Four-Tensors, the Mother Tongue of Classical Physics V/Dlf

Reinhart Frosch

Four-Tensors, the Mother Tongue of Classical Physics

v/dlf vdf Hochschulverlag AG an der ETH Zurich Contents

PART ONE: SPECIAL RELATIVITY 15 1 Events 17 2 Time Measurements 18 3 Measurements of Spatial Coordinates 19 4 Laws of Nature 24 5 The Electromagnetic Field 26 6 The Microscopic Maxwell Equations 29 7 Electromagnetic Waves 34 8 Wave Groups 39 9 The Electromagnetic Wave Equation 45 10 The Michelson-Morley Experiment 51 11 The Relativity Principle 58 12 The Lorentz Transformation 64 13 The Lorentz Transformation (Continued) 72 14 Test of the Standard Lorentz Transformation 74 15 Conventional Formulation of the Standard Lorentz Transformation 77 16 Lorentz or Fitzgerald Contraction 79 17 Time Dilatation 81 18 The Twin "Paradox" 86 19 The Invariant Line Element 92 20 Modified Definition of the Lorentz Transformation 95 21 Representation of homogeneous Lorentz Transformations by Matrices 99 22 Lorentz Boost in an Arbitrary Direction 106

PART TWO: MATHEMATICS OF FOUR-TENSORS 113 23 The Four-Nabla Operator 115 24 Representation of Lorentz Transformations by Means of the Four-Nabla-Operator 117 25 The Distance Four-Vector 120

11 26 Definition of the Contravariant Four-Vector 122 27 Scalar Functions of Event Coordinates 124 28 The Four-Gradient 126 29 Definition of the Covariant Four-Vector 128 30 The Scalar Product of a Contravariant Four-Vector and a Covariant Four-Vector 130 31 The Direct Product of Four-Vectors 133 32 Transformation of the Four-Nabla Operator 135 33 Four-Tensors of Higher Rank 138 34 The Metric Tensor 143 35 The Free Parameters of the Lorentz Transformation 147 36 The Kronecker Tensor 151 37 The Scalar Product of Two Contravariant Four-Vectors ... 154 38 The Contravariant Analog of the Metric Tensor 157 39 The Raising and Lowering of Tensor Indices 160 40 The Contraction of Mixed Tensors 164

PART THREE: PSEUDOTENSORS 167 41 The Area Defined by Two Two-Vectors 169 42 Definition of Pseudoscalars 174 43 Definition of Pseudotensors 182 44 Pseudotensors in Three-Dimensional Space 188 45 Pseudotensors in Four-Dimensional Spacetime 191

PART FOUR: FOUR-TENSORS AND LAWS OF NATURE . . 195 46 Four-Velocity 197 47 Relation of Four-Velocity to Three-Velocity 201 48 Relation of Four-Velocity to Three-Velocity (Continued) . . 204 49 Four-Acceleration 206 50 Relation of Four-Acceleration to Three-Acceleration 207 51 Relation of Four-Acceleration to Three-Acceleration (Continued) 210 52 Rest Masses 212 53 Four-Momentum 217 54 Relation of Four-Momentum to Three-Momentum 219 55 Energy 222

12 56 Relation of Four-Momentum to Energy and Three- Momentum 227 57 Rules for Laws of Nature 230 58 The Four-Momentum Conservation Law 233 59 Mass Defects 239 60 The Electric Charge 245 61 The Four-Vector of Electric Current Density 248 62 Relation of Four-Vector of Electric Current Density to Corresponding Three-Vector 250 63 The Continuity Equation 252 64 The Electromagnetic Four-Potential 254 65 Relation of Four-Potential to Three-Vector and Coulomb Potentials 257 66 The Electromagnetic Field-Strength Four-Tensor 259 67 Gauge Invariance of the Field-Strength Four-Tensor 260 68 The Lorentz Condition 263 69 Relation of the Field-Strength Four-Tensor to the Field- Strength Three-Vectors 267 70 The "New" Maxwell Equations 269 71 The Number of Maxwell Equations 271 72 Consistency of "New" and "Old" Maxwell Equations 273 73 The Wave Equation for the Four-Potential 276 74 The Dual Field-Strength Pseudotensor 278 75 Alternative Form of "New" Maxwell Equations 280 76 The Maxwell Equations and the Continuity Equation 282 77 The Electromagnetic Four-Force on an Electrically Charged Particle 284 78 Relation of the Four-Force to the Three-Force 285 79 Newton's Second Law for Particles of Constant Rest Mass 287 80 "Old" and "New" Forms of Newton's Second Law 288 81 Alternative "New" Form of Newton's Second Law 290 82 "Old" and "New" Forms of Newton's Second Law (Continued) 292

REFERENCES 295 INDEX 297