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SGC - 90

100 1998 (Maldacena) ads/cft 100 2011 4 1 2012 3

1) μ, ν 2) i i, j, k 3) a h a,,h 1.6 4) μ μ = / x μ 5) μ 6) g μν (, +, +, +, ) g μν 7) Γ α μν Γμν α = 1 2 gαβ ( μ g βν + ν g βμ β g μν ) 8) R μναβ ( μ ν ν μ )V α = R μναβ V β

1 1 1.1.............................. 1 1.2.................................. 1 1.2.1....................... 1 1.2.2........................ 2 1.3.................................. 3 1.3.1..................... 3 1.3.2........................ 3 1.3.3.................................... 4 1.3.4............................ 4 1.4.................................... 6 1.4.1.................................. 6 1.4.2.............................. 8 1.4.3.................................. 12 1.4.4..................................... 13 1.4.5.................................... 15 1.4.6............................... 16 1.4.7.............................. 16 1.4.8........................... 18 1.5.................................... 20 1.5.1.............................. 20 1.5.2.............................. 22 1.5.3.............................. 22 1.5.4............................. 24 1.5.5.............................. 25 1.5.6............................... 25 1.6.................................. 26 2 27 2.1................................ 27 2.1.1................................ 27 2.1.2........................ 28

2.1.3............................ 28 2.2............................ 30 2.3................................. 31 2.3.1............................... 31 2.3.2................................ 32 2.4........................................ 34 2.4.1.................................. 34 2.4.2......................... 35 2.4.3.................................. 36 2.5.................................. 37 3 39 3.1 4................................ 39 3.1.1.............................. 40 3.1.2..................... 45 3.1.3................................ 48 3.1.4..................................... 49 3.2....................................... 52 3.2.1.................................. 52 3.2.2............................... 56 3.2.3............................... 59 3.3.............................. 60 3.3.1............................ 60 3.3.2............................. 63 3.3.3....................... 64 3.4........................................ 67 4 70 4.1............................. 70 4.2................................ 73 4.3....................................... 75 4.4................................. 78 4.5.............................. 80 5 84 5.1.................................... 84 5.2........................................ 85 5.3....................................... 86 5.4............................ 90 iv

5.4.1................................ 90 5.4.2.................................... 91 5.4.3..................... 94 6 96 6.1....................................... 96 6.2................................... 99 7 100 7.1........................... 100 7.2....................................... 101 7.2.1.............................. 101 7.2.2................................... 105 7.3.................................... 108 7.3.1............................ 108 7.3.2......................... 108 7.3.3.................................. 109 7.4.................................... 111 7.4.1............................. 111 7.4.2............................. 111 7.4.3.................. 112 7.5................................... 116 7.5.1........................ 116 7.5.2............................. 118 7.5.3............................... 124 7.6............................. 125 7.6.1...................... 125 7.6.2..................... 131 7.6.3................................... 135 138 139 v

1 1.1 1905 1.2 1.3 1.3 1.2 1.2.1 x, y, z 2

2 2.1 2.1.1 g μν (x) g μν R μναβ 2

3 3.1 4 4 1 1 3.3.1 2 3 3 log 3 [D. Ida, Phys. Rev. Lett. 85, 3758 (2000)] BTZ [M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)]

4 1 1.6 c =1 4.1 D (M,g) (M,g) g ab M D 1 Σ n a { 1 g ab n a n b = ɛ = 1. Σ q ab = g ab ɛn a n b (4.1) n a q ab n b =(g ab ɛn a n b )n b = n a ɛ 2 n a = 0 (4.2) 1 ADM [R. Arnowitt, S. Deser, C. W. Misner, Phys. Rev. 116, 1322 (1959).]

5 2 2 2 4 5 c =1 5.1 Σ (M,g ) (Σ +,g + ) Σ Σ y Σ ds 2 = dy 2 + q μν (y, x)dx μ dx ν (5.1) x μ = y dx μ dy = d2 x μ dy 2 = 0 (5.2)

6 c =1 6.1 ds 2 = (1 2GM/r)dt 2 + (6.1) M g μν = η μν + h μν (6.2) η μν h μν 4 h μν = O(1/r) (6.3)

7 c =1 7.1 100

[1] R. M. Wald: General Relativity (University of Chicago Press, Chicago, 1984). [2] S. W. Hawking and G. F. R. Ellis: Large scale structure of space-time (Cambridge University Press, Cambridge, 1973). [3] ( )( 1992). [4] E. Poisson: A Relativist s Toolkit (Cambridge University Press, Cambridge, 2004). [5] : ( = )( 1978). [6] : ( 1996). [7] : ( )( 1997). [8] : ( 15)( 1987). [9] : ( ) ( 1990). [10] : ( )( 1996). [11] : ( 2005) ( 2010). [12] : ( 8)( 1987). [13] : ( )( 2010). [14] C. W. Misner, K.S.Thorne and J. A. Wheeler: Gravitation (Freeman, San Francisco, 1973). [15] (SGC )( 2006).

2 55 14 93 86 6 1 4 108 36 106 80 35 52 52 49 71 84 73 2 81 135 67 19 8 6 104 104 62 16 16 4 69 8, 10 43 6 104 102 4 69 105 73 101 4 106 44, 51 75 121 13 42 53 25 72 14 101 30 106 105 26 103 1, 27 64 101 4 56 135

49 44 80 59 6 12 107 54 95 93 90 8 52 125 107 111 44 67 111 3 11 49, 125 70 46 102 75 15 12 7 3 18 ADM 97 FLRW 53 140

著者略歴 白水 しろみず 徹也てつや 1991 年山口大学理学部物理学科卒業 1996 年京都大学大学院理学研究科物理学第二専攻修了博士 ( 理学 ) 東京大学大学院理学系研究科物理学専攻助手, 同研究科附属ビッグバン宇宙国際研究センター助手, 東京工業大学大学院理工学研究科基礎物理学専攻助教授 / 准教授 2005 年第 20 回西宮湯川記念賞 2006 年平成 18 年度文部科学大臣表彰若手科学者賞 2008 年京都大学大学院理学研究科物理学 宇宙物理学専攻准教授 2014 年名古屋大学大学院多元数理科学研究科教授 ( 名古屋大学素粒子宇宙起源研究機構兼任 ) 専門分野相対論, 宇宙論主要著書 DOJIN 選書 026 宇宙の謎に挑むブレーンワールド 化学同人,2009 年. 臨時別冊 数理科学 SGC ライブラリ -90 アインシュタイン方程式一般相対性理論のよりよい理解のために ( 電子版 ) 著者白水徹也 2018 年 3 月 25 日初版発行 ISBN 978 4 7819 9937 1 この電子書籍は 2012 年 5 月 25 日初版発行の同タイトルを底本としています. 数理科学編集部発行人森平敏孝 TEL.(03)5474 8816 FAX.(03)5474 8817 ホームページ http : //www.saiensu.co.jp ご意見 ご要望は sk@saiensu.co.jp まで. 発行所 株式会社サイエンス社 TEL.(03)5474 8500( 代表 ) 151 0051 東京都渋谷区千駄ヶ谷 1 3 25 本誌の内容を無断で複写複製 転載することは, 著作者および出版者の権利を侵害することがありますので, その場合にはあらかじめサイエンス社著作権担当者あて許諾をお求めください. 組版ビーカム

2018.2.26 version,,,.. 1. 2,. 2. 2,...... 3. 2, (1.4) (1.6). 4. 2, (1.5). V (x ) V (x ) 5. 5, (1.23). x µ /c 1 x i /c 1. 6. 8, (1.48).. 7. 9, (1.54) ξ x. ϕ (x ) x µ. Γ σ µν = 2 x ρ x σ x µ x ν x ρ. 8. 11, (1.71) x ν. A µ (x) Γ µ αν(x)a α (x) x ν. 9. 12 1. (1.75), A α f fa α ( µ ν ν µ )(fa α ) = f( µ ν ν µ )A α. 10. 13, (1.90). R µν, R,, R µν, R,,

11. 17, (1.20) 2.. 12. 18, (1.125).. x = x 1 = 13. 19, (1.133). δγ α ρβ δγρ να δγ µ ρβ δγρ να. 14. 21, 6, 4. 15. 21, (1.150). e µ = xν x µ e ν. 16. 24, (1.74) (1.175). D D 1. 17. 29, t τ. 18. 33, (2.41). 19. 33,,. 20. 34, (2.54). 4πG c 2 ρ 4πG c 2 ρδ ij. d g. 21. 34, (2.55). g g. 22. 35, (2.55). (2.56). µ F µν µ F µν. 23. 35, (2.56). A µ A µ. 24. 36, (2.69).. 25. 37, 2.5..,. 26. 38, (2.81).. (2.81),.,,.

27. 40, (3.4). (3.5). r 2 (dθ 2 + sin 2 θϕ 2 ) r 2 (dθ 2 + sin 2 θdϕ 2 ). 28. 40, (3.6). Γ A BC. Γ A BC = 1 2 σad ( B σ DC + C σ DB D σ BC ) =: (2) Γ A BC(σ). 29. 40, (3.7). 30. 42, (3.20). = f 2h + h f f = 4h2 2h h f 4h 2 1 r 4 1 r 6. 31. 43, (3.28) t <. 32. 43, (3.30)., r = r g, t <, r = r g 33. 44, 3.1, 45 3.2.. 34. 46,.. 35. 47, (3.57), (3.58) 4 2. 36. 48, (3.65) 37. 49, (3.75). Γ 0 f 00 = f Γ0 00 = f 2f, Γ0 rr = h 2f Γ0 rr = ḣ 2f. ds 2 = 1 h(r) dt2 + ds 2 = 1 h(r) c2 dt 2 +. 38. 49, (3.79). M a... M, a... 39. 50, (3.80). (4GMa sin 2 θ/rc 2 )cdtdϕ + r 2 (dθ 2 + sin 2 θdϕ 2 ).

40. 50, (3.81). 41. 51, (3.90). M > a GM/c 2 > a. dt 2 c 2 dt 2. 42. 51, (3.90). 43. 52, (3.93). r + r r e r + < r < r e. 44. 59, (3.130). 1 r 2 /l 2 > 0 1 r 2 /l 2 > 0. 45. 59, (3.130), (3.134).. 46. 59, (3.132). a(τ) a(t). 47. 59, (3.135). ( ct ) ( ct ) Y = l sin sinh χ sin θ cos ϕ Y = l sin sinh χ sin θ sin ϕ. l l 48. 60, (3.138). sinh 2 ρω 2 2 sinh 2 (ρ/l)dω 2 2. 49. 61, (3.147). 50. 61, (3.148). (D 2)f rfh (D 2)h rh 2 (D 2)f. 2rfh (D 2)h 2rh 2. 51. 61, (3.149). (D 2) R AB (D 2) R AB. 52. 65, (3.173),. 53. 67, 11 2.. 1 m2 Q 2 ρ 2(D 3) 1 m2 Q 2 4ρ 2(D 3).

54. 75, 4.2.. Σ ξ+σξ Σ ξ+δξ. 55. 76,. 2, 3 3, 4. 56. 77, 2.............. 57. 78, (4.47). q ij (dx i + N i dt)(dx + N j dt) q ij (dx i + N i dt)(dx j + N j dt). 58. 78, (4.51). n a ( c a a c )n c n a ( c a a c )n c. 59. 80, (4.65). 60. 82, (4.72). dσ 2 dσ 1. 61. 83, (4.80). dtl L. 62. 83, (4.81), (4.82). K ij q ij K K j i δj i K. 63. 84, 5.1 2. (Σ +, g + ) (M +, g + ). 64. 86, (5.13). [T ab ] [T ab n a n b ]. 65. 88, (5.32), σ ϵ 66. 89, (5.41) (5.47). ȧ2 + f ȧ 2 + f +. 67. 90, (5.48). ä + f +/2 ȧ2 + f + = κ2 ρ 4.

68. 91, (5.60).. 69. 91, (5.60). 70. 92, 3. κ 2[ Λg ab + S ab ] Λg ab + κ 2 S ab... 71. 93, (5.76). ( (3.109), (3.110)) ( (3.109), (3.111)). 72. 93, (5.76). + Λc2 3 8πG 3 ρ +Λ 3 + 8πG 3 ρ. 73. 93, (5.77) (5.81). 4πG 3 74. 93, (5.81) 11. (ρ + P ) 4πG(ρ + 3P ). 3. 75. 102, (7.13). 76. 103, (7.24). k a l b + l a k b k a l b + l a k b. ( ) ( ) 2(u a ξ a ) 2 + 2 P 2(u a ξ a ) 2 + 1 P. 77. 103, (7.25).. 78. 106, 3/ C 3/ C. 79. 106, 7.. 80. 107, T T. 81. 107,. T T. 82. 108, 7.3.1. 83. 111,..

84. 112, (7.33).. 85. 113, (7.35). 114, (7.40).. 86. 114, (7.46).. R ab h ab = (2) R + 2ˆθ + ˆθ e 2f 2D 2 f 2(Df) 2 + 2e 2f v ˆθ+. 87. 114, (7.47).. (2) R = 2ˆθ + ˆθ e 2f + 2D 2 f + 2(Df) 2 2e 2f v ˆθ+ + 16πGe 2f T ab n a +n b. 88. 115, (7.48). e 2f v ˆθ+ 2e 2f v ˆθ+. 89. 116, 7.. 90. 117, 3. f(y) = 8M/r 3 4πr 2 8GM/r 3 4πr 2. 91. 118, (7.61) k ab. k ab := k ab 1 2 h abk. 92. 119, (7.70) 2. ϵ, ξ µ,. Dirac spinor Weyl spinor, Weyl spinor 2., Weyl spinor,. 93. 120, (7.74)-(7.76). ϵ 1. ϵ 1 ϵ 1 = ϵ ϵ 0. 94. 120, (7.75), (7.76). A Â

95. 120, (7.79). (3) γ i (3) i. 96. 120, (7.80) Γ j Γĵ. δ ij δîĵ. 97. 120, (7.81). j h k î k h j î. 98. 121, (7.84). γ i γ k i k. 99. 121, (7.88). 1 2 Gµ ν ξ µ 1 2 Gν µξ µ. 100. 121, (7.90). t α t ν. 101. 123, (7.100). î i. 102. 123, (7.104). D A DÂ. 103. 124, 7.5.2.. 104. 126, (7.116) = 0. 105. 127, (7.133). (3) R (3) ijkl R ijkl (3) R (3) ijkl R ijkl. 106. 128, (7.141). (7.140) (7.140). 107. 129, (7.145).., ρ 0 4m. 108. 129, (7.146). h 1/2 V (2) RdV Σ 109. 129, (7.148). M m. Σ V (2) RdΣ. ρ

110. 129, (7.149), (7.150). (7.149) (7.150). 111. 129, (7.150). 22 (7.149). 112. 129, 22.. 113. 131, (7.156). 114. 132, (7.160). da 2 ds 2. 1 V 2 4 V 2.. 115. 132, (7.164). (D 1) R ± (D 1) R 116. 133, 7.7. ( Σ ±, g ± ) ( Σ ±, g ± ). 117. 134, (7.173) 4 5. Σ Σ. 118. 134, (7.173) 8.. 119. 135, 7.6.2.,., M. Rogatko, Phys.Rev. D67 (2003) 084025. 120., [3] ( )(, 2000).. 121. 140,.

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