Relativity Demystified. McGraw-Hill

DAVID McMAHON

Relativity Demystified

A Quick Review of Special Relativity
Fundamentally, our commonsense intuition about howthe universeworks is tied
up in notions about space and time. In 1905, Einstein stunned the physics world
with the special theory of relativity, a theory of space and time that challenges
many of these closely held commonsense assumptions about how the world
works. By accepting that the speed of light in vacuum is the same constant value
for all observers, regardless of their state of motion, we are forced to throw away
basic ideas about the passage of time and the lengths of rigid objects.

This book is about the general theory of relativity, Einstein’s theory of
gravity. Therefore our discussion of special relativity will be a quick overview
of concepts needed to understand the general theory. For a detailed discussion
of special relativity, please see our list of references and suggested reading at
the back of the book.

The theory of special relativity has its origins in a set of paradoxes that were
discovered in the study of electromagnetic phenomena during the nineteenth
century. In 1865, a physicist named James Clerk Maxwell published his famous
set of results we now call Maxwell’s equations. Through theoretical studies
alone, Maxwell discovered that there are electromagnetic waves and that they
travel at one speed—the speed of light c. Let’s take a quick detour to get a
glimpse into the way this idea came about. We will work in SI units.
In careful experimental studies, during the first half of the nineteenth century,

PREFACE

The theory of relativity stands out as one of the greatest achievements in science.
The “special theory”,which did not include gravity,was put forward by Einstein
in 1905 to explain many troubling facts that had arisen in the study of electricity
and magnetism. In particular, his postulate that the speed of light in vacuum is the
same constant seen by all observers forced scientists to throwaway many closely
held commonsense assumptions, such as the absolute nature of the passage of
time. In short, the theory of relativity challenges our notions of what reality is,
and this is one of the reasons why the theory is so interesting.

Einstein published the “general” theory of relativity, which is a theory about
gravity, about a decade later. This theory is far more mathematically daunting,
and perhaps this is why it took Einstein so long to come up with it. This theory is
more fundamental than the special theory of relativity; it is a theory of space and
time itself, and it not only describes, it explains gravity. Gravity is the distortion
of the structure of spacetime as caused by the presence of matter and energy,
while the paths followed by matter and energy (think of bending of passing light
rays by the sun) in spacetime are governed by the structure of spacetime. This
great feedback loop is described by Einstein’s field equations.

This is a book about general relativity. There is no getting around the fact
that general relativity is mathematically challenging, so we cannot hope to
learn the theory without mastering the mathematics. Our hope with this book
is to “demystify” that mathematics so that relativity is easier to learn and more
accessible to a wider audience than ever before. In this book we will not skip
any of the math that relativity requires, but we will present it in what we hope
to be a clear fashion and illustrate how to use it with many explicitly solved
examples. Our goal is to make relativity more accessible to everyone. Therefore
we hope that engineers, chemists, and mathematicians or anyone who has had
basic mathematical training at the college level will find this book useful. And

of course the book is aimed at physicists and astronomers

who want to learn the theory.

The truth is that relativity looks much harder than it is. There is a lot to learn,

but once you get comfortable with the new math and new notation, you will

actually find it a bit easier than many other

technical areas you have studied in the past.

This book is meant to be a self-study guide or a supplement, and not a fullblown

textbook. As a result we may not cover every detail and will not provide

lengthly derivations or detailed physical explanations. Those can be found in any

number of fine textbooks on the market. Our focus here is also in “demystifying”

the mathematical framework of relativity, and so we will not include lengthly

descriptions of physical arguments. At the end of the bookwe provide a listing of

references used in the development of this manuscript, and you can select books

from that list to find the details we are leaving out. In much of the material, we

take the approach in this book of stating theorems and results, and then applying

them in solved problems. Similar problems in the end-of chapter quiz help you

try things out yourself.

So if you are taking a relativity course, you might want to use this book to

help you gain better understanding of your main textbook, or help you to see

how to accomplish certain tasks. If you are interested in self-study, this book

will help you get started in your own mastery of the subject and make it easier

for you to read more advanced books.

While this book is taking a lighter approach than the textbooks in the field,

we are not going to cut corners on using advanced mathematics. The bottom

line is you are going to need some mathematical background to find this book

useful. Calculus is a must, studies of differential equations, vector analysis and

linear algebra are helpful. A background in basic physics is also helpful.

Relativity can be done in different ways using a coordinate-based approach

or differential forms and Cartan’s equations.We much prefer the latter approach

and will use it extensively. Again, it looks intimidating at first because there are

lots of Greek characters and fancy symbols, and it is a new way of doing things.

When doing calculations it does require a bit of attention to detail. But after a

bit of practice, you will find that its not really so hard. So we hope that readers

will invest the effort necessary to master this nice mathematical way of solving

physics problems.

Screenshot

Purchase Now !
Just with Paypal

Product details

Price
File Size	5,353 KB
Pages	370 p
File Type	PDF format
Print Version	0-07-145545-0
Copyright	2006 by The McGraw-Hill Companies, Inc

Contents

Preface xi

CHAPTER 1 A Quick Review of Special Relativity 1

Frame of Reference 5

Clock Synchronization 5

Inertial Frames 6

Galilean Transformations 7

Events 7

The Interval 8

Postulates of Special Relativity 9

Three Basic Physical Implications 13

Light Cones and Spacetime Diagrams 17

Four Vectors 19

Relativistic Mass and Energy 20

Quiz 21

CHAPTER 2 Vectors, One Forms, and the Metric 23

Vectors 23

New Notation 25

Four Vectors 27

The Einstein Summation Convention 28

Tangent Vectors, One Forms, and the

Coordinate Basis 29

Coordinate Transformations 31

For more information about this title, click here

The Metric 32

The Signature of a Metric 36

The Flat Space Metric 37

The Metric as a Tensor 37

Index Raising and Lowering 38

Index Gymnastics 41

The Dot Product 42

Passing Arguments to the Metric 43

Null Vectors 45

The Metric Determinant 45

Quiz 45

CHAPTER 3 More on Tensors 47

Manifolds 47

Parameterized Curves 49

Tangent Vectors and One Forms, Again 50

Tensors as Functions 53

Tensor Operations 54

The Levi-Cevita Tensor 59

Quiz 59

CHAPTER 4 Tensor Calculus 60

Testing Tensor Character 60

The Importance of Tensor Equations 61

The Covariant Derivative 62

The Torsion Tensor 72

The Metric and Christoffel Symbols 72

The Exterior Derivative 79

The Lie Derivative 81

The Absolute Derivative and Geodesics 82

The Riemann Tensor 85

The Ricci Tensor and Ricci Scalar 88

The Weyl Tensor and Conformal Metrics 90

Quiz 91

CHAPTER 5 Cartan’s Structure Equations 93

Introduction 93

Holonomic (Coordinate) Bases 94

Nonholonomic Bases 95

Commutation Coefficients 96

Commutation Coefficients and Basis

One Forms 98

Transforming between Bases 100

A Note on Notation 103

Cartan’s First Structure Equation and the

Ricci Rotation Coefficients 104

Computing Curvature 112

Quiz 120

CHAPTER 6 The Einstein Field Equations 122

Equivalence of Mass in Newtonian Theory 123

Test Particles 126

The Einstein Lift Experiments 126

The Weak Equivalence Principle 130

The Strong Equivalence Principle 130

The Principle of General Covariance 131

Geodesic Deviation 131

The Einstein Equations 136

The Einstein Equations with Cosmological

Constant 138

An Example Solving Einstein’s Equations

in 2 + 1 Dimensions 139

Energy Conditions 152

Quiz 152

CHAPTER 7 The Energy-Momentum Tensor 155

Energy Density 156

Momentum Density and Energy Flux 156

Stress 156

Conservation Equations 157

Dust 158

Perfect Fluids 160

Relativistic Effects on Number Density 163

More Complicated Fluids 164

Quiz 165

CHAPTER 8 Killing Vectors 167

Introduction 167

Derivatives of Killing Vectors 177

Constructing a Conserved Current

with Killing Vectors 178

Quiz 178

CHAPTER 9 Null Tetrads and the Petrov Classification 180

Null Vectors 182

A Null Tetrad 184

Extending the Formalism 190

Physical Interpretation and the Petrov

Classification 193

Quiz 201

CHAPTER 10 The Schwarzschild Solution 203

The Vacuum Equations 204

A Static, Spherically Symmetric Spacetime 204

The Curvature One Forms 206

Solving for the Curvature Tensor 209

The Vacuum Equations 211

The Meaning of the Integration Constant 214

The Schwarzschild Metric 215

The Time Coordinate 215

The Schwarzschild Radius 215

Geodesics in the Schwarzschild Spacetime 216

Particle Orbits in the Schwarzschild

Spacetime 218

The Deflection of Light Rays 224

Time Delay 229

Quiz 230

CHAPTER 11 Black Holes 233

Redshift in a Gravitational Field 234

Coordinate Singularities 235

Eddington-Finkelstein Coordinates 236

The Path of a Radially Infalling Particle 238

Eddington-Finkelstein Coordinates 239

Kruskal Coordinates 242

The Kerr Black Hole 244

Frame Dragging 249

The Singularity 252

A Summary of the Orbital Equations

for the Kerr Metric 252

Relativity Demystified. McGraw-Hill

DAVID McMAHON

Contact Form