(Ebook-Pdf) - Mathematics - Physics - General Relativity And Cosmology For Undergraduates

(Ebook-Pdf) - Mathematics - Physics - General Relativity And Cosmology For Undergraduates

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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
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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. ................ 1 1 2 APPLICATIONS 13 2.1 Conservation laws . ....................... 1 3 2.2 Age of the Universe . ...................... 1 4 2.3 Inflation. ............................. 1 5 2.4 Quantum Cosmology. 1 6 2.4.1 Derivation of the Schr¨ odinger equation. ........ 1 6 2.4.2 Wheeler-DeWitt equation. ............... 1 7 2.5 Summary . ............................ 1 8 2.6 Problems . 1 9 2.7 Answers. 2 0 2.8 Solutions . 2 1 3 TENSORS 23 3.1 Contravariant and Covariant Vectors. 2 3 3.2 Higher Rank Tensors. 2 6 3.3 Review of Cartesian Tensors. .................. 2 7 3.4 Metric Tensor. 2 8 3.4.1 Special Relativity. .................... 3 0 3.5 Christoffel Symbols. 3 1 1
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2 CONTENTS 3.6 Christoffel Symbols and Metric Tensor. ............ 3 6 3.7 Riemann Curvature Tensor . .................. 3 8 3.8 Summary . ............................ 3 9 3.9 Problems . 4 0 3.10 Answers. ............................. 4 1 3.11 Solutions . 4 2 4 ENERGY-MOMENTUM TENSOR 45 4.1 Euler-Lagrange and Hamilton’s Equations. .......... 4 5 4.2 Classical Field Theory. ..................... 4 7 4.2.1 Classical Klein-Gordon Field. ............. 4 8 4.3 Principle of Least Action . ................... 4 9 4.4 Energy-Momentum Tensor for Perfect Fluid. ......... 4 9 4.5 Continuity Equation . ...................... 5 1 4.6 Interacting Scalar Field . .................... 5 1 4.7 Cosmology with the Scalar Field . ............... 5 3 4.7.1 Alternative derivation . ................. 5 5 4.7.2 Limiting solutions . 5 6 4.7.3 Exactly Solvable Model of Inflation. 5 9 4.7.4 Variable Cosmological Constant. 6 1 4.7.5 Cosmological constant and Scalar Fields. ....... 6 3 4.7.6 Clarification. ....................... 6 4 4.7.7 Generic Inflation and Slow-Roll Approximation. ... 6 5 4.7.8 Chaotic Inflation in Slow-Roll Approximation. .... 6 7 4.7.9 Density Fluctuations. 7 2 4.7.10 Equation of State for Variable Cosmological Constant 73 4.7.11 Quantization . 7 7 4.8 8 0 5 EINSTEIN FIELD EQUATIONS 83 5.1 Preview of Riemannian Geometry. 8 4 5.1.1 Polar Coordinate. 8 4 5.1.2 Volumes and Change of Coordinates. 8 5 5.1.3 Differential Geometry . 8 8 5.1.4 1-dimesional Curve. 8 9 5.1.5 2-dimensional Surface . 9 2 5.1.6 3-dimensional Hypersurface. .............. 9 6 5.2 Friedmann-Robertson-Walker Metric. 9 9 5.2.1 Christoffel Symbols. 1 0 1
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CONTENTS 3 5.2.2 Ricci Tensor. ....................... 1 0 2 5.2.3 Riemann Scalar and Einstein Tensor. ......... 1 0 3 5.2.4 Energy-Momentum Tensor . .............. 1 0 4 5.2.5 Friedmann Equations . ................. 1 0 4 5.3 Problems . ............................ 1 0 5 6 Einstein Field Equations 107 7 Weak Field Limit 109 8 Lagrangian Methods 111
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4 CONTENTS
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Chapter 1 NEWTONIAN COSMOLOGY 1.1 Introduction Many of the modern ideas in cosmology can be explained without the need to discuss General Relativity. The present chapter represents an attempt to do this based entirely on Newtonian mechanics. The equations describing the velocity (called the Friedmann equation) and acceleration of the universe are derived from Newtonian mechanics and also the cosmological constant is introduced within a Newtonian framework. The equations of state are also derived in a very simple way. Applications such as conservation laws, the age of the universe and the inflation, radiation and matter dominated epochs are discussed.
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(Ebook-Pdf) - Mathematics - Physics - General Relativity And Cosmology For Undergraduates

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