Special relativity

Preface to the First Edition......Page 7
Preface to the Second Edition......Page 10
Contents......Page 13
About the Author......Page 23
 1.1 Introduction......Page 24
 1.2 Elements From the Theory of Linear Spaces......Page 25
  1.2.1 Coordinate Transformations......Page 26
 1.3 Inner Product: Metric......Page 30
 1.4 Tensors......Page 34
  1.4.1 Operations of Tensors......Page 37
 1.5 The case of Euclidian Geometry......Page 38
 1.6 The Lorentz Geometry......Page 41
  1.6.1 Lorentz Transformations......Page 42
 1.7 Algebraic Determination of the Lorentz Transformation L(β)......Page 51
  1.8.1 Relativistic Parallelism of Space Axes......Page 65
  1.8.2 The Kinematic Interpretation of Lorentz Transformation......Page 67
 1.9 The Geometry of the Boost......Page 68
  1.10.1 Proper Frame of a Timelike Four-Vector......Page 73
  1.10.2 Characteristic Frame of a Spacelike Four-Vector......Page 74
 1.11 Particle Four-Vectors......Page 75
 1.12 The Center System (CS) of a System of Particle Four-Vectors......Page 78
 2.1 Introduction......Page 79
 2.2 The Role of Physics......Page 80
 2.3 The Structure of a Theory of Physics......Page 83
 2.4 Physical Quantities and Reality of a Theory of Physics......Page 84
 2.5 Inertial Observers......Page 87
  2.6.1 Principle of Inertia......Page 88
  2.6.2 The Covariance Principle......Page 89
 2.7 Relativity and the Predictions of a Theory......Page 90
 3.1 Introduction......Page 92
  3.2.1 Mass Point......Page 93
  3.2.2 Space......Page 94
  3.2.3 Time......Page 97
 3.3 Newtonian Inertial Observers......Page 100
  3.3.1 Determination of Newtonian Inertial Observers......Page 101
   3.3.1.2 Axiom on the Dimension of Space......Page 102
 3.4 Galileo Principle of Relativity......Page 103
  3.5.2 Galileo Principle of Communication......Page 105
 3.6 Newtonian Physical Quantities: The Covariance Principle......Page 106
 3.7 Newtonian Composition Law of Vectors......Page 107
 3.8 Newtonian Dynamics......Page 109
  3.8.1 Law of Conservation of Linear Momentum......Page 110
 4.1 Introduction......Page 112
  4.2.1 The Existence of Non-Newtonian Physical Quantities......Page 113
  4.2.2 The Limit of Special Relativity to Newtonian Physics......Page 114
 4.3 The Physical Role of the Speed of Light......Page 117
 4.4 The Physical Definition of Spacetime......Page 118
  4.4.2 The Geometry of Spacetime......Page 119
  4.5.1 The Light Cone......Page 121
  4.5.2 World Lines......Page 122
  4.5.3 Curves in Minkowski Space......Page 123
  4.5.4 Geometric Definition of Relativistic Inertial Observers (RIO)......Page 124
  4.5.5 Proper Time......Page 125
  4.5.7 Proper or Rest Space......Page 126
 4.6 Spacetime Description of Motion......Page 127
   4.6.1.1 Axiom of Relativistic Inertia......Page 129
  4.6.2 Relativistic Measurement of the Position Vector......Page 130
 4.7 The Einstein Principle of Relativity......Page 131
  4.7.1 The Equation of Lorentz Isometry......Page 132
  4.8.1 Lorentz Covariance Principle: Part I......Page 134
  4.8.3 Rules for Constructing Lorentz Tensors......Page 135
 4.9 Universal Speeds and the Lorentz Transformation......Page 136
 5.2 The Concepts of Space and Time in Special Relativity......Page 142
 5.3 Measurement of Spatial and Temporal Distance in Special Relativity......Page 143
 5.4 Relativistic Definition of Spatial and Temporal Distances......Page 145
 5.5 Timelike Position Four-Vector: Measurement of Temporal Distance......Page 146
 5.6 Spacelike Position Four-Vector: Measurement of Spatial Distance......Page 151
 5.7 The General Case......Page 155
 5.8 The Reality of Length Contraction and Time Dilation......Page 156
 5.9 The Rigid Rod......Page 158
 5.10 Optical Images in Special Relativity......Page 160
  5.11.1 A Brief Summary of the Lorentz Transformation......Page 167
  5.11.2 Parallel and Normal Decomposition of Lorentz Transformation......Page 169
   Case 1......Page 170
   Case 2......Page 171
  5.11.4 The Algebraic Method......Page 172
  5.11.5 The Geometric Method......Page 177
 6.2 Relativistic Mass Point......Page 182
 6.3 Relativistic Composition of 3-Vectors......Page 187
 6.4 Relative Four-Vectors......Page 196
 6.5 The 3-Velocity Space......Page 202
 6.6 Thomas Precession......Page 205
 7.1 Introduction......Page 212
 7.2 The Four-Acceleration......Page 213
 7.3 Calculating Accelerated Motions......Page 220
 7.4 Hyperbolic Motion of a Relativistic Mass Particle......Page 224
  7.4.1 Geometric Representation of Hyperbolic Motion......Page 228
 7.5 Synchronization......Page 231
  7.5.1 Einstein Synchronization......Page 232
  7.5.2 K-Calculus......Page 234
 7.6 Rigid Motion of Many Relativistic Mass Points......Page 236
 7.7 Rigid Motion and Hyperbolic Motion......Page 237
  7.7.1 Born Synchronization of LRIO......Page 240
  7.7.2 Synchronization of Chronometry......Page 241
  7.7.3 The Kinematics in the LCF......Page 242
  7.7.4 The Case of the Gravitational Field......Page 246
 7.8 General One Dimensional Rigid Motion......Page 248
  7.8.1 The Case of Hyperbolic Motion......Page 249
 7.9 Rotational Rigid Motion......Page 251
  7.9.1 The Transitive Property of the Rigid Rotational Motion......Page 256
  7.10.1 The Kinematics of Relativistic Observers......Page 257
  7.10.2 Chronometry and the Spatial Line Element......Page 259
  7.10.3 The Rotating Disk......Page 261
  7.10.4 Definition of the Rotating Disk for a RIO......Page 262
  7.10.5 The Locally Relativistic Inertial Observer (LRIO)......Page 264
   7.10.5.1 Observation of the Disk in A......Page 265
   7.10.5.2 Calculation of the Path of B in A......Page 266
   7.10.5.3 Calculation of the Velocity of B in A......Page 267
  7.10.6 The Accelerated Observer......Page 269
   7.10.6.2 Spatial Geometry......Page 271
   7.10.6.3 The Velocity of Light for the Accelerating Observer ̃......Page 272
 7.11 The Generalization of Lorentz Transformation and the Accelerated Observers......Page 274
  7.11.1 The Generalized Lorentz Transformation......Page 275
  7.11.2 The Special Case u0(l,x)=u1(l,x)=u(x)......Page 277
   7.11.2.1 Conclusion......Page 282
  7.11.3 Equation of Motion in a Gravitational Field......Page 283
  7.12.1 Experiment 1: The Gravitational Redshift......Page 284
  7.12.2 Experiment 2: The Gravitational Time Dilation......Page 286
  7.12.3 Experiment 3: The Curvature of Spacetime......Page 288
 8.1 Introduction......Page 289
 8.2 Various Paradoxes......Page 290
 9.1 Introduction......Page 302
 9.2 The (Relativistic) Mass......Page 303
 9.3 The Four-Momentum of a ReMaP......Page 304
 9.4 The Four-Momentum of Photons (Luxons)......Page 313
 9.6 The System of Natural Units......Page 316
 10.1 Introduction......Page 321
 10.3 Relativistic Reactions......Page 323
  10.3.1 The Sum of Particle Four-Vectors......Page 325
 10.4 Working With Four-Momenta......Page 327
 10.5 Special Coordinate Frames in the Study of Relativistic Collisions......Page 329
  10.6.1 The Physics of the Generic Reaction......Page 331
  10.6.2 Threshold of a Reaction......Page 336
 10.7 Transformation of Angles......Page 343
  10.7.1 Radiative Transitions......Page 347
  10.7.2 Reactions with Two Photon Final State......Page 351
  10.7.3 Elastic Collisions: Scattering......Page 356
 11.2 The Four-Force......Page 364
 11.3 Inertial Four-Force and Four-Potential......Page 380
  11.3.1 The Vector Four-Potential......Page 382
 11.4 The Lagrangian Formalism for Inertial Four-Forces......Page 383
 11.5 Motion in a Central Potential......Page 390
 11.6 Motion of a Rocket......Page 395
 11.7 The Frenet-Serret Frame in Minkowski Space......Page 404
  11.7.1 The Physical Basis......Page 409
  11.7.2 The Generic Inertial Four-Force......Page 413
  11.7.3 Applications......Page 414
   11.7.3.2 The Lorentz-Dirac Four-Force......Page 415
   11.7.3.3 Bonnor Four-Force......Page 417
 12.1 Decompositions......Page 418
  12.1.1 Writing a Tensor of Valence (0,2) as a Matrix......Page 419
 12.2 The Irreducible Decomposition wrt a Non-null Vector......Page 420
   12.2.1.1 Decomposition of a Euclidean Vector......Page 421
   12.2.1.2 Decomposition of a Euclidean Second Order Tensor......Page 422
  12.2.2 1+3 Decomposition in Minkowski Space......Page 425
 12.3 1+1+2 Decomposition wrt  a Pair of a Timelike Four-Vector and a Non-null Four-Vector......Page 431
 13.1 Introduction......Page 437
 13.2 Maxwell Equations in Newtonian Physics......Page 439
 13.3 The Electromagnetic Potential......Page 441
 13.4 The Equation of Continuity......Page 447
 13.5 The Electromagnetic Four-Potential......Page 455
  13.6.1 The Transformation of the Fields......Page 459
  13.6.2 Maxwell Equations in Terms of Fij......Page 461
  13.6.3 The Invariants of the Electromagnetic Field......Page 462
 13.7 The Physical Significance of the Electromagnetic Invariants......Page 465
  13.7.1 The Case Y=0......Page 466
  13.7.2 The Case Y=0......Page 468
 13.8 Motion of a Charge in an Electromagnetic Field: The Lorentz Force......Page 470
 13.9 Motion of a Charge in a Homogeneous Electromagnetic Field......Page 473
  13.9.1 The Case of a Homogeneous Electric Field......Page 474
  13.9.2 The Case of a Homogeneous Magnetic Field......Page 478
  13.9.3 The Case of Two Homogeneous Fields of Equal Strength and Perpendicular Directions......Page 480
  13.9.4 The Case of Homogeneous and Parallel FieldsEB......Page 482
 13.10 The Relativistic Electric and Magnetic Fields......Page 484
  13.10.1 The Levi Civita Tensor Density......Page 485
  13.10.2 The Case of Vacuum......Page 487
  13.10.3 The Electromagnetic Theory for a General Medium......Page 490
  13.10.4 The Electric and Magnetic Moments......Page 492
  13.10.5 Maxwell Equations for a General Medium......Page 493
  13.10.6 The 1+3 Decomposition of Maxwell Equations......Page 494
 13.11 The Four-Current of Conductivity and Ohm's Law......Page 499
  13.11.1 The Continuity Equation Ja;a=0 for an Isotropic Material......Page 503
 13.12 The Electromagnetic Field in a Homogeneous and Isotropic Medium......Page 504
 13.13 Electric Conductivity and the Propagation Equation for Da......Page 509
 13.14 The Generalized Ohm's Law......Page 510
 13.15 The Energy Momentum Tensor of the Electromagnetic Field......Page 513
 13.16 The 1+3 Decomposition of the Tensor Tab......Page 521
 13.17 The Electromagnetic Field of a Moving Charge......Page 522
  13.17.1 The Invariants......Page 524
  13.17.3 The Liénard–Wiechert Potentials and the Fields E, B......Page 525
 13.18 Special Relativity and Practical Applications......Page 536
 13.19 The Systems of Units SI and Gauss in Electromagnetism......Page 540
 14.2 Mathematical Preliminaries......Page 543
  14.2.1 1+3 Decomposition of a Bivector Xab......Page 544
 14.3 The Derivative of the Bivector Xab along the Vector pa......Page 546
  14.4.1 The Angular Momentum in Newtonian Theory......Page 549
  14.4.2 The Angular Momentum of a Particle in Special Relativity......Page 551
  14.5.1 The Magnetic Dipole......Page 555
  14.5.2 The Relativistic Spin......Page 559
  14.5.3 Motion of a Particle with Spin in a Homogeneous Electromagnetic Field......Page 564
  14.5.4 Transformation of Motion in......Page 566
 15.1 Introduction......Page 569
  15.2.1 Definition of the Lorentz Transformation......Page 571
  15.2.2 Computation of the Covariant Lorentz Transformation......Page 573
  15.2.3 The Action of the Covariant Lorentz Transformation on the Coordinates......Page 577
  15.2.4 The Invariant Length of a Four-Vector......Page 582
 15.3 The Four Types of the Lorentz Transformation Viewed as Spacetime Reflections......Page 583
 15.4 Relativistic Composition Rule of 4-Vectors......Page 586
  15.4.1 Computation of the Composite Four-Vector......Page 590
  15.4.2 The Relativistic Composition Rule for 3-Velocities......Page 591
  15.4.3 Riemannian Geometry and Special Relativity......Page 593
  15.4.4 The Relativistic Rule for the Composition of 3-Accelerations......Page 599
 15.5 The Composition of Lorentz Transformations......Page 600
 16.1 Tetrads and the Lorentz Transformation......Page 605
 16.2 The Null Triad......Page 607
  16.2.1 The Allcock Approach......Page 608
  16.2.2 The Two Null Vector Approach (Synge)......Page 609
  16.2.3 The Characteristic Tetrad of a Null Triad......Page 613
 16.3 The Null Tetrad......Page 617
  16.3.1 The Proper Orthochronous Lorentz Transformation in Terms of the Null Tetrad......Page 620
  16.3.2 The Special Lorentz Transformation in Terms of Bivectors......Page 625
 17.1 Collisions and Geometry......Page 627
 17.2 Geometric Description of Collisions in Newtonian Physics......Page 628
 17.4 The General Geometric Results......Page 631
  17.4.1 The 1+3 Decomposition of a Particle Four-Vector wrt a Timelike Four-Vector......Page 634
 17.5 The System of Two to One Particle Four-Vectors......Page 635
  17.5.1 The Triangle Function of a System of Two Particle Four-Vectors......Page 637
  17.5.2 Extreme Values of the Four-Vectors (AB)2......Page 639
  17.5.3 The System Aa,Ba,(A+B)a of Particle Four-Vectors in CS......Page 641
  17.5.4 The System Aa,Ba,(A+B)a in the Lab......Page 642
 17.6 The Relativistic System Aa+Ba→Ca+Da......Page 647
  17.6.1 The Reaction B-3mu→C+D......Page 660
 18.2 The Disturbance......Page 662
 18.3 Waves in Newtonian Physics......Page 664
 18.4 Plane Waves......Page 666
 18.5 Spherical Waves......Page 668
 18.6 Linear Superposition of Waves......Page 670
 18.7 Period and Wavelength of a Wave......Page 671
  18.8.1 The Frequency Four-Vector......Page 672
  18.8.2 The Doppler Shift......Page 676
   18.8.2.1 The Radial Doppler Shift......Page 678
   18.8.2.2 The Transverse Doppler Effect......Page 683
  18.8.3 The Aberration of the Wave Vector......Page 685
 18.9 Electromagnetic Waves in a Homogeneous and Isotropic Medium......Page 691
 18.10 Center of Momentum of a System of Frequency Four Vectors......Page 695
 18.11 Waves and Particles......Page 699
  18.11.1 deBroglie Waves and Light......Page 700
  18.11.2 deBroglie Waves and Particles......Page 702
 19.1 The Manifold Structure......Page 718
 19.2 The Geometry Defined by the Metric......Page 720
 19.4 Geometry and the Covariance Principle......Page 721
 19.5 Kinematics: The Connecting Vector......Page 723
  19.6.1 Vector Formulation......Page 724
  19.6.2 Tensor Formulation......Page 725
  20.1.1 The Relative Motion in Vector Formulation......Page 727
  20.1.2 The Relative Velocity of Linear Continua in Tensor Notation......Page 729
  20.1.3 The Kinematic Interpretation of Newtonian Relative Velocity......Page 730
 20.2 Newtonian Kinematics of a Linear Deformable Body in Terms of the Connecting Vector......Page 731
  20.3.1 The Geometrization of Rigid Motion......Page 734
  20.3.2 The Geometrization of Strain Motion......Page 737
 20.4 The Geometry of a General Newtonian Deformable Body......Page 738
 20.5 The Stress Tensor of a Linear Deformable Continuum......Page 740
  20.5.1 Body and Surface Forces......Page 741
  20.5.2 The Perfect Deformable Body......Page 742
  20.5.3 The Imperfect Deformable Body......Page 744
 20.6 Classification of the Stress Tensor of Linear Newtonian Deformable Bodies......Page 745
  20.6.2 Shear Stress......Page 746
  20.6.3 Bulk or Volume Stress......Page 747
 20.7 Worked Examples on the Newtonian Stress Tensor......Page 748
21 The Stress: Strain Relation for Elastic Newtonian Deformable Bodies......Page 755
 21.1 The Stress Rate - Strain Relation for a Linear Elastic Isotropic Body......Page 757
  21.1.1 The Coefficients n,k of a Linear Elastic Isotropic Body......Page 759
  21.1.2 A Second Derivation of the Rate of Strain Stress Relation form Hooke's Law......Page 760
  21.2.1 The Force Due to a Stress Tensor......Page 762
  21.2.2 Newton's Second Law in Terms of the Displacement Vector......Page 763
  21.2.3 Kinematic Interpretation of the Equation of Motion of the Strain Vector in a Linear Elastic Isotropic Newtonian Body......Page 765
  21.2.4 The Dynamic Equation of Motion for a Linear Elastic Isotropic Newtonian Body......Page 769
22 Newtonian Fluids......Page 771
 22.1 Mathematical Concepts Relevant to Fluids......Page 772
 22.2 Classification of Fluids and Fluid Flows......Page 775
  22.2.1 Geometric Description of the Flow of a Fluid......Page 776
  22.2.2 Calculating the Streamlines......Page 777
 22.3 The Equation of Continuity: Conservation of Mass......Page 780
 22.4 The Equation of Motion of a Perfect Fluid......Page 783
 22.5 The Navier Stokes Equation......Page 788
 22.6 The Electromagnetic Field as a Newtonian Viscous Fluid......Page 792
 22.7 A Short Detour to Thermodynamics and Hydrodynamics......Page 795
 23.1 General Definitions......Page 799
 23.2 Relativistic Fluids in Special Relativity......Page 801
 23.3 The 1+3 Decomposition wrt a General Non-null Vector qa......Page 804
 23.4 The Extended Metric and the Bivector Metric......Page 806
 23.5 The 1+3 Decomposition wrt the Four-Velocity ua......Page 808
 23.6 The Kinematics of a Relativistic Fluid......Page 809
 23.7 The Dynamics of a Relativistic Fluid......Page 810
  23.7.1 The Case of a Single Particle......Page 811
  23.7.2 The Case of a Single Particle Under the Action of Forces......Page 815
 23.8 The Equation of Motion of a Relativistic Fluid......Page 817
  23.8.1 The Dynamical Physical Quantities of a Relativistic Fluid......Page 819
 23.9 The Dynamical Equations of Motion of a Relativistic Fluid: A Simplified Approach......Page 820
  23.9.1 The Energy Momentum Tensor of a Relativistic Viscous Fluid......Page 823
 23.10 The Electromagnetic Field in Vacuum as a Relativistic Fluid......Page 824