Contents 1 Minkowski Spacetime .......................................... 1 1.1 Introduction ............................................... 1 1.2 The Four Dimensions ....................................... 2 1.2.1 Spacetime as an Affine Space .......................... 2 1.2.2 A Few Notations ..................................... 3 1.2.3 Affine Coordinate System ............................. 4 1.2.4 Constant c .......................................... 4 1.2.5 Newtonian Spacetime ................................ 5 1.3 Metric Tensor .............................................. 6 1.3.1 Scalar Product on Spacetime ........................... 7 1.3.2 Matrix of the Metric Tensor ........................... 9 1.3.3 Orthonormal Bases ................................... 10 1.3.4 Classification of Vectors with Respect to g ............... 11 1.3.5 Norm of a Vector .................................... 11 1.3.6 Spacetime Diagrams ................................. 12 1.4 Null Cone and Time Arrow .................................. 15 1.4.1 Definitions .......................................... 15 1.4.2 Two Useful Lemmas ................................. 15 1.4.3 Classification of Unit Vectors .......................... 16 1.5 Spacetime Orientation....................................... 19 1.6 Vector / Linear Form Duality ................................. 21 1.6.1 Linear Forms and Dual Space .......................... 21 1.6.2 Metric Duality....................................... 22 1.7 Minkowski Spacetime ....................................... 23 1.8 Before Going Further... ...................................... 26 2 Worldlines and Proper Time .................................... 29 2.1 Introduction ............................................... 29 2.2 Worldline of a Particle ...................................... 29 2.3 Proper Time ............................................... 31 2.3.1 Definition .......................................... 31 xvii xviii Contents 2.3.2 Ideal Clock ......................................... 34 2.4 Four-Velocity and Four-Acceleration .......................... 35 2.4.1 Four-Velocity ....................................... 35 2.4.2 Four-Acceleration .................................... 37 2.5 Photons ................................................... 39 2.5.1 Null Geodesics ...................................... 39 2.5.2 Light Cone ......................................... 40 2.6 Langevin’s Traveller and Twin Paradox ........................ 40 2.6.1 Twins’ Worldlines ................................... 41 2.6.2 Proper Time of each Twin ............................. 43 2.6.3 The “Paradox” ...................................... 44 2.6.4 4-Velocity and 4-Acceleration.......................... 46 2.6.5 A Round Trip to the Galactic Centre .................... 50 2.6.6 Experimental Verifications ............................ 53 2.7 Geometrical Properties of a Worldline ......................... 56 2.7.1 Timelike Geodesics .................................. 56 2.7.2 Vector Field along a Worldline ......................... 57 2.7.3 Curvature and Torsions ............................... 58 3 Observers ..................................................... 63 3.1 Introduction ............................................... 63 3.2 Simultaneity and Measure of Time ............................ 64 3.2.1 The Problem ........................................ 64 3.2.2 Einstein-Poincare´ Simultaneity ......................... 65 3.2.3 Local Rest Space .................................... 66 3.2.4 Nonexistence of Absolute Time ........................ 69 3.2.5 Orthogonal Projector onto the Local Rest Space .......... 70 3.2.6 Euclidean Character of the Local Rest Space ............. 73 3.3 Measuring Spatial Distances ................................. 73 3.3.1 Synge Formula ...................................... 74 3.3.2 Born’s Rigidity Criterion .............................. 75 3.4 Local Frame ............................................... 77 3.4.1 Local Frame of an Observer ........................... 77 3.4.2 Coordinates with Respect to an Observer ................ 79 3.4.3 Reference Space of an Observer ........................ 81 3.5 Four-Rotation of a Local Frame............................... 82 3.5.1 Variation of the Local Frame along the Worldline ......... 82 3.5.2 Orthogonal Decomposition of Antisymmetric Bilinear Forms 84 3.5.3 Application to the Variation of the Local Frame ........... 87 3.5.4 Inertial Observers .................................... 90 3.6 Derivative of a Vector Field along a Worldline .................. 91 3.6.1 Absolute Derivative .................................. 91 3.6.2 Derivative with Respect to an Observer .................. 91 3.6.3 Fermi-Walker Derivative .............................. 92 3.7 Locality of an Observer’s Frame .............................. 93 Contents xix 4 Kinematics 1: Motion with Respect to an Observer . 97 4.1 Introduction ............................................... 97 4.2 Lorentz Factor ............................................. 98 4.2.1 Definition .......................................... 98 4.2.2 Expression in Terms of the 4-Velocity and the 4-Acceleration101 4.2.3 Time Dilation ....................................... 103 4.3 Velocity Relative to an Observer .............................. 104 4.3.1 Definition .......................................... 104 4.3.2 4-Velocity and Lorentz Factor in Terms of the Velocity ..... 106 4.3.3 Maximum Relative Velocity ........................... 109 4.3.4 Component Expressions .............................. 110 4.4 Experimental Verifications of Time Dilation .................... 111 4.4.1 Atmospheric Muons .................................. 111 4.4.2 Other Tests ......................................... 113 4.5 Acceleration Relative to an Observer .......................... 113 4.5.1 Definition .......................................... 113 4.5.2 Relation to the Second Derivative of the Position Vector .... 114 4.5.3 Expression of the 4-acceleration ........................ 117 4.6 Photon Motion ............................................. 121 4.6.1 Propagation Direction of a Photon ...................... 121 4.6.2 Velocity of Light ..................................... 122 4.6.3 Experimental Tests of the Invariance of the Velocity of Light 125 5 Kinematics 2: Change of Observer ...............................133 5.1 Introduction ............................................... 133 5.2 Relations between Two Observers ............................. 133 5.2.1 Reciprocity of the Relative Velocity ..................... 134 5.2.2 Length Contraction................................... 136 5.3 Law of Velocity Composition ................................ 139 5.3.1 General Form ....................................... 139 5.3.2 Decomposition in Parallel and Transverse Parts ........... 141 5.3.3 Collinear Velocities .................................. 144 5.3.4 Alternative Formula .................................. 145 5.3.5 Experimental Verification: Fizeau Experiment ............ 146 5.4 Law of Acceleration Composition ............................. 148 5.5 Doppler Effect ............................................. 150 5.5.1 Derivation .......................................... 151 5.5.2 Experimental Verifications ............................ 153 5.6 Aberration ................................................ 154 5.6.1 Theoretical Expression ............................... 155 5.6.2 Distortion of the Celestial Sphere ....................... 158 5.6.3 Experimental Verifications ............................ 158 5.7 Images of Moving Objects ................................... 160 5.7.1 Image and Instantaneous Position....................... 160 5.7.2 Apparent Rotation ................................... 161 xx Contents 5.7.3 Image of a Sphere .................................... 163 5.7.4 Superluminal Motions ................................ 166 6 Lorentz Group .................................................169 6.1 Introduction ............................................... 169 6.2 Lorentz Transformations .................................... 169 6.2.1 Definition and characterization ......................... 169 6.2.2 Lorentz Group....................................... 171 6.2.3 Properties of Lorentz Transformations ................... 172 6.3 Subgroups of O(3,1) ........................................ 174 6.3.1 Proper Lorentz Group SO(3,1) ......................... 174 6.3.2 Orthochronous Lorentz Group ......................... 175 6.3.3 Restricted Lorentz Group ............................. 176 6.3.4 Reduction of the Lorentz Group to SOo(3,1) ............. 176 6.4 Classification of Restricted Lorentz Transformations ............. 178 6.4.1 Invariant Null Direction ............................... 178 6.4.2 Decomposition with Respect to an Invariant Null Direction . 179 6.4.3 Spatial Rotations..................................... 183 6.4.4 Lorentz Boosts ...................................... 185 6.4.5 Null Rotations ....................................... 187 6.4.6 Four-Screws ........................................ 189 6.4.7 Eigenvectors of a Restricted Lorentz Transformation ...... 190 6.4.8 Summary ........................................... 191 6.5 Polar Decomposition ........................................ 193 6.5.1 Statement and Demonstration .......................... 193 6.5.2 Explicit Forms ...................................... 194 6.6 Properties of Lorentz Boosts ................................. 196 6.6.1 Kinematical