Untangling General Relativity

Simon Sherwood is the author of Quantum Untangling (Wiley, 2024). Previously he was Chairman of Elegant Hotels PLC and the CEO of Orient-Express Hotels. He holds an MBA from Harvard Business School and is also a former strategy consultant with the Boston Consulting Group.

• Part I The Essentials1 Overview 1.1 Einstein Field Equations 1.2 Gravity as Curved Spacetime 1.3 The Equivalence Principle 1.4 Working Out the Details 1.5 Gimme, Gimme, Gimme... Some Hard Evidence 1.6 The Cosmological Constant 1.7 Vacuum Curvature 1.8 Cosmology 1.8.1 The Expanding Universe 1.8.2 An Accelerating Expansion 1.8.3 The Big Picture 1.9 The Field Equations in Full Form 2 Special Relativity 2.1 Relativity 2.2 The Speed of Light is Constant: So What? 2.3 The Invariant Interval Equation 2.4 Time Dilation Quantified 2.5 Length Contraction 2.6 Leading Clocks Lag 2.7 Adding Things Up: An Apparent Paradox 2.8 Energy and Momentum 2.9 Energy, Momentum, Time and Space 2.10 Summary 3 The Metric 3.1 The Minkowski Metric 3.2 Einstein’s Tensor and the Metric 3.3 Distortion in the Metric 3.4 Curvature, Dung Balls and a First Hint of Gravity 3.5 A Mathematical Challenge 3.6 Upper and Lower Indices 3.7 Raising/Lowering Indices With Wonky Metrics (Off-Diagonal Terms) 3.8 Summary 4 Covariant Derivatives and Christoffel Symbols 4.1 Covariant Derivatives 4.2 Christoffel Symbols 4.2.1 What are Christoffel Symbols? 4.2.2 Calculating the Value of Christoffel Symbols 4.3 Summary 5 The Geodesic Equation and Gravity 5.1 A 2-D Model of Time Dilation and Gravitational Acceleration5.2 The Geodesic Equation5.3 What Happens to the Dung Beetle? 5.4 Albert Versus Isaac: Differences Emerge 5.5 Albert Versus Isaac: Seeing the Light5.6 A Victory for Einstein 5.7 Time Dilation: Hafele-Keating and GPS 5.8 Geodesic Summary5.9 Tensors: Why...? What...? How...?5.10 Where’s the Fridge?6 The Equivalence Principle and Ricci Tensor 6.1 The Equivalence Principle 6.1.1 A Planet With a Hole 6.1.2 Light in a Gravitational Field 6.2 From Newton’s Gravity to Geodesic Separation 6.3 The Magnificent Ricci Tensor 6.4 An Intuitive Explanation of the Ricci Tensor6.5 Vacuum Curvature: An Apparent Paradox 6.6 The Ricci Scalar 6.7 Summary 7 The Maths of Curvature 7.1 Parallel Transport 7.2 The Riemann Tensor 7.2.1 Indices of the Riemann Tensor 7.2.2 Calculating Components of the Riemann Tensor 7.2.3 Deriving the Formula for the Riemann Tensor (Optional) 7.3 Calculating the Ricci Tensor 7.4 Calculating the Ricci Scalar 7.5 Example Calculations: Aarrgghh! 7.5.1 Symmetry Shortcut for Diagonal Metrics 7.5.2 Flat Space with Polar Coordinates 7.6 Hunting for Vacuum Solutions 7.7 Summary 8 The Energy-Momentum Tensor 8.1 Tensor Indices 8.2 Introduction to the Energy-Momentum Tensor 8.3 Mass Density Flow of Dust 8.4 Energy-Momentum Tensor of Dust 8.5 Symmetry of the Energy-Momentum Tensor 8.6 Covariant Derivative of the Energy-Momentum Tensor 8.7 Energy-Momentum Tensor of a Perfect Fluid 8.8 Summary 9 Deriving the Einstein Field Equations (EFEs) 9.1 Why Does Energy-Momentum Curve Spacetime? 9.2 Generalising Coordinates 9.3 The Ricci Tensor: Why So Complicated? 9.3.1 Riemann Symmetries Within the Ricci Tensor9.3.2 The Symmetries Complicate Things 9.4 Deriving the Ricci Relationship 9.4.1 Finding the Right Ricci Function: Tensor Contraction (Optional) 9.5 What Does This Tell Us About Spacetime Curvature? 9.5.1 Vacuum 9.5.2 Rest Mass Energy 9.5.3 Relating the Ricci Tensor to Energy-Density 9.6 Curvature Footprints 9.7 Summary 10 Einstein Field Equations: The Full Story 10.1 Einstein’s Weak Field Metric 10.1.1 Refresher: Gravitational Potential 10.1.2 Weak Field Geodesic Equation 10.1.3 Weak Field Ricci Tensor 10.2 Energy-Momentum, Curvature and the Vacuum 10.3 Calculating the Value of Einstein’s Gravitational Constant10.4 The Poisson Equation (Optional Refresher) 10.5 The EFEs in Full Form (Almost!) 10.6 The Cosmological Constant 10.7 Summary 11 Module Summary and Conventions 11.1 Module Summary 11.2 The Field Equations in Full Form (Finally!) 11.3 Why Does Energy-Momentum Distort Spacetime? 11.4 Conventions (Optional) 11.4.1 Conventions (1): Metric Signature 11.4.2 Conventions (2): Definition of Curvature Tensors 11.4.3 Conventions (3): Definition of Energy-Momentum Tensor 11.5 Stationary Action Derivation of the EFEs 11.6 Final Thoughts on This Module 11.7 Module Memory Jogger Part II Vacuum Curvature12 The Schwarzschild Metric: Derivation 12.1 Metric Symmetries: A Diagonal Metric 12.2 Ricci and Riemann Symmetries 12.3 Simon’s Ricci Cheat Sheet 12.4 Deriving the Schwarzschild Metric: Relating Time and Space 12.5 Birkoff’s Theorem 12.6 The Schwarzschild Metric 12.7 Summary 12.8 Why Do We Care? 12.9 Schwarzschild with Cosmological Constant (Optional) 13 Schwarzschild and Black Holes 13.1 Schwarzschild Revisited 13.2 Black Holes: Overview 13.3 Minky and Schwart 13.4 Proper Acceleration 13.5 White Dwarfs and Neutron Stars 13.6 Falling into a Black Hole 13.7 Time: For Minky the Clock Stops 13.8 A Simple Illustrative Model 13.9 Space: Schwarzschild Radial Coordinate 13.10 Inside the Event Horizon13.11 Summary 14 Orbits and Conserved Quantities 14.1 Noether, Killing Vectors and Conservation Laws14.2 Conserved Quantities Along Geodesics14.3 Conserved Quantities of the Schwarzschild Metric 14.4 Radial Plunge 14.4.1 Plunge Time from Horizon to Singularity 14.5 Angular Momentum and Rotational Energy 14.6 A Few Words 14.7 Orbits and Trajectories 14.7.1 Newton’s Circular Orbits 14.7.2 Schwarzschild’s Circular Orbits 14.7.3 Innermost Stable Circular Orbit (ISCO) 14.7.4 The Photon Sphere14.8 Quasars14.9 Summary15 Revisiting Einstein’s Success (Optional) 15.1 The Deflection of Light and Gravitational Lensing 15.2 The Precession of Mercury 15.2.1 Binet’s Equation 15.2.2 Binet with the Schwarzschild Metric 15.2.3 The Missing Precession 15.3 The Aftermath 16 Schwarzschild: Other Coordinates 16.1 Introduction for Dummies 16.2 Eddington-Finkelstein (EF) Coordinates 16.3 Intuitive EF16.3.1 EF Maths Step 1 (Optional) 16.3.2 EF Maths Step 2: Along Comes a Tortoise (Optional) 16.4 Crossing the Black Hole Event Horizon 16.5 White Holes 16.6 Kruskal-Szekeres (KS) Coordinates 16.6.1 KS Maths (Optional) 16.7 The KS Big Picture Schwarzschild Diagram 16.8 Penrose-Carter Diagrams 16.9 Schwarzschild Metric: Final Thoughts 17 Kerr Metric: An Intuitive Introduction 17.1 The Kerr Metric Using BL Coordinates17.2 Why Angular Momentum Matters17.3 An Oblate Spheroid17.4 BL Radial Coordinate Mathematics (Optional)17.5 Minkowski Spacetime Using the BL Radial Coordinate17.6 The Other BL Coordinates17.7 Summary18 Kerr Black Holes 18.1 The Outer Event Horizon18.2 The Inner Event Horizon 18.3 Underlying Riemann Curvature18.4 The Kerr Singularity (Ringularity)18.5 Frame-Dragging18.6 The Ergosphere 18.7 Penrose, Blandford–Znajek and Quasars (Revisited)18.8 Extremal Black Holes and Cosmic Censorship 18.9 Conserved Quantities and Contorted Orbits18.10 Maximal Extension of the Kerr Metric 18.11 Summary18.12 Cosmic Censorship and the Kerr Metric (Optional)18.12.1 Effective Potential of the Kerr Metric (Equatorial)18.12.2 Characterising the MAMO 18.12.3 Tracking the MAMO 19 Gravitational Waves 19.1 Einstein’s Flip-Flop 19.2 The Maths of GW Radiation 19.2.1 Massless Graviton 19.2.2 Transverse Wave 19.2.3 Light Speed GW Radiation in the EFEs 19.2.4 Two Distinct Polarisations 19.3 Tell Me More About Gravitational Waves 19.4 Chirp GW150914: A Case Study 19.5 Chirp GW170817 19.6 Pulsar Timing Arrays 19.7 A Note on Hawking Radiation19.7.1 Taking the Temperature of a Black Hole 19.8 Summary20 Module Summary: Vacuum Curvature 20.1 Schwarzschild Metric 20.2 Kerr Metric 20.3 Gravitational Waves and Hawking Radiation 20.4 Module Memory Jogger Part III Cosmology21 The Friedmann–Robertson–Walker (FRW) Metric 21.1 The Cosmological Principle 21.2 The Hubble Parameter 21.3 The Expanding Universe: A Newtonian View 21.4 General Relativity (GR) View: A Co-Moving Frame 21.5 Introduction to 3-D Spatial Curvature 21.6 Spatial Curvature in the FRW Metric (Optional) 21.7 The FRW Metric 21.8 Ricci Curvature and the Cosmological Principle 21.9 Summary 22 The Friedmann Equations 22.1 FRW Metric: Ricci Calculation 22.1.1 Calculation Step 1 22.1.2 Calculation Step 2 22.1.3 Ricci Tensor Components of FRW Metric 22.2 Deriving the Friedmann Equations 22.3 The Cosmic Rest Frame 22.4 Energy-Density and Expansion 22.4.1 An Intuitive Introduction 22.4.2 A Bit More Rigour (Optional) 22.5 Dominant Relationships 22.6 The Accelerating Effect of Vacuum Energy 22.7 Critical Energy-Density 22.8 Summary 23 Welcome to the Dark Side 23.1 Spatial Curvature: Feeling Flat 23.2 Dark Matter 23.3 Modelling the Universe23.4 Radiation’s Trivial Contribution 23.5 The Cosmic Age Problem: Globular Clusters 23.6 The Accelerating Universe 23.6.1 Type 1a Supernovae 23.6.2 Evidence of Acceleration 23.7 Summary: The Energy Mix of the Universe 23.8 What is Dark (Vacuum) Energy? 24 After the Big Bang 24.1 Dating the Early Universe 24.2 Baryogenesis (Protons and Neutrons Form) 24.3 Nuclear Fusion (Light Atomic Nuclei Form) 24.4 Cosmic Microwave Background (CMB) 24.5 A Star is Born 24.6 Our Place in the Cosmic Web 24.7 Horizons and the Fate of the Universe 24.8 Summary 25 Inflation 25.1 Arguments for Inflation25.1.1 The Flatness Problem25.1.2 Where are the Magnetic Monopoles? 25.1.3 The Horizon/Homogeneity Problem 25.2 Introduction to Inflation 25.3 The Inflaton Field25.4 How Much Inflation Had to Occur?25.5 The Maths Behind the Inflaton Field (Optional) 25.6 Quantum Field Fluctuations 25.7 Evidence for Inflation in the CMB25.8 The Inflationary Multiverse 25.9 Summary 26 Interpreting the CMB (Optional) 26.1 Underlying Causes of CMB Temperature Variation 26.2 The CMB Power Spectrum 26.3 Super-Horizon Anisotropy 26.4 Effect of Baryon Acoustic Oscillations (BAO) 26.5 Peak-1 and Measuring Flatness 26.6 Comparing Peaks: Another Measure of Baryon Density27 Module Summary: Cosmology 27.1 Theory 27.2 Observation 27.2.1 Flatness 27.2.2 Dark Matter 27.2.3 Dark (Vacuum) Energy 27.3 From the Big Bang to Today 27.4 Some Bits That Might Not Fit 27.5 Inflation 27.5.1 The Rationale for Inflation 27.5.2 The Mechanism of Inflation 27.5.3 Evidence for Inflation from the CMB 27.6 Cosmology: Watch the News 27.7 Module Memory Jogger 28 The Big Challenge 28.1 General Relativity Versus Quantum Mechanics 28.2 The Challenge 28.3 String Theory 28.3.1 Gravity in String Theory 28.3.2 Difficulties with String Theory 28.4 Loop Quantum Gravity (LQG) 28.4.1 LQG Space as a Quantum Entity 28.4.2 Difficulties with LQG 28.5 Spacetime is Doomed 28.6 Entropic Gravity 28.7 Postquantum Gravity 28.8 Toodle-Pip! Index