Contents
Preface vii
Part 1: Black Holes and Quantum Mechanics 1
1. The Schwarzschild Black Hole 3
1.1 Schwarzschild Coordinates 3
1.2 Tortoise Coordinates 7
1.3 Near Horizon Coordinates (Rindler space) 8
1.4 Kruskal-Szekeres Coordinates 10
1.5 Penrose Diagrams 14
1.6 Formation of a Black Hole 15
1.7 Fidos and Frefos and the Equivalence Principle 21
2. Scalar Wave Equation in a Schwarzschild Background 25
2.1 Near the Horizon 28
3. Quantum Fields in Rindler Space 31
3.1 Classical Fields 31
3.2 Entanglement 32
3.3 Review of the Density Matrix 34
3.4 The Unruh Density Matrix 36
3.5 Proper Temperature 39
4. Entropy of the Free Quantum Field in Rindler Space 43
4.1 Black Hole Evaporation 48
5. Thermodynamics of Black Holes 51
6. Charged Black Holes 55
7. The Stretched Horizon 61
8. The Laws of Nature 69
8.1 Information Conservation 69
8.2 Entanglement Entropy 71
8.3 Equivalence Principle 77
8.4 Quantum Xerox Principle 79
9. The Puzzle of Information Conservation in Black Hole Environments 81
9.1 A Brick Wall? 84
9.2 Black Hole Complementarity 85
9.3 Baryon Number Violation 89
10. Horizons and the UV/IR Connection 95
Part 2: Entropy Bounds and Holography 99
11. Entropy Bounds 101
11.1 Maximum Entropy 101
11.2 Entropy on Light-like Surfaces 105
11.3 Friedman-Robertson-Walker Geometry 110
11.4 Bousso's Generalization 114
11.5 de Sitter Cosmology 119
11.6 Anti de Sitter Space 123
12. The Holographic Principle and Anti de Sitter Space 127
12.1 The Holographic Principle 127
12.2 AdS Space 128
12.3 Holography in AdS Space 130
12.4 The AdS/CFT Correspondence 133
12.5 The Infrared Ultraviolet Connection 135
12.6 Counting Degrees of Freedom 138
13. Black Holes in a Box 141
13.1 The Horizon 144
13.2 Information and the AdS Black Hole 144
Part 3: Black Holes and Strings 149
14. Strings 151
14.1 Light Cone Quantum Mechanics 153
14.2 Light Cone String Theory 156
14.3 Interactions 159
14.4 Longitudinal Motion 161
15. Entropy of Strings and Black Holes 165
Conclusions 175
Bibliography 179
Index 181