2rg;
R=r
rg 11=2
er=(2rg)coshct
2rg: (4.3.179)
197
Region III is mathematically equivalent to the time-reversed region I. Region IV in Fig. 23 ( V <
0; U > 0) represents the interior of a white hole ( r < rg), where the transformation from ( t;r) to
(T;R) is given by
T=
1 r
rg1=2
er=(2rg)coshct
2rg;
R=
1 r
rg1=2
er=(2rg)sinhct
2rg: (4.3.180)
Region IV is mathematically equivalent to the time-reversed region II. These 4 regions represent
amaximally extended Schwarzschild solution . The Schwarzschild metric (4.3.118) is covered only
by region I or region III. The ingoing Eddington-Finkelstein and Painlev e-Gullstrand metrics are
covered by regions I and II. The outgoing Eddington-Finkelstein and Painlev e-Gullstrand metrics
are covered by regions III and IV.
Figure 23: Kruskal coordinates.
4.3.7 Weyl isotropic metric
The interval of the static, spherically symmetric gravitational eld in vacuum, expressed in isotropic
spherical coordinates ( ct;;; ), is given by the Weyl isotropic metric (4.3.138):
ds2=(1 rg=(4))2
(1 +rg=(4))2c2dt2 (1 +rg=(4))4(d2+2d
2); (4.3.181)
where 0<1is the radial coordinate in Weyl's constraint. This metric does not change its form
under the radial coordinate transformation
!0=r2
g
16; (4.3.182)
and is Galilean for !1 . Therefore it is also Galilean for !0, describing an Einstein-Rosen
bridge or wormhole: two Schwarzschild solutions of the Einstein eld equations (a black hole and
white hole) connected at the common event horizon r=rg=4. The metric tensor at the horizon is
singular, det gik= 0. The nonzero components of the Riemann curvature tensor for this metric are
given by
R02
02=R03
03=R12
12=R13
13=rg
23(1 +rg=(4))6;
R01
01=R23
23= rg
3(1 +rg=(4))6; (4.3.183)
198
so the Kretschmann scalar (4.3.128) is nite everywhere,
I= 12r2
g 6(1 +rg=(4)) 12; (4.3.184)
going to zero as !1 and!0. For rg=4, the interval (4.3.181) reduces to
ds2=
1 rg
c2dt2
1 +rg
(dx2+dy2+dz2): (4.3.185)
The Weyl metric for > rg=4 describes the exterior sheet of a Schwarzschild black hole. The
transformation of the radial coordinate,
!r=(1 +rg=(4))2; (4.3.186)
brings the interval (4.3.181) into the standard Schwarzschild form (4.3.118). The spacetime given
by the metric (4.3.181) for < rg=4 is regarded by observers at > rg=4 as the interior of a
black hole. Because of the invariance of the metric (4.3.181) under the transformation (4.3.182),
this interior is an image of the other exterior sheet. Since the motion through the event horizon
of an Einstein-Rosen bridge is unidirectional, this interior is equivalent to the exterior sheet of a
Schwarzschild white hole. The spacetime of an Einstein-Rosen bridge is regular everywhere. The
surface of rotation representing embedding of the non-Euclidean, two-dimensional equatorial plane
of the Einstein-Rosen geometry into the three-dimensional Euclidean space is the combination of
two Flamm's paraboloids, as shown in Fig. 24.
Figure 24: Einstein-Rosen bridge.
We de ne a new radial coordinate such that
r=rg+2
4rg: (4.3.187)
The Schwarzschild metric (4.3.118) becomes
ds2=2
4r2g+2c2dt2 4r2
g+2
4r2gd2
rg+2
4rg2
d
2; (4.3.188)
which is equivalent to the Weyl metric (4.3.181), describing an Einstein-Rosen bridge. Near the
horizon, where 0, (4.3.188) reduces to
ds2=2
4r2gc2dt2 d2 r2
gd
2: (4.3.189)
Introducing new coordinates ~t;~rsuch that
c~t=sinhct
2rg;~r=coshct
2rg; (4.3.190)
199
turns (4.3.189) into
ds2=c2d~t2 d~r2 r2
gd
2: (4.3.191)
Thus the ( ~t;~r) subspace of the spacetime represented by the interval (4.3.191) is
at. Accordingly,
the (t;) subspace of the spacetime represented by the interval (4.3.181) near the event horizon is
nearly
at. The interval (4.3.189) is analogous to the Rindler metric (4.1.41) and the coordinates
ct; in (4.3.189) are analogous to cT;X (4.1.40). If =c2=a0, wherea0= const, then (4.3.190)
gives a hyperbola ~ r2 (c~t)2=c4=a2
0, which is analogous to (4.1.36). Therefore (4.3.187) gives the
proper acceleration of a massive particle near the event horizon:
a01
2rgp
1 rg=r: (4.3.192)
4.3.8 Interior Schwarzschild solution
We consider a spherically symmetric gravitational eld in spacetime lled with an ideal
uid. The
general form of the metric in such a eld is given by the interval (4.3.101). We can apply a coordinate
transformation from t;r;; to (t;r);R(t;r);; such that
g01= 0; u1=dR
d = 0: (4.3.193)
The second condition means that no radial motion takes place in this frame of reference. Since
the spherical symmetry also requires u2=u3= 0, this frame of reference is a comoving frame.
Accordingly, the interval (4.3.101) becomes
ds2=e( ;R)c2d 2 e( ;R)dR2 e( ;R)R2(d2+ sin2d 2); (4.3.194)
where,andare real functions. We can still apply coordinate transformations ! 0( ) and
R!R0(R) without changing the form of the interval (4.3.194). The components of the Einstein
tensor corresponding to (4.3.194) that do not vanish identically are
G0
0= e
00 2
R2+3
4
0+2
R2
1
2
0+2
R
0!
+1
2e
__+_2
2
+e
R2;
G1
1= 1
2e
1
2
0+2
R2
+
0+2
R
0!
+e
1
2__+3
4_2
+e
R2;
G2
2=G3
3= 1
4e
200+02+ 200 4
R2+
0+2
R2
00
+
0+2
R
(0 0)!
+1
4e (2+_2+ 2+ _2 __ __+__);
G1
0=1
2e
2 _0+
0+2
R
( _ _) 0_!
; (4.3.195)
where dot denotes di erentiation with respect to and prime denotes di erentiation with respect
toR. The Einstein equations in this frame of reference are
G0
0=; G1
1=G2
2=G3
3= p; G1
0= 0: (4.3.196)
The corresponding conservation law Tik
:k= 0 gives
_+ 2 _= 2_
+p; 0= 2p0
+p; (4.3.197)
where the constants of integration depend on the allowed transformations ! 0( ) andR!R0(R).
For a spherically symmetric gravitational eld in spacetime lled with a spin
uid, !~andp!~p.
200
If a spherically symmetric sphere of radius ais in hydrostatic equilibrium then the metric does
not depend on the time coordinate and the interval (4.3.194) reduces to
ds2=e(r)c2dt2 e(r)dr2 r2(d2+ sin2d 2); (4.3.198)
where we used the freedom of applying a coordinate transformation r!r0(r) to set(r) = 0, as in
(4.3.107). The Einstein equations (4.3.195) reduce to (cf. (4.3.109))
1
r2 e 1
r2 0
r
=;
1
r2 e 1
r2+0
r
= p: (4.3.199)
Integrating the rst equation in (4.3.199) gives
e=
1
rZr
0r2dr 1
=
1 rg(r)
r 1
; (4.3.200)
where
rg(r) =Zr
0(r)r2dr (4.3.201)
is the gravitational radius of the sphere of radius rcentered at the origin (cf. (4.3.117), (4.3.119)
and (4.3.121)). The second equation in (4.3.199) and the second equation in (4.3.197) yield
p0= 1
2(+p)
e1
r+pr
1
r!
= 1
2r2(+p)e
rg(r) +pr3
; (4.3.202)
which, upon substituting (4.3.200), gives the Tolman-Oppenheimer-Volko equation :
dp
dr= 1
2r2(+p)
rg(r) +pr3
1 rg(r)
r: (4.3.203)
In the nonrelativistic limit, (4.3.203) reduces to
dp
dr= Gm(r)(r)
r2; (4.3.204)
whereis the mass density and
m(r) = 4Zr
0(r)r2dr (4.3.205)
is the mass of the sphere of radius rcentered at the origin. The hydrostatic condition (4.3.204) can
also be derived for a spherically symmetric system from (2.5.100) for v= 0, using (4.3.56).
If the energy density inside the sphere is homogeneous, = const, then (4.3.200) gives
e(r)=
1 1
3r2 1
; (4.3.206)
which, with (4.3.202), leads to
p0= 1
2(+p)
1 1
3r2 1
3+p
r: (4.3.207)
Integrating this equation with the condition that the pressure vanishes at the boundary of the sphere,
p(a) = 0, gives
p(r) =h(r) h(a)
3h(a) h(r); (4.3.208)
201
where
h(r) =
1 1
3r21=2
: (4.3.209)
The pressure (4.3.208) is a decreasing function of r. The condition that p(0) be nite gives
a>9
8rg: (4.3.210)
Physically realistic equations of state obey p=3. Thus we have p(0)=3, which yields
a9
5rg: (4.3.211)
The second equation in (4.3.197) integrates to
e(r)=e(a)
1 +p(r)
2
: (4.3.212)
The continuity of the metric at the boundary of the sphere, r=a, requires (cf. (4.3.112))
e(a)=e (a): (4.3.213)
Thus the interval for the interior Schwarzschild metric for a constant energy density is given by
ds2=9
4
h(a) 1
3h(r)2
c2dt2 h 2(r)dr2 r2(d2+ sin2d 2): (4.3.214)
4.3.9 Tolman solution
4.3.10 Kerr metric
The spacetime geometry in the region surrounding a mass mrotating with angular momentum Lis
given by the Kerr metric :
ds2=
1 rgr
2
c2dt2 2
dr2 2d2
r2+a2+a2sin2rgr
2
sin2d 2
+2asin2rgr
2cdtd ; (4.3.215)
where
a=L
mc; (4.3.216)
2=r2+a2cos2; (4.3.217)
=r2 rgr+a2: (4.3.218)
A coordinate transformation,
x+iy= (r+ia)sinei ;
z=rcos; (4.3.219)
brings the coordinates ( r;; ), referred to as the Boyer-Lindquist coordinates , to the Kerr-Schild
coordinates (x;y;z ), in which the Kerr metric becomes
gij=ij+fkikj; (4.3.220)
where
k0= 1; kx=rx+ay
r2+a2; ky=ry ax
r2+a2; kz=z
r;
f=rgr3
r4+a2z2: (4.3.221)
202
The vector kis a unit vector. The relation between the Boyer-Lindquist and Kerr-Schild coordinates
is the same as the relation between the oblate spheroidal and Cartesian coordinates. The Kerr
metric is the stationary ( @gij=@t= 0), axisymmetric ( @gij=@ = 0), asymptotically
at ( gij!ij
asr!1 ), vacuum ( Tij= 0) solution of the Einstein equations. !
4.3.11 Motion in Kerr eld
4.3.12 Reissner-Nordstr om metric
4.3.13 Kerr-Newman metric
The spacetime geometry in the region surrounding a mass mof chargeqrotating with angular
momentum Lis given in the Boyer-Lindquist coordinates ( r;; ) (4.3.219) by the Kerr-Newman
metric :
ds2=
1 rgr r2
q
2
c2dt2 2
dr2 2d2
r2+a2+a2sin2rgr r2
q
2
sin2d 2+ 2asin2rgr r2
q
2cdtd ; (4.3.222)
whereaand2are given by (4.3.216) and (4.3.217), and
=r2 rgr+a2+r2
q; (4.3.223)
where
r2
q=Gq2
c4: (4.3.224)
The electromagnetic potential in the Kerr-Newman eld is given by
Aidxi=qrcdt
2 qrasin2d
2: (4.3.225)
This metric can be written in the Kerr-Schild form (4.3.220), where fis generalized to
f=r2
r4+a2z2(rgr r2
q): (4.3.226)
The electromagnetic potential in these coordinates is given by
Ai=qr3
r4+a2z2ki: (4.3.227)
The Kerr-Newman metric is the stationary ( @gij=@t= 0), axisymmetric ( @gij=@ = 0), asymp-
totically
at ( gij!ijasr!1 ) solution of the Einstein-Maxwell equations without sources
(ji= 0).
At large distances r a, the electric potential in (4.3.225) reduces to the Coulomb potential
=q
rand the magnetic potential in (4.3.225) reduces to (4.2.49) for the magnetic moment
m=qa^z: (4.3.228)
Thus we have
m=q
mcM; (4.3.229)
so the gyromagnetic ratio (4.2.55) for the Kerr-Newman eld is g= 2 which is twice larger than
that for a nonrelativistic particle or a system of identical nonrelativistic particles.
!
203
4.3.14 Weak gravitational eld
The component h03=2GM
rc2sin2satis es the eld equation P03= 0 linearized in h03. Thus, for a
slowly rotating body, h03=2GM
rc2sin2is valid even if the gravitational eld is strong.
4.3.15 Gravitational waves
4.3.16 Kottler-de Sitter metric
In the presence of the cosmological constant , the Lagrangian density (2.5.21) gives the eld
equations
Pik 1
2Pgik=Tik+ gik: (4.3.230)
For a spherically symmetric gravitational eld around a massive sphere, these equations give the
Kottler metric :
ds2=
1 rg
r r2
3
c2dt2
1 rg
r r2
3 1
dr2 r2(d2+ sin2d 2); (4.3.231)
which generalizes the Schwarzschild metric (4.3.118). If rg= 0, the Kottler metric (4.3.231) reduces
to the de Sitter metric :
ds2=
1 r2
3
c2dt2
1 r2
3 1
dr2 r2(d2+ sin2d 2): (4.3.232)
For the de Sitter metric (4.3.232), the curvature is given by
Rijkl=
3(gikgjl gilgjk): (4.3.233)
4.3.17 Friedmann-Lema^ tre-Robertson-Walker metric
4.4 Spinors
4.4.1 Free spinors
4.4.2 Dirac equation in central electric eld
4.4.3 Schr odinger equation in central electric eld
4.4.4 Dirac equation in uniform magnetic eld
204
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