Table of Contents

The most beautiful physical theory Carlo Rovelli 1

Astrophysical black holes Andrew C. Fabian Anthony N. Lasenby 7

1 Introduction 7

2 A brief history of astrophysical black holes 8

2.1 Early history 8

2.2 The Schwarzschild metric 9

3 Relativistic astrophysics emerges 11

3.1 Rotating black holes 13

3.2 Black holes as energy sources 13

3.3 Motion in the Schwarzschild metric 14

3.4 Circular orbits 16

3.5 Stability of circularorbits 17

4 Evidence from X-rays, quasars and AGN 20

5 The black hole at the centre of the Milky Way 23

5.1 Sgr A* 23

5.2 GR effects on orbits 25

6 Galaxies and black holes 29

6.1 AGN feedback 29

6.2 Jets, Gamma-Ray Bursts and the birth of black holes 32

7 Current observations of accreting black holes 33

7.1 Particle motion in the Kerr metric 34

7.2 Current observations of accreting black holes continued 37

7.3 Velocities and frequencies 39

7.4 Further AGN properties 40

8 Measurements of the masses of black holes 41

9 Measurements of black hole spin 42

9.1 Equations for photon motion and redshift 43

9.2 Light bending around a Kerr black hole 46

9.3 Iron line emission 46

10 Future observations of astrophysical black holes 49

11 More general spherically symmetric black holes 52

12 Primordial black holes 54

12.1 Hawking radiation 54

12.2 Link with surface gravity 55

12.3 Astrophysical aspects of black hole evaporation 56

12.4 Black hole entropy 57

12.5 Laws of black hole thermodynamics and the Penrose process 58

12.6 Adiabatic (reversible) changes 60

12.7 Other processes for extracting energy from a spinning black hole 60

13 Conclusions 62

Observations of General Relativity at strong and weak limits Gene G. Byrd Arthur Chernin Pekka Teerikorpi Mauri Valtonen 67

1 Introduction 67

2 Tests in the solar system and binary systems 69

2.1 Orbit precession 69

2.2 Gravitational waves 70

2.3 Lense-Thirring effect and relativistic spin-orbit coupling 71

2.4 Bendingof light rays and gravitational redshift 71

2.5 Massive spinning black hole test 72

3 Observational discovery of a non-zero cosmological constant (dark energy) 73

4 Weak limit test near zero gravity surface 74

4.1 Dark energy antigravity as a test of General Relativity 74

4.2 Local dark energy test via outflow 74

4.3 Dynamical structure of a gravitating system within dark energy 77

5 Estimating cosmologically nearby dark energy: the Local Group 79

6 Mass, dark energy density and the lost gravity effect 81

7 Dark energy in the Coma Cluster of galaxies 82

8 Testing the constancy of Λ 82

9 Strong limit: Spinning black holes and no-hairtheorem 85

10 OJ287 binary system 87

10.1 The binary model 87

10.2 OJ287 flares and jet 88

10.3 OJ287 orbit parameters (without using outburst times) 92

11 Modeling binaries with Post Newtonian methods (with outburst times) 94

12 OJ287 results at the strong field limit 94

13 Conclusions 95

14 Appended section; mass, dark energy density and the "lost gravity" effect 97

15 Appended section: dark energy in the Coma cluster of galaxies 101

15.1 Three mass estimates of the cluster 101

15.2 Matter mass profile 102

15.3 Upper limits and beyond 104

16 Appended section: modeling binaries with Post Newtonian methods 105

General Relativity and dragging of inertial frames Ignazio Ciufolini 125

1 Frame-dragging: the theory 126

1.1 Dragging of inertial frames and the origin of inertia 126

1.2 Dragging of inertial frames and the gravitomagnetic analogy 127

1.3 The gravitomagnetic formal analogy of General Relativity with electrodynamics 128

1.4 Dragging of inertial frames inside a hollow sphere 131

1.5 Frame-dragging phenomena on clocks and photons 132

1.6 Frame-dragging, time-delay and gravitational lensing 134

1.7 [An invariant characterisation of frame-dragging] 138

2 The need to further test General Relativity 140

2.1 The universe and the triumph General Relativity 140

2.2 The riddle of dark energy and dark matter 141

2.3 Unified theories, alternative gravitational theories and some limits of General Relativity 141

2.4 [Frame-dragging, Chern-Simons gravity and String theory] 142

3 The holy grail of experimental General Relativity and its observation with the LAG EOS satellites and Gravity Probe-B 145

4 The LARES space experiment 151

4.1 The LARES satellite, its structure and its orbit 151

4.2 The LARES satellite, General Relativity and geodesic motion 153

4.3 The LARES satellite and its preliminary orbital results 154

4.4 [LARES error analyses] 155

5 Conclusions 158

GNSS and other applications of General Relativity Neil Ashby 165

1 Introduction 165

2 Relativity principles 169

3 Astronomical and geocentric time scales 170

4 Earth-based time scales TT, TAI, UTC 174

5 Gravitational frequency shifts 176

6 Sagnac effect; realizing coordinate time 178

7 Relativistic effects on orbiting clocks 180

8 The eccentricity effect 183

9 Navigation on the rotating earth 183

10 Emission coordinates 186

11 JUNO and other missions 187

12 Summary 187

The strange world of quantum spacetime Carlo Rovelli 189

1 A world with no space 189

2 A world without time 191

3 Loop gravity 194

4 Quantum spacetime 198

5 Empirical evidence 198

Index 203

List of contributors 209