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# Black Holes: An Introduction / Edition 2

Black Holes: An Introduction / Edition 2 available in Hardcover, Paperback

## Overview

This introduction to the fascinating subject of black holes fills a significant gap in the literature which exists between popular, non-mathematical expositions and advanced textbooks at the research level. It is designed for advanced undergraduates and first year postgraduates as a useful stepping-stone to the advanced literature.

The book provides an accessible introduction to the exact solutions of Einstein's vacuum field equations describing spherical and axisymmetric (rotating) black holes. The geometry and physical properties of these spacetimes are explored through the motion of particles and light. The use of different coordinate systems, maximal extensions and Penrose diagrams is explained. The association of the surface area of a black hole with its entropy is discussed and it is shown that with the introduction of quantum mechanics black holes cease to be black and can radiate. This result allows black holes to satisfy the laws of thermodynamics and thus be consistent with the rest of physics.

In this new edition the problems in each chapter have been revised and solutions are provided. The text has been expanded to include new material on wormholes and clarify various other issues.

## Product Details

ISBN-13: | 9781848163829 |
---|---|

Publisher: | Imperial College Press |

Publication date: | 10/28/2009 |

Pages: | 212 |

Product dimensions: | 6.00(w) x 9.10(h) x 0.80(d) |

## Read an Excerpt

This introduction to the fascinating subject of black holes fills a significant gap in the literature which exists between popular, non-mathematical expositions and advanced textbooks at the research level. It is designed for advanced undergraduates and first year postgraduates as a useful stepping-stone to the advanced literature.

The book provides an accessible introduction to the exact solutions of Einstein's vacuum field equations describing spherical and axisymmetric (rotating) black holes. The geometry and physical properties of these spacetimes are explored through the motion of particles and light. The use of different coordinate systems, maximal extensions and Penrose diagrams is explained. The association of the surface area of a black hole with its entropy is discussed and it is shown that with the introduction of quantum mechanics black holes cease to be black and can radiate. This result allows black holes to satisfy the laws of thermodynamics and thus be consistent with the rest of physics.

In this new edition the problems in each chapter have been revised and solutions are provided. The text has been expanded to include new material on wormholes and clarify various other issues.

## First Chapter

This introduction to the fascinating subject of black holes fills a significant gap in the literature which exists between popular, non-mathematical expositions and advanced textbooks at the research level. It is designed for advanced undergraduates and first year postgraduates as a useful stepping-stone to the advanced literature.

The book provides an accessible introduction to the exact solutions of Einstein's vacuum field equations describing spherical and axisymmetric (rotating) black holes. The geometry and physical properties of these spacetimes are explored through the motion of particles and light. The use of different coordinate systems, maximal extensions and Penrose diagrams is explained. The association of the surface area of a black hole with its entropy is discussed and it is shown that with the introduction of quantum mechanics black holes cease to be black and can radiate. This result allows black holes to satisfy the laws of thermodynamics and thus be consistent with the rest of physics.

In this new edition the problems in each chapter have been revised and solutions are provided. The text has been expanded to include new material on wormholes and clarify various other issues.

## Table of Contents

Preface v

1 Relativistic Gravity 1

1.1 What is a black hole? 1

1.2 Why study black holes? 3

1.3 Elements of general relativity 3

1.3.1 The principle of equivalence 3

1.3.2 The Newtonian affine connection 4

1.3.3 Newtonian gravity 5

1.3.4 Metrics in relativity 6

1.3.5 The velocity and momentum 4-vector 7

1.3.6 General vectors and tensors 8

1.3.7 Locally measured physical quantities 9

1.3.8 Derivatives in relativity 9

1.3.9 Acceleration 4-vector 11

1.3.10 Paths of light 11

1.3.11 Einstein's field equations 11

1.3.12 Symmetry and Killing's equation 12

2 Spherical Black Holes 15

2.1 The Schwarzschild metric 15

2.1.1 Coordinates 16

2.1.2 Proper distance 17

2.1.3 Proper time 17

2.1.4 Redshift 17

2.1.5 Interpretation of M and geometric units 18

2.1.6 The Schwarzschild radius 18

2.1.7 The event horizon 19

2.1.8 Birkoff's theorem 19

2.1.9 Israel's theorem 19

2.2 Orbits in Newtonian gravity 20

2.2.1 Newtonian Energy 20

2.2.2 Angular momentum 20

2.2.3 The Newtonian effective potential 20

2.2.4 Classification of Newtonian orbits 21

2.3 Particle orbits in the Schwarzschild metric 21

2.3.1 Constants of the motion 22

2.3.2 Conserved Energy 22

2.3.3 Angular momentum 23

2.3.4 The effective potential 24

2.3.5 Newtonian approximation to the metric 24

2.3.6 Classification of orbits 25

2.3.7 Radial infall 25

2.3.8 The locally measured energy of a particle 26

2.3.9 Circular orbits 27

2.3.10 Comparison with Newtonian orbits 28

2.3.11 Orbital velocity in the frame of a hovering observer 29

2.3.12 Energy in the last stable orbit 30

2.4 Orbits of light rays 31

2.4.1 Radial propagation of light 32

2.4.2 Capturecross-section for light 33

2.4.3 The view of the sky for a stationary observer 34

2.5 Classical tests 36

2.6 Palling into a black hole 36

2.6.1 Free-fall time for a distant observer 37

2.6.2 Light-travel time 38

2.6.3 What the external observer sees 38

2.6.4 An infalling observer's time 39

2.6.5 What the infalling observer feels 40

2.7 Capture by a black hole 41

2.7.1 Case I: Capture of high angular momentum particles 41

2.7.2 Case II: Capture of low energy particles 42

2.8 Surface gravity of a black hole 42

2.8.1 The proper acceleration of a hovering observer 42

2.8.2 Surface gravity 43

2.8.3 Rindler coordinates 44

2.9 Other coordinates 46

2.9.1 Null coordinates 46

2.9.2 Eddington-Finkelstein coordinates 47

2.10 Inside the black hole 47

2.10.1 The infalling observer 49

2.11 White holes 50

2.12 Kruskal coordinates 50

2.12.1 The singularities at r = 0 and cosmic censorship 52

2.12.2 The spacetime of a collapsing star 52

2.13 Embedding diagrams 53

2.14 Asymptotic flatness 55

2.14.1 The Penrose-Carter diagram for the Schwarzschild metric 57

2.14.2 The Penrose-Carter diagram for the Newtonian metric 57

2.15 Non-isolated black holes 58

2.15.1 The infinite redshift surface 58

2.15.2 Trapped surfaces 58

2.15.3 Apparent horizon 58

2.16 The membrane paradigm 59

3 Rotating Black Holes 61

3.1 The Kerr metric 62

3.2 The event horizon 62

3.2.1 The circumference of the event horizon 63

3.2.2 The area of the event horizon 63

3.3 Properties of the Kerr metric coefficients 64

3.3.1 Identities 64

3.3.2 Contravariant components 64

3.4 Interpretation of m, a and geometric units 65

3.5 Extreme Kerr black hole 66

3.6 Robinson's theorem 66

3.7 Particle orbits in the Kerr geometry 66

3.7.1 Constants of the motion 67

3.7.2 Energy 68

3.7.3 Angular momentum 69

3.7.4 The Carter integral 69

3.7.5 The radial equation 70

3.7.6 The effective potential 71

3.8 Frame-dragging 72

3.8.1 Free fall with zero angular momentum 73

3.8.2 Orbits with non-zero angular momentum 74

3.9 Zero angular momentum observers (ZAMOs) 75

3.9.1 Some applications of ZAMOs 76

3.10 Photon orbits 77

3.10.1 The photon effective potential 77

3.10.2 Azimuthal motion 78

3.10.3 Photon capture cross-section 78

3.11 The static limit surface 79

3.12 The infinite redshift surface 81

3.13 Circular orbits in the equatorial plane 81

3.13.1 Innermost (marginally) stable circular orbit 82

3.13.2 Period of a circular orbit 84

3.13.3 Energy of the innermost stable orbit 85

3.13.4 Angular momentum of the innermost stable orbit 85

3.13.5 Marginally bound orbits 86

3.13.6 Unbound orbits 86

3.14 Polar orbits 87

3.14.1 Orbital period 90

3.15 The ergosphere 90

3.15.1 Negative energy orbits 90

3.15.2 Energy and angular momentum 92

3.15.3 The Penrose process 93

3.15.4 Realising the Penrose process 93

3.16 Spinning up a black hole 95

3.16.1 From Schwarzschild to extreme Kerr black hole 96

3.17 Other coordinates 97

3.18 Penrose-Carter diagram 99

3.18.1 Interior solutions and collapsing stars 102

3.19 Closed timelike lines 103

3.20 Charged black holes 104

4 Black Hole Thermodynamics 107

4.1 Black hole mechanics 107

4.1.1 Surface gravity 107

4.1.2 Redshift 109

4.1.3 Conservation of energy 110

4.2 The area of a Kerr black hole horizon cannot decrease 111

4.2.1 Area change by accretion 111

4.2.2 Area change produced by the Penrose process 112

4.2.3 The area theorem 113

4.2.4 Irreducible mass 113

4.2.5 Maximum energy extraction 114

4.2.6 Naked singularities 114

4.3 Scattering of waves 115

4.3.1 Superradiance 115

4.4 Thermodynamics 120

4.4.1 Horizon temperature 122

4.4.2 The four laws of black hole thermodynamics 124

4.5 Hawking radiation 125

4.5.1 Introduction 125

4.5.2 Casimir effect 126

4.5.3 Thermal vacua in accelerated frames 127

4.5.4 Hawking radiation 131

4.6 Properties of radiating black holes 135

4.6.1 Entropy and temperature 135

4.6.2 Radiating black holes 136

4.6.3 Black hole in a box 137

4.7 Entropy and microstates 139

5 Wormholes and Time Travel 141

5.1 Introduction 141

5.2 Wormholes 141

5.3 Traversible wormholes 143

5.4 Creating a wormhole 145

5.5 Weak energy condition 146

5.6 Exotic matter 147

5.7 Time machines 148

5.8 Chronology protection 150

6 Astrophysical Black Holes 151

6.1 Introduction 151

6.2 Stellar mass black holes 152

6.2.1 Formation 152

6.2.2 Finding stellar mass black holes 154

6.2.3 The black hole at the centre of the Galaxy 155

6.3 Supermassive black holes in other galaxies 156

6.3.1 Intermediate mass black holes 157

6.3.2 Mini black holes 158

6.4 Further evidence for black hole spin 159

6.5 Conclusions 160

Solutions to Problems 163

References 189

Bibliography 193

Index 195

## Reading Group Guide

Preface v

1 Relativistic Gravity 1

1.1 What is a black hole? 1

1.2 Why study black holes? 3

1.3 Elements of general relativity 3

1.3.1 The principle of equivalence 3

1.3.2 The Newtonian affine connection 4

1.3.3 Newtonian gravity 5

1.3.4 Metrics in relativity 6

1.3.5 The velocity and momentum 4-vector 7

1.3.6 General vectors and tensors 8

1.3.7 Locally measured physical quantities 9

1.3.8 Derivatives in relativity 9

1.3.9 Acceleration 4-vector 11

1.3.10 Paths of light 11

1.3.11 Einstein's field equations 11

1.3.12 Symmetry and Killing's equation 12

2 Spherical Black Holes 15

2.1 The Schwarzschild metric 15

2.1.1 Coordinates 16

2.1.2 Proper distance 17

2.1.3 Proper time 17

2.1.4 Redshift 17

2.1.5 Interpretation of M and geometric units 18

2.1.6 The Schwarzschild radius 18

2.1.7 The event horizon 19

2.1.8 Birkoff's theorem 19

2.1.9 Israel's theorem 19

2.2 Orbits in Newtonian gravity 20

2.2.1 Newtonian Energy 20

2.2.2 Angular momentum 20

2.2.3 The Newtonian effective potential 20

2.2.4 Classification of Newtonian orbits 21

2.3 Particle orbits in the Schwarzschild metric 21

2.3.1 Constants of the motion 22

2.3.2 Conserved Energy 22

2.3.3 Angular momentum 23

2.3.4 The effective potential 24

2.3.5 Newtonian approximation to the metric 24

2.3.6 Classification of orbits 25

2.3.7 Radial infall 25

2.3.8 The locally measured energy of a particle 26

2.3.9 Circular orbits 27

2.3.10 Comparison with Newtonian orbits 28

2.3.11 Orbital velocity in the frame of a hovering observer 29

2.3.12 Energy in the last stable orbit 30

2.4 Orbits of light rays 31

2.4.1 Radial propagation of light 32

2.4.2 Capturecross-section for light 33

2.4.3 The view of the sky for a stationary observer 34

2.5 Classical tests 36

2.6 Palling into a black hole 36

2.6.1 Free-fall time for a distant observer 37

2.6.2 Light-travel time 38

2.6.3 What the external observer sees 38

2.6.4 An infalling observer's time 39

2.6.5 What the infalling observer feels 40

2.7 Capture by a black hole 41

2.7.1 Case I: Capture of high angular momentum particles 41

2.7.2 Case II: Capture of low energy particles 42

2.8 Surface gravity of a black hole 42

2.8.1 The proper acceleration of a hovering observer 42

2.8.2 Surface gravity 43

2.8.3 Rindler coordinates 44

2.9 Other coordinates 46

2.9.1 Null coordinates 46

2.9.2 Eddington-Finkelstein coordinates 47

2.10 Inside the black hole 47

2.10.1 The infalling observer 49

2.11 White holes 50

2.12 Kruskal coordinates 50

2.12.1 The singularities at r = 0 and cosmic censorship 52

2.12.2 The spacetime of a collapsing star 52

2.13 Embedding diagrams 53

2.14 Asymptotic flatness 55

2.14.1 The Penrose-Carter diagram for the Schwarzschild metric 57

2.14.2 The Penrose-Carter diagram for the Newtonian metric 57

2.15 Non-isolated black holes 58

2.15.1 The infinite redshift surface 58

2.15.2 Trapped surfaces 58

2.15.3 Apparent horizon 58

2.16 The membrane paradigm 59

3 Rotating Black Holes 61

3.1 The Kerr metric 62

3.2 The event horizon 62

3.2.1 The circumference of the event horizon 63

3.2.2 The area of the event horizon 63

3.3 Properties of the Kerr metric coefficients 64

3.3.1 Identities 64

3.3.2 Contravariant components 64

3.4 Interpretation of m, a and geometric units 65

3.5 Extreme Kerr black hole 66

3.6 Robinson's theorem 66

3.7 Particle orbits in the Kerr geometry 66

3.7.1 Constants of the motion 67

3.7.2 Energy 68

3.7.3 Angular momentum 69

3.7.4 The Carter integral 69

3.7.5 The radial equation 70

3.7.6 The effective potential 71

3.8 Frame-dragging 72

3.8.1 Free fall with zero angular momentum 73

3.8.2 Orbits with non-zero angular momentum 74

3.9 Zero angular momentum observers (ZAMOs) 75

3.9.1 Some applications of ZAMOs 76

3.10 Photon orbits 77

3.10.1 The photon effective potential 77

3.10.2 Azimuthal motion 78

3.10.3 Photon capture cross-section 78

3.11 The static limit surface 79

3.12 The infinite redshift surface 81

3.13 Circular orbits in the equatorial plane 81

3.13.1 Innermost (marginally) stable circular orbit 82

3.13.2 Period of a circular orbit 84

3.13.3 Energy of the innermost stable orbit 85

3.13.4 Angular momentum of the innermost stable orbit 85

3.13.5 Marginally bound orbits 86

3.13.6 Unbound orbits 86

3.14 Polar orbits 87

3.14.1 Orbital period 90

3.15 The ergosphere 90

3.15.1 Negative energy orbits 90

3.15.2 Energy and angular momentum 92

3.15.3 The Penrose process 93

3.15.4 Realising the Penrose process 93

3.16 Spinning up a black hole 95

3.16.1 From Schwarzschild to extreme Kerr black hole 96

3.17 Other coordinates 97

3.18 Penrose-Carter diagram 99

3.18.1 Interior solutions and collapsing stars 102

3.19 Closed timelike lines 103

3.20 Charged black holes 104

4 Black Hole Thermodynamics 107

4.1 Black hole mechanics 107

4.1.1 Surface gravity 107

4.1.2 Redshift 109

4.1.3 Conservation of energy 110

4.2 The area of a Kerr black hole horizon cannot decrease 111

4.2.1 Area change by accretion 111

4.2.2 Area change produced by the Penrose process 112

4.2.3 The area theorem 113

4.2.4 Irreducible mass 113

4.2.5 Maximum energy extraction 114

4.2.6 Naked singularities 114

4.3 Scattering of waves 115

4.3.1 Superradiance 115

4.4 Thermodynamics 120

4.4.1 Horizon temperature 122

4.4.2 The four laws of black hole thermodynamics 124

4.5 Hawking radiation 125

4.5.1 Introduction 125

4.5.2 Casimir effect 126

4.5.3 Thermal vacua in accelerated frames 127

4.5.4 Hawking radiation 131

4.6 Properties of radiating black holes 135

4.6.1 Entropy and temperature 135

4.6.2 Radiating black holes 136

4.6.3 Black hole in a box 137

4.7 Entropy and microstates 139

5 Wormholes and Time Travel 141

5.1 Introduction 141

5.2 Wormholes 141

5.3 Traversible wormholes 143

5.4 Creating a wormhole 145

5.5 Weak energy condition 146

5.6 Exotic matter 147

5.7 Time machines 148

5.8 Chronology protection 150

6 Astrophysical Black Holes 151

6.1 Introduction 151

6.2 Stellar mass black holes 152

6.2.1 Formation 152

6.2.2 Finding stellar mass black holes 154

6.2.3 The black hole at the centre of the Galaxy 155

6.3 Supermassive black holes in other galaxies 156

6.3.1 Intermediate mass black holes 157

6.3.2 Mini black holes 158

6.4 Further evidence for black hole spin 159

6.5 Conclusions 160

Solutions to Problems 163

References 189

Bibliography 193

Index 195

## Interviews

Preface v

1 Relativistic Gravity 1

1.1 What is a black hole? 1

1.2 Why study black holes? 3

1.3 Elements of general relativity 3

1.3.1 The principle of equivalence 3

1.3.2 The Newtonian affine connection 4

1.3.3 Newtonian gravity 5

1.3.4 Metrics in relativity 6

1.3.5 The velocity and momentum 4-vector 7

1.3.6 General vectors and tensors 8

1.3.7 Locally measured physical quantities 9

1.3.8 Derivatives in relativity 9

1.3.9 Acceleration 4-vector 11

1.3.10 Paths of light 11

1.3.11 Einstein's field equations 11

1.3.12 Symmetry and Killing's equation 12

2 Spherical Black Holes 15

2.1 The Schwarzschild metric 15

2.1.1 Coordinates 16

2.1.2 Proper distance 17

2.1.3 Proper time 17

2.1.4 Redshift 17

2.1.5 Interpretation of M and geometric units 18

2.1.6 The Schwarzschild radius 18

2.1.7 The event horizon 19

2.1.8 Birkoff's theorem 19

2.1.9 Israel's theorem 19

2.2 Orbits in Newtonian gravity 20

2.2.1 Newtonian Energy 20

2.2.2 Angular momentum 20

2.2.3 The Newtonian effective potential 20

2.2.4 Classification of Newtonian orbits 21

2.3 Particle orbits in the Schwarzschild metric 21

2.3.1 Constants of the motion 22

2.3.2 Conserved Energy 22

2.3.3 Angular momentum 23

2.3.4 The effective potential 24

2.3.5 Newtonian approximation to the metric 24

2.3.6 Classification of orbits 25

2.3.7 Radial infall 25

2.3.8 The locally measured energy of a particle 26

2.3.9 Circular orbits 27

2.3.10 Comparison with Newtonian orbits 28

2.3.11 Orbital velocity in the frame of a hovering observer 29

2.3.12 Energy in the last stable orbit 30

2.4 Orbits of light rays 31

2.4.1 Radial propagation of light 32

2.4.2 Capturecross-section for light 33

2.4.3 The view of the sky for a stationary observer 34

2.5 Classical tests 36

2.6 Palling into a black hole 36

2.6.1 Free-fall time for a distant observer 37

2.6.2 Light-travel time 38

2.6.3 What the external observer sees 38

2.6.4 An infalling observer's time 39

2.6.5 What the infalling observer feels 40

2.7 Capture by a black hole 41

2.7.1 Case I: Capture of high angular momentum particles 41

2.7.2 Case II: Capture of low energy particles 42

2.8 Surface gravity of a black hole 42

2.8.1 The proper acceleration of a hovering observer 42

2.8.2 Surface gravity 43

2.8.3 Rindler coordinates 44

2.9 Other coordinates 46

2.9.1 Null coordinates 46

2.9.2 Eddington-Finkelstein coordinates 47

2.10 Inside the black hole 47

2.10.1 The infalling observer 49

2.11 White holes 50

2.12 Kruskal coordinates 50

2.12.1 The singularities at r = 0 and cosmic censorship 52

2.12.2 The spacetime of a collapsing star 52

2.13 Embedding diagrams 53

2.14 Asymptotic flatness 55

2.14.1 The Penrose-Carter diagram for the Schwarzschild metric 57

2.14.2 The Penrose-Carter diagram for the Newtonian metric 57

2.15 Non-isolated black holes 58

2.15.1 The infinite redshift surface 58

2.15.2 Trapped surfaces 58

2.15.3 Apparent horizon 58

2.16 The membrane paradigm 59

3 Rotating Black Holes 61

3.1 The Kerr metric 62

3.2 The event horizon 62

3.2.1 The circumference of the event horizon 63

3.2.2 The area of the event horizon 63

3.3 Properties of the Kerr metric coefficients 64

3.3.1 Identities 64

3.3.2 Contravariant components 64

3.4 Interpretation of m, a and geometric units 65

3.5 Extreme Kerr black hole 66

3.6 Robinson's theorem 66

3.7 Particle orbits in the Kerr geometry 66

3.7.1 Constants of the motion 67

3.7.2 Energy 68

3.7.3 Angular momentum 69

3.7.4 The Carter integral 69

3.7.5 The radial equation 70

3.7.6 The effective potential 71

3.8 Frame-dragging 72

3.8.1 Free fall with zero angular momentum 73

3.8.2 Orbits with non-zero angular momentum 74

3.9 Zero angular momentum observers (ZAMOs) 75

3.9.1 Some applications of ZAMOs 76

3.10 Photon orbits 77

3.10.1 The photon effective potential 77

3.10.2 Azimuthal motion 78

3.10.3 Photon capture cross-section 78

3.11 The static limit surface 79

3.12 The infinite redshift surface 81

3.13 Circular orbits in the equatorial plane 81

3.13.1 Innermost (marginally) stable circular orbit 82

3.13.2 Period of a circular orbit 84

3.13.3 Energy of the innermost stable orbit 85

3.13.4 Angular momentum of the innermost stable orbit 85

3.13.5 Marginally bound orbits 86

3.13.6 Unbound orbits 86

3.14 Polar orbits 87

3.14.1 Orbital period 90

3.15 The ergosphere 90

3.15.1 Negative energy orbits 90

3.15.2 Energy and angular momentum 92

3.15.3 The Penrose process 93

3.15.4 Realising the Penrose process 93

3.16 Spinning up a black hole 95

3.16.1 From Schwarzschild to extreme Kerr black hole 96

3.17 Other coordinates 97

3.18 Penrose-Carter diagram 99

3.18.1 Interior solutions and collapsing stars 102

3.19 Closed timelike lines 103

3.20 Charged black holes 104

4 Black Hole Thermodynamics 107

4.1 Black hole mechanics 107

4.1.1 Surface gravity 107

4.1.2 Redshift 109

4.1.3 Conservation of energy 110

4.2 The area of a Kerr black hole horizon cannot decrease 111

4.2.1 Area change by accretion 111

4.2.2 Area change produced by the Penrose process 112

4.2.3 The area theorem 113

4.2.4 Irreducible mass 113

4.2.5 Maximum energy extraction 114

4.2.6 Naked singularities 114

4.3 Scattering of waves 115

4.3.1 Superradiance 115

4.4 Thermodynamics 120

4.4.1 Horizon temperature 122

4.4.2 The four laws of black hole thermodynamics 124

4.5 Hawking radiation 125

4.5.1 Introduction 125

4.5.2 Casimir effect 126

4.5.3 Thermal vacua in accelerated frames 127

4.5.4 Hawking radiation 131

4.6 Properties of radiating black holes 135

4.6.1 Entropy and temperature 135

4.6.2 Radiating black holes 136

4.6.3 Black hole in a box 137

4.7 Entropy and microstates 139

5 Wormholes and Time Travel 141

5.1 Introduction 141

5.2 Wormholes 141

5.3 Traversible wormholes 143

5.4 Creating a wormhole 145

5.5 Weak energy condition 146

5.6 Exotic matter 147

5.7 Time machines 148

5.8 Chronology protection 150

6 Astrophysical Black Holes 151

6.1 Introduction 151

6.2 Stellar mass black holes 152

6.2.1 Formation 152

6.2.2 Finding stellar mass black holes 154

6.2.3 The black hole at the centre of the Galaxy 155

6.3 Supermassive black holes in other galaxies 156

6.3.1 Intermediate mass black holes 157

6.3.2 Mini black holes 158

6.4 Further evidence for black hole spin 159

6.5 Conclusions 160

Solutions to Problems 163

References 189

Bibliography 193

Index 195

## Recipe

Preface v

1 Relativistic Gravity 1

1.1 What is a black hole? 1

1.2 Why study black holes? 3

1.3 Elements of general relativity 3

1.3.1 The principle of equivalence 3

1.3.2 The Newtonian affine connection 4

1.3.3 Newtonian gravity 5

1.3.4 Metrics in relativity 6

1.3.5 The velocity and momentum 4-vector 7

1.3.6 General vectors and tensors 8

1.3.7 Locally measured physical quantities 9

1.3.8 Derivatives in relativity 9

1.3.9 Acceleration 4-vector 11

1.3.10 Paths of light 11

1.3.11 Einstein's field equations 11

1.3.12 Symmetry and Killing's equation 12

2 Spherical Black Holes 15

2.1 The Schwarzschild metric 15

2.1.1 Coordinates 16

2.1.2 Proper distance 17

2.1.3 Proper time 17

2.1.4 Redshift 17

2.1.5 Interpretation of M and geometric units 18

2.1.6 The Schwarzschild radius 18

2.1.7 The event horizon 19

2.1.8 Birkoff's theorem 19

2.1.9 Israel's theorem 19

2.2 Orbits in Newtonian gravity 20

2.2.1 Newtonian Energy 20

2.2.2 Angular momentum 20

2.2.3 The Newtonian effective potential 20

2.2.4 Classification of Newtonian orbits 21

2.3 Particle orbits in the Schwarzschild metric 21

2.3.1 Constants of the motion 22

2.3.2 Conserved Energy 22

2.3.3 Angular momentum 23

2.3.4 The effective potential 24

2.3.5 Newtonian approximation to the metric 24

2.3.6 Classification of orbits 25

2.3.7 Radial infall 25

2.3.8 The locally measured energy of a particle 26

2.3.9 Circular orbits 27

2.3.10 Comparison with Newtonian orbits 28

2.3.11 Orbital velocity in the frame of a hovering observer 29

2.3.12 Energy in the last stable orbit 30

2.4 Orbits of light rays 31

2.4.1 Radial propagation of light 32

2.4.2 Capturecross-section for light 33

2.4.3 The view of the sky for a stationary observer 34

2.5 Classical tests 36

2.6 Palling into a black hole 36

2.6.1 Free-fall time for a distant observer 37

2.6.2 Light-travel time 38

2.6.3 What the external observer sees 38

2.6.4 An infalling observer's time 39

2.6.5 What the infalling observer feels 40

2.7 Capture by a black hole 41

2.7.1 Case I: Capture of high angular momentum particles 41

2.7.2 Case II: Capture of low energy particles 42

2.8 Surface gravity of a black hole 42

2.8.1 The proper acceleration of a hovering observer 42

2.8.2 Surface gravity 43

2.8.3 Rindler coordinates 44

2.9 Other coordinates 46

2.9.1 Null coordinates 46

2.9.2 Eddington-Finkelstein coordinates 47

2.10 Inside the black hole 47

2.10.1 The infalling observer 49

2.11 White holes 50

2.12 Kruskal coordinates 50

2.12.1 The singularities at r = 0 and cosmic censorship 52

2.12.2 The spacetime of a collapsing star 52

2.13 Embedding diagrams 53

2.14 Asymptotic flatness 55

2.14.1 The Penrose-Carter diagram for the Schwarzschild metric 57

2.14.2 The Penrose-Carter diagram for the Newtonian metric 57

2.15 Non-isolated black holes 58

2.15.1 The infinite redshift surface 58

2.15.2 Trapped surfaces 58

2.15.3 Apparent horizon 58

2.16 The membrane paradigm 59

3 Rotating Black Holes 61

3.1 The Kerr metric 62

3.2 The event horizon 62

3.2.1 The circumference of the event horizon 63

3.2.2 The area of the event horizon 63

3.3 Properties of the Kerr metric coefficients 64

3.3.1 Identities 64

3.3.2 Contravariant components 64

3.4 Interpretation of m, a and geometric units 65

3.5 Extreme Kerr black hole 66

3.6 Robinson's theorem 66

3.7 Particle orbits in the Kerr geometry 66

3.7.1 Constants of the motion 67

3.7.2 Energy 68

3.7.3 Angular momentum 69

3.7.4 The Carter integral 69

3.7.5 The radial equation 70

3.7.6 The effective potential 71

3.8 Frame-dragging 72

3.8.1 Free fall with zero angular momentum 73

3.8.2 Orbits with non-zero angular momentum 74

3.9 Zero angular momentum observers (ZAMOs) 75

3.9.1 Some applications of ZAMOs 76

3.10 Photon orbits 77

3.10.1 The photon effective potential 77

3.10.2 Azimuthal motion 78

3.10.3 Photon capture cross-section 78

3.11 The static limit surface 79

3.12 The infinite redshift surface 81

3.13 Circular orbits in the equatorial plane 81

3.13.1 Innermost (marginally) stable circular orbit 82

3.13.2 Period of a circular orbit 84

3.13.3 Energy of the innermost stable orbit 85

3.13.4 Angular momentum of the innermost stable orbit 85

3.13.5 Marginally bound orbits 86

3.13.6 Unbound orbits 86

3.14 Polar orbits 87

3.14.1 Orbital period 90

3.15 The ergosphere 90

3.15.1 Negative energy orbits 90

3.15.2 Energy and angular momentum 92

3.15.3 The Penrose process 93

3.15.4 Realising the Penrose process 93

3.16 Spinning up a black hole 95

3.16.1 From Schwarzschild to extreme Kerr black hole 96

3.17 Other coordinates 97

3.18 Penrose-Carter diagram 99

3.18.1 Interior solutions and collapsing stars 102

3.19 Closed timelike lines 103

3.20 Charged black holes 104

4 Black Hole Thermodynamics 107

4.1 Black hole mechanics 107

4.1.1 Surface gravity 107

4.1.2 Redshift 109

4.1.3 Conservation of energy 110

4.2 The area of a Kerr black hole horizon cannot decrease 111

4.2.1 Area change by accretion 111

4.2.2 Area change produced by the Penrose process 112

4.2.3 The area theorem 113

4.2.4 Irreducible mass 113

4.2.5 Maximum energy extraction 114

4.2.6 Naked singularities 114

4.3 Scattering of waves 115

4.3.1 Superradiance 115

4.4 Thermodynamics 120

4.4.1 Horizon temperature 122

4.4.2 The four laws of black hole thermodynamics 124

4.5 Hawking radiation 125

4.5.1 Introduction 125

4.5.2 Casimir effect 126

4.5.3 Thermal vacua in accelerated frames 127

4.5.4 Hawking radiation 131

4.6 Properties of radiating black holes 135

4.6.1 Entropy and temperature 135

4.6.2 Radiating black holes 136

4.6.3 Black hole in a box 137

4.7 Entropy and microstates 139

5 Wormholes and Time Travel 141

5.1 Introduction 141

5.2 Wormholes 141

5.3 Traversible wormholes 143

5.4 Creating a wormhole 145

5.5 Weak energy condition 146

5.6 Exotic matter 147

5.7 Time machines 148

5.8 Chronology protection 150

6 Astrophysical Black Holes 151

6.1 Introduction 151

6.2 Stellar mass black holes 152

6.2.1 Formation 152

6.2.2 Finding stellar mass black holes 154

6.2.3 The black hole at the centre of the Galaxy 155

6.3 Supermassive black holes in other galaxies 156

6.3.1 Intermediate mass black holes 157

6.3.2 Mini black holes 158

6.4 Further evidence for black hole spin 159

6.5 Conclusions 160

Solutions to Problems 163

References 189

Bibliography 193

Index 195