Untangling General Relativity

An easy-to-read introduction to Einstein’s theory of general relativity

In Untangling General Relativity, Simon Sherwood explains the details of general relativity with clarity, enthusiasm, and a sense of fun. Designed to be accessible to non-experts, the book combines intuitive explanations with the essential mathematics needed for a deep understanding of the subject. Sherwood introduces that maths gradually and clearly, in a step-by-step program designed to expand your appreciation and grasp of general relativity.

Untangling General Relativity serves as an effective springboard for more in-depth studies. It lays the groundwork for mastering the advanced topics covered in relativity textbooks and university physics courses.

Readers will find:

• A thorough introduction to general relativity, including the interpretation of gravity as curved spacetime and a full derivation of Einstein’s field equations
• Comprehensive explanations of the spacetime metric, the equivalence principle, the geodesic equation, and the energy-momentum and curvature tensors
• Vacuum curvature: the Schwarzschild and Kerr metrics, black holes, white holes, event horizons, and gravitational waves
• Cosmology: the Friedmann equations, dark matter and energy, the Big Bang, inflation and an overview of current efforts to develop a quantum theory of gravity

Perfect for undergraduate students preparing to take a university-level physics course dealing with general relativity for the first time, Untangling General Relativity will also benefit students of the natural sciences and instructors and educators with a professional or academic interest in the subject.

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

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.