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1 | (54) |
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1 | (1) |
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1.2 Elements From the Theory of Linear Spaces |
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2 | (4) |
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1.2.1 Coordinate Transformations |
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2 | (4) |
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1.3 Inner Product - Metric |
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6 | (4) |
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10 | (4) |
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1.4.1 Operations of Tensors |
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13 | (1) |
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1.5 The Case of Euclidean Geometry |
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14 | (3) |
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17 | (9) |
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1.6.1 Lorentz Transformations |
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18 | (8) |
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1.7 Algebraic Determination of the General Vector Lorentz Transformation |
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26 | (14) |
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1.8 The Kinematic Interpretation of the General Lorentz Transformation |
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40 | (3) |
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1.8.1 Relativistic Parallelism of Space Axes |
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40 | (2) |
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1.8.2 The Kinematic Interpretation of Lorentz Transformation |
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42 | (1) |
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1.9 The Geometry of the Boost |
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43 | (5) |
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1.10 Characteristic Frames of Four-Vectors |
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48 | (2) |
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1.10.1 Proper Frame of a Time like Four-Vector |
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48 | (1) |
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1.10.2 Characteristic Frame of a Space like Four-Vector |
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49 | (1) |
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1.11 Particle Four-Vectors |
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50 | (2) |
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1.12 The Center System (CS) of a System of Particle Four-Vectors |
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52 | (3) |
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2 The Structure of the Theories of Physics |
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55 | (12) |
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55 | (1) |
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56 | (3) |
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2.3 The Structure of a Theory of Physics |
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59 | (1) |
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2.4 Physical Quantities and Reality of a Theory of Physics |
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60 | (2) |
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62 | (1) |
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2.6 Geometrization of the Principle of Relativity |
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63 | (3) |
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2.6.1 Principle of Inertia |
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63 | (1) |
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2.6.2 The Covariance Principle |
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64 | (2) |
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2.7 Relativity and the Predictions of a Theory |
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66 | (1) |
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67 | (20) |
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67 | (1) |
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68 | (6) |
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68 | (1) |
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69 | (2) |
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71 | (3) |
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3.3 Newtonian Inertial Observers |
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74 | (4) |
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3.3.1 Determination of Newtonian Inertial Observers |
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75 | (2) |
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3.3.2 Measurement of the Position Vector |
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77 | (1) |
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3.4 Galileo Principle of Relativity |
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78 | (1) |
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3.5 Galileo Transformations for Space and Time - Newtonian Physical Quantities |
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79 | (2) |
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3.5.1 Galileo Covariant Principle: Part I |
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79 | (1) |
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3.5.2 Galileo Principle of Communication |
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80 | (1) |
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3.6 Newtonian Physical Quantities. The Covariance Principle |
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81 | (1) |
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3.6.1 Galileo Covariance Principle: Part II |
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81 | (1) |
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3.7 Newtonian Composition Law of Vectors |
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82 | (1) |
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83 | (4) |
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3.8.1 Law of Conservation of Linear Momentum |
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84 | (3) |
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4 The Foundation of Special Relativity |
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87 | (30) |
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87 | (1) |
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4.2 Light and the Galileo Principle of Relativity |
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88 | (4) |
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4.2.1 The Existence of Non-Newtonian Physical Quantities |
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88 | (1) |
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4.2.2 The Limit of Special Relativity to Newtonian Physics |
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89 | (3) |
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4.3 The Physical Role of the Speed of Light |
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92 | (1) |
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4.4 The Physical Definition of Spacetime |
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93 | (3) |
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94 | (1) |
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4.4.2 The Geometry of Spacetime |
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94 | (2) |
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4.5 Structures in Minkowski Space |
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96 | (6) |
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96 | (1) |
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97 | (1) |
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4.5.3 Curves in Minkowski Space |
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98 | (1) |
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4.5.4 Geometric Definition of Relativistic Inertial Observers (RIO) |
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99 | (1) |
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99 | (1) |
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4.5.6 The Proper Frame of a RIO |
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100 | (1) |
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4.5.7 Proper or Rest Space |
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101 | (1) |
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4.6 Spacetime Description of Motion |
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102 | (3) |
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4.6.1 The Physical Definition of a RIO |
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103 | (1) |
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4.6.2 Relativistic Measurement of the Position Vector |
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104 | (1) |
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4.6.3 The Physical Definition of an LRIO |
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105 | (1) |
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4.7 The Einstein Principle of Relativity |
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105 | (3) |
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4.7.1 The Equation of Lorentz Isometry |
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106 | (2) |
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4.8 The Lorentz Covariance Principle |
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108 | (2) |
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4.8.1 Rules for Constructing Lorentz Tensors |
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109 | (1) |
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4.8.2 Potential Relativistic Physical Quantities |
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110 | (1) |
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4.9 Universal Speeds and the Lorentz Transformation |
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110 | (7) |
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5 The Physics of the Position Four-Vector |
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117 | (38) |
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117 | (1) |
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5.2 The Concepts of Space and Time in Special Relativity |
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117 | (1) |
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5.3 Measurement of Spatial and Temporal Distance in Special Relativity |
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118 | (2) |
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5.4 Relativistic Definition of Spatial and Temporal Distances |
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120 | (1) |
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5.5 Timelike Position Four-Vector - Measurement of Temporal Distance |
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121 | (5) |
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5.6 Spacelike Position Four-Vector - Measurement of Spatial Distance |
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126 | (3) |
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129 | (1) |
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5.8 The Reality of Length Contraction and Time Dilation |
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130 | (2) |
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132 | (2) |
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5.10 Optical Images in Special Relativity |
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134 | (7) |
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5.11 How to Solve Problems Involving Spatial and Temporal Distance |
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141 | (14) |
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5.11.1 A Brief Summary of the Lorentz Transformation |
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141 | (1) |
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5.11.2 Parallel and Normal Decomposition of Lorentz Transformation |
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142 | (1) |
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5.11.3 Methodologies of Solving Problems Involving Boosts |
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143 | (3) |
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5.11.4 The Algebraic Method |
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146 | (4) |
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5.11.5 The Geometric Method |
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150 | (5) |
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6 Relativistic Kinematics |
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155 | (30) |
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155 | (1) |
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6.2 Relativistic Mass Point |
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155 | (4) |
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6.3 Relativistic Composition of Three-Vectors |
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159 | (7) |
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6.4 Relative Four-Vectors |
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166 | (8) |
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6.5 The three-Velocity Space |
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174 | (3) |
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177 | (8) |
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185 | (68) |
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185 | (1) |
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7.2 The Four-Acceleration |
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186 | (7) |
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7.3 Calculating Accelerated Motions |
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193 | (4) |
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7.4 Hyperbolic Motion of a Relativistic Mass Particle |
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197 | (7) |
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7.4.1 Geometric Representation of Hyperbolic Motion |
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200 | (4) |
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204 | (1) |
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7.5.1 Einstein Synchronization |
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204 | (1) |
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7.6 Rigid Motion of Many Relativistic Mass Points |
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205 | (1) |
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7.7 Rigid Motion and Hyperbolic Motion |
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206 | (10) |
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7.7.1 The Synchronization of LRIO |
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208 | (1) |
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7.7.2 Synchronization of Chronometry |
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209 | (2) |
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7.7.3 The Kinematics in the LCF Σ |
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211 | (3) |
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7.7.4 The Case of the Gravitational Field |
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214 | (2) |
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7.8 General One-Dimensional Rigid Motion |
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216 | (3) |
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7.8.1 The Case of Hyperbolic Motion |
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217 | (2) |
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7.9 Rotational Rigid Motion |
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219 | (5) |
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7.9.1 The Transitive Property of the Rigid Rotational Motion |
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222 | (2) |
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224 | (15) |
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7.10.1 The Kinematics of Relativistic Observers |
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224 | (1) |
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7.10.2 Chronometry and the Spatial Line Element |
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225 | (3) |
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228 | (1) |
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7.10.4 Definition of the Rotating Disk for a RIO |
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229 | (1) |
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7.10.5 The Locally Relativistic Inertial Observer (LRIO) |
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230 | (5) |
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7.10.6 The Accelerated Observer |
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235 | (4) |
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7.11 The Generalization of Lorentz Transformation and the Accelerated Observers |
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239 | (9) |
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7.11.1 The Generalized Lorentz Transformation |
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240 | (2) |
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7.11.2 The Special Case u0(l1, x1) = u1(l1, x1) = u(x1) |
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242 | (5) |
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7.11.3 Equation of Motion in a Gravitational Field |
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247 | (1) |
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7.12 The Limits of Special Relativity |
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248 | (5) |
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7.12.1 Experiment 1: The Gravitational Redshift |
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249 | (2) |
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7.12.2 Experiment 2: The Gravitational Time Dilation |
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251 | (1) |
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7.12.3 Experiment 3: The Curvature of Spacetime |
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252 | (1) |
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253 | (12) |
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253 | (1) |
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254 | (11) |
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265 | (18) |
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265 | (1) |
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9.2 The (Relativistic) Mass |
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266 | (1) |
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9.3 The Four-Momentum of a ReMaP |
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267 | (8) |
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9.4 The Four-Momentum of Photons (Luxons) |
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275 | (3) |
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9.5 The Four-Momentum of Particles |
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278 | (1) |
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9.6 The System of Natural Units |
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278 | (5) |
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10 Relativistic Reactions |
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283 | (42) |
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283 | (1) |
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10.2 Representation of Particle Reactions |
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284 | (1) |
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10.3 Relativistic Reactions |
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285 | (4) |
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10.3.1 The Sum of Particle Four-Vectors |
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286 | (2) |
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10.3.2 The Relativistic Triangle Inequality |
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288 | (1) |
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10.4 Working with Four-Momenta |
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289 | (2) |
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10.5 Special Coordinate Frames in the Study of Relativistic Collisions |
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291 | (1) |
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10.6 The Generic Reaction A + B → C |
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292 | (12) |
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10.6.1 The Physics of the Generic Reaction |
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293 | (4) |
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10.6.2 Threshold of a Reaction |
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297 | (7) |
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10.7 Transformation of Angles |
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304 | (21) |
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10.7.1 Radiative Transitions |
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308 | (4) |
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10.7.2 Reactions With Two-Photon Final State |
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312 | (5) |
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10.7.3 Elastic Collisions - Scattering |
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317 | (8) |
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325 | (52) |
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325 | (1) |
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325 | (15) |
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11.3 Inertial Four-Force and Four-Potential |
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340 | (3) |
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11.3.1 The Vector Four-Potential |
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342 | (1) |
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11.4 The Lagrangian Formalism for Inertial Four-Forces |
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343 | (7) |
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11.5 Motion in a Central Potential |
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350 | (5) |
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355 | (8) |
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11.7 The Frenet-Serret Frame in Minkowski Space |
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363 | (14) |
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11.7.1 The Physical Basis |
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368 | (4) |
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11.7.2 The Generic Inertial Four-Force |
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372 | (5) |
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12 Irreducible Decompositions |
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377 | (18) |
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377 | (2) |
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12.1.1 Writing a Tensor of Valence (0, 2) as a Matrix |
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378 | (1) |
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12.2 The Irreducible Decomposition wrt a Non-null Vector |
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379 | (10) |
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12.2.1 Decomposition in a Euclidean Space En |
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379 | (4) |
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12.2.2 1 + 3 Decomposition in Minkowski Space |
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383 | (6) |
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12.3 1 + 1 + 2 Decomposition wrt a Pair of Timelike Vectors |
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389 | (6) |
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13 The Electromagnetic Field |
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395 | (100) |
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395 | (1) |
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13.2 Maxwell Equations in Newtonian Physics |
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396 | (3) |
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13.3 The Electromagnetic Potential |
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399 | (6) |
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13.4 The Equation of Continuity |
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405 | (7) |
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13.5 The Electromagnetic Four-Potential |
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412 | (3) |
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13.6 The Electromagnetic Field Tensor Fij |
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415 | (6) |
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13.6.1 The Transformation of the Fields |
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415 | (2) |
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13.6.2 Maxwell Equations in Terms of Fij |
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417 | (1) |
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13.6.3 The Invariants of the Electromagnetic Field |
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418 | (3) |
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13.7 The Physical Significance of the Electromagnetic Invariants |
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421 | (5) |
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422 | (1) |
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423 | (3) |
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13.8 Motion of a Charge in an Electromagnetic Field - The Lorentz Force |
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426 | (3) |
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13.9 Motion of a Charge in a Homogeneous Electromagnetic Field |
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429 | (11) |
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13.9.1 The Case of a Homogeneous Electric Field |
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430 | (4) |
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13.9.2 The Case of a Homogeneous Magnetic Field |
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434 | (2) |
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13.9.3 The Case of Two Homogeneous Fields of Equal Strength and Perpendicular Directions |
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436 | (2) |
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13.9.4 The Case of Homogeneous and Parallel Fields E || B |
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438 | (2) |
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13.10 The Relativistic Electric and Magnetic Fields |
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440 | (14) |
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13.10.1 The Levi-Civita Tensor Density |
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440 | (2) |
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13.10.2 The Case of Vacuum |
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442 | (3) |
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13.10.3 The Electromagnetic Theory for a General Medium |
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445 | (3) |
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13.10.4 The Electric and Magnetic Moments |
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448 | (1) |
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13.10.5 Maxwell Equations for a General Medium |
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448 | (1) |
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13.10.6 The 1 + 3 Decomposition of Maxwell Equations |
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449 | (5) |
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13.11 The Four-Current of Conductivity and Ohm's Law |
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454 | (5) |
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13.11.1 The Continuity Equation Jaa = 0 for an Isotropic Material |
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458 | (1) |
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13.12 The Electromagnetic Field in a Homogeneous and Isotropic Medium |
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459 | (4) |
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13.13 Electric Conductivity and the Propagation Equation for Ea |
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463 | (2) |
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13.14 The Generalized Ohm's Law |
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465 | (2) |
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13.15 The Energy Momentum Tensor of the Electromagnetic Field |
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467 | (8) |
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13.16 The Electromagnetic Field of a Moving Charge |
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475 | (14) |
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477 | (1) |
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13.16.2 The Fields E1, B1 |
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478 | (1) |
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13.16.3 The Lienard-Wiechert Potentials and the Fields E, B |
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478 | (11) |
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13.17 Special Relativity and Practical Applications |
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489 | (3) |
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13.18 The Systems of Units SI and Gauss in Electromagnetism |
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492 | (3) |
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14 Relativistic Angular Momentum |
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495 | (26) |
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495 | (1) |
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14.2 Mathematical Preliminaries |
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495 | (3) |
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14.2.1 1 + 3 Decomposition of a Bivector Xab |
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495 | (3) |
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14.3 The Derivative of Xab Along the Vector pa |
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498 | (2) |
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14.4 The Angular Momentum in Special Relativity |
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500 | (6) |
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14.4.1 The Angular Momentum in Newtonian Theory |
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500 | (2) |
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14.4.2 The Angular Momentum of a Particle in Special Relativity |
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502 | (4) |
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14.5 The Intrinsic Angular Momentum - The Spin Vector |
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506 | (15) |
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14.5.1 The Magnetic Dipole |
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506 | (4) |
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14.5.2 The Relativistic Spin |
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510 | (5) |
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14.5.3 Motion of a Particle with Spin in a Homogeneous Electromagnetic Field |
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515 | (2) |
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14.5.4 Transformation of Motion in σ |
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517 | (4) |
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15 The Covariant Lorentz Transformation |
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521 | (34) |
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521 | (2) |
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15.2 The Covariant Lorentz Transformation |
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523 | (11) |
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15.2.1 Definition of the Lorentz Transformation |
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523 | (1) |
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15.2.2 Computation of the Covariant Lorentz Transformation |
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524 | (5) |
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15.2.3 The Action of the Covariant Lorentz Transformation on the Coordinates |
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529 | (5) |
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15.2.4 The Invariant Length of a Four-Vector |
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534 | (1) |
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15.3 The Four Types of the Lorentz Transformation Viewed as Spacetime Reflections |
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534 | (3) |
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15.4 Relativistic Composition Rule of Four-Vectors |
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537 | (13) |
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15.4.1 Computation of the Composite Four-Vector |
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540 | (2) |
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15.4.2 The Relativistic Composition Rule for Three-Velocities |
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542 | (2) |
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15.4.3 Riemannian Geometry and Special Relativity |
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544 | (4) |
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15.4.4 The Relativistic Rule for the Composition of Three-Accelerations |
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548 | (2) |
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15.5 The Composition of Lorentz Transformations |
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550 | (5) |
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16 Geometric Description of Relativistic Interactions |
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555 | (34) |
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16.1 Collisions and Geometry |
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555 | (1) |
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16.2 Geometric Description of Collisions in Newtonian Physics |
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556 | (2) |
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16.3 Geometric Description of Relativistic Reactions |
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558 | (1) |
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16.4 The General Geometric Results |
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559 | (4) |
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16.4.1 The 1 + 3 Decomposition of a Particle Four-Vector wrt a Timelike Four-Vector |
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561 | (2) |
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16.5 The System of Two to One Particle Four-Vectors |
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563 | (11) |
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16.5.1 The Triangle Function of a System of Two Particle Four-Vectors |
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565 | (2) |
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16.5.2 Extreme Values of the Four-Vectors (A ± B) 2 |
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567 | (1) |
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16.5.3 The System Aa, Ba, (A + Ba) of Particle Four-Vectors in CS |
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568 | (2) |
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16.5.4 The System Aa, Ba, (A + Ba) in the Lab |
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570 | (4) |
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16.6 The Relativistic System Aa + Ba → Ca + Da |
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574 | (15) |
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16.6.1 The Reaction B → C + D |
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587 | (2) |
Bibliography |
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589 | (2) |
Index |
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591 | |