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Contents
Part I The Essentials
1 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 Acceleration
5.2 The Geodesic Equation
5.3 What Happens to the Dung Beetle?
5.4 Albert Versus Isaac: Differences Emerge
5.5 Albert Versus Isaac: Seeing the Light
5.6 A Victory for Einstein
5.7 Time Dilation: Hafele-Keating and GPS
5.8 Geodesic Summary
5.9 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 Tensor
6.5 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 Tensor
9.3.2 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 Constant
10.4 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 Curvature
12 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 Horizon
13.11 Summary
14 Orbits and Conserved Quantities
14.1 Noether, Killing Vectors and Conservation Laws
14.2 Conserved Quantities Along Geodesics
14.3 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 Sphere
14.8 Quasars
14.9 Summary
15 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 EF
16.3.1 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 Coordinates
17.2 Why Angular Momentum Matters
17.3 An Oblate Spheroid
17.4 BL Radial Coordinate Mathematics (Optional)
17.5 Minkowski Spacetime Using the BL Radial Coordinate
17.6 The Other BL Coordinates
17.7 Summary
18 Kerr Black Holes
18.1 The Outer Event Horizon
18.2 The Inner Event Horizon
18.3 Underlying Riemann Curvature
18.4 The Kerr Singularity (Ringularity)
18.5 Frame-Dragging
18.6 The Ergosphere
18.7 Penrose, Blandford-Znajek and Quasars (Revisited)
18.8 Extremal Black Holes and Cosmic Censorship
18.9 Conserved Quantities and Contorted Orbits
18.10 Maximal Extension of the Kerr Metric
18.11 Summary
18.12 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 Radiation
19.7.1 Taking the Temperature of a Black Hole
19.8 Summary
20 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 Cosmology
21 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 Universe
23.4 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 Inflation
25.1.1 The Flatness Problem
25.1.2 Where are the Magnetic Monopoles?
25.1.3 The Horizon/Homogeneity Problem
25.2 Introduction to Inflation
25.3 The Inflaton Field
25.4 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 CMB
25.8 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 Density
27 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
