橡超弦理論はブラックホールの謎を解けるか？

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1999 3 (Can String Theory Solve the Puzzles of Black Holes?) 305-0801 1-1 makoto.natsuume@kek.jp D-brane 1 Schwarzschild 60 80 2 [1] 1 1 1

2 2 [2] 25 2.2 2 2.1 [7,8] Schwarzschild 2GM/c 2 Schwarzschild G M c 2 2

T κ de = TdS dm = κ 8πG da ds 0 da 0 T = 0 κ = 0 1: Schwarzschild A E M κ T κ S A 3

Bekenstein 74 Hawking Hawking κ κ Hawking T = hκ/2π 3 S BH = A 4G h Bekenstein - Hawking (1) S BH Hawking S BH S BH h h 0 S BH ultraviolet catastrophe Rayleigh - Jeans S BH 2.2 [9] Hawking Hawking pure state Hawking mixed state Einstein - Podolsky - Rosen 3 c = k B = 1 h = 1 (1) 4

Hawking 4 Hawking 3 Schwarzschild 4 A 5

[8] fundamental string g s G G gs 2l2 s l s g s Schwarzschild 2GM g s g s Schwarzschild Schwarzschild M? (2) M d string [10] d string e lsm (3) d BH d BH e A G e GM 2 (4) (1) 2GM M 6

2: Schwarzschild 7

g s 1 g s (3) Schwarzschild Schwarzschild l s l s well-defined GM l s S string l s M GM M GM 2 (5) G (3) (4) 10 58 (1) 10 77 10 80 [11] 93 Susskind 8

[12] Susskind Horowitz Polchinski [13] 4 Susskind 5 (5) 4.1 BPS Q Q M Q = M BPS BPS Q = M BPS BPS BPS Schwarzschild 9

Reissner-Nordström Q M Q < M Hawking Q = M Q = M Hawking extremal black hole BPS BPS BPS (2)? Q BPS (6) BPS BPS (6) Susskind 95 Sen [14] Sen 5 BPS BPS D-brane 4.2 D-brane D-brane [2, 15] D-brane p p = 0 p = 1 p = 2 membrane.. Dp-brane D-brane 10

D-brane BPS D-brane BPS 3: D-brane D-brane D-brane [2, 15] D-brane D-brane D-brane 1/(g s l s ) D-brane Q Q/(g s l s ) D-brane Schwarzschild 2GM O(g s Q) g s D-brane D-brane BPS? Q D-brane BPS (7) Strominger Vafa Reissner-Nordström BPS (1) 5 11

Strominger Vafa [16, 17] Reissner-Nordström 10 g s 1 φ g s = e φ Reissner-Nordström D-brane 4 Sen D-brane Reissner-Nordström BPS g s 1 BPS D-brane Q D-brane g s Q D-brane Schwarzschild O(g s Q) 4 D-brane D-brane Q g s g s Q 1 12

4: Callan Maldacena D-brane 5 8 R IIB D1 D1 D5 D5 D-brane 5 Q 1 D1 Q 5 D5 D1 D5 D1 10 x 5, x 6, x 7, x 8, x 9 5 dominant [18] 13

T 5 = T 4 S 1 D1 x 9 R n/r D-brane Q 1, Q 5, n ds 2 = λ 2/3 dt 2 + λ 1/3 (dr 2 + r 2 dω 2 3 ) (8) dω 2 3 λ = (1 + c 1Q 1 )(1 + c 5Q 5 )(1 + c nn r 2 r 2 ) (9) r2 c 1, c 5, c n G 5 c 1 c 5 c n = (4G 5 /π) 2 c 1 Q 1 = c 5 Q 5 = c n n = ρ 2 0 (8) Reissner-Nordström ds 2 = (1 ρ2 0 ρ 2)2 dt 2 + dρ2 (1 ρ2 0) + ρ 2 ρ2 dω 2 3 (10) 2 ρ 2 = r 2 +ρ 2 0 r = 0 6 S 3 2π 2 ( ) 3 (1) S BH = A = 2π Q 1 Q 5 n (11) 4G 5 D-brane D1 x 9 R D1 D5 4Q 1 Q 5 D-brane Q 1 = Q 5 = 1 D1 D5 D5 D1 D1 x 5, x 6, x 7, x 8 6 r = 0 r = 0 dω 2 3 r = 0 λ 1 + Q/r 2 14

d (3) 2πR n/r n d exp(2π cn 6 ) (12) c 1/2 c = 6Q 1 Q 5 S = 2π Q 1 Q 5 n (13) (11) 6 5 Hawking [16] D-brane Hawking D-brane D-brane BPS 6.1 15

1. 2. 3. Bekenstein - Hawking [2] Feynman 6.2 3 (1) Hawking [19] 16

[20] [21] Strominger D-brane Hawking D-brane D-brane Hawking D-brane BPS g s D-brane D-brane D-brane D-brane Hawking 17

[1] A. Strominger and C. Vafa: Phys. Lett. B379 (1996) 99, hepth/9601029 [A]. 7 [2] 53, 312, May 1998 [B]. [3] J. Polchinski: Rev. Mod. Phys. 68 (1996) 1245, hep-th/9607050 [B]. [4] Search and Discovery, Physics Today, Vol. 50, No. 3, 19, March 1997 [B]. [5] G. T. Horowitz: gr-qc/9604051; gr-qc/9704072 [C]; J. M. Maldacena: hep-th/9607235[c]; Nucl. Phys. Proc. Suppl. 61A (1998) 111, hep-th/9705078 [C]; A. W. Peet: hep-th/9712253 [C]; A. Sen: hep-th/9802051 [C];, No. 416, 51, Feb. 1998 [C]. [6] 52, 161, March 1997 [B]. [7], No. 422, 36, Aug. 1998 [B]. [8] R. M. Wald: gr-qc/9702022 [C]. [9] L. Susskind: Scientific American, Vol. 276, No. 4, 52, April 1997 [B]; J. Preskill: hep-th/9209058 [C]; D. N. Page: hep-th/9305040 [C]. [10] M. B. Green, J. H. Schwarz, and E. Witten: Superstring theory (Cambridge Univ. Press, 1987); J. Polchinski: String theory (Cambridge Univ. Press, 1998). 7 18

[11] G. t Hooft: Nucl. Phys. B335 (1990) 138 [A]. [12] L. Susskind: hep-th/9309145 [A]. [13] G. T. Horowitz and J. Polchinski: Phys. Rev. D57 (1998) 2557, hepth/9707170 [A]; Phys. Rev. D55 (1997) 6189, hep-th/9612146 [A]. [14] A. Sen: Mod. Phys. Lett. A10 (1995) 2081, hep-th/9504147 [A]. [15] J. Polchinski: Phys. Rev. Lett. 75 (1995) 4724, hep-th/9510017 [A]; J. Polchinski, S. Chaudhuri, and C. V. Johnson: hep-th/9602052 [C]; J. Polchinski: hep-th/9611050 [C]. [16] C. G. Callan and J. M. Maldacena: Nucl. Phys. B472 (1996) 591, hep-th/9602043 [A]. [17] G. Horowitz and A. Strominger: Phys. Rev. Lett. 77 (1996) 2368, hepth/9602051 [A]. [18] J. M. Maldacena and L. Susskind: Nucl. Phys. B475 (1996) 679, hepth/9604042 [A]. [19] G. W. Gibbons and S. W. Hawking: Phys. Rev. D15 (1977) 2752 [A]. [20] A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov: Phys. Rev. Lett. 80 (1998) 904, gr-qc/9710007 [A]. [21] A. Strominger: J. High Energy Phys. 2 (1998) 9, hep-th/9712251 [A]. 19

[1] [1] Hawking 8 Feynman 8 Weinberg [2] Polchinski Fermi liquid 20

[1] private communications. [2] S. Weinberg: Prog. Theor. Phys. Suppl. 86 (1986) 43. [3] J. Polchinski: in Recent Directions in Particle Theory, Proceedings of the 1992 TASI, eds. J. Harvey and J. Polchinski (World Scientific, Singapore, 1993) hep-th/9210046. [4] J. Fröhlich : private communications. [3] [4] 21

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