An introduction to the relativistic theory of gravitation / P. Hájíček ; translated by Frank Meyer and Jan Metzger.

Format:

Book

Author/Creator:

Hájíček, P. (Petr)

Contributor:

Meyer, Frank.

Metzger, Jan.

Series:

Lecture notes in physics 0075-8450 ; 750.

Lecture notes in physics, 0075-8450 ; 750

Language:

English

German

Subjects (All):

General relativity (Physics).

Physical Description:

xiii, 280 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2008]

Summary:

The geometric interpretation of gravitation is one of the major foundations of modern theoretical physics. This primer introduces classical general relativity with emphasis on the clarity of conceptual structure and on the basic mathematical methods to build up systematically application skills. The wealth of physical phenomena entailed by the Einstein's equations is revealed with the help of specific models describing gravitomagnetism, gravitational waves, cosmology, gravitational collapse and black holes. End-of-chapter exercises complete the main text.

This book is based on class-tested notes for courses that have been held by the author over many years at the University of Bern, where Einstein worked at the local patent office and where the foundations of special relativity were laid.

Contents:

1 Geometrization of Mechanics 1

1.1 Selected Facts 1

1.1.1 The Cavendish Experiment 1

1.1.2 Eotvos Experiment 2

1.1.3 Deflection of Light 2

1.1.4 Redshift 3

1.2 Equivalence Principle 3

1.3 Newton-Galilei Space-Time 4

1.4 Free Motion 6

1.5 Affine Connection 7

1.5.1 Curvilinear Coordinates 7

1.5.2 Curves and Tangential Vectors 9

1.5.3 Affine Connection of a Manifold 10

1.5.4 The Metric Affine Connection 13

1.6 Cartan-Friedrichs Space-Time 22

1.7 Curvature Tensor 24

1.8 The Equivalence Principle 27

1.8.1 Galilei Equivalence Principle 27

1.8.2 Geodesic Systems 28

1.8.3 Local Inertial Frames 28

1.8.4 Formulation of the Principle 29

1.9 Parallel Transport 30

2 Relativistic Particle Dynamics in Gravitational Fields 39

2.1 Relativistic Gravity 39

2.2 Geometry of Minkowski Space-Time 39

2.3 Particle Dynamics in General Relativity 44

2.4 Local Measurements 50

2.4.1 General Reference Frame 50

2.4.2 Proper Time 52

2.4.3 Radar Measurement 52

2.4.4 Simultaneity 53

2.4.5 Distances 54

2.4.6 Spectra and Directions 54

2.5 Stationary Space-Time 55

2.5.1 The 3-Space 58

2.5.2 Free Falls 58

2.5.3 The Gravitoelectric and Gravitomagnetic Force 59

2.5.4 Redshift 61

2.5.5 Gravitational Time Dilation 63

2.6 Isometry (A Mathematical Intermezzo) 65

2.6.1 Rotation in E[superscript 2] 65

2.6.2 Diffeomorphisms 66

2.6.3 Lie Derivative 67

2.6.4 Killing Vector Field 70

2.7 Rotationally Symmetric Space-Times 72

2.7.1 Rotation Surfaces 72

2.7.2 Space-Times 73

2.7.3 Geodesic Equation in the Static Case 75

2.8 Asymptotically Flat Space-Times 76

2.8.1 Eddington-Robertson Expansion 77

2.8.2 Energy and Momentum Balance 78

2.9 Motion of Planets 81

2.9.1 Comparison with Newton's Theory 82

2.9.2 Perihelion Shift 83

2.10 Light Signals in the Solar System 85

2.10.1 Deflection of Light 86

2.10.2 Radar Echo Delay 88

3 Field Dynamics 93

3.1 Electrodynamics 93

3.1.1 Equivalence Principle 94

3.1.2 The Maxwell Equations 96

3.1.3 The Stress-Energy Tensor 97

3.2 Variation Principle 100

3.2.1 Transformation Properties of Tensor Fields 100

3.2.2 The Action 101

3.2.3 Variation Formula 102

3.2.4 Field Equations of Matter 105

3.3 Covariant Derivative 105

3.3.1 Definition of the Covariant Derivative 106

3.3.2 Direct Expression for the Covariant Derivative 107

3.3.3 Algebraic Properties 109

3.3.4 Metric Affine Connections 111

3.4 The Stress-Energy Tensor 114

3.4.2 Properties 116

3.4.3 Interpretation of the Divergence Formula 118

3.4.4 Ideal Fluids 120

4 Dynamics of Gravity 125

4.1 The Action 125

4.2 The Einstein Equations 127

4.2.1 Properties of the Curvature Tensor 129

4.3 General Covariance of the Einstein Equations 131

4.4 Weak Gravitational Field 133

4.4.1 Auxiliary Metrics and Gauge Transformations 135

4.4.2 Affine Connection and Curvature 136

4.4.3 The Cosmological Constant 140

4.4.4 The Linearized Einstein Equations 141

4.4.5 Stationary Fields 143

4.4.6 Gravitomagnetic Phenomena 147

4.4.7 Plane Waves 149

4.4.8 Measurable Properties of Plane Waves 152

5 Cosmological Models 159

5.1 Homogeneous Isotropic 3-Spaces 159

5.1.1 The Cosmological Principle 159

5.1.2 Euclidean Space 160

5.1.3 The Sphere S[superscript 3] 160

5.1.4 The Pseudo-sphere P[superscript 3] 162

5.2 Robertson-Walker Space-Times 163

5.2.1 Metric 163

5.2.2 Cosmic Rest System 165

5.2.3 Cosmological Redshift 166

5.2.4 Cosmological Horizons 168

5.2.5 Einstein Tensor of the Robertson-Walker Space-Time 171

5.3 Cosmic Dynamics 171

5.3.1 Friedmann-Lemaitre Equations 171

5.3.2 Cosmic Acceleration 172

5.3.3 Linear Equations of State 174

5.4 Parameterization of Physically Distinct Models 176

5.4.1 Qualitative Discussion of the Dynamics 176

5.4.2 Density Parameters 180

5.4.3 The [Omega]-Diagram 183

5.4.4 Luminosity Distance and the Measurement of [Lambda] 187

5.4.5 The Friedmann Models 191

5.5 Space-Times with Maximal Symmetry (10 Killing Fields) 191

5.5.1 Minkowski Space-Time 191

5.5.2 de Sitter Space-Time 192

5.5.3 Anti-de Sitter Space-Time 199

5.6 The Early Universe 202

5.6.1 Horizon Problem 202

5.6.2 Flatness Problem 203

5.6.3 Entropy Problem 204

5.6.4 Cosmic Inflation: Orders of Magnitude 204

5.6.5 Quantum Cosmology 206

6 Rotationally Symmetric Models of Stars 209

6.1 Hydrostatic Equilibrium of Non-rotating Stars 209

6.1.1 Equations of the Hydrostatic Equilibrium 209

6.1.2 Conditions at the Center 210

6.1.3 Conditions at the Surface 212

6.1.4 The Metric Outside the Star 212

6.1.5 Comparison to Newtonian Gravity 213

6.1.6 Mass Limits 215

6.1.7 Junction Conditions 217

6.2 Properties of the Schwarzschild Solution 218

6.2.1 The Birkhoff Theorem 218

6.2.2 Radial Light Rays 219

6.2.3 Eddington-Finkelstein Coordinates 220

6.2.4 The Horizon 221

6.3 Oppenheimer-Snyder Collapse Model 224

6.3.1 The Interior 224

6.3.2 The Outside 225

6.3.3 The Surface 225

6.3.4 Radial Light-Like Geodesics 229

7 Stationary Black Holes 237

7.1 Hypersurfaces 237

7.1.2 Tangential Vectors 238

7.1.3 Induced Metric 239

7.1.4 Normal 239

7.1.5 Classification of Hypersurfaces 239

7.2 Rotating Charged Black Holes 241

7.2.1 First Look at Kerr-Newman Space-Time 243

7.3 Dynamics of Charged Particles 249

7.3.1 Integrals of Motion 249

7.3.2 The Equatorial Plane and the Axes of Symmetry 252

7.4 Energetics of Black Holes 254

7.4.1 Available Energy of a Black Hole 255

7.4.2 Energy of Particles in the Field of a Black Hole 262.

Notes:

Includes bibliographical references and index.

ISBN:

9783540786580

3540786589

OCLC:

221218012