## The origin of black hole entropy by Mukohyama, S

By Mukohyama, S

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Here N1 , N¯1 , N5 , N¯5 , nL and nR are defined by Q1 = N1 − N¯1 , 34 CHAPTER 3. BLACK HOLE ENTROPY Q5 = N = Pψ = Pχ = E = N5 − N¯5 , nL − nR , RV 1 R (N5 + N¯5 ) + (nL + nR ), − (N1 + N¯1 ) − 2g 2g R RV R (N1 + N¯1 ) − (N5 + N¯5 ), g g R RV 1 (N1 + N¯1 ) + (N5 + N¯5 ) + (nL + nR ). 9) The scales of compactification are written in terms of these quantities as R = g 2 nL nR N1 N¯1 V = N1 N¯1 N5 N¯5 1/4 , 1/2 . 10) The black brane solution characterized by (N1 , N¯1 , N5 , N¯5 , nL , nR ) can be interpreted as a configuration of strings and solitons in the type IIB superstring theory [68], provided that interactions among string and soliton can be neglected.

84) j=1 pi and Aij ≡ ˜i|T (|j j|)|˜i . Aij has the following properties: ∞ 0 ≤ Aij ≤ 1. Aij = 1, i=1 Similarly it is shown that n Aij qj /bn , q˜i = lim n→∞ where bn ≡ n i=1 qi . 84) and the continuity of f , it is shown that   n pj /an  f (˜ pi /˜ qi ) = lim f  Aij . 85) Next define Cin and C˜in by n Cin ≡ Aij qj /˜ qi , j=1 C˜in ≡ Cin /an , then the convex property of f means n f (˜ pi /˜ qi ) ≤ lim n→∞ since n j=1 j=1 Aij qj ˜ n f (Ci pj /qj ) Cin q˜i Aij qj Aij qj = 1, n ≥ 0. 86) CHAPTER 2.

3 Canonical ensemble of open strings The Hawking process can be described in the D-brane picture as a decay process of non-BPS excitations of D-branes [70, 72, 73]. A collision of a right-moving open string with a left-moving one on the D-branes results in an emission of a closed string leaving away from the D-branes, which is interpreted as Hawking radiation. The spectrum of the emission of the closed string can, in principle, be obtained from decay rates of the string, provided that the initial state of the open strings on the D-branes is given.