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Part 1: Black Holes and Quantum Mechanics 1

1.T he 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 PenroseDiagrams . . . . . . . . . . . . . . . . . . . . . . . 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 DensityMatrix . . . . . . . . . . . . . . . . . 34
3.4 The Unruh DensityMatrix . . . . . . . . . . . . . . . . . . 36
3.5 Proper Temperature . . . . . . . . . . . . . . . . . . . . . . 39
4.Entropy of the Free Quantum Field in Rindler Space 43
4.1 Black Hole Evaporation .. .. .. . .. .. .. . .. .. .. 48
xiii
xiv Black Holes, Information, and the String Theory Revolution
5.Thermodynamics of Black Holes 51
6.C harged 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 QuantumXerox Principle . . . . . . . . . . . . . . . . . . 79
9.The Puzzle of Information Conservation in Black Hole
Environments 81
9.1 A BrickWall? . . . . . . . . . . . . . . . . . . . . . . . . . 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.E ntropy 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
Contents xv
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



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