Spacetime without Reference Frames

Description

Reference frames are not necessary. It is true for Galilean relativistic (nonrelativistic) spacetime and for special relativistic one as well. The absolute, geometric, reference frame free formalism provides deep insight and clear understanding. PART ONE SPACETIME MODELS INTRODUCTION --- 21 1. The principles of covariance and of relativity --- 21 2. Measure lines --- 22 3. Spacetime heuristics --- 24 I. NONRELATIVISTIC SPACETIME MODEL --- 31 1. Fundamentals --- 31 1.1. Absolute time progress --- 31 1.2. The spacetime model --- 31 1.3. Structure of world vectors and covectors --- 34 1.4. The arithmetic spacetime model --- 38 1.5. Classification of physical quantities --- 39 1.6. Comparison of spacetime models --- 42 1.7. The split spacetime model --- 44 1.8. Exercises --- 45 2. World lines --- 47 2.1. History of a masspoint: world line --- 47 2.2. A characterization of world lines --- 49 2.3. Classification of world lines --- 49 2.4. Newtonian equation --- 50 2.5. Exercises --- 52 3. Observers --- 52 3.1. The notion of an observer and its space --- 52 3.2. Classification of observers --- 54 3.3. Reference frames, splitting of spacetime --- 56 3.4. Exercise --- 57 4. Rigid observers --- 57 4.1. Inertial observers --- 57 4.2. Characterization of rigid observers --- 60 4.3. About the spaces of rigid observers --- 65 4.4. Observers with origin --- 67 4.5. Exercises --- 69 5. Some special observers --- 70 5.1. Why the inertial observers are better than the others --- 70 5.2. Uniformly accelerated observer --- 70 5.3. Uniformly rotating observer --- 72 5.4. Exercises --- 75 6. Kinematics --- 77 6.1. The history of a masspoint is observed as a motion --- 77 6.2. Relative velocities --- 78 6.3. Motions relative to a rigid observer --- 81 6.4. Some motions relative to an inertial observer --- 82 6.5. Some motions relative to a uniformly accelerated observer- 84 6.6. Some motions relative to a uniformly rotating observer --- 85 6.7. Exercise --- 86 7. Some kinds of observation --- 86 7.1. Vectors observed by inertial observers --- 86 7.2. Measuring rods --- 88 8. Vector splittings --- 88 8.1. What is a splitting? --- 88 8.2. Splitting of vectors --- 89 8.3. Splitting of covectors --- 91 8.4. Vectors and covectors are split in a different way --- 93 8.5. Splitting of vector fields and covector fields according to inertial observers --- 94 8.6. Splitting of vector fields and covector fields according to rigid observers --- 96 8.7. Exercises --- 97 9. Tensor splittings --- 98 9.1. Splitting of tensors, cotensors, etc. --- 98 9.2. Splitting of antisymmetric tensors --- 99 9.3. Splitting of antisymmetric cotensors --- 100 9.4. Splitting of cotensor fields --- 101 9.5. Exercises --- 104 10. Reference systems --- 105 10.1. The notion of a reference system --- 105 10.2. Galilean reference systems --- 109 10.3. Subscripts and superscripts --- 111 10.4. Reference systems associated with global rigid observers --- 113 10.5. Equivalent reference systems --- 115 10.6. Exercises --- 117 11. Spacetime groups --- 119 11.1. The three-dimensional orthogonal group --- 119 11.2. Exercises --- 124 11.3. The Galilean group --- 125 11.4. The split Galilean group --- 130 11.5. Exercises --- 132 11.6. The Noether group --- 134 11.7. The vectorial Noether group --- 138 11.8. The split Noether group --- 139 11.9. Exercises --- 142 II. SPECIAL RELATIVISTIC SPACETIME MODELS --- 145 1. Fundamentals --- 145 1.1. Absolute light propagation --- 145 1.2. The spacetime model --- 147 1.3. Structure of world vectors and covectors --- 149 1.4. The arithmetic spacetime model --- 155 1.5. Classification of physical quantities --- 157 1.6. Comparison of spacetime models --- 158 1.7. The u-split spacetime model --- 160 1.8. Exercises --- 161 2. World lines --- 163 2.1. History of a masspoint: world line --- 163 2.2. Proper time of world lines --- 166 2.3. World line functions --- 168 2.4. Classification of world lines --- 170 2.5. World horizons --- 173 2.6. Newtonian equation --- 175 2.7. Exercises --- 176 3. Observers and synchronizations --- 177 3.1. The notions of an observer and its space --- 177 3.2. The notions of a synchronization and its time --- 178 3.3. Global inertial observers and their spaces --- 180 3.4. Inertial reference frames --- 181 3.5. Standard inertial frames --- 183 3.6. Standard splitting of spacetime --- 184 3.7. Exercise --- 185 4. Kinematics --- 186 4.1. Motions relative to a standard inertial frame --- 186 4.2. Relative velocities --- 187 4.3. Addition of relative velocities --- 189 4.4. History regained from motion --- 190 4.5. Relative accelerations --- 191 4.6. Some particular motions --- 192 4.7. Standard speed of light --- 193 4.8. Motions relative to a nonstandard inertial reference frame- 195 4.9. Exercises --- 196 5. Some comparison between different spaces and times --- 197 5.1. Physically equal vectors in different spaces --- 197 5.2. How to perceive spaces of other reference systems? --- 198 5.3. Lorentz contraction --- 200 5.4. The tunnel paradox --- 204 5.5. No measuring rods --- 204 5.6. Time dilation --- 205 5.7. The twin paradox --- 206 5.8. Experiments concerning time --- 207 5.9. Exercises --- 209 6. Some special noninertial observers --- 210 6.1. General reference frames --- 210 6.2. Distances in observer spaces --- 211 6.3. A method of finding the observer space --- 213 6.4. Uniformly accelerated observer I --- 214 6.5. Uniformly accelerated observer II --- 218 6.6. Uniformly rotating observer I --- 222 6.7. Uniformly rotating observer II --- 226 6.8. Exercises --- 231 7. Vector splittings --- 234 7.1. Splitting of vectors --- 234 7.2. Splitting of covectors --- 237 7.3. Splitting of vector fields --- 239 7.4. Exercises --- 240 8. Tensor splittings --- 240 8.1. Splitting of tensors --- 240 8.2. Splitting of antisymmetric tensors --- 241 8.3. Splitting of tensor fields --- 243 8.4. Exercises --- 244 9. Reference systems --- 244 9.1. The notion of a reference system --- 244 9.2. Lorentzian reference systems --- 246 9.3. Equivalent reference systems --- 248 9.4. Curve lengths calculated in coordinates --- 250 9.5. Exercises --- 251 10. Spacetime groups --- 253 10.1. The Lorentz group --- 253 10.2. The u-split Lorentz group --- 257 10.3. Exercises --- 259 10.4. The Poincar´e group --- 260 10.5. The vectorial Poincar´e group --- 262 10.6. The u-split Poincar´e group --- 263 10.7. Exercises --- 265 11. Relation between the two types of spacetime models --- 266 III. FUNDAMENTAL NOTIONS OF GENERAL RELATIVISTIC SPACETIME MODELS --- 267 PART TWO MATHEMATICAL TOOLS IV. TENSORIAL OPERATIONS --- 275 0. Identifications --- 275 1. Duality --- 275 2. Coordinatization --- 279 3. Tensor products --- 280 4. Tensor quotients --- 293 5. Tensorial operations and orientation --- 296 V. PSEUDO-EUCLIDEAN VECTOR SPACES --- 299 1. Pseudo-Euclidean vector spaces --- 299 2. Tensors of pseudo-Euclidean vector spaces --- 303 3. Euclidean vector spaces --- 308 4. Minkowskian vector spaces --- 321 VI. AFFINE SPACES --- 333 1. Fundamentals --- 333 2. Affine maps --- 337 3. Differentiation --- 339 4. Submanifolds in affine spaces --- 345 5. Coordinatization --- 352 6. Differential equations --- 361 7. Integration on curves --- 363 VII. LIE GROUPS --- 367 1. Groups of linear bijections --- 367 2. Groups of affine bijections --- 369 3. Lie groups --- 371 4. The Lie algebra of a Lie group --- 376 5. Pseudo-orthogonal groups --- 378 6. Exercises --- 379 SUBJECT INDEX --- 381 LIST OF SYMBOLS --- 385 COMMENTS AND BIBLIOGRAPHY --- 389