# 我今年写的一本书 A Concise Introduction to General Relativity

This book introduces the basics of general relativity. In terms of content arrangement, the International System of Units (SI) is used to derive all formulas. In addition, this book also attempts to use low-dimensional Riemannian geometric forms to promote the understanding of curved space-time. A more detailed introduction to Einstein summation convention is also one of the features of this book. This will help readers better understand the calculation and derivation process of tensors. In each chapter, there is a detailed description of the learning goals, along with exercises and inquiry topics. In this way, readers can deepen their understanding of the key and difficult issues in these chapters, and closely integrate theory and practice. To read this book, readers only need to have knowledge of college advanced mathematics and general physics.

# Preface

Beginners always find it difficult to learn general relativity because the knowledge of Riemannian geometry and tensor analysis methods used in it are very different from the content learned in college physics and advanced mathematics. Especially after using Einstein summation convention, most of the textbooks did not explain it well, leading to many confusing problems in the understanding of formula derivation. The use of the natural system of units has caused many readers to be unable to verify the formula derived from the general theory of relativity with specific observational data.

In order to solve these problems, the writing of this book has made some new attempts.

First, the entire book adopts the International System of Units, which ensures that the knowledge learned in general relativity can be effectively linked with college physics knowledge. If you need to understand the results of academic papers, you can easily convert the formula of the International System of Units into the natural system of units. The reverse is not so easy.

Second, in order to understand abstract Riemannian geometry knowledge, this book uses low-dimensional space-time coordinates as much as possible for direct mathematical derivation, so that readers can obtain more intuitive physical images. On the basis of understanding the low-dimensional (two-dimensional, three-dimensional) space-time, and then transition to the use of tensor operations to solve the four-dimensional space-time problems, the whole process can maintain a smooth connection, and let the specific physical image always run through the tensor calculation process.

Third, considering that the general theory of relativity is currently able to deal with spherical symmetry problems, this book only focuses on polar coordinates, cylindrical coordinates, and spherical-cylindrical coordinate. Repeated formula derivation is believed to help readers understand this coordinate system. The metric tensor is firmly stored in the knowledge system of the brain.

Fourth, considering that Einstein summation convention occupies a very important position in the entire theory of general relativity, in the third chapter of this book, a section is dedicated to the introduction, and it is carried out in the exercises at the end of some chapters.

Most of the formulas in this book have a detailed derivation process, and only need basic knowledge of college advanced mathematics to understand. Each chapter also provides an inquiry topic. There are no standard answers to these topics. Readers are required to search for relevant information through the Internet to explore and discover.

At the beginning of the writing of this book, ten chapters were originally planned. During the writing of the first draft, I felt that the content should be as concise and clear as possible, so I compressed some of the content and chapters, and finally formed the current six chapters. The thickness is also greatly reduced. I believe this can also make it easier for beginners.

The first chapter of this book gives a brief review of Newtonian mechanics, electrodynamics, and relativity, hoping to establish a connection between these mechanics’ knowledge.

The second chapter introduces the mathematical knowledge of Riemannian geometry. In order to be able to use basic mathematical knowledge to understand Christoffel symbols, Riemann curvature, etc., this chapter gives a “two-dimensional space plus one-dimensional time” space-time. So that the use of cylindrical coordinates can deal with complex Riemannian geometric problems. After obtaining a general conclusion, a smooth transition to the spherical-cylindrical coordinate system of “three-dimensional space plus one-dimensional time”. This makes the relationship between vectors, matrices and tensors very clear.

The third chapter involves Einstein's field equations, and approximate calculations for weak fields are done to compare with Newton's equations of motion. It also introduces some important rules of Einstein summation convention.

The fourth chapter analyzes the spherically symmetric gravitational field and obtains the Schwarzschild solution. And conducted a more in-depth analysis of black holes, involving Hawking radiation and Penrose process. This chapter also introduces topics such as taking photos of black holes.

The fifth chapter obtains the gravitational wave solution and analyzes the nature of the gravitational wave. This chapter also naturally involves the measurement of gravitational waves.

The sixth chapter involves some topics in cosmology. In terms of content arrangement, important evidences of general relativity such as Mercury's perihelion precession, light bending, and gravitational redshift are also included in this chapter. This is also a new attempt. Including the universe model, Robertson-Walker metric and Friedmann equations are also introduced in this chapter. This chapter will also cover topics such as Hubble's law, gravitational lensing, and dark matter.

In order to better grasp some of the key and difficult issues in general relativity, the beginning of each chapter of this book has learning goals. At the end of each chapter, a certain number of exercises are provided so that they can be used to consolidate the content learned and to check whether there are still omissions in the process of understanding the content. Another feature of this book is that most of the reference books and materials involved in the related content can be found on the Internet, which brings great convenience for readers to learn the knowledge of general relativity.

Zhi Cheng

March 2021. Guangzhou, China.

# Contents

1 Introduction to the theory of relativity1

1.1 Mechanics theories1

1.1.1 Newtonian mechanics1

1.1.2 Electrodynamics1

1.1.3 Special theory of relativity2

1.1.4 General relativity3

1.2 Minkowski spacetime4

1.2.1 Minkowski coordinates4

1.2.2 Time-like space-time and space-like space-time5

1.3 Exercises6

Inquiry topic: General Relativity and Quantum Mechanics6

2 Riemannian geometry8

2.1 The distance between two points8

2.1.1 The distance between two points on rectangular coordinates8

2.1.2 Use dual coordinates to represent the distance between two points9

2.2 Metric in cylindrical coordinates12

2.2.1 Cylindrical coordinate with time axis12

2.2.2 Curved space-time15

2.3 Tensor and curvature16

2.3.1 From matrix to tensor16

2.3.2 The shortest path16

2.3.3 Covariant differential20

2.3.4 Riemann curvature22

2.4 Spherical-cylindrical coordinate27

2.4.1 Four-dimensional spherical-cylindrical coordinate27

2.4.2 Representation of spherical-cylindrical coordinate27

2.4.3 Geodesic equations under spherical-cylindrical coordinate28

2.4.4 Riemann curvature in spherical-cylindrical coordinate system32

2.5 Exercises40

Inquiry topic: Riemann curvature and curved space-time41

3 Einstein Field Equation42

3.1 Einstein's theory42

3.1.1 Einstein summation convention42

3.1.2 Curved Space-time and Einstein Field Equation44

3.2 Weak gravitational field48

3.2.1 Flat space-time48

3.2.2 Newton's law of motion and space-time curvature49

3.2.3 Newton's approximate solution of Einstein's field equation53

3.3 Exercises56

Inquiry topic: General relativity and Newton's laws of motion57

4 Schwarzschild solution and black hole59

4.1 Solution of Spherically Symmetric Gravitational Field59

4.1.1 Spherically Symmetric Gravitational Field59

4.1.2 Schwarzschild solution62

4.1.3 Schwarzschild metric67

4.2 Black hole solution and motion in the gravitational field68

4.2.1 Black hole solution68

4.2.2 The motion in a spherically symmetric gravitational field69

4.2.3 Comparison the relativistic and Newton's equation of motion75

4.3 The horizon of black holes and Schwarzschild space-time76

4.3.1 Space-time boundary of black hole76

4.3.2 Schwarzschild space-time78

4.4 The radiation of black hole82

4.4.1 The entropy of black hole82

4.5 Types of black holes and Penrose process86

4.5.1 Types of black holes86

4.5.2 Penrose Process87

4.6 Photos of black hole91

4.6.1 Analyze the important characteristics of black hole photos91

4.6.2 How to take photos of black holes92

4.7 Exercises94

Inquiry topic: Explore the black hole photos96

5 Gravitational waves97

5.1 Gravitational wave solution97

5.1.1 Wave equation of gravitational waves97

5.1.2 Lorentz gauge of gravitational potential99

5.2 Gravitational Waves properties104

5.2.1 Plane wave solutions of gravitational waves104

5.2.2 The energy of gravitational waves106

5.3 Measurement of gravitational waves108

5.3.1 The principle of gravitational waves measurement108

5.3.2 Gravitational wave measurement method109

5.3.3 LIGO gravitational wave measuring device110

5.4 Exercises111

Inquiry topic: Explore the space gravitational wave detection devices113

6 Cosmology114

6.1 Some important evidences of general relativity114

6.1.1 Mercury perihelion precession114

6.1.2 The bending of light117

6.1.3 Gravitational redshift120

6.2 Space-time model of the universe123

6.2.1 Robertson-Walker metric123

6.2.2 Units and their conversion in cosmology126

6.2.3 Natural System of Units128

6.3 The expansion of the universe129

6.3.1 Friedmann equations129

6.3.2 Changes in the curvature of the universe130

6.3.3 Redshift of galaxies132

6.4.1 Big Bang Universe Model133

6.5 The cosmological constant and the gravitational lens135

6.5.1 Cosmological Constant135

6.5.2 Gravitational lens135

6.6 Dark matter136

6.6.1 Evidence for the existence of dark matter136

6.6.2 The composition of dark matter138

6.7 Exercises139

Inquiry topic: Explore dark matter and galaxy rotation curve140

References141

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