A general relativity workbook:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Mill Valley, California
University Science Books
[2013]
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xx, 476 Seiten Illustrationen, Diagramme |
| ISBN: | 9781891389825 |
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| 245 | 1 | 0 | |a A general relativity workbook |c Thomas A. Moore, Pomona College |
| 264 | 1 | |a Mill Valley, California |b University Science Books |c [2013] | |
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Datensatz im Suchindex
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| adam_text | Titel: A general relativity workbook
Autor: Moore, Thomas Andrew
Jahr: 2013
CONTENTS
Preface jcv
1. INTRODUCTION 1
Concept Summary 2
Homework Problems 9
General Relativity in a Nutshell 11
2. REVIEW OF SPECIAL RELATIVITY 13
Concept Summary 14
Box 2.1 Overlapping IRFs Move with Constant Relative Velocities 19
Box 2.2 Unit Conversions Between SI and GR Units 20
Box 2.3 One Derivation of the Lorentz Transformation 21
Box 2.4 Lorentz Transformations and Rotations 25
Box 2.5 Frame-Independence of the Spacetime Interval 26
Box 2.6 Frame-Dependence of the Time Order of Events 26
Box 2.7 Proper Time Along a Path 27
Box 2.8 Length Contraction 27
Box 2.9 The Einstein Velocity Transformation 28
Homework Problems 29
3. FOUR-VECTORS 31
Concept Summary 32
Box 3.1 The Frame-Independence of the Scalar Product 36
Box 3.2 The Invariant Magnitude of the Four-Velocity 36
Box 3.3 The Low-Velocity Limit of u 37
Box 3.4 Conservation of Momentum or Four-momentum? 38
Box 3.5 Example: The GZK Cosmic-Ray Energy Cutoff 40
Homework Problems 42
4. INDEX NOTATION 43
Concept Summary 44
Box 4.1 Behavior of the Kronecker Delta 48
Box 4.2 EM Field Units in the GR Unit System 48
Box 4.3 Electromagnetic Equations in Index Notation 49
Box 4.4 Identifying Free and Bound Indices 50
Box 4.5 Rule Violations 50
Box 4.6 Example Derivations 51
Homework Problems 52
VII
viii CONTENTS
5. ARBITRARY COORDINATES 53
Concept Summary 54
Box 5.1 The Polar Coordinate Basis 58
Box 5.2 Proof of the Metrie Transformation Law 59
Box 5.3 A 2D Example: Parabolic Coordinates 60
Box 5.4 The LTEs as an Example General Transformation 62
Box 5.5 The Metrie Transformation Law in Fiat Space 62
Box 5.6 A Metrie for a Sphere 63
Homework Problems 63
6. TENSOR EQUATIONS 65
Concept Summary 66
Box 6.1 Example Gradient Covectors 70
Box 6.2 Lowering Indices 71
Box 6.3 The Inverse Metrie 72
Box 6.4 The Kronecker Delta Is a Tensor 73
Box 6.5 Tensor Operations 73
Homework Problems 75
7. MAXWELL S EQUATIONS 77
Concept Summary 78
Box 7.1 Gauss s Law in Integral and Differential Form 82
Box 7.2 The Derivative ofm2 83
Box 7.3 Raising and Lowering Indices in Cartesian Coordinates 83
Box 7.4 The Tensor Equation for Conservation of Charge 84
Box 7.5 The Antisymmetry of F Implies Charge Conservation 85
Box 7.6 The Magnetic Potential 86
Box 7.7 Proof of the Source-Free Maxwell Equations 87
Homework Problems 88
8. GEODESICS 89
Concept Summary 90
Box 8.1 The Worldline of Longest Proper Time in Fiat Spacetime 93
Box 8.2 Derivation of the Euler-Lagrange Equation 94
Box 8.3 Deriving the Second Form of the Geodesic Equation 95
Box 8.4 Geodesics for Fiat Space in Parabolic Coordinates 96
Box 8.5 Geodesics for the Surface of a Sphere 98
Box 8.6 The Geodesic Equation Does Not Determine the Scale of T 100
Box 8.7 Light Geodesics in Fiat Spacetime 101
Homework Problems 102
9. THE SCHWARZSCHILD METRIC 105
Concept Summary 106
Box 9.1 Radial Distance HO
Box 9.2 Falling from Rest in Schwarzschild Spacetime 111
Box 9.3 GM for the Earth and the Sun 112
Box 9.4 The Gravitational Redshift for Weak Fields 112
Homework Problems 114
CONTENTS ix
10. PARTICLE ORBITS 115
Concept Summary 116
Box 10.1 Schwarzschild Orbits Must Be Planar 120
Box 10.2 The Schwarzschild Conservation of Energy Equation 121
Box 10.3 Deriving Conservation of Newtonian Energy for Orbits 722
Box 10.4 The Radii of Circular Orbits 122
Box 10.5 Kepler s Third Law 124
Box 10.6 The Innermost Stable Circular Orbit (ISCO) 125
Box 10.7 The Energy Radiated by an Inspiraling Particle 726
Homework Problems 727
11. PRECESSION OF THE PERIHELION 129
Concept Summary 130
Box 11.1 Verifying the Orbital Equation for u( b) 135
Box 11.2 Verifying the Newtonian Orbital Equation 135
Box 11.3 Verifying the Equation for the Orbital Wobble 136
Box 11.4 Application to Mercury 136
Box 11.5 Constructing the Schwarzschild Embedding Diagram 757
Box 11.6 Calculating the Wedge Angle 8 138
Box 11.7 A Computer Model for Schwarzschild Orbits 138
Homework Problems 141
12. PHOTON ORBITS 143
Concept Summary 144
Box 12.1 The Meaning of the Impact Parameter b 148
Box 12.2 Derivation of the Equation of Motion for a Photon 148
Box 12.3 Features of the Effective Potential Energy Function for Light 149
Box 12.4 Photon Motion in Fiat Space 149
Box 12.5 Evaluating 4-Vector Components in an Observer s Frame 750
Box 12.6 An Orthonormal Basis in Schwarzschild Coordinates 750
Box 12.7 Derivation of the Critical Angle for Photon Emission 757
Homework Problems 752
13. DEFLECTION OF LIGHT 153
Concept Summary 154
Box 13.1 Checking Equation 13.2 759
Box 13.2 The Differential Equation for the Shape of a Photon Orbit 760
Box 13.3 The Differential Equation for the Photon Wobble 760
Box 13.4 The Solution for u((b) in the Large-r Limit 767
Box 13.5 The Maximum Angle of Light Deflection by the Sun 767
Box 13.6 The Lens Equation 762
Box 13.7 The Ratio of Image Brightness to the Source Brightness 765
Homework Problems 764
14. EVENT HORIZON 167
Concept Summary 765
Box 14.1 Finite Distance to r = 2GM 172
Box 14.2 ProperTimeforFreeFallfromr = 7?tor = 0 174
CONTENTS
Box 14.3 The Future Is Finite Inside the Event Horizon 775
Homework Problems 776
15. ALTERNATIVE COORDINATES 179
Concept Summary 780
Box 15.1 Calculating dt/dr 184
Box 15.2 The Global Rain Metrie 785
Box 15.3 The Limits on drldt Inside the Event Horizon 785
Box 15.4 Transforming to Kruskal-Szekeres Coordinates 186
Homework Problems 188
16. BLACK HOLE THERMODYNAMICS 189
Concept Summary 790
Box 16.1 Free-Fall Time to the Event Horizon from r = 2GM + e 194
Box 16.2 Calculating Ex 195
Box 16.3 Evaluating kB, ft, and Tfor a Solar-Mass Black Hole 796
Box 16.4 Lifetime of a Black Hole 797
Homework Problems 198
17. THE ABSOLUTE GRADIENT 199
Concept Summary 200
Box 17.1 Absolute Gradient of a Vector 204
Box 17.2 Absolute Gradient of a Covector 204
Box 17.3 Symmetry of the Christoffel Symbols 205
Box 17.4 The Christoffel Symbols in Terms of the Metrie 205
Box 17.5 Checking the Geodesic Equation 206
Box 17.6 A Trick for Calculating Christoffel Symbols 206
Box 17.7 The Local Flatness Theorem 207
Homework Problems 270
18. GEODESIC DEVIATION 211
Concept Summary 272
Box 18.1 Newtonian Tidal Deviation Near a Spherical Object 276
Box 18.2 Proving Equation 18.9 277
Box 18.3 The Absolute Derivative of f» 277
Box 18.4 Proving Equation 18.14 275
Box 18.5 An Example of Calculating the Riemann Tensor 278
Homework Problems 220
19. THE RIEMANN TENSOR 221
Concept Summary 222
Box 19.1 The Riemann Tensor in a Locally Inertial Frame 224
Box 19.2 Symmetries of the Riemann Tensor 225
Box 19.3 Counting the Riemann Tensor s Independent Components 226
Box 19.4 The Bianchi Identity 227
Box 19.5 The Ricci Tensor Is Symmetrie 228
Box 19.6 The Riemann and Ricci Tensors and R for a Sphere 228
Homework Problems 230
CONTENTS xi
20. THE STRESS-ENERGY TENSOR 231
Concept Summary 232
Box 20.1 Why the Source of Gravity Must Be Energy, Not Mass 236
Box 20.2 Interpretation of V in a Locally Inertial Frame 236
Box 20.3 The Stress-Energy Tensor for a Perfect Fluid in Its Rest LIF 237
Box 20.4 Equation 20.16 Reduces to Equation 20.15 239
Box 20.5 Fluid Dynamics from Conservation of Four-Momentum 239
Homework Problems 247
21. THE EINSTEIN EQUATION 243
Concept Summary 244
Box 21.1 The Divergence of the Ricci Tensor 248
Box 21.2 Finding the Value ofb 249
Box 21.3 Showing that -7? + 4A = KT 250
Homework Problems 257
22. INTERPRETING THE EQUATION 253
Concept Summary 254
Box 22.1 Conservation of Four-Momentum Implies 0 = Vv(p0M ) 258
Box 22.2 The Inverse Metrie in the Weak-Field Limit 258
Box 22.3 The Riemann Tensor in the Weak-Field Limit 259
Box 22.4 The Ricci Tensor in the Weak-Field Limit 260
Box 22.5 The Stress-Energy Sources of the Metrie Perturbation 267
Box 22.6 The Geodesic Equation for a Slow Particle in a Weak Field 262
Homework Problems 263
23. THE SCHWARZSCHILD SOLUTION 265
Concept Summary 266
Box 23.1 Diagonalizing the Spherically Symmetrie Metrie 270
Box 23.2 The Components ofthe Ricci Tensor 277
Box 23.3 Solving for B 274
Box 23.4 Solving for a(r) 275
Box 23.5 The Christoffel Symbols with t-t as Subscripts 275
Homework Problems 276
24. THE UNIVERSE OBSERVED 279
Concept Summary 280
Box 24.1 Measuring Astronomical Distances in the Solar System 284
Box 24.2 Determining the Distance to Stellar Clusters 286
Box 24.3 How the Doppler Shift Is Connected to Radial Speed 287
Box 24.4 Values of the Hubble Constant 288
Box 24.5 Every Point Is the Expansion s Center 288
Box 24.6 The Evidence for Dark Matter 289
Homework Problems 290
25. A METRIC FOR THE COSMOS 293
Concept Summary 294
Box 25.1 The Universal Ricci Tensor 298
xii CONTENTS
Box 25.2 Raising One Index of the Universal Ricci Tensor 298
Box 25.3 The Stress-Energy Tensor with One Index Lowered 298
Box 25.4 The Einstein Equation with One Index Lowered 301
Box 25.5 Verifying the Solutions for q 302
Homework Problems 303
26. EVOLUTION OF THE UNIVERSE 305
Concept Summary 306
Box 26.1 The Other Components of the Einstein Equation 310
Box 26.2 Consequences of Local Energy /Momentum Conservation 577
Box 26.3 Deriving the Density/Scale Relationship for Radiation 572
Box 26.4 Deriving the Friedman Equation 572
Box 26.5 The Friedman Equation for the Present Time 575
Box 26.6 Deriving the Friedman Equation in Terms of the Omegas 575
Box 26.7 The Behavior of a Matter-Dominated Universe 574
Homework Problems 575
27. COSMIC IMPLICATIONS 317
Concept Summary 578
Box 27.1 Connecting the Redshift z to the Hubble Constant 322
Box 27.2 Deriving the Hubble Relation in Terms of Redshift z 322
Box 27.3 The Luminosity Distance 323
Box 27.4 The Differential Equation for a T]) 323
Box 27.5 How to Generate a Numerical Solution for Equation 27.18 324
Homework Problems 325
28. THE EARLY UNIVERSE 327
Concept Summary 328
Box 28.1 Single-Component Universes 332
Box 28.2 The Transition to Matter Dominance 333
Box 28.3 The Time-Temperature Relation 333
Box 28.4 Neutrino Decoupling 335
Box 28.5 The Number Density of Photons 337
Homework Problems 338
29. CMB FLUCTUATIONS AND INFLATION 339
Concept Summary 340
Box 29.1 The Angular Width of the Largest CMB Fluctuations 345
Box 29.2 The Equation for Qk(t) 346
Box 29.3 Cosmic Flatness at the End of Nucleosynthesis 347
Box 29.4 The Exponential Inflation Formula 347
Box 29.5 Inflation Calculations 348
Homework Problems 349
30. GAUGE FREEDOM 351
Concept Summary 352
Box 30.1 The Weak-Field Einstein Equation in Terms of hMV 355
Box 30.2 The Trace-Reverse of hMV 356
CONTENTS xiii
Box 30.3 The Weak-Field Einstein Equation in Terms of HßV 357
Box 30.4 Gauge Transformations of the Metrie Perturbations 358
Box 30.5 A Gauge Transformation Does Not Change RaßMV 359
Box 30.6 Lorentz Gauge 360
Box 30.7 Additional Gauge Freedom 567
Homework Problems 567
31. DETECTING GRAVITATIONAL WAVES 363
Concept Summary 364
Box 31.1 Constraints on Our Trial Solution 368
Box 31.2 The Transformation to Transverse-Traceless Gauge 369
Box 31.3 A Particle at Rest Remains at Rest in TT Coordinates 577
Box 31.4 The Effect of a Gravitational Wave on a Ring of Particles 372
Homework Problems 373
32. GRAVITATIONAL WAVE ENERGY 375
Concept Summary 376
Box 32.1 The Ricci Tensor 379
Box 32.2 The Averaged Curvature Scalar 379
Box 32.3 The General Energy Density of a Gravitational Wave 379
Homework Problems 382
33. GENERATING GRAVITATIONAL WAVES 383
Concept Summary 384
Box 33.1 H,ß for a Compact Source Whose CM is at Rest 388
Box 33.2 A Useful Identity 388
Box 33.3 The Transverse-Traceless Components of A v 390
Box 33.4 How to Find P^ for Waves Moving in the h Direction 597
Box 33.5 Flux in Terms of Pk 393
Box 33.6 Evaluating the Integrals in the Power Calculation 394
Homework Problems 395
34. GRAVITATIONAL WAVE ASTRONOMY 397
Concept Summary 398
Box 34.1 The Dumbbell Pk 402
Box 34.2 The Power Radiated by a Rotating Dumbbell 403
Box 34.3 The Total Energy of an Orbiting Binary Pair 404
Box 34.4 The Time-Rate-of-Change of the Orbital Period 404
Box 34.5 Characteristics offBoötis 405
Homework Problems 406
35. GRAVITOMAGNETISM 407
Concept Summary 408
Box 35.1 The Lorentz Condition for the Potentials 472
Box 35.2 The Maxwell Equations for the Gravitational Field 475
Box 35.3 The Gravitational Lorentz Equation 474
Box 35.4 The Gravitomagnetic Moment of a Spinning Object 474
Box 35.5 Angular Speed of Gyroscope Precession 475
Homework Problems 476
xiv CONTENTS
36. THE KERR METRIC 417
Concept Summary 478
Box 36.1 Expanding IR -r V to First Order in r/R 421
Box 36.2 The Integral for ha 422
Box 36.3 Why the Other Terms in the Expansion Integrate to Zero 423
Box 36.4 Transforming the Weak-Field Solution to Polar Coordinates 424
Box 36.5 The Weak-Field Limit of the Kerr Metrie 425
Homework Problems 426
37. PARTICLE ORBITS IN KERR SPACETIME 427
Concept Summary 428
Box 37.1 Calculating Expressions for dtldx and dcbldr 431
Box 37.2 Verify the Value of{g4f~gngH 432
Box 37.3 The Energy-Conservation-Like Equation of Motion 433
Box 37.4 Kepler s Third Law 434
Box 37.5 The Radii of ISCOs When a = GM 435
Homework Problems 436
38. ERGOREGION AND HORIZON 437
Concept Summary 438
Box 38.1 The Radii Where g = 0 447
Box 38.2 The Angular Speed Range When dr and/or dO * 0 442
Box 38.3 Angular-Speed Limits in the Equatorial Plane 443
Box 38.4 The Metrie of the Event Horizon s Surface 444
Box 38.5 The Area of the Outer Kerr Event Horizon 445
Box 38.6 Transformations Preserve the Metrie Determinant s Sign 445
Homework Problems 447
39. NEGATIVE-ENERGY ORBITS 449
Concept Summary 450
Box 39.1 Quadratic Form for Conservation of Energy 454
Box 39.2 The Square Root Is Zero at the Event Horizon 454
Box 39.3 Negative e Is Possible Only in the Ergoregion 456
Box 39.4 The Fundamental Limit on SM in Terms of SS 457
Box 39.5 5Mir 0 458
Box 39.6 The Spin Energy Contribution to a Black Hole s Mass 45P
Homework Problems 460
Appendix: A Diagonal Metrie Worksheet 463
Index 467
|
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| spelling | Moore, Thomas A. Verfasser (DE-588)1165544628 aut A general relativity workbook Thomas A. Moore, Pomona College Mill Valley, California University Science Books [2013] xx, 476 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier General relativity (Physics) Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Allgemeine Relativitätstheorie (DE-588)4112491-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027373622&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moore, Thomas A. A general relativity workbook General relativity (Physics) Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| subject_GND | (DE-588)4112491-1 (DE-588)4143389-0 |
| title | A general relativity workbook |
| title_auth | A general relativity workbook |
| title_exact_search | A general relativity workbook |
| title_full | A general relativity workbook Thomas A. Moore, Pomona College |
| title_fullStr | A general relativity workbook Thomas A. Moore, Pomona College |
| title_full_unstemmed | A general relativity workbook Thomas A. Moore, Pomona College |
| title_short | A general relativity workbook |
| title_sort | a general relativity workbook |
| topic | General relativity (Physics) Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| topic_facet | General relativity (Physics) Allgemeine Relativitätstheorie Aufgabensammlung |
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