General Relativity and Cosmology for Undergraduates - J. Norbury.pdf

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GENERAL RELATIVITY &
COSMOLOGY
for Undergraduates
Professor John W. Norbury
Physics Department
University of Wisconsin-Milwaukee
P.O. Box 413
Milwaukee, WI 53201
1997
Contents
1 NEWTONIAN COSMOLOGY 5
1.1 Introduction ............................ 5
1.2 Equation of State ......................... 5
1.2.1 Matter ........................... 6
1.2.2 Radiation ......................... 6
1.3 Velocity and Acceleration Equations .............. 7
1.4 Cosmological Constant ...................... 9
1.4.1 Einstein Static Universe ................. 11
2 APPLICATIONS 13
2.1 Conservation laws ........................ 13
2.2 Age of the Universe ....................... 14
2.3 In°ation .............................. 15
2.4 Quantum Cosmology ....................... 16
2.4.1 Derivation of the Schrodinger equation ......... 16
2.4.2 Wheeler-DeWitt equation ................ 17
2.5 Summary ............................. 18
2.6 Problems ............................. 19
2.7 Answers .............................. 20
2.8 Solutions ............................. 21
3 TENSORS 23
3.1 Contravariant and Covariant Vectors .............. 23
3.2 Higher Rank Tensors ....................... 26
3.3 Review of Cartesian Tensors ................... 27
3.4 Metric Tensor ........................... 28
3.4.1 Special Relativity ..................... 30
3.5 Christo®el Symbols ........................ 31
1
2
CONTENTS
3.6 Christo®el Symbols and Metric Tensor ............. 36
3.7 Riemann Curvature Tensor ................... 38
3.8 Summary ............................. 39
3.9 Problems ............................. 40
3.10 Answers .............................. 41
3.11 Solutions ............................. 42
4 ENERGY-MOMENTUM TENSOR 45
4.1 Euler-Lagrange and Hamilton's Equations ........... 45
4.2 Classical Field Theory ...................... 47
4.2.1 Classical Klein-Gordon Field .............. 48
4.3 Principle of Least Action .................... 49
4.4 Energy-Momentum Tensor for Perfect Fluid .......... 49
4.5 Continuity Equation ....................... 51
4.6 Interacting Scalar Field ..................... 51
4.7 Cosmology with the Scalar Field ................ 53
4.7.1 Alternative derivation .................. 55
4.7.2 Limiting solutions .................... 56
4.7.3 Exactly Solvable Model of In°ation ........... 59
4.7.4 Variable Cosmological Constant ............. 61
4.7.5 Cosmological constant and Scalar Fields ........ 63
4.7.6 Clari¯cation ........................ 64
4.7.7 Generic In°ation and Slow-Roll Approximation .... 65
4.7.8 Chaotic In°ation in Slow-Roll Approximation ..... 67
4.7.9 Density Fluctuations ................... 72
4.7.10 Equation of State for Variable Cosmological Constant 73
4.7.11 Quantization ....................... 77
4.8 Problems ............................. 80
5 EINSTEIN FIELD EQUATIONS 83
5.1 Preview of Riemannian Geometry ................ 84
5.1.1 Polar Coordinate ..................... 84
5.1.2 Volumes and Change of Coordinates .......... 85
5.1.3 Di®erential Geometry .................. 88
5.1.4 1-dimesional Curve .................... 89
5.1.5 2-dimensional Surface .................. 92
5.1.6 3-dimensional Hypersurface ............... 96
5.2 Friedmann-Robertson-Walker Metric .............. 99
5.2.1 Christo®el Symbols .................... 101
CONTENTS
3
5.2.2 Ricci Tensor ........................ 102
5.2.3 Riemann Scalar and Einstein Tensor .......... 103
5.2.4 Energy-Momentum Tensor ............... 104
5.2.5 Friedmann Equations .................. 104
5.3 Problems ............................. 105
6 Einstein Field Equations
107
7 Weak Field Limit
109
8 Lagrangian Methods
111
4
CONTENTS

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