This textbook equips Masters' students studying Physics and Astronomy with the necessary mathematical tools to understand the basics of General Relativity and its applications. It begins by reviewing classical mechanics with a more geometrically oriented language, continues with Special Relativity and, then onto a discussion on the pseudo-Riemannian space-times. Applications span from the inner and outer Schwarzschild solutions to gravitational wave, black holes, spherical relativistic hydrodynamics, and Cosmology. The goal is to limit the abstract formalization of the problems, to favor a hands-on approach with a number of exercises, without renouncing to a pedagogical derivation of the main mathematical tools and findings.
Features
• Provides a self-contained introduction to General Relativity and to its standar applications.
• Presents readers with all the tools necessary for further learning and research in the field.
• Accessible to readers with just foundational knowledge of linear algebra and Lagrangian mechanics.

List of Figures xiii

List of Boxes xv

Part I From Forces to Curvature

Chapter 1 Space and Time: The Classical View 3

1.1 Introduction 3

1.2 Metric Space 3

1.3 Homogeneity and Isotropy of Space 4

1.4 Covariant or Contravariant Vector Components 6

1.5 Motion of a Free Test-Particle 8

1.6 Inertial Frames and Galileo's Relativity Principle 10

1.7 The Derivative of a Vector 10

1.8 Velocity of Interactions and Second Newton's Law 13

1.9 Homogeneity of Time: Energy Conservation 14

1.10 Homogeneity of Space: The Third Newton's Law 14

1.11 Planetary Motions 15

1.12 Non-Inertial Reference Frames 17

Chapter 2 From Space and Time to Space-Time 19

2.1 Introduction 19

2.2 A Metric Space-Time 19

2.3 Homogeneity and Isotropy of the Minkowski Space-Time 21

2.4 Ordinary vs. Hyperbolic Rotations 23

2.5 Hyperbolic Rotations vs. Drag Velocities 24

2.6 Lorentz Transformations: A Graphical Approach 25

2.7 Proper Time and Proper Length 27

2.8 Motion of a Free Test-Particle 29

2.9 Four-Velocity and Four-Acceleration Vectors 30

2.10 Geodesics in Minkowski Space-Time 31

2.11 Four-Momentum 32

2.12 Relativistic Velocity Composition Law 33

Chapter 3 From Inertial to Non-Inertial Reference Frames 35

3.1 Introduction 35

3.2 Linear vs. Non-Linear Coordinate Transformations 35

3.3 Motion of a Free Test-Particle in a Rotating Frame 37

3.4 Geodesics in a Generic Space-Time 39

3.5 Geodesic Motion in the Rotating Reference Frame 40

3.6 Something More on Proper Time 41

3.7 Clock Synchronization 42

3.8 Proper Spatial Distances 44

Chapter 4 Pseudo-Riemannian Spaces 47

4.1 Introduction 47

4.2 Manifolds 47

4.3 How to Move to The Tangent Plane? 50

4.4 Vector Fields 51

4.5 Tensors 53

4.6 How to Recognize Tensors 54

4.7 General Covariance 56

Chapter 5 The Riemann-Christoffel Curvature Tensor 57

5.1 Introduction 57

5.2 Parallel Transport in Curved Space: The 2D Case 57

5.3 More About the Christoffel Symbols 59

5.4 Parallel Transport in Curved Manifolds 60

5.5 Covariant Derivative and Covariant Differential 61

5.6 Covariant Derivatives are Tensors 61

5.7 More on Geodesics 62

5.8 More on Covariant Derivative 63

5.9 Covariant Derivative of a Tensor 64

5.10 The Riemann-Christoffel Curvature Tensor 65

5.11 Second Covariant Derivatives 67

5.12 Symmetries of the Riemann-Christoffel Tensor 67

Chapter 6 From Non-inertial Frames to Gravity: The Equivalence Principle 71

6.1 Introduction 71

6.2 The Equivalence Principle 71

6.3 Switching Off Gravity: The Free-Fall 72

6.4 "Creating Gravity": Non-Inertial Frames 73

6.5 The Gravity Case: A Metric Space 75

6.6 The Motion of a Test-Particle in a Gravitational Field 77

6.7 Geodesic Deviation 78

6.8 Link Between Geometry and Dynamics 79

6.9 Riemann-Christoffel Tensor in the Rotating Frame 81

6.10 Riemann-Christoffel Tensor and Gravity 81

Part II From Curvature to Observations

Chapter 7 Observational Test of the Equivalence Principle 85

7.1 Introduction 85

7.2 Inertial vs. Gravitational Masses 85

7.3 Gravitational Time Dilation 86

7.4 The Global Positioning System - GPS 86

7.5 Gravitational Redshift 87

7.6 The Long Gravitational Redshift Hunt 88

7.7 The Nordtvedt Rffect 89

Chapter 8 Field Equations in the "vacuum" 95

8.1 Introduction 95

8.2 Field Equations in the "Vacuum": Requirements 95

8.3 The Ricci Tensor 96

8.4 Gravitational Field Equations in the "Vacuum" 97

8.5 The Einstein Tensor 99

8.6 Hilbert's Action 100

8.7 The Action for a Cosmological Constant 102

8.8 The Geometry of Space-Time in the "Vacuum" 103

Chapter 9 Test-Particles in the Schwarzschild Space-Time 109

9.1 Introduction 109

9.2 The Schwarzschild Solution for a Point Mass 109

9.3 The "Embedding" Procedure 110

9.4 The Jebsen-Birkhoff Theorem 111

9.5 First Integrals in the Schwarzschild Space-Time 112

9.6 Energy Conservation in GR 113

9.7 The Radial Infall 114

9.8 Orbits in a Schwarzschild Geometry 115

9.9 Stable Circular Orbits: α >√3 118

9.10 The Case of a Non-Radial Infall: α >√3 119

9.11 Photons in the Schwarzschild Space-Time 119

Chapter 10 The Classical Tests of General Relativity 123

10.1 Introduction 123

10.2 Planetary Motion 123

10.3 The Perihelion Shift of Mercury 125

10.4 Light Ray's Deflection 126

10.5 Gravitational Lensing 130

10.6 The Schwarzschild Metric in Totally Isotropic Form 133

10.7 Light Travel Time in a Schwarzschild Geometry 134

Chapter 11 Gravitational Waves in the "Vacuum" 139

11.1 Introduction 139

11.2 Linearized Gravity 139

11.3 Gauge Transformations 140

11.4 The Lorentz Gauge 141

11.5 Gravitational Waves 142

11.6 TT Gauge 144

11.7 Gauge Invariant Approach 145

11.8 Field Equations 147

11.9 Effects of a Gravitational Wave 149

11.10 Interferometers 151

11.11 The Direct Detection of Gravitational Waves 153

11.12 The Indirect Evidence for Gravitational Waves 154

Part III From Singularities to Cosmological Scales

Chapter 12 Schwarzschild Black Holes 159

12.1 Introduction 159

12.2 Singularities of the Schwarzschild Metric 159

12.3 Conformally-Flat Coordinates 161

12.4 Kruskal-Szekeres vs. Schwarzschild Coordinates 164

12.5 The Kruskal-Szekeres Plane 166

12.6 The Schwarzschild Black Hole 170

Chapter 13 Field Equations in Non-"Empty" Space-Times 173

13.1 Introduction 173

13.2 Field Equations: Requirements 173

13.3 Conservation Laws for a Relativistic Fluid 174

13.4 The Matter Energy-Momentum Tensor 174

13.5 The EM Energy-Momentum Tensor 177

13.6 An Isotropic Radiation Field 179

13.7 Field Equations Inside a Matter/Energy Distribution 180

13.8 The Reissner-Nordstrom Solution 181

13.9 Orbits in a Reissner-Nordstrom Geometry 184

13.10 Free-Fall on the Reissner-Nordstrom Black Hole 185

13.11 The Kerr Solution 188

Chapter 14 Further Applications of the Field Equations 193

14.1 Introduction 193

14.2 The Inner Schwarzschild Space-Time 193

14.3 Proper vs. Observable Mass 198

14.4 Spherical Relativistic Hydrodynamics 200

14.5 The Gravitational Collapse 205

Chapter 15 Theoretical Cosmology 209

15.1 Introduction 209

15.2 An Isotropic and Homogeneous Matter Distribution 209

15.3 The FLRW Metric 210

15.4 The Spatial Sector of the FLRW Space-Time 211

15.5 The Friedmann Equations 212

15.6 Equation of Motions 213

15.7 Cosmological Parameters 214

15.8 The De Sitter Model 215

15.9 The Closed Friedmann Universe 215

15.10 The Open Friedmann Universe 216

15.11 The Einstein-De Sitter Universe 217

15.12 A Flat, Λ-Dominated Universe 217

15.13 H0 and the Age of the Universe 218

15.14 The Cosmological Redshift 219

15.15 Comoving Distances 221

15.16 The Proper Angular Diameter Distance 222

15.17 The Proper Luminosity Distance 223

15.18 SNe La and Dark Energy 225

Chapter 16 The Hot Big-Bang 227

16.1 Introduction 227

16.2 The CMB 227

16.3 A Radiation-Dominated Universe 228

16.4 The Neutron-to-Baryon Ratio 229

16.5 A ν's Cosmic Background 230

16.6 Primordial Nucleosynthesis 231

16.7 Primordial Abundance of Light Nuclei 232

16.8 The Recombination of the Primordial Plasma 234

16.9 The Puzzles of the Standard Model 235

16.10 An Early Accelerated Phase? 237

16.11 Cosmic Inflation 238

Appendix A Exercises 241

References 247

Index 253

Nicola Vittorio is full professor of Astronomy and Astrophysics at the Physics Department of the University of Rome ``Tor Vergata''. He carried out theoretical studies in cosmology and in the formation and evolution of the large-scale structure of the universe. He published 250 articles on refereed journals, and also was the organizer and the editor of several conferences and conference proceedings. Nicola Vittorio is the Scientific Coordinator of the national ASI/LiteBIRD Project, financed by the Italian Space Agency-ASI to support the participation of the Italian community to the Phase A activities of the JAXA-led LiteBIRD satellite. He is the President of the Board that coordinates the PhD activities of ``Tor Vergata'' University, the coordinator of the PhD program in Astronomy, Astrophysics and Space Science (run jointly by the Universities of Rome ``Tor Vergata'' and Sapienza with the National Institute of Astrophysics-INAF) and the Coordinator of the Erasmus Mundus Joint Master program in Astrophysics and Space Science-MASS (run jointly by the universities of Rome "Tor Vergata", Belgrade, Bremen, and Côte d'Azur). Nicola Vittorio is National Fellow of the Academy of Sciences of Turin, Distinguished Member of the Italian Physical Society, Member of the European Academy of Sciences and Arts, of the European Physical Society, of the International Astronomical Union, of the Italian Astronomical Society, and of the Italian Society for the Advancement of Science.

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