String Theory Demystified

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CHAPTER 14 Black Holes 249 Entropy Calculations for Black Holes with String Theory We will consider two cases, a Schwarzschild black hole where we use a somewhat loose heuristic estimate and a calculation for a fi ve-dimensional charged black hole. In string theory, the calculation of the entropy of a black hole is easiest in fi ve dimensions. This is due to an amazing property of fi ve-dimensional black holes that arises from supersymmetry. It turns out that supersymmetry allows us to count up string states while taking the string-coupling constant gs = 0 , which amounts to considering a set of noninteracting strings—something not really possible in a black hole. This greatly simplifi es the calculation and what is remarkable is that the result obtained with no coupling is valid for any coupling strength g . s Remember the adiabatic theorem? The procedure used here is something you already know about from ordinary quantum mechanics. In quantum mechanics, you can disturb a system adiabatically so that the energy levels are not disturbed. The adiabatic method is used here—the string-coupling constant is varied adiabatically so that large gravitational forces transition to a weak regime. Entropy, however, is an adiabatic invariant. So while we weaken the string coupling, the entropy remains the same as long as things are done adiabatically. In string theory, we start with a collection of highly coupled strings and then let the coupling gs → 0 slowly. We work in the usual D = 10 space-time of superstring theory, and need to compactify some extra dimensions to get an effective fi vedimensional space-time. Supersymmetry remains unbroken if we compactify 5 9 dimensions into tiny circles. We compactify the dimensions x , ..., x leaving us with the remaining fi ve-dimensional space-time described by the coordinates 0 1 2 3 4 x , x , x , x , x . The black hole can actually be thought of as two objects—a string carrying a charge Q 1 and a 5-brane with charge Q 2 . These charges are winding modes as we will see below. First we consider an adiabatic process g s → 0 applied to a Schwarzschild black hole in D dimensions. A straightforward calculation using the laws of black hole mechanics shows that the entropy is given by D−2 A D 3 S = = mBG 4G D D D 1 − −3 Ω (14.33) where m B is the mass of the black hole. The entropy can be estimated quite simply from string theory considerations. For an excited string, the entropy is proportional to the product of the mass times its length: D−2 S ∝ m (14.34) s
250 String Theory Demystifi ed Now let’s use the string theory considerations to fi nd the entropy. This is done by applying the adiabatic procedure to the string coupling g s → 0 and noting that the entropy remains unchanged during the process. We follow the procedure outlined in Susskind (please see References). What is found is that as g s → 0 , the black hole turns into a single string which has the same entropy as the black hole. The black hole transitions into a single string when the radius of the horizon is the same as the fundamental string length s. The Planck length p can be related to the fundamental string length using: 2 p s 2 s (14.35) D− = g The Schwarzschild radius of the black hole is given in terms of its mass as r = ( m G ) s B D We can approximately take the gravitational constant to be So that: rs s 1 D−3 (14.36) 2 D−2 G ≈ g (14.37) D s s 1 2 D 3 (14.38) s B s = ( ) − mg We are going to be interested in r → 1, that is, the point where the s s Schwarzschild radius approaches the string length. During the adiabatic process, the mass becomes a function of the string-coupling constant, m= m( g ). If we take s the initial coupling to be gB then m = m( g ). The entropy is a function of the B B product mg ( ) , and since the entropy is an adiabatic invariant, this product must s p be a constant. Using Eq. (14.35) we can write mg m g ⎛ ⎞ ( )= s B⎜ ⎝ g ⎟ ⎠ 2 B 2 s 1 D−2 D−2 D−3 Now, r → 1 when mg ( ) = which gives s s s p s (14.39) mg ( ) = s s g 1 2 (14.40) s
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Accounting Demystified Advanced Cal
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Copyright © 2009 by The McGraw-Hil
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For more information about this tit
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Contents vii BRST in String Theory-
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Contents ix CHAPTER 14 Black Holes
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PREFACE String theory is the greate
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Preface xiii time as this one to he
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CHAPTER 1 Introduction General rela
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CHAPTER 1 Introduction 5 we say tha
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CHAPTER 1 Introduction 11 So if α
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CHAPTER 1 Introduction 13 Figure 1.
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CHAPTER 1 Introduction 15 • Type
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CHAPTER 1 Introduction 17 Close up,
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CHAPTER 1 Introduction 19 5. The mi
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CHAPTER 2 The Classical String I: E
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CHAPTER 2 Equations of Motion 23 To
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CHAPTER 2 Equations of Motion 25 No
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CHAPTER 2 Equations of Motion 29 ct
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CHAPTER 2 Equations of Motion 31 Th
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CHAPTER 2 Equations of Motion 33 is
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CHAPTER 2 Equations of Motion 35 Si
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CHAPTER 2 Equations of Motion 37 ho
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CHAPTER 2 Equations of Motion 39 Th
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CHAPTER 2 Equations of Motion 41 wh
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CHAPTER 2 Equations of Motion 43 In
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CHAPTER 2 Equations of Motion 47 We
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CHAPTER 2 Equations of Motion 49 In
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CHAPTER 3 The Classical String II:
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CHAPTER 3 Symmetries and Worldsheet
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CHAPTER 4 String Quantization At th
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CHAPTER 4 String Quantization Here,
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CHAPTER 4 String Quantization That
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CHAPTER 4 String Quantization Using
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CHAPTER 4 String Quantization expec
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CHAPTER 4 String Quantization † A
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CHAPTER 4 String Quantization To ob
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CHAPTER 4 String Quantization lower
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CHAPTER 4 String Quantization this)
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CHAPTER 4 String Quantization Norma
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CHAPTER 5 Conformal Field Theory Pa
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CHAPTER 6 BRST Quantization So far
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CHAPTER 6 BRST Quantization 117 It
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CHAPTER 6 BRST Quantization 119 Cal
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CHAPTER 6 BRST Quantization 121 The
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CHAPTER 6 BRST Quantization 123 To
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CHAPTER 6 BRST Quantization 125 The
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CHAPTER 7 RNS Superstrings The real
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CHAPTER 7 RNS Superstrings 129 EXAM
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CHAPTER 7 RNS Superstrings 131 SOLU
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CHAPTER 7 RNS Superstrings 133 In t
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CHAPTER 7 RNS Superstrings 135 Now,
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CHAPTER 7 RNS Superstrings 137 We c
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CHAPTER 7 RNS Superstrings 139 We o
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CHAPTER 7 RNS Superstrings 141 OPEN
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CHAPTER 7 RNS Superstrings 143 If w
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CHAPTER 7 RNS Superstrings 145 The
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CHAPTER 7 RNS Superstrings 147 From
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CHAPTER 7 RNS Superstrings 151 3. A
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CHAPTER 8 Compactifi cation and T-D
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CHAPTER 9 Superstring Theory Contin
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CHAPTER 10 A Summary of Superstring
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CHAPTER 11 Type II String Theories
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CHAPTER 11 Type II String Theories
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Page 222 and 223: CHAPTER 12 Heterotic String Theory
Page 236 and 237: CHAPTER 13 D-Branes One of the most
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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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Page 312 and 313: Books References Becker, K., M. Bec
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|NS〉 ⊕ |NS〉 Sector, 203 |NS
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INDEX 301 Cyclic model, 277 Cylinde
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INDEX 303 Mechanics, quantum. See Q
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INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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