Contents
Preface vii
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.S calar Wave Equation in a Schwarzschild Background 25
2.1 Near the Horizon . . . . . . . . . . . . . . . . . . . . . . . . 28
3.Q uantum 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.E ntropy of the Free Quantum Field in Rindler Space 43
4.1 Black Hole Evaporation .. .. .. . .. .. .. . .. .. .. 48
5.T hermodynamics of Black Holes 51
6.C harged Black Holes 55
7.The Stretched Horizon 61
8.T he 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.T he 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.Ho rizons 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.T he 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.Bl ack 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.S trings 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.En tropy of Strings and Black Holes 165
Conclusions 175
Bibliography 179
Index 181
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