Table of Contents
Introduction to General Relativity and Cosmology gives undergraduate students an overview of the fundamental ideas behind the geometric theory of gravitation and spacetime. Through pointers on how to modify and generalise Einstein's theory to enhance understanding, it provides a link between standard textbook content and current research in the field. Chapters present complicated material practically and concisely, initially dealing with the mathematical foundations of the theory of relativity, in particular differential geometry. This is followed by a discussion of the Einstein field equations and their various properties. Also given is analysis of the important Schwarzschild solutions, followed by application of general relativity to cosmology. Questions with fully worked answers are provided at the end of each chapter to aid comprehension and guide learning. This pared down textbook is specifically designed for new students looking for a workable, simple presentation of some of the key theories in modern physics and mathematics.
Preface vii
About the Author xi
1 Differential Geometry 1
1.1 The Concept of a Vector 1
1.1.1 Vector operations 2
1.1.2 Projections and basis vectors 3
1.1.3 Towards tangent space and all that 5
1.1.4 Index notation 7
1.2 Manifolds and Tensors 9
1.2.1 Tangent space and vector fields 9
1.2.2 Tensors 16
1.2.3 Manifolds and metric 19
1.2.4 Examples of metrics 23
1.2.5 Geodesies 26
1.2.6 Covariant derivative 33
1.2.7 Parallel transport and geodesies 38
1.3 Curvature 42
1.3.1 Infinitesimal parallelogram 43
1.3.2 Riemann, Ricci and Weyl tensors 47
1.3.3 Geodesic deviation equation 55
1.4 Euler-Lagrange Equations 57
1.5 Further Reading 60
1.6 Exercises 61
2 Einstein Field Equations 67
2.1 Some Physics Background 67
2.1.1 Newton's theory of gravity 67
2.1.2 Special relativity 70
2.1.3 Maxwell equations 75
2.1.4 Matter tensors 78
2.2 Geometry and Gravity 82
2.2.1 Geodesies and Newton's law 82
2.2.2 Curvature and the Poisson equation 85
2.2.3 Field equations of General Relativity 87
2.2.4 The principle of minimal gravitational coupling 89
2.3 Weak Gravity 91
2.3.1 Linearised Riemann and Ricci tensors 92
2.3.2 Gauge transformations 94
2.3.3 Linearised Einstein field equations 95
2.3.4 Gravitational waves 97
2.4 Variational Approach to General Relativity 100
2.5 Further Reading 103
2.6 Exercises 107
3 Schwarzschild Solutions 111
3.1 Spherical Symmetry and Birkhoff's Theorem 111
3.2 The Schwarzschild Solution 113
3.3 The Schwarzschild Interior Solution 117
3.4 Geodesies in Schwarzschild Spacetime 123
3.5 Testing General Relativity - The Classical Tests 127
3.5.1 Perihelion precession of Mercury 127
3.5.2 Light deflection by the Sun 131
3.5.3 Gravitational redshift of light 135
3.5.4 Radar echo or gravitational time delay 138
3.6 The Schwarzschild Radius 140
3.6.1 Radial null geodesies 141
3.6.2 Eddington-Finkelstein coordinates 142
3.6.3 Kruskal-Szekeres coordinates 143
3.6.4 Black holes 145
3.7 Further Reading 145
3.8 Exercises 147
4 Cosmology 153
4.1 Classical and Modern Cosmology 153
4.1.1 Cosmological principle 153
4.1.2 Geometry of constant time hypersurfaces 155
4.1.3 Friedmann-Lemaître Robertson-Walker metric 157
4.1.4 Particle horizons 158
4.1.5 Field equations 160
4.2 Cosmological Solutions 163
4.2.1 Matter-dominated universe 163
4.2.2 Radiation-dominated universe 165
4.2.3 The Einstein static universe 166
4.2.4 De Sitter universe 167
4.3 Physical Cosmology 169
4.3.1 Cosmological parameters 169
4.3.2 Redshift 172
4.3.3 Distances in cosmology 174
4.3.4 Distance redshift relationships 177
4.3.5 The universe today 181
4.3.6 Shortcomings in cosmology 185
4.4 Inflation 187
4.4.1 Accelerated expansion 188
4.4.2 Scalar fields in cosmology 190
4.4.3 Slow-roll inflation 192
4.5 Further Reading 199
4.6 Exercises 200
5 Solutions to Exercises 205
5.1 Solutions: Differential Geometry 205
The concept of a vector 205
Manifolds and tensors 208
Curvature 212
5.2 Solutions: Einstein Field Equations 218
Some physics background 218
Geometry and gravity 224
Weak gravity 226
Variational approach to General Relativity 227
5.3 Solutions: Schwarzschild Solutions 230
The Schwarzschild solution 230
The Schwarzschild interior solution 239
Geodesies in Schwarzschild spacetime 242
Testing General Relativity - the classical tests 246
The Schwarzschild radius 249
5.4 Solutions: Cosmology 252
Classical and Modern Cosmology 252
Physical cosmology 257
Inflation 260
Bibliography 265
Index 267
