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Inhaltsverzeichnis Table of Contents Part I The Essentials 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1 Einstein Field Equations 17 1.2 Gravity as Curved Spacetime 18 1.3 The Equivalence Principle 20 1.4 Working Out the Details 20 1.5 Gimme, Gimme, Gimme... Some Hard Evidence 21 1.6 The Cosmological Constant 22 1.7 Vacuum Curvature 22 1.8 Cosmology 23 1.8.1 The Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.8.2 An Accelerating Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.8.3 The Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.9 The Field Equations in Full Form 25 2 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 Relativity 26 2.2 The Speed of Light is Constant: So What? 27 2.3 The Invariant Interval Equation 28 2.4 Time Dilation Quantified 30 2.5 Length Contraction 31 2.6 Leading Clocks Lag 32 2.7 Adding Things Up: An Apparent Paradox 33 2.8 Energy and Momentum 34 2.9 Energy, Momentum, Time and Space 36 2.10 Summary 37 3 The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1 The Minkowski Metric 38 3.2 Einstein's Tensor and the Metric 40 3.3 Distortion in the Metric 40 3.4 Curvature, Dung Balls and a First Hint of Gravity 43 3.5 A Mathematical Challenge 44 3.6 Upper and Lower Indices 46 3.7 Raising/Lowering Indices With Wonky Metrics (Off-Diagonal Terms) 48 3.8 Summary 50 4 Covariant Derivatives and Christoffel Symbols . . . . . . . . . . . . . . . . . . 53 4.1 Covariant Derivatives 53 4.2 Christoffel Symbols 55 4.2.1 What are Christoffel Symbols? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.2 Calculating the Value of Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Summary 60 5 The Geodesic Equation and Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1 A 2-D Model of Time Dilation and Gravitational Acceleration 62 5.2 The Geodesic Equation 64 5.3 What Happens to the Dung Beetle? 65 5.4 Albert Versus Isaac: Differences Emerge 67 5.5 Albert Versus Isaac: Seeing the Light 68 5.6 A Victory for Einstein 70 5.7 Time Dilation: Hafele-Keating and GPS 70 5.8 Geodesic Summary 71 5.9 Tensors: Why...? What...? How...? 72 5.10 Where's the Fridge? 73 6 The Equivalence Principle and Ricci Tensor . . . . . . . . . . . . . . . . . . . . . 75 6.1 The Equivalence Principle 75 6.1.1 A Planet With a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.1.2 Light in a Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 From Newton's Gravity to Geodesic Separation 78 6.3 The Magnificent Ricci Tensor 80 6.4 An Intuitive Explanation of the Ricci Tensor 81 6.5 Vacuum Curvature: An Apparent Paradox 83 6.6 The Ricci Scalar 84 6.7 Summary 85 7 The Maths of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1 Parallel Transport 87 7.2 The Riemann Tensor 89 7.2.1 Indices of the Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.2.2 Calculating Components of the Riemann Tensor . . . . . . . . . . . . . . . . . . . . . ...
