General Relativity : Basics and Beyond

264 Pages·2014·3.839 MB·English

by  Ghanashyam Date

General
Physics
G
General Relativity: Basics and Beyond familiarizes you with the basic features
of the theory of general relativity as well as some of its more advanced aspects. e
Relativity
Employing the pedagogical style of a textbook, it includes essential ideas and just
n
enough background material needed for you to appreciate the issues and current
research.
e
The first five chapters form the core of an introductory course on general relativity. r
The author traces Einstein’s arguments and presents examples of space-times a Basics and Beyond
corresponding to different types of gravitational fields. He discusses the adaptation
l
of dynamics in a Riemannian geometry framework, the Einstein equation and its

elementary properties, and different phenomena predicted or influenced by general R
relativity.
e
Moving on to more sophisticated features of general relativity, the book presents
l
the physical requirements of a well-defined deterministic framework for non-
a
gravitational dynamics and describes the characterization of asymptotic space-
times. After covering black holes, gravitational waves, and cosmological space- t
i
times, the book examines the evolutionary interpretation for the class of globally
v
hyperbolic space-times, explores numerical relativity, and discusses approaches
i
that address the challenges of general relativity.
t
Features y
• Emphasizes the physical ideas and motivations of the theory while refraining
from exhaustive details
• Covers the mathematical aspects that are important in understanding
the scope and limitations of the theory, such as the mathematical model
for space-time and basic physical quantities related to space-time
measurements
• Explores current research topics, including the quasi-local generalization of
black holes, the challenge of directly detecting gravitational waves, and the
nature of cosmological singularities
D
• Discusses an emergent gravity viewpoint and the main approaches to a
a
quantum theory of gravity
t
• Provides the necessary background information, including differential e
geometry
Ghanashyam Date
K15233
www.crcpress.com
K15233_cover.indd 1 10/2/14 2:10 PM
General
Relativity
Basics and Beyond

General
Relativity
Basics and Beyond
Ghanashyam Date
The Institute of Mathematical Sciences
Chennai, India
CRC Press
Taylor & Francis Group
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Contents
Preface ix
I The Basics 1
1 From Newton to Einstein: Synthesis of General Relativity 3
1.1 Space, Time, Observers . . . . . . . . . . . . . . . . . . . . . 3
1.2 General Relativity and Space-Time Arenas . . . . . . . . . . 7
2 Examples of Space-Times 11
2.1 No Gravity (Minkowski Space-Time) . . . . . . . . . . . . . 18
2.2 Uniform Gravity (Rindler Space-Time) . . . . . . . . . . . . 19
2.3 Centrifugal Gravity (Uniformly Rotating Platform) . . . . . 21
2.4 Spherical Gravity (The Schwarzschild Space-Time) . . . . . . 23
2.5 Cosmological Gravity (Robertson–Walker Space-Times) . . . 25
2.6 Undulating Gravity (Gravitational Waves) . . . . . . . . . . 27
3 Dynamics in Space-Time 29
3.1 Particle Motion Including Spin . . . . . . . . . . . . . . . . . 29
3.2 Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Dynamics of Space-Time 37
4.1 Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Elementary Properties and Peculiarities . . . . . . . . . . . 40
4.3 The Stress Tensor and Fluids . . . . . . . . . . . . . . . . . 42
4.4 Operational Determination of the Metric . . . . . . . . . . . 46
5 Elementary Phenomenology 49
5.1 Geodesics and the Classic Tests . . . . . . . . . . . . . . . . 49
5.1.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.2 Deflection of Light . . . . . . . . . . . . . . . . . . . . 53
5.1.3 Precession of Perihelia . . . . . . . . . . . . . . . . . 55
5.2 Relativistic Cosmology . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 Friedmann–Lamaitre–Robertson–Walker Cosmologies 58
5.2.2 Digression on Big-Bang Cosmology: Thermal History 62
5.2.3 Cosmic Microwave Background Radiation . . . . . . . 68
5.3 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . 72
5.3.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . 73
v
vi Contents
5.3.2 Gravitational Radiation . . . . . . . . . . . . . . . . . 75
5.3.3 Radiated Energy and the Quadrupole Formula . . . . 78
5.4 Black Holes—Elementary Aspects . . . . . . . . . . . . . . . 83
5.4.1 Static Black Holes . . . . . . . . . . . . . . . . . . . . 83
5.4.1.1 Schwarzschild Black Hole . . . . . . . . . . . 83
5.4.1.2 The Reissner–Nordstrom Black Hole . . . . . 86
5.4.2 Stationary (Non-Static) Black Holes . . . . . . . . . . 88
5.4.2.1 Kerr–Newman Black Holes . . . . . . . . . . 88
5.4.3 Observational Status . . . . . . . . . . . . . . . . . . 90
5.5 Stars in GR . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
II The Beyond 95
6 The Space-Time Arena 97
6.1 Preliminary Notions and Results . . . . . . . . . . . . . . . 98
6.2 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Determinism and Global Hyperbolicity . . . . . . . . . . . . 105
6.4 Geodesics and Congruences . . . . . . . . . . . . . . . . . . 108
6.5 Singularity Theorems . . . . . . . . . . . . . . . . . . . . . . 117
7 Asymptotic Structure 123
7.1 Vicinity of the Null Infinity . . . . . . . . . . . . . . . . . . . 128
7.2 Vicinity of the Spatial Infinity . . . . . . . . . . . . . . . . . 136
8 Black Holes 139
8.1 Examples of Extended Black Hole Solutions . . . . . . . . . 139
8.2 General Black Holes and Uniqueness Theorems . . . . . . . 144
8.3 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . 147
8.4 Quasi-Local Definitions of Horizons . . . . . . . . . . . . . . 150
9 Cosmological Space-Times 159
10 Gravitational Waves 167
10.1 Conceptual Issues . . . . . . . . . . . . . . . . . . . . . . . . 167
10.2 Observational Issues . . . . . . . . . . . . . . . . . . . . . . 169
11 Field Equation: Evolutionary Interpretation 173
11.1 The 3 + 1 Decomposition . . . . . . . . . . . . . . . . . . . 173
11.2 Initial Value Formulation . . . . . . . . . . . . . . . . . . . . 176
11.3 Hamiltonian Formulation (ADM) . . . . . . . . . . . . . . . 179
12 Numerical Relativity 185
Contents vii
13 Into the Quantum Realm 191
13.1 Gravity Is “Emergent” . . . . . . . . . . . . . . . . . . . . . 192
13.2 The Quantum Gravity Paradigm . . . . . . . . . . . . . . . 194
13.2.1 String Theory: The Unification Paradigm . . . . . . . 195
13.2.2 LoopQuantumGravity:TheBackgroundIndependence
Demand . . . . . . . . . . . . . . . . . . . . . . . . . 196
14 Mathematical Background 199
14.1 Basic Differential Geometry . . . . . . . . . . . . . . . . . . 199
14.2 Sets, Metric Spaces and Topological Spaces . . . . . . . . . 199
14.3 Manifolds and Tensors . . . . . . . . . . . . . . . . . . . . . 202
14.4 Affine Connection and Curvature . . . . . . . . . . . . . . . 207
14.5 Metric Tensor and Pseudo–Riemannian Geometry . . . . . . 211
14.6 Summary of Differential Geometry . . . . . . . . . . . . . . . 212
14.7 Theorems on Initial Value Problem . . . . . . . . . . . . . . 232
14.8 Petrov Classification . . . . . . . . . . . . . . . . . . . . . . . 234
Epilogue 237
Bibliography 239
Index 251

Preface
It fills me with a sense of joy and humility to present this book on the eve
of the centenary year of the publication of Albert Einstein’s General Theory
of Relativity. When general relativity arrived, it had an aura of mystery due
to its sophisticated view of space, time and gravitational phenomena. From
the early phase when primary elaboration of the theory was mathematical in
nature,ithasevolvedintoaphasewhereitisbeingconfrontedbyincreasingly
sophisticatedexperimentsthathavebeensuccessfulsofar.Studentsareoften
attracted to the theory and want to know what yet can be done with it.
The book is envisaged as an attempt to familiarize students and prospective
researcherswiththebasicfeaturesofthetheoryandofferaperspectiveonits
more advance features.
There are many excellent textbooks from the classics by Misner–Thorne–
Wheeler, Weinberg and Wald to the more recent ones by Sean Carroll, James
HartleandThanuPadmanabhan,withdifferingstylesandemphasisandthere
are excellent review articles on frontline topics. The idea here is to combine
the ‘textbook’ and ‘the review’. Thus, I have tried to adopt the pedagogical
styleofatextbookwhileavoidinganemphasisondetailedtreatments,andat
thesametime,triedtopresenttheessentialideasandjustenoughbackground
material needed for students to appreciate the issues and current research.
There was also a conscious effort to emphasize the physical ideas and
motivations, contrasting the mathematical idealizations which are important
inappreciatingthescopeandlimitationsofthetheory.Consequently,requisite
mathematical background of differential geometry is summarized in the last
chapter while the main text emphasizes the physical aspects.
The first five chapters usually form the core of an introductory course
on General Relativity (GR) and constitute the “Basics” part of the book.
The first chapter traces Einstein’s arguments and informally motivates the
mathematical model for space-time. In the second chapter, we first discuss
the basic physical quantities related to space-time measurements and their
relation to a metric in an arbitrary coordinate system. This is followed by
examplesofspace-timescorrespondingtodifferenttypesofgravitationalfields.
Some of these are revisited subsequently for further elaboration. Chapter 3
discussesadaptationof dynamicsinaRiemanniangeometryframeworkwhile
the next chapter presents the Einstein equation together with its elementary
properties. The fifth chapter discusses different phenomena either predicted
ix