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This text explains special relativity and the foundations of general relativity theory in terms understandable to undergraduates in mathematics and physics coming to the subject for the first time. Space-time geometry is emphasized throughout, providing a basic understanding of the special relativity effects of time dilation, length contraction, and the relativity of simultaneity. Bondi's k-calculus is introduced as a simple means of calculating the magnitudes of these effects, and this leads to a derivation of the Lorentz Transformation as a way of unifying these results. The invariant interval of flat space-time is compared to that of curved space-times, and the basic properties of simple cosmological models are presented. Appendices enable the advanced student to master the application of four tensors to the relativistic study of energy, momentum and electromagnetism.
| Introduction | 1 | |
| 1. | Space-time diagrams and the foundations of special relativity | 6 |
| 1.1 | The concept of a space-time | 6 |
| 1.2 | Causality and the speed of light | 13 |
| 1.3 | Relative motion in special relativity | 26 |
| 2. | Fundamentals of measurement | 38 |
| 2.1 | Time | 38 |
| 2.2 | Distance | 40 |
| 2.3 | Simultaneity | 44 |
| 2.4 | World maps, world pictures, and radar maps | 48 |
| 3. | Measurements in flat space-times | 53 |
| 3.1 | The Doppler effect | 53 |
| 3.2 | Relative velocity | 62 |
| 3.3 | Simultaneity | 72 |
| 3.4 | Time dilation | 80 |
| 3.5 | Length contraction | 92 |
| 3.6 | The whole package of kinematic effects | 101 |
| 3.7 | Relativistic dynamics | 107 |
| 3.8 | The consistency of physics | 127 |
| 4. | The Lorentz transformation and the invariant interval | 129 |
| 4.1 | The Lorentz transformation | 129 |
| 4.2 | Space-time separation invariants | 147 |
| 4.3 | Some flat-space universes | 172 |
| 5. | Curved space-times | 197 |
| 5.1 | The general concept | 197 |
| 5.2 | Acceleration and gravitation: the principle of equivalence | 200 |
| 5.3 | Freely falling motion and the meaning of geodesics | 207 |
| 5.4 | The metric form and the metric tensor | 213 |
| 5.5 | Light rays | 220 |
| 5.6 | Causality | 227 |
| 5.7 | Parallel propagation along a curve | 229 |
| 6. | Spherical and stellar collapse | 234 |
| 6.1 | The Schwarzschild solution | 234 |
| 6.2 | Spherical collapse to black holes | 244 |
| 7. | Simple cosmological models | 254 |
| 7.1 | Space-time geometry | 254 |
| 7.2 | The evolution of the universe | 261 |
| 7.3 | Observable quantities | 268 |
| 7.4 | The light cone, observational limits, and horizons | 275 |
| 7.5 | Steady state and inflationary universes | 283 |
| 7.6 | Small universes | 285 |
| Afterword | 290 | |
| Appendices | 293 | |
| A. | Line integrals | 293 |
| B. | Four-vectors and relativistic dynamics | 300 |
| C. | Four-tensors, electromagnetism, and energy-momentum conservation | 317 |
| Symbols used | 345 | |
| Index | 346 |
Overview
This text explains special relativity and the foundations of general relativity theory in terms understandable to undergraduates in mathematics and physics coming to the subject for the first time. Space-time geometry is emphasized throughout, providing a basic understanding of the special relativity effects of time dilation, length contraction, and the relativity of simultaneity. Bondi's k-calculus is introduced as a simple means of calculating the magnitudes of these effects, and this leads to a derivation of ...