D.dvi

Transcription

1 2005 3

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3 Brane AdS/CFT black hole Kaluza-Klein Kazula-Klein Kaluza-Klein Brane Brane Bulk brane Kaluza-Klein brane Einstein-Yang-Mills Einstein-Yang-Mills Einstein-Yang-Mills Einstein-Yang-Mills Bartnik-McKinnon Colored black hole Einstein-Yang-Mills Einstein-Yang-Mills Horizon

4 black hole big bang Friedmann big bang domain wall Schwarzschild-AdS domain wall domain wall bulk brane Schwarzschild-AdS bulk brane U(1) gauge bulk brane SU(2) gauge bulk brane black hole I (inner horizon ) II (inner horizon ) Energy A 85 A.1, Bianchi A B gauge 87 B.1 gauge B.1.1 gauge B.1.2 gauge B.1.3 tensor B.1.4 gauge... 91

5 5 B.2 SU(2) gauge B SU(2) gauge B SU(2) gauge C5 101 C.1 5 Einstein-Yang-Mills C C.2.1 ɛ = C.2.2 ɛ =

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7 SU(2) U(1) gauge (Weinberg-Salam ) 3 SU(5), O(10) gauge ( ) 4 [1] [2] M 2 M scale anomaly anomaly 10 M 11 M 4 Einstein Einstein

8 Einstein [3] 4 Einstein big bang Hubble 3K [4] M 0 Einstein 4 Einstein potential r r 1 Newton Einstein n potential r n+3 Newton 1.2 Brane Newton compact Kaluza-Klein [5] 1 compact [6] Brane M [34] energy scale energy scale [7, 8] energy Newton [9] [10] brane 2 Kaluza-Klein brane 4 Brane energy Newton black hole brane 4 Einstein [11] brane brane P. Kraus brane 1 Randall-Sundrum Schwarzschild-anti de Sitter 4 domain wall 2 [12] domain wall Friedmann energy 2 big bang inflation [13] inflation 1 Einstein 4 Einstein 2 anti de Sitter AdS

9 1.3. AdS/CFT black hole 9 1 [14] [15] brane P. Kraus Schwarzschild-AdS domain wall [16] AdS/CFT black hole 1 AdS/CFT [17] AdS D-brane M D-brane M gauge gauge [18] black hole 1 black hole [19, 20] Black hole black hole black hole D-brane black hole 4 4 black hole brane [12] black hole black hole Schwarzschild black hole [21] Yang-Mills gauge Kaluza-Klein compact [22] gauge brane gauge brane [23] gauge gauge 4 [24] black hole [25] [26, 27] black hole [20] black hole 4 black hole gauge 5 black hole [28] 4 4 brane 3

10 Kaluza-Klein brane 4 3 gauge Einstein-Yang-Mills black hole 4 Einstein-Yang-Mills black hole 5 Einstein-Yang-Mills black hole 4 brane P. Kraus 5 Schwarzschild-AdS domain wall 5 domain wall 5 black hole 5 5 c = =1 c Planck [29, 30]

11 Kaluza-Klein Brane 2 4 Einstein Kaluza-Klein Kaluza-Klein 20 T. Kaluza, O. Klein [5] 4 compact Kazula-Klein 4 [5, 31, 32] Kazula-Klein 4 Kazula-Klein 5 Einstein 4 5 (M 5,g ab ) 4 (M 4,g αβ ) M 5 M 4 S 1 M 5 = M 4 S 1 (2.1) 5 Kaluza-Klein 2.1 S 1 5 g ab M 4 x α (α =1,, 4) S 1 y g ab (x α,y) = g ab (x α ) (2.2) 5 g ab 4 1 g ab dx a dx b = g αβ dx α dx β +Φ 2 (dy + A α dx α )(dy + A β dx β ) (2.3) g αβ, A α Φ x α A α Φ 5 y = y + ξ(x α ) (2.4)

12 : Kaluza-Klein 5 M 5 4 x α 1 y A α = A α α ξ, (2.5) Φ = Φ (2.6) A α Φ 5 U(1) gauge scalar gauge 5 Einstein 5 Einstein 5 Einstein-Hilbelt S = M5 dx 5 g 5 R (5) 16πG 5 (2.7) g 5 5 g ab R (5) g ab 5 Ricci scalar G (2.3) S = dx 4 [ Φ g 4 R (4) 1 M 4 16πG 4 4 Φ2 F αβ F αβ 2 α Φ α ] Φ 3 Φ 2 (2.8) g 4 4 g αβ R (4) g αβ 4 Ricci scalar F αβ A α F αβ = α A β β A α (2.9) y 4 G 4 G 4 = G 5 L (2.10)

13 2.1. Kaluza-Klein 13 L S 1 L = dy (2.11) S 1 5 Einstein (2.7) (2.8) 4 Einstein U(1) gauge scalar (2.8) scalar Φ ΦR (4) scalar Φ g αβ = Φ 2 3 gαβ (2.12) φ = 2lnΦ 3 (2.13) (2.8) S = M 4 dx 4 g πG 4 [ R (4) 1 4 e 3φ F αβ F αβ 1 ] 2 αφ α φ (2.14) [33] g 4 4 g αβ R (4) g αβ 4 Ricci scalar Kaluza-Klein 5 Einstein 4 Einstein n Einstein 4 Einstein [32, 22] n (M n,g ab ) 4 (M 4,g αβ ) compact (n 4) K n 4 M n = M 4 K n 4 (2.15) n K n 4 n Einstein 4 Einstein Lie gauge scalar 4 n (n 4) K n 4 L G 4 = G n L n 4 (2.16) n =5 5 K 1 = S 1 U(1) n>5 K n gauge scalar gauge Yang-Mills 3 1 K n 4 torus T n 4 gauge

14 Brane Kaluza-Klein Planck scale 4 brane [6] compact 3 (brane) M [34] [7, 8] brane 4 [11, 35] Brane 4 5 Einstein 4 5 (M 5,g ab ) 4 M 5 4 M 4 δ- 2 M 4 Z 2-5 Einstein 5 5 Einstein R (5) ab 1 2 g abr (5) = Λ 5 g ab +8πG 5 T (5) ab (2.17) R (5) ab R(5) g ab 5 Ricci tensor Ricci scalar G 5 5 Λ energy Λ 5 Λ 5 < 0 T (5) ab 5 energy- tensor M 4 y =0 T (5) ab = δ(y)s ab (2.18) δ- S ab brane energy- tensor S ab S ab = σq ab + τ ab (2.19) brane energy σ energy- tensor τ ab σ brane τ ab n a = 0 (2.20) n a M 4 vector Brane 2.2 (2.17) brane M 4 q ab = g ab n a n b (2.21) 2 M 5 bulk M 4 brane

15 2.2. Brane : Brane 5 M 5 4 x α 2 (brane) τ ab brane σ y brane brane Z 2-3 Gauss (A.10)-(A.12) R (4) ab 1 [ 2 q abr (4) = R (5) cd 1 ] 2 g cdr (5) q c a q d b + R (5) cd nc n d q ab + KK ab K c a K cb 1 2 q ab[k 2 K ab K ab ] R (5)c def n c q a d n e q b f (2.22) K ab M 4 K = K a a (2.17) Weyl tensor (A.4) R (4) ab 1 2 q abr (4) = 1 2 Λ 5q ab + 16πG 5 3 { [ T (5) cd q a c q d b + T (5) cd nc n d 1 ] } 4 T (5) q ab + KK ab K a c K cb 1 2 q ab[k 2 K ab K ab ] E ab (2.23) T (5) = T (5) a a E ab 5 Weyl tensor E ab = C (5) cdef nc q a d n e q b f (2.24) 5 energy- tensor T (5) ab (2.18) M 4 5 Einstein (2.17) δ- 1 Israel [36] [K ab Kq ab ] ± = 8πG 5 S ab (2.25) 3 A.2 [11]

16 16 2 [K ab ] ± = 8πG 5 [ S ab 1 3 S c c q ab ] (2.26) M 4 Z 2 - [X] ± = lim y +0 X lim y 0 X (2.27) lim K ab = lim K ab (2.28) y +0 y 0 (2.23) y 0 (2.26) brane R (4) ab 1 2 q abr (4) = Λ 4 q ab +8πG 4 τ ab +(8πG 5 ) 2 π ab E ab (2.29) π ab energy- tensor 2 π ab = 1 4 τ acτ c b ττ ab q abτ cd τ cd 1 24 q abτ 2 (2.30) τ = τ a a δ- lim y 0 T (5) ab =0 (2.17) 4 lim y 0 E ab 4 G 4 4 Λ 4 5 G 5 5 Λ 5 brane σ G 4 = 4πG2 5 σ (2.31) 3 Λ 4 = Λ πG 4σ (2.32) 4 G 4 brane σ 5 Einstein (2.17) brane (2.29) 4 Einstein energy- tensor 2 5 Weyl tensor Einstein (2.17) π ab E ab E ab 5 brane (2.17) π ab E ab 4 brane Codacci (A.13) 5 Einstein (2.17) D b K ab D a K = 8πG 5 T (5) cd nc q a d (2.33) y 0 (2.26) D b τ ab = 0 (2.34) brane

17 2.3. Kaluza-Klein brane Bulk brane brane 5 bulk bulk brane bulk [35] 5 energy- tensor (2.18) T (5) ab = δ(y)s ab + T ab (2.35) T ab bulk energy- tensor (2.29) brane R (4) ab 1 2 q abr (4) = Λ 4 q ab +8πG 4 τ ab +(8πG 5 ) 2 π ab E ab (8πG 5) 2 f ab (2.36) f ab bulk energy- f ab = T cd q a c q d b + [T cd nc n d 14 ] T q ab (2.37) T = T a a (2.34) brane D b τ ab = 8πG 5 T cd nc q a d (2.38) (2.38) brane energy bulk energy bulk Kaluza-Klein brane Kaluza-Klein Brane Kaluza-Klein brane 3 100GeV energy scale l cm (2.39) scale l 10 2 cm (2.40) scale 4 Newton scale [7] Kaluza-Klein brane

18 18 2 Kaluza-Klein L energy scale energy scale Kaluza-Klein L cm (2.41) energy scale Planck L cm (2.42) brane brane brane 4 energy 5 Λ 5 scale 5 AdS Newton energy [9] 5 AdS l5 scale ds 2 = g (0) ab dxa dx b = e y /l η αβ dx α dx β + dy 2 (2.43) 1 l 2 = Λ 5 6 (2.44) η αβ 4 Minkowski Cartesian brane y =0 g ab = g (0) ab + h ab (2.45) 5 (y ) h 55 = 0 (2.46) h 5α = 0 (2.47) 5 y gauge α h αβ = 0 (2.48) h α α = 0 (2.49)

19 2.3. Kaluza-Klein brane 19 gauge Einstein (2.17) 1 order [ 2y m2 e y /l + 1l 2 2l δ(y) ] ψ(y) = 0 (2.50) 4 h αβ h αβ (x α,y) = ψ(y)e ip γx γ (2.51) m m 2 = p γ p γ (2.52) 5 y =0 Z 2 - z = 2sgn(y)l(e y /2l 1) (2.53) Ψ(z) = ψ(y)e y /4l (2.54) (2.50) [ 1 ] 2 2 z + V (z) Ψ(z) = m 2 Ψ(z) (2.55) potential V (z) V (z) = 15 8( z +2l) 2 3 δ(z) (2.56) 4l (2.56) 2.3 z V (z) 0 z 0 V (z) z =0 δ- m =0 mode z =0 brane 4 mode m >0 mode mode Kaluza-Klein mode Kaluza-Klein mode compact spectrum z =0 potential energy Kaluza-Klein mode Green M potential U(r) U(r) = G ) 4M (1+ 2l2 r 3r 2 (2.57) G 4 G 4 = 2G 5 l (2.58) 4 Λ 4 0 (2.31) 5 scale l l 10 2 cm (2.59)

20 : Potential (2.56) z =0 δ(z) energy scale Planck scale M Pl = GeV (2.60) energy scale M EW = 10 3 GeV (2.61) Kaluza-Klein brane Kaluza-Klein 4 n (2.16) 4 G 4 G n L n 4 Kaluza-Klein energy scale brane brane scale 5 scale (2.58) l energy scale energy scale Planck scale energy scale G 5 l L. Randall R. Sundrum 5 2 brane G 5 l Planck scale energy scale [8] compact 5 (y ) S 1 compact 2 brane Z AdS (2.43) y =0 4 S 1 /Z 2 compact M [34]

21 2.3. Kaluza-Klein brane 21 y = y c brane y = y c y =0 brane y = y c brane y =0 brane y = y c brane AdS (2.43) 4 warp factor e yc/l 4 5 G 4 = 2G 5 l (1 e y c/l ) 1 (2.62) y = y c brane y c (2.58) y = y c energy scale energy scale m 0 warp factor y = y c brane energy scale m m = e y c/l m 0 (2.63) G 5, l, y c G 5, l y c G 4 m Planck scale energy scale y c /l 36 e yc/l energy scale m Planck scale m 0 G 1/3 5 l cm m GeV brane m =0 mode brane

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23 23 3 Einstein-Yang-Mills Einstein-Yang-Mills 3.1 Einstein-Yang-Mills black hole Einstein-Yang-Mills 3.1 Einstein-Yang-Mills Einstein-Yang-Mills Yang-Mills gauge Yang-Mills Maxwell Maxwell gauge vector potential scalar potential A α =(φ, A) A = A grad ξ (3.1) φ = φ + ξ (3.2) t A α = A α α ξ (3.3) gauge gauge gauge gauge gauge Maxwell Yang-Mills gauge Weinberg-Salam Yang-Mills gauge (gauge ) U(1) U(1) 2 U 1, U 2 U 1 U 2 U(1) Yang-Mills U(1) Weinberg-Salam SU(2) U(1) SU(3) SU(5), O(10) 2.1 Kaluza-Klein 5 4

24 24 3 Einstein-Yang-Mills gauge U(1) 4 gauge gauge gauge ( gauge ) gauge Lie G G Lie g τ i i (i =1,, dim(g)) τ i g ijk τ i [τ i,τ j ] τ i τ j τ j τ i = g ijk τ k (3.4) 1 G gauge gauge A a Lie g M 1 A a = A i aτ i (3.5) gauge gauge gauge gauge g G A a = ga a g 1 +( a g)g 1 (3.6) Gauge Lie g M scalar V = V i τ i A a = A a + D a V (3.7) D a V = a V [A a,v] (3.8) g = expv = 1+V (3.9) 4 Maxwell gauge field strength Lie g M 2 F ab F ab = a A b b A a [A a,a b ] (3.10) 3 gauge Field strength F ab gauge (3.6) F ab = gf ab g 1 = F ab [F ab,v] (3.11) 1 Lie Einstein

25 Einstein-Yang-Mills 25 field strength (3.10) D [a F bc] = 0 (3.12) gauge Lagrangian Maxwell Tr 2 Lie L YM = 1 16πg 2 Tr(F abf ab ) (3.13) Tr(τ i τ j ) = δ ij (3.14) δ ij Kronecker s delta g 2 Yang-Mills L YM gauge Lagrangian L YM A a D a F ab = 0 (3.15) (3.12) (3.15) Yang-Mills Yang-Mills Yang-Mills Einstein Einstein-Yang-Mills Einstein-Hilbert Yang-Mills S = dx n [ R g 1 ] M n 16πG n 16πg 2 Tr(F abf ab ) (3.16) M n n S = dx n [ 1 g (R 2Λ) 1 ] M n 16πG n 16πg 2 Tr(F abf ab ) g ab Einstein (3.17) R ab 1 2 g abr +Λg ab = 8πG n T ab (3.18) Yang-Mills energy- tensor T ab 1 T ab = [F 4πg 2 Tr c a F bc 14 ] g abf cd F cd (3.19) Einstein-Yang-Mills 4 gauge SU(2) gauge SU(2) gauge SU(2) Lie su(2) [τ i,τ j ] = ε ijk τ k (3.20)

26 26 3 Einstein-Yang-Mills τ i (i =1, 2, 3) ε ijk τ i Pauli σ i τ i = 1 2i σi (3.21) 4 Einstein-Yang-Mills (3.16) [24] black hole black hole [25] [24, 25, 37] Einstein-Yang-Mills 4 dσ ds 2 = f(r)e 2δ(r) dt 2 + dr2 f(r) + r2 dσ 2 2 (3.22) dσ 2 2 = dθ 2 +sin 2 θdϕ 2 (3.23) f(r) f(r) = 1 2μ(r) r (3.24) r 2 δ(r), μ(r) μ(r) δ(r) lapse λ = ( G4 g 2 ) 1/2 (3.25) 4 SU(2) gauge B (B.88)-(B.91) gauge A = A t τ 3 dt + A r τ 3 dr +( φ 1 τ 1 φ 2 τ 2 )dθ +(φ 2 τ 1 φ 1 τ 2 cot θτ 3 )sinθdϕ (3.26) [38] gauge A r =0 Yang-Mills φ 2 =0 (3.26) A t φ 1 A t, φ 1 Yang-Mills part part part Reissner-Nordström black hole part part Einstein-Yang-Mills [39] A t =0 part SU(2) gauge A A = w(r)τ 1 dθ (w(r)τ 2 +cotθτ 3 )sinθdϕ (3.27)

27 Einstein-Yang-Mills 27 w(r) =φ 1 Field strength (3.10) F = w (r)τ 1 dr dθ w (r)τ 2 dr (sin θdϕ) (1 w(r) 2 )τ 3 dθ (sin θdϕ) (3.28) r Einstein (3.18) Yang-Mills (3.15) μ = fw r 2 (1 w2 ) (3.29) δ = 2w 2 r (3.30) (fe δ w ) + 2 r 2 e δ w(1 w 2 ) = 0 (3.31) r (3.29)-(3.31) w w w (3.32) w(r) w(r) Einstein-Yang-Mills (3.29)-(3.31) (3.29)-(3.31) 2 1 Schwarzschild 1 w = ±1 (3.33) μ = M (3.34) δ = 0 (3.35) w = 0 (3.36) μ = M 1 2r (3.37) δ = 0 (3.38) 1/g Reissner-Nordström Bartnik-McKinnon (3.29)-(3.31) 1989 R. Bartnik J. McKinnon [24] Bartnik-McKinnon Yang-Mills (3.12), (3.15) Einstein [40] Bartnik-McKinnon [24, 37]

28 28 3 Einstein-Yang-Mills 3.1: 4 Bartnik-McKinnon 1( [37] ) μ(r)( ) lapse δ(r)( ) n node Einstein-Yang-Mills (3.29)-(3.31) r =0,r = f(r h )=0 r = r h singular μ, w r =0 parameter b μ = 2b 2 r 3 + O(r 5 ) (3.39) w = 1+br 2 +( 4 5 b b2 )r 4 + O(r 6 ) (3.40) f(r) > 0 (3.41) r = μ, w parameter a, M μ = M a2 r 3 + O(r 4 ) (3.42) [ w = ± 1 a ] r + 3a2 6aM 4r 2 + O(r 3 ) (3.43) Parameter M μ ADM δ Einstein- Yang-Mills (3.29)-(3.31) δ μ, w t δ r = r = δ = 0 (3.44)

29 Einstein-Yang-Mills : 4 Bartnik-McKinnon 2( [37] ) Gauge potential w(r)( ) Q(r)( ) n node Einstein-Yang-Mills (3.29)-(3.31) parameter b 3.1, 3.2 n 2 n gauge w(r) node n w(r) n 0 node n Bartnik-McKinnon parameter b, M 3.1 Q(r) f(r) = 1 2M r + Q2 (r) r 2 (3.45) Q(r) Reissner-Nordström 3.2 Q(r) 3.1: 4 Bartnik-McKinnon node n parameter b ADM M n =5 n b M [41]

30 30 3 Einstein-Yang-Mills 3.3: 4 colored black hole ( [37] ) μ(r), lapse δ(r), gauge potential w(r)( ) Q(r)( ) n node Event horizon r h =1 1 r R n gauge Reissner-Nordström (3.36)-(3.37) Q(r) 0 r 1 R n r Schwarzschild Node n Q(r) ADM Colored black hole R. Bartnik J. McKinnon M. S. Volkov, D. V. Galt sov P. Bizon black hole [25] colored black hole Einstein-Maxwell black hole Kerr-Newmann black hole 3 (black hole [42]) Colored black hole [25, 37] Black hole [29] r =0 event horizon event horizon r = r h 2μ(r h ) = r h (3.46)

31 Einstein-Yang-Mills : 4 colored black hole ADM horizon r h ( [37] ) n node n =0 Schwarzschild black hole n = Reissner-Nordström black hole (3.31) w(r) w w h (1 wh 2 r=rh = ) r h [1 (1 wh 2)/r2 h ] (3.47) w h = w(r h ) event horizon r >r h (3.41) r = (3.42), (3.43) event horizon Einstein-Yang-Mills (3.29)-(3.31) parameter w h 3.3 event horizon r h =1 Colored black hole n n gauge w(r) event horizon node node n colored black hole parameter w h, M 3.2 (3.45) Q(r) 3.2: 4 colored black hole node n parameter w h ADM M n =4 n w h M

32 32 3 Einstein-Yang-Mills 3.5: 4 colored black hole event horizon ( [37] ) μ(r) gauge potential w(r) n node n =3 colored black hole ln μ(r) lapse δ(r) gauge potential w (r)/(20r) 3.3 Event horizon event horizon Q(r) gauge Reissner-Nordström (3.36)-(3.37) Event horizon Q(r) 0 Schwarzschild black hole ADM M horizon r h [20] Black hole entropy S Einstein black hole A A/4 r h entropy 3.4 M r h M colored black hole horizon r h Schwarzschild black hole colored black hole Schwarzschild black hole entropy colored black hole Schwarzschild black hole colored black hole horizon r h node horizon node colored black hole entropy event horizon event horizon parameter 2 [43] [37, 43] (3.46), (3.47) r<r h 3.5 μ, δ w event horizon μ r 0 μ 0

33 Einstein-Yang-Mills : (3.51), (3.52) ( [37] ) (a) (b) μ(r) (c) μ(r) (d) μ(r) (e) μ(r) r 0(τ ) cycle μ 0 r =0 μ, w r e x = w (3.48) e y = (1 w2 ) 2 fr 2 (3.49) ( ) r τ = ln (3.50) r 0 w (3.32) w > 0 (3.29)-(3.31) 2 ẋ = e y 1 (3.51) ẏ = 1+e y 2e 2x (3.52) τ (x, y) =(0, 0) τ attractor 3.6 r =0 τ x, y inner horizon inner horizon r = r (3.46), (3.47) 3 3 (3.46), (3.47) r h w h r w = w(r ) r 0

34 34 3 Einstein-Yang-Mills r <r<r h parameter r h Reissner-Nordström event horizon event horizon inner horizon Bartnik-McKinnon colored black hole SU(2) Einstein-Yang-Mills black hole black hole black hole [44, 45] Einstein-Yang-Mills-Higgs black hole [46] Einstein-Skyrme black hole [47] (Kaluza-Klein ) Einstein-Yang-Millsdilaton black hole [48] black hole [49] Einstein-Maxwell Einstein-Born-Infeld black hole [50] Brans-Dicke Yang-Mills-Higgs black hole [51] SU(2) gauge black hole [52] black hole [26, 27] black hole μ(r, t) = μ(r)+ɛμ 1 (r)e iωt (3.53) δ(r, t) = δ(r)+ɛδ 1 (r)e iωt (3.54) w(r, t) = w(r)+ɛw 1 (r)e iωt (3.55) ɛ gauge (3.26) φ 2 (r) =0 [26] (3.53)-(3.55) even-parity φ 2 (r) odd-parity (3.53)-(3.55) ɛ 1 order Schrödinger d2 w 1 dr 2 r tortoise + V (r )w 1 = ω 2 w 1 (3.56) dr dr = eδ f (3.57)

35 Einstein-Yang-Mills 35 r 0 r < black hole <r < (3.56) ω 2 < 0 4 mode Bartnik-McKinnon colored black hole n node n mode Bartnik-McKinnon colored black hole even-parity odd-parity n node n mode [27] colored black hole black hole black hole [20] even-parity 1 mode n =1 Bartnik-McKinnon intermediate attractor [53] black hole black hole gauge parameter n =1 Bartnik-McKinnon black hole colored black hole n node 2n mode 2000 J. Bjoraker Y. Hosotani Einstein-Yang-Mills black hole node [45] node mode Einstein-Yang-Mills 4 Einstein-Yang-Mills 5 Einstein-Yang-Mills M 11 1 S 1 /Z 2 compact S 1 /Z 2 compact 1 10 E 8 E 8 heterotic [34] brane 6 Kaluza-Klein compact 4 brane Kaluza-Klein 2 compact A. Lukas 6 Calabi-Yau 5 [21] 5 bulk dilaton scalar gauge A. Lukas gauge U(1) gauge compact 6 topology compact U(1) gauge 5 gauge ( SU(5) gauge ) brane gauge gauge ( SU(3) SU(2) 4 ω 2

36 36 3 Einstein-Yang-Mills U(1) gauge ) [23] brane gauge gauge brane 5 Einstein-Yang-Mills black hole [28] Einstein-Yang-Mills 5 Einstein-Yang-Mills 5 dσ f(t, r) ds 2 = f(t, r)e 2δ(t,r) dt 2 + dr2 f(t, r) + r2 dσ 2 3 (3.58) dσ 2 3 = dψ 2 +sin 2 ψ(dθ 2 +sin 2 θdϕ 2 ) (3.59) f(t, r) = 1 μ(t, r) r 2 + ɛ r2 l 2 (3.60) t, r 2 δ(t, r), μ(t, r) μ(t, r) δ(t, r) lapse black hole 4 λ = ( G5 g 2 ) 1/2 (3.61) Einstein-Yang-Mills (3.17) (3.60) 3 5 Λ 5 ɛ =0 5 Λ 5 Λ 5 > 0 ɛ = 1 Λ 5 < 0 ɛ =1 l scale parameter 6 Λ 5 = ɛ (λl) 2 (3.62) 5 gauge gauge SU(2) gauge B (B.127)-(B.131) (B.140)-(B.144) (B.127)-(B.131) SU(2) gauge part (B.140)-(B.144) part 4 SU(2) gauge part part 5 l scale λ

37 Einstein-Yang-Mills 37 SU(2) gauge part Einstein-Yang-Mills (B.127)-(B.131) A = A t τ 3 dt + A r τ 3 dr (3.63) gauge A r =0 Einstein (3.18) Yang-Mills (3.15) μ = 2 3 r3 (A t eδ ) 2 (3.64) μ = 0 (3.65) δ = 0 (3.66) [ (A t e δ) 2 ] [ (A t e δ) ] 2 6 ( + A r t e δ) 2 t r = 0 (3.67) = 0 (3.68) μ = M 2Q2 3r 2 (3.69) δ = 0 (3.70) A t = Q r 2 (3.71) Reissner-Nordström black hole 4 part Reissner-Nordström black hole SU(2) gauge part Einstein-Yang-Mills (B.140)-(B.144) A = τ 3 (Ẋdt + X dr + φdψ +cosθdϕ) +cosψ [(τ 1 sin X τ 2 cos X)dθ (τ 1 cos X + τ 2 sin X)sinθdϕ] + φ sin ψ [(τ 1 cos X + τ 2 sin X)dθ +(τ 1 sin X τ 2 cos X)sinθdϕ] (3.72) gauge X =0 6 SU(2) gauge A A = τ 3 (w(t, r)dψ +cosθdϕ) cos ψ [τ 2 dθ + τ 1 sin θdϕ]+w(t, r)sinψ [τ 1 dθ τ 2 sin θdϕ] (3.73) w(t, r) =φ field strength (3.10) F = [ẇ(t, τ 3 r)dt dψ + w (t, r)dr dψ (1 w(t, r) 2 )(sin ψdθ) (sin ψ sin θdϕ) ] [ẇ(t, + τ 1 r)dt (sin ψdθ)+w (t, r)dr (sin ψdθ)+(1 w(t, r) 2 )dψ (sin ψ sin θdϕ) ] [ẇ(t, τ 2 r)dt (sin ψ sin θdϕ)+w (t, r)dr (sin ψ sin θdϕ) (1 w(t, r) 2 )dψ (sin ψdθ) ] (3.74) 6 τ 1 -τ 2 X X

38 38 3 Einstein-Yang-Mills Einstein (3.18) μ = 2r [fw 2 + f 1 e 2δ ẇ 2 + (1 w2 ) 2 ] Yang-Mills (3.15) r 2 (3.75) μ = 4rfw ẇ (3.76) δ = 2 r [w 2 + f 2 e 2δ ẇ 2 ] (3.77) 1 r (rfe δ w ) + 2 r 2 e δ w(1 w 2 ) = ( f 1 e δ ẇ ) (3.78) t, r 4 Einstein-Yang-Mills (3.75)-(3.78) w w w (3.79) w(t, r) w(t, r) (3.75)-(3.78) 4 (3.29)-(3.31) (3.75), (3.78) r (3.75)-(3.78) t μ = 2r [fw 2 + (1 w2 ) 2 ] (3.80) r 2 δ = 2 r w 2 (3.81) 1 r (rfe δ w ) + 2 r 2 e δ w(1 w 2 ) = 0 (3.82) (3.80)-(3.82) 4 (3.29)-(3.31) 2 1 w = ±1 (3.83) μ = M (3.84) δ = 0 (3.85) ɛ =0 Schwarzschild ɛ = 1 Schwarzschild-dS ɛ =1 Schwarzschild-AdS 1 w = 0 (3.86) μ = M +2lnr (3.87) δ = 0 (3.88) 4 w =0 Yang-Mills part Reissner-Nordström (3.36)-(3.38) 5 w =0 Reissner-Nordström horizon

39 Einstein-Yang-Mills (3.86)-(3.88) r ɛ =0 Kretschmann R abcd R abcd ( ) 2 ln r R abcd R abcd 288 (as r ) (3.89) 0 ɛ = ±1 f(r) r 4 R abcd R abcd 40 l 2 (as r ) (3.90) f(r) 1+ɛ r2 l 2 (as r ) (3.91) ɛ =0 ɛ = ±1 AdS ds μ(r) r energy 1 [ G 5 M ADM = ds i j h ij η ij j h k ] k (3.92) 16π I 0 Arnowitt-Deser-Misner energy [54] η ab Minkovski Cartesian h ab r =0 (A.8) i 5 i =1,, 4 ds i I 0 ɛ =0 (3.86)-(3.88) M ADM 3π G 5 M ADM = lim r 8 λ2 (M +2lnr) (3.93) M ADM ln r order 3π/8 (3.75) μ = 3π dv[ T 0 0] (3.94) 8 factor ds Abbott-Deser energy [55] ɛ = 1 (3.86)-(3.88) M AD = M ADM (3.95) AdS energy AdS timelike Killing vector ξ I cross section C G 5 M ξ [C] = λ2 l E ab ξ a ds b (3.96) 16π C

40 40 3 Einstein-Yang-Mills [56] E ab Weyl tensor leading order part E ab = l2 Ω 2 C acbdn c n d (3.97) ds b cross section C Ω factor n a n a = a Ω (3.98) I vector ɛ = 1 (3.86)-(3.88) (3.96) [ 3π G 5 M ξ [C] = lim r 8 λ2 M +2lnr 7 ] 6 (3.99) energy (3.86)-(3.88) ( ɛ =0 ɛ = 1 ds ɛ =1 AdS ) energy 4 global monopole [57] global monopole O(3) scalar ds 2 = f m (r)dt 2 + dr2 f(r) + r2 dσ 2 2 (3.100) f m (r) r f m (r) = 1 α 2M r ( ) 1 + O r 2 (3.101) α M μ(r) (3.24) r μ(r) = M + αr ( ) O (3.102) r ADM energy t r (3.100) f m (r) r t = (1 α) 1/2 t (3.103) r = (1 α) 1/2 r (3.104) ds 2 = f m ( r)d t 2 + d r2 f m ( r) +(1 α) r2 dσ 2 2 (3.105) f m ( r) = 1 2 M r (3.106)

41 Einstein-Yang-Mills 41 M = (1 α) 3/2 M (3.107) (3.106) α U. Nucamendi D. Sudarsky ADM global monopole energy [58] 5 μ(r) r 2 order energy (3.86)-(3.88) μ(r) ln r order (3.86)-(3.88) r ( (A)dS ) energy Horizon (3.86)-(3.88) horizon (3.86)-(3.88) Riemann tensor scalar r =0 r =0 f(r) =0 1 M +2lnr r 2 + ɛ r2 l 2 = 0 (3.108) horizon ɛ horizon ɛ =0 (3.108) M>0 2 r ± black hole r <r + r + event horizon r inner horizon r =0 timelike (3.108) M =1 1 r = r d r d event horizon inner horizon horizon extreme black hole M <1 horizon r =0 ɛ =1 AdS (3.108) M>M cr 2 r ± M cr M cr = 1 2 ( 1+r 2 +cr ) 2lnr+cr (3.109) r +cr ( ) 1/2 r +cr = l l 2 2 (3.110) r <r + r + event horizon r inner horizon M cr l 1 l Λ 0 1 r =0 ɛ =0 timelike (3.108) M = M cr 1 r = r d r d event horizon inner horizon horizon extreme black hole M <M cr horizon r =0

42 42 3 Einstein-Yang-Mills ɛ = 1 ds DS cosmological horizon 1 horizon l>2 2 M min <M<M max M min, M max r ±cr M min = r cr 2 2lnr cr r4 cr l 2 (3.111) M max = r+cr 2 2lnr +cr r4 +cr l 2 (3.112) ( ) 1/2 r ±cr = l 1 ± 1 8l 2 2 (3.113) (3.108) 3 r = r ±,r c r (<r cr ) <r + (<r +cr ) <r c r + event horizon r inner horizon r c cosmological horizon M = M min r = r + event horizon inner horizon M = M max r + = r c event horizon cosmological horizon M <M min M>M max (3.108) 1 r =0 l 2 2 M min, M max M cr = 3 ln (3.114) 2 M = M cr 3 horizon l 2 2 M (3.108) 1 r =0 horizon parameter M black hole black hole black hole event horizon (3.86)-(3.88) Black hole Hawking T BH event horizon [ 1 T BH = 1 1 ] 2πr + r+ 2 +2ɛ r2 + l 2 (3.115) [59] black hole entropy 3 event horizon S = 1 2 π2 r 3 + (3.116) 7 black hole energy ADM π 2 dm T = T BH ds +ΦdQ (3.117)

43 Einstein-Yang-Mills : (3.86)-(3.88) horizon IH, EH, CH DH inner horizon, event horizon, cosmological horizon horizon ɛ, l M Horizon M<1 None ɛ =0 M =1 DH M>1 IH, EH M<M cr None ɛ =1 M = M cr DH M cr <M IH, EH M<M min IH ɛ = 1, l>2 2 M = M min IH, DH M min <M<M max IH, EH, CH M = M max DH, CH CH M>M max ɛ = 1, l 2 2 CH energy M T [58] Φ Q gauge potential 4 Yang-Mills gauge parameter dq =0 M T (3.115), (3.116) (3.117) M T = 3π 8 M (3.118) M =0 M T =0 energy M energy M T event horizon r + (3.108) M T = 3π [ ( ) ] r+ 2 1+ɛ r2 + 8 l 2 2lnr + (3.119) 3.7 (3.86)-(3.88) Reissner-Nordström M T r + energy black hole entropy 3.7 M T T BH parameter ɛ, l Hawking M T black hole Hawking energy (3.108), (3.115) dt BH = 2 1 3/r+ 2 2ɛr+/l 2 2 dm T 3π 2 r /r+ 2 +2ɛr+/l 2 2 (3.120) ɛ =0 1 <r + < 3 M r + > 3 M ɛ =1 l 2 6

44 44 3 Einstein-Yang-Mills 3.7: (3.86)-(3.88) energy M T event horizon r + Hawking T BH M T r + ɛ =0 ɛ =1 Reissner-Nordström Reissner-Nordström-AdS M T T BH ɛ =0 ɛ =1 l =6 l =4 ɛ = 1 l =5 l >2 6 r +cr <r + <r ch r + >r +ch M r ch <r + <r +ch M r ±ch ( r ±ch = l ) 1/2 1 ± l 2 (3.121) ɛ = 1 r cr <r + <r ch M r ch <r + r +cr M r ch ( r ch = l ) 1/ l 2 (3.122) Einstein-Yang-Mills 5 black hole ɛ =0 ɛ =1 AdS

45 Einstein-Yang-Mills 45 4 r =0 (3.75)-(3.78) r =0 parameter b μ(r) = 4b 2 r 4 + O(r 5 ) (3.123) δ(r) = 4b 2 r ( ) 4ɛ 3 b2 3b 8b2 r 4 + O(r 5 ) (3.124) 3l2 w(r) = 1+br 2 b ( ) 4ɛ 3b 8b2 r 4 + O(r 5 ) (3.125) 6 3l2 δ r =0 r =0 δ =0 r = t t = e δ( ) t (3.126) scale 4 horizon (3.41) r = 4 Einstein-Yang-Mills (3.75)-(3.78) r = μ(r) 8 C.1 5 energy r = ( AdS ) μ(r) O(r) order (3.75)-(3.78) ( AdS) ɛ =0 ɛ =1 ɛ =0 parameter b b min b min <b<0 (3.127) b min (3.128) parameter 4 Bartnik-McKinnon 5 parameter parameter 3.8 gauge potential w(r) Parameter b 0 w(r) 1 Yang-Mills instanton [61] gauge w node w(r) μ(r) ln r order μ(r) ln r order 8 Yang-Mills field strength r = black hole [60]

46 46 3 Einstein-Yang-Mills 3.8: 5 1 ɛ =0 b = 0.01, 0.1, 0.5 gauge potential w(r) ɛ =1 l =10 b = 0.01, 0.1, 0.5 gauge potential w(r) w(r) w(r) ±1 μ(r) w(r) 0 r C.2.1 w μ (C.52) (C.63) lapse Bartnik-McKinnon ɛ =1 ɛ =0 AdS parameter b b min <b<0 (3.129) b min scale parameter l l ( ) b min l b min Einstein-Yang-Mills parameter b [45] l =10 parameter b 3.8 gauge potential w(r) ɛ =0 0 W 0 1 W 0 parameter b b W 0 1 b min parameter b node 4 Einstein-Yang-Mills r 0 W 0 1 ɛ =1 4 5

47 Einstein-Yang-Mills : 5 2 ɛ =0 b = 0.01, 0.1, 0.5 μ(r) ɛ =1 l =10 b = 0.01, 0.1, 0.5 μ(r) gauge potential w(r) 3.9 μ(r) ln r order 4 ɛ =0 r μ(r) μ = M 0 +2(1 W0 2 )2 ln r (3.130) C black hole 5 black hole black hole event horizon 4 colored black hole black hole horizon event horizon r = r h ( ) μ(r h ) = rh 2 1+ɛ r2 h l 2 (3.131)

48 48 3 Einstein-Yang-Mills 3.10: 5 3 ɛ =0 b = 0.01, 0.1, 0.5 lapse δ(r) ɛ =1 l =10 b = 0.01, 0.1, 0.5 lapse δ(r) 3.4: ɛ =1 parameter b b min b min <b<0 (3.3.3 ) b s b s <b<0 (3.3.5 ) b crit domain wall b min <b<b crit domain wall ( ) l b min b s b crit l b min b s b crit b min b min b min b min b min b min b min

49 Einstein-Yang-Mills : 5 black hole 1 ɛ =0 black hole w h =0.99, 0.9, 0.5 gauge potential w(r) ɛ =1 l =10 black hole w h =0.99, 0.9, 0.5 gauge potential w(r) Yang-Mills (3.78) w w h (1 wh 2 (r h ) = ) r h [1 + 2ɛrh 2/l2 (1 wh 2)2 /rh 2] (3.132) w h = w(r h ) δ(r h ) = 0 (3.133) (3.126) r = 4 event horizon r>r h (3.41) black hole (3.75)-(3.78) r = μ(r) ( AdS ) black hole μ(r) O(r) order (3.75)-(3.78) ( AdS) black hole parameter r h 1 <w h < black hole ɛ =0 ɛ =1 black hole ɛ =0 gauge potential w(r) z =2lnr μ(r) w(r) ln r order ɛ =1 gauge potential w(r) W 0 μ(r) ln r order lapse δ(r) 4

50 50 3 Einstein-Yang-Mills 3.12: 5 black hole 2 ɛ =0 black hole w h =0.99, 0.9, 0.5 μ(r) ɛ =1 l =10 black hole w h =0.99, 0.9, 0.5 μ(r) black hole M T black hole Hawking T BH = eδ( ) 2πr h [1 (1 w2 h )2 r 2 h ] +2ɛ r2 h l 2 (3.134) [59] δ( ) = lim r δ(r) δ(r) event horizon (3.133) event horizon black hole entropy (3.116) black hole gauge Q gauge potential w(r) ɛ =1 w(r) w(r) W 0 dq =0 black hole M T 1 (3.117) ɛ =1 M T M T 3π [ M T = lim μ 2(1 w 2 ) 2 ln r ] (3.135) r 8 M T μ (3.130) (3π/8)M 0 ɛ =0 w(r) ɛ =1 μ (C.63) (3π/8)M 0 M T μ(r) ln r order

51 Einstein-Yang-Mills : 5 black hole 3 ɛ =0 black hole w h =0.99, 0.9, 0.5 lapse δ(r) ɛ =1 l =10 black hole w h =0.99, 0.9, 0.5 lapse δ(r) black hole [16] Event horizon μ, δ, w 3.14, 3.15 ɛ =0 ɛ = [43] inner horizon 2 event horizon μ r 0 μ 0 bounce μ 0 bounce r =0 bounce μ inner horizon inner horizon μ r 0 μ 0 bounce 0 f r = r 0 4 colord black hole black hole colord black hole event horizon r h parameter b inner horizon event horizon r h 5 black hole event horizon r h parameter b inner horizon event horizon

52 52 3 Einstein-Yang-Mills 3.14: 5 black hole 1 Event horizon r h =1 ɛ =1,w h =0.4 5 black hole event horizon μ(r) lapse δ(r) ( ) inner horizon ɛ =1,w h = black hole event horizon μ(r) lapse δ(r) inner horizon r h parameter b inner horizon black hole black hole even-parity μ(r, t) = μ 0 (r)+ɛμ 1 (r)e iωt (3.136) δ(r, t) = δ 0 (r)+ɛδ 1 (r)e iωt (3.137) w(r, t) = w 0 (r)+ɛw 1 (r)e iωt (3.138) ɛ μ 0 (r), δ 0 (r) w 0 (r) (3.53)-(3.55) Einstein-Yang-Mills (3.75)-(3.78) μ 0 (r), δ 0 (r) w 0 (r) Einstein-Yang-Mills (3.80)-(3.82)

53 Einstein-Yang-Mills : 5 black hole 1 Event horizon r h =1 ɛ =1,w h =0.4 ɛ =1,w h = black hole gauge potential w(r) ɛ 1 order μ 1 = 0 w 2 0 2r [2f 0 w w 1 r 2 μ 1 4(1 w2 0 )w ] 0 r 2 w 1 (3.139) μ 1 = 4rf 0 w 0 w 1 (3.140) δ 1 = 4 r w 0 w 1 (3.141) 1 ( ) r 3 (rf 0e δ 0 w 0 ) f0 1 μ 1 f 0 e δ 0 1 w 0 r 2 f 0 1 μ 1 + δ r (rf 0e δ 0 w 1 ) + 2 r 2 e δ 0 (1 3w 2 0 )w 1 = ω 2 f 1 0 eδ 0 w 1 (3.142) f 0 f 0 = 1 μ 0 r 2 + ɛr2 l 2 (3.143) (3.139) (3.140) (3.139), (3.141) (3.142) 4 tortoise r (3.57) w 1 χ = w 1 r 1/2 (3.144)

54 54 3 Einstein-Yang-Mills (3.142) (3.139), (3.140) (3.141) Schrödinger d2 χ dr 2 + V (r )χ = ω 2 χ (3.145) Potential V (r ) (3.139)-(3.142) V (r ) = f 0 e δ 0 { 2 r 2 e δ 0 (3w0 2 1) + r 1/2 (r 1/2 f 0 e δ 0 ) r } [ f0 e δ 0 w 0 2 ] (3.146) r ɛ =0 0 r < ɛ =1 0 r <r,max ɛ =0 black hole <r < ɛ =1 black hole <r <r,max r, <r <r, r, r =0 black hole event horizon r = r + r, r = 4 (3.145) ω 2 < 0 9 mode ω 2 > 0 Potential V (r ) (3.145) χ 10 r, <r <r, [ χ dχ ] r, [ r, dχ + dr r, r, dr 2 + V (r) χ 2 ] dr = ω 2 r, r, χ 2 dr (3.147) black hole potential V (r ) r = r, w 1 0 gauge energy flux χ dχ dr r =r, 0 (3.148) r = r, =0 w 1 =0 w 1 =0 χ dχ dr 0 (3.149) r =r, (3.145) ω 2 V (r ) > 0 ω 2 > 0 V (r ) mode black hole r = r, (3.148) r = r, = V (r ) 0 (3.145) 9 ω 2 10 χ χ

55 Einstein-Yang-Mills 55 event horizon r = r, = χ e iωr (3.150) event horizon ω 2 < 0 ω ω Im ω<0 (3.150) χ dχ dr 0 (3.151) r =r, (3.145) ω 2 V (r ) > 0 ω 2 > 0 ω 2 < 0 ω 2 > 0 V (r ) black hole mode (3.86)-(3.88) potential V (r ) V (r ) = f ] 0 [5M 4r lnr 9r 2 +3ɛ r4 l 2 (3.152) ɛ =0 ɛ = 1 V (r ) r ɛ =1 ( ) M > 1 5 l > ln r p +9r 2 p 3 r4 p l 2 M l r p ( ) r p = l l 2 12 (3.153) (3.154) (3.155) ɛ =1 V (r ) potential 3.16 ɛ =0 ɛ =1 potential potential (3.145) ω 2 [16] ɛ =0 mode ɛ =1 mode b s <b<0 b s l 3.4 l parameter b l (3.156) b s = b min w b s <b<0 gauge potential w node

56 56 3 Einstein-Yang-Mills 3.16: 5 potential V (r ) ɛ =0 potential V (r ) b = 0.01 b = 0.1 potential ɛ =1 l =10 potential V (r ) b = 0.01 b = 0.1 potential potential V (r ) r 5 node mode 4 [45, 26, 27] node ɛ =0 black hole mode ɛ =1 black hole node Einstein-Yang-Mills [16] 5 (3.58) r =const. 3 SO(4) 5 5 r r =const. 3 Euclid 3 SO(3, 1) E(3) 11 5 Einstein-Yang-Mills k 1, 0, 1 dσ 2 3,k 11 E(3) 3 Euclid ds 2 = f k (r)e 2δ(r) dt 2 + dr2 f k (r) + r2 dσ 2 3,k (3.157) dσ 2 3,k = dψ 2 + S k (ψ) 2 (dθ 2 +sin 2 θdϕ 2 ) (3.158)

57 Einstein-Yang-Mills 57 S k (ψ) S k (ψ) = sin ψ k =1 ψ k =0 sinh ψ k = 1 (3.159) dσ3,k 2 k 3 k =1 3 (3.59) 5 k =0 3 Euclid 5 k = f k (r) f k (r) = k μ(r) r 2 + ɛ r2 l 2 (3.160) r 2 δ(r), μ(r) μ(t, r) δ(t, r) lapse (3.61) λ l (3.62) SU(2) gauge B part A i t = 0 (3.161) A i r = 0 (3.162) A i ψ = (0, 0,w) (3.163) ( A i θ = ws k (ψ), ds ) k(ψ) dψ, 0 (3.164) ( A i ϕ = ds ) k(ψ) dψ sin θ, ws k(ψ)sinθ,cos θ (3.165) E(n) SO(3, 1) compact B.1 gauge (3.161)-(3.165) gauge (3.161)-(3.165) A = τ 3 (w(r)dψ +cosθdϕ) ds k(ψ) dψ field strength (3.10) [τ 2dθ + τ 1 sin θdϕ]+w(r)s k (ψ)[τ 1 dθ τ 2 sin θdϕ](3.166) F = [ τ 3 w (r)dr dψ (k w(r) 2 )(sin ψdθ) (sin ψ sin θdϕ) ] [ + τ 1 w (r)dr (sin ψdθ)+(k w(r) 2 )dψ (sin ψ sin θdϕ) ] [ τ 2 w (r)dr (sin ψ sin θdϕ) (k w(r) 2 )dψ (sin ψdθ) ] (3.167) k =1 w(r) =±1 k =0 w(r) =0 F =0 k = 1 F =0 w(r) k = 1 pure gauge gauge ansatz Gauge potential w(r) w(r) =±i pure gauge w(r)

58 58 3 Einstein-Yang-Mills 3.17: 5 (k =0) black hole 1 Event horizon r h =1 ɛ =1,w h =0.7 5 (k =0) black hole event horizon μ(r) lapse δ(r) ( ) inner horizon ɛ =1,w h = (k =0) black hole event horizon μ(r) lapse δ(r) inner horizon Einstein-Yang-Mills Einstein μ = 2r [f k w 2 + (k w2 ) 2 ] r 2 (3.168) δ = 2 r w 2 (3.169) Yang-Mills 1 r (rf ke δ w ) + 2 r 2 e δ w(k w 2 ) = 0 (3.170) k =1 (3.80)-(3.82) k =0 k = 1 k =0 (3.168)-(3.170) 1 μ = M (3.171) δ = 0 (3.172) w = 0 (3.173) (3.173) pure gauge black hole 1999 D. Birmingham Schwarzschild(-AdS) black hole k =0 [62] k =0 k =1 black hole (3.86)-(3.88) ( )

59 Einstein-Yang-Mills : 5 (k =0) black hole 2 Event horizon r h =1 ɛ =1,w h =0.7 ɛ =1,w h = (k =0) black hole gauge potential w(r) k = 1 (3.168)-(3.170) 1 μ = M +2lnr (3.174) δ = 0 (3.175) w = 0 (3.176) k =1 black hole (3.86)-(3.88) k = 1 μ = M black hole gauge (3.161)-(3.165) pure gauge Einstein black hole [62] μ, w k =0 k = 1 black hole horizon r = r h ( ) μ(r h ) = rh 2 k + ɛ r2 h l 2 (3.177) r = r h Yang-Mills (3.78) w (3.75)-(3.78) k =0 ( AdS) black hole parameter r h 1 <w h < , 3.18 black hole k = 1 ( AdS) black hole

60

61 energy scale 4 Einstein energy scale big bang Einstein big bang brane big bang brane 4.1 big bang big bang 4 Einstein big bang big bang [4] Friedmann big bang 3 (3.158) 4 ds 2 = dτ 2 + a 2 (τ) [ dψ 2 + S k (ψ) 2 (dθ 2 +sin 2 θdϕ 2 ) ] (4.1) Freidmann-Robertson-Walker k 1, 0, 1 S k (ψ) (3.159) τ a(τ) scale scale factor energy- tensor T αβ = (ρ + P )U α U β + Pg αβ (4.2) ρ, P U α energy 4 Einstein 2 H 2 + k a 2 = Λ πG 4 ρ (4.3) 3 ρ +3H(ρ + P ) = 0 (4.4)

62 62 4 H Hubble H = ȧ a Λ 4 τ (4.3) scale factor Friedmann (4.4) energy (4.4) energy ρ (4.5) P = (γ 1)ρ (4.6) ρ = ρ 0 ( a a 0 ) 3γ (4.7) a 0 ρ 0 = ρ(a 0 ) energy (dust ) γ =1 energy ρ a 3 ( ) γ =4/3 energy ρ a 4 γ =0 energy energy ρ (4.6) Friedmann (4.3) potential U(a) 1 2ȧ2 + U(a) = k 2 U(a) = 1 [ Λ πG ] 4C 0 a 2 3γ 3 (4.8) (4.9) scale factor a potential U(a) energy k/2 4.1 potential U(a) Λ 4 =0 dust (ρ a 4 ) k =0 k = 1 k =1 Friedmann 1929 Hubble scale factor energy 1965 A. Penzias R. W. Wilson big bang big bang big bang

63 4.1. big bang : Potential U(a) Λ 4 > 0, Λ 4 < 0 Λ 4 =0 potential ρ a 4 Friedmann (4.3) Hubble H 0 energy ρ 0 Λ 4 k Λ 4 =0 energy ρ 0 ρ cr =3H0 2/(8πG 4) ρ 0 /ρ cr 1% order 1 Friedmann (4.3) ρ/ρ cr 1 Planck scale order 1 horizon scale scale horizon scale horizon monopole scale inflation [13] big bang inflation 1 Friedmann (4.3) scale (4.6) k =0 scale factor a a t 2/(3γ) (4.10) (γ 2/3) 0

64 64 4 (4.7) scale factor energy ρ Ricci scalar R a =0 big bang Big bang Freidmann-Robertson-Walker [29, 14] big bang [15] domain wall Brane 1 brane big bang Brane [63] 5 bulk domain wall brane [16, 12] Schwarzschild-AdS domain wall brane 5 bulk [9] domain wall [12] big bang brane 3 brane 5 bulk bulk 3 ds 2 = f k (r)dt 2 + dr2 f k (r) + r2 dσ 2 3,k (4.11) dσ 2 3,k (3.158) f k(r) l 5 Λ 5 f k (r) = k M r 2 + r2 l 2 (4.12) l 2 = 6 Λ 5 (4.13) 1 Schwarzschild-AdS black hole [62] 2 M black hole k big bang k 5 bulk r =const. 1 l l = l/λ λ (3.61) 2 Schwarzschild-AdS r = a(τ) (4.14)

65 domain wall : Z 2-2 Schwarzschild-AdS r >a(τ) 2 r = a(τ) domain wall τ domain wall a(τ) scale factor brane 1 Z 2 - (4.11) 5 2 domain wall (4.14) 2 domain wall 4.2 domain wall Z 2 - domain wall energy- tensor S αβ = (ρ + P )U α U β + Pγ αβ (4.15) ρ, P U α energy 4 brane tensor energy ρ m P m brane σ ρ P ρ = ρ m + σ (4.16) P = P m σ (4.17) domain wall Einstein domain wall Domain wall Israel (2.26) [36] 5 (4.11) domain wall Gaussian normal ds 2 = dη 2 + γ αβ dx α dx β (4.18) domain wall γ αβ (2.26) [ [K αβ ] ± = 8πG 5 S αβ 1 ] 3 S μ μ γ αβ (4.19)

66 66 4 K αβ K αβ = 1 2 nμ μ γ αβ (4.20) n μ domain wall vector Domain wall 5 u a domain wall vector n a u a ( ) fk (a)+ȧ =, ȧ, 0, 0, 0 (4.21) f k (a) n a = ( ȧ ) f k (a), f k (a)+ȧ 2, 0, 0, 0 (4.22) τ n a domain wall domain wall K αβ K + ij (= K ij ) = fk (a)+ȧ 2 γ ij (4.23) a K + ττ (= K ττ ) = d da fk (a)+ȧ 2 (4.24) 3 (4.19) (4.15) (4.23) (4.15) (4.24) H 2 + f k(a) a 2 = (8πG 5) 2 ρ 2 (4.25) 36 ( ) d 2 fk (a)+ȧ da 2 = 4πG 5 3 ρ + P (4.26) domain wall H Hubble big bang (4.5) (4.25) (4.12), (4.16) (4.17) H 2 + k a 2 = Λ πG 4 ρ m + 4πG 4 3 3σ ρ2 m + M a 4 (4.27) G 4 Λ 4 (2.31) (2.32) 4 4 (2.31) brane σ (4.27) Friedmann (4.3) 2 big bang Friedmann (4.27) 3 domain wall energy brane Einstein (2.29) π ab brane Friedmann brane energy inflation [65] (4.27) 3 [64]

67 domain wall bulk Schwarzschild-AdS black hole brane Einstein (2.29) E ab 4 5 bulk a 4 big bang scale factor order 4.3 (4.26) (4.26) (4.25) ρ +3H(ρ + P ) = 0 (4.28) big bang energy (4.4) domain wall energy bulk domain wall energy domain wall (4.6) energy ρ domain wall (4.7) big bang domain wall 5 bulk brane 3.3 M dilaon scalar gauge [21, 23] 5 domain wall 5 bulk Schwarzschild-AdS (4.11) 5 bulk Schwarzschild-AdS domain wall [16] 5 bulk 3 ds 2 = f k (r)e 2δ(r) dt 2 + dr2 f k (r) + r2 dω 2 3,k (4.29) f k (r) Schwarzschild-AdS (4.12) k big bang k μ(r) f k (r) = k μ(r) r 2 + r2 l 2 (4.30) l 5 Λ 5 (4.13) 5 5 bulk 2 μ(r) δ(r) k = ɛ =1 k ɛ =1 μ(r) δ(r) lapse μ = M δ =0 (4.29) 4 5 bulk Weyl tensor part M 5 1

68 68 4 (4.14) domain wall (4.29) 5 2 domain wall (4.14) 2 domain wall domain wall Z 2 - Domain wall energy- tensor (4.15) domain wall Gaussian normal (4.18) Israel (4.19) domain wall Domain wall 5 u a domain wall vector n a ( ) u a = e δ(a) fk (a)+ȧ 2, ȧ, 0, 0, 0 (4.31) f k (a) ( ) n a = e δ(a) ȧ f k (a), f k (a)+ȧ 2, 0, 0, 0 (4.32) domain wall τ domain wall K αβ K + ij (= K ij ) = fk (a)+ȧ 2 γ ij a (4.33) K ττ + (= K ττ ) = d [e δ(a) ] eδ(a) f k (a)+ȧ da (4.34) (4.19) (4.15) (4.33) (4.15) (4.34) H 2 + f k(a) a 2 = (8πG 5) 2 ρ 2 (4.35) 36 e δ(a) d [e δ(a) ] f k (a)+ȧ da 2 = 4πG 5 ( 2 3 ρ + P ) (4.36) domain wall H Hubble big bang (4.5) (4.35) (4.30), (4.16) (4.17) H 2 + k a 2 = Λ πG 4 ρ m + 4πG 4 3 3σ ρ2 m + μ(a) a 4 (4.37) G 4 Λ 4 (2.31) (2.32) 4 4 (4.37) 3 Schwarzschild-AdS domain wall (4.27) 4 μ(r) μ(a) =M (4.27) (2.31) brane σ (4.37) Friedmann domain wall Friedmann domain wall energy 2

69 domain wall 69 5 bulk μ(a) 4.3 (4.36) Schwarschild-AdS domain wall brane energy bulk brane energy bulk energy bulk 5 bulk domain wall energy (4.36) (4.35) ρ +3H(ρ + P ) = ρȧ dδ(a) da (4.38) (4.28) ( (4.4)) δ(a) domain wall energy energy domain wall energy ρ domain wall big bang Schwarzschild-AdS domain wall brane σ 0 (4.38) P m = (γ 1)ρ m (4.39) ρ m +3γHρ m = δρ m (4.40) ρ m = ρ m,0 e δ δ 0 ( ) 3γ a (4.41) domain wall a 0 ρ m,0 = ρ m (a 0 ) δ 0 = δ(a 0 ) (4.7) lapse δ(a) a 6 energy domain wall 5 bulk AdS AdS δ(a) a domain wall energy energy ρ (4.7) case γ =0 ρ m energy big bang Schwarzschild-AdS brane energy ρ m lapse energy energy lapse δ(a) a brane σ 0 (4.39) (4.38) 6 3 a 0 ρ m +3γHρ m = δ(ρ m + σ) (4.42)

70 70 4 ( ρ m = e δ δ 0 a a 0 ) 3γ [ a ρ m,0 + σ da dδ(a) ] a 0 da e δ(a) a 3γ (4.43) domain wall brane (4.43) 2 energy lapse 5 bulk AdS AdS δ(a) a domain wall energy energy ρ (4.7) bulk brane 5 bulk domain wall big bang Friedmann (4.37) energy (4.38) bulk 5 Schwarzschild-AdS brane bulk U(1) gauge brane bulk SU(2) gauge brane domain wall (4.37) potential 1 2ȧ2 + U(a) = 0 (4.44) Potential U(a) U(a) = 1 [ k μ(a) 2 a 2 Λ 4 3 a2 8πG ( 4 ρ m (a)a )] 3 2σ ρ m(a) (4.45) potential U(a) energy ρ m (a) (4.43) Schwarzschild-AdS bulk brane 5 bulk 5 bulk Schwarzschild-AdS μ(r) δ(r) μ(r) = M (4.46) δ(r) = 0 (4.47) Lapse δ(a) domain wall energy (4.28) energy ρ m (4.7) potential U(a) (4.45) U(a) = 1 2 [ k M a 2 Λ 4 3 a2 8πG 4C 0 3 ( a (2 3γ) 1+ C )] 0 2σ a 3γ (4.48)

71 bulk brane : Brane potential U(a) 1 5 Schwarzschild-AdS bulk potential (4.48) bulk parameter k =1 k =0 k = 1 M =1 bulk parameter k = 1 M = 1 5 Reissner-Nordström-AdS bulk potential (4.52) bulk parameter k =1 k =0 k = 1 M =1 Q 2 =0.1 C 0 ρ m,0 a 0 domain wall energy ρ m 0 4 Λ 4 0 potential U(a) bulk event horizon 7 parameter M k =1 k =0 k = 1 M > l2 4 (4.49) event horizon M [62] 4.3 k = 1 M<0 potential U(a) a 0 k = 1 M<0 potential U(a) a 0 scale factor a bounce domain wall bounce catenary-type domain wall ρ m bounce 5 bulk bulk Schwarzschild-AdS k = 1 M<0 domain wall k = 1 M<0 catenary-type 7 Event horizon black hole brane

72 bulk M =0 r =0 black hole 5 bulk brane ( black hole ) U(1) gauge bulk brane 5 bulk U(1) gauge domain wall [66, 67] 5 bulk Reissner-Nordström-AdS μ(r) δ(r) μ(r) = M Q2 r 2 (4.50) δ(r) = 0 (4.51) lapse δ(a) bulk domain wall energy (4.28) energy ρ m (4.7) potential U(a) (4.45) U(a) = 1 Ma Q2 [k a 4 Λ 4 3 a2 8πG ( 4C 0 a (2 3γ) 1+ C )] 0 3 2σ a 3γ (4.52) potential (4.48) 3 5 Reissner-Nordström-AdS Q big bang γ =2 scale factor order domain wall energy ρ m 0 4 Λ 4 0 potential U(a) 4.3 Reissner-Nordström-AdS Q domain wall domain wall a bounce k domain wall k =0 k = 1 bounce catenary-type k =1 bounce Λ 4 =0 Reissner-Nordström-AdS black hole inner horizon (4.30) (4.52) Λ 4 =0 f k (a) > U(a) (4.53) f k (a) (4.30) domain wall bounce a = a b U(a b )=0 inner horizon a = a f(a )=0 f k (a) U(a) a a b < a (4.54) domain wall bounce inner horizon k =1 Penrose 4.4 inner horizon

73 bulk brane : Reissner-Nordström-AdS black hole Penrose domain wall domain wall Event horizon domain wall inner horizon inner horizon domain wall [67] domain wall 5 bulk U(1) gauge bulk Reissner-Nordström-AdS domain wall k =0 k = 1 catenary-type k =1 bounce inner horizon 5 bulk Schwarzschild-AdS bulk r =0 black hole 5 bulk U(1) gauge brane ( black hole )

74 SU(2) gauge bulk brane 5 bulk SU(2) gauge domain wall [16] brane Einstein-Yang-Mills black hole Λ 5 < 0 bulk domain wall 1 r μ(r) r domain wall domain wall (ɛ =1 ) black hole 5 bulk domain wall domain wall domain wall energy SU(2) gauge part Reissner-Nordström-AdS bulk domain wall k =1 domain wall k =0 k = 1 k =1 parameter b = λ 2 d2 w dr 2 (4.55) r=0 b min <b<0 λ (3.61) w(r) (3.73) gauge potential (b min ) b s <b<0 b min b s Λ 5 < 0 scale l Einstein-Yang-Mills scale λ l = l λ (4.56) domain wall Λ 4 =0 σ ρ m potential (4.45) domain wall U(a) < 0 a parameter b min <b<b crit (4.57) parameter domain wall b crit l 3.4 l domain wall parameter domain wall

75 bulk brane : Brane potential U(a) 2 5 bulk potential (4.45) 4 Λ bulk μ(a) lapse δ(a) k =1,l =0.1 b = 7.3 ( b s <b crit ) l (4.58) (2.31) (2.32) Λ 4 =0 G 5 = G 4l 2 (4.59) (4.58) g G 4 l (4.60) g 2 Yang-Mills g 2 l 4 order (4.60) domain wall parameter b s <b<b crit 4.5 domain wall bulk potential (4.45) Λ 4 =0 a 0.1 a 1 domain wall ρ m bounce 4 Λ 4 a Λ 4 < 0 8 Λ 4 > 0 catenary-type 8 parameter b

76 black hole 5 bulk black hole domain wall black hole k =1 domain wall black hole k =0 k = 1 domain wall k =0 black hole k = 1 black hole bulk domain wall k =1 k = 1 5 bulk μ(r) lapse δ(r) (3.87) (3.88) Lapse δ(a) bulk domain wall energy (4.28) energy ρ m (4.7) potential U(a) (4.45) U(a) = 1 [ k M 2 a 2 2lna a 2 Λ 4 3 a2 8πG ( 4C 0 a (2 3γ) 1+ C )] 0 3 2σ a 3γ (4.61) Schwarzschild-AdS bulk potential (4.48) 3 ln a/a 2 big bang (4.6) a potential a 0 domain wall potential U(a) domain wall inner horizon bulk I (inner horizon ) k =1 k = inner horizon 2 inner horizon I inner horizon II I 5 bulk μ(r) lapse δ(r) event horizon k = k = Λ 4 =0 bulk potential (4.45) potential U(a) a inner horizon black hole μ(r) r μ(r) μ 0 cycle potential U(a) 4.6 μ(a) 0 U(a) 0 1 m(r) 0 bounce U(a) domain wall bulk du(a)/da du(a)/da domain wall

77 bulk brane : Brane potential U(a) 3 I (inner horizon ) bulk potential 4 Λ 4 0 k =1,w h =0.4 k =0,w h =0.7 black hole potential (4.45) l =1 r h =1 a U(a) 1 U(a) 0 U(a) 0 II (inner horizon ) bulk potential 4 Λ 4 0 k =1,w h =0.701 k =0,w h =0.56 black hole potential (4.45) l =1 r h =1 a U(a) 1 U(a) 0 k =0 k =1 U(a max )=0 a max II (inner horizon ) II 5 black hole inner horizon 5 bulk μ(r) lapse δ(r) event horizon k = k = μ(r) (3.86)-(3.88) a ln r 4.6 Λ 4 =0 bulk potential (4.45) Inner horizon potential U(a) a 0 bulk domain wall a bounce domain wall k =0 bounce catenary-type k =1 domain wall bounce inner horizon 5 bulk U(1) SU(2) gauge

78 : bulk domain wall Bulk field 5 bulk k domain wall Bulk spacetime 5 bulk Sch-AdS 5 Schwarzschild-AdS (4.3.1 ) RN-AdS 5 Reissner-Nordström (4.3.2 ) particle-like 5 ( ) analytic BH 5 numerical BH I II I II ( ) Horizon 5 bulk event horizon (EH) inner horizon (IH) Evolution domain wall BB BC (big bang ) BB (big bang ) O C catenary-type Λ 4 =0 Bulk field k Bulk spacetime Horizon Evolution 1 EH BB BC 0 Sch-AdS BH EH BB -1 EH BB, C 1 EH + IH O U(1) or SU(2) part 0 RN-AdS BH EH + IH C -1 EH + IH C analytic BH EH + IH O 1 numerical BH I EH BB BC numerical BH II EH + IH O SU(2) part particle-like None O 0 numerical BH I EH BB numerical BH II EH + IH C -1 analytic BH EH + IH C domain wall Energy black hole lapse δ(r) r domain wall energy domain wall energy 5 bulk domain wall 5 5 bulk 4.5 lapse δ(a) du(a)/da (4.38) brane energy 5 bulk du(a)/da lapse δ(a) domain wall energy 5 black hole ( ) 5 bulk event horizon a lapse δ(a) domain wall energy

79 bulk brane 79 event horizon lapse δ(a) domain wall energy 5 bulk I lapse δ(a) domain wall energy 5 bulk domain wall domain wall energy 5 bulk domain wall energy 5 bulk energy domain wall a ( a ) domain wall energy (4.43) Einstein- Yang-Mills (3.81) gauge potential w (C.69) (4.43) 2 γ 2 8σW (γ 2) a 6 (4.62) γ =2 scale factor order γ =2 8σW 2 1 ln a a 6 (4.63) big bang (4.6) a Friedmann (4.37) μ(a)/a 4 μ (C.71) μ(a) a 4 = M 0 a 4 +(k W 0 2 ln a )2 a 4 (4.64) 1 a 4 2 a 4 ln a big bang (4.6) Brane γ =4/3 γ =1 dust

80

81 Einstein gauge Einstein-Yang-Mills black hole 4 1 brane 5 domain wall brane 5 Einstein-Yang-Mills brane Einstein-Yang-Mills 4 Einstein-Yang-Mills SU(2) gauge part colored black hole [24, 25] 5 Einstein-Yang-Mills [28] 4 Reissner-Nordström(-(A)dS) black hole horizon horizon Reissner-Nordström(-(A)dS) r 4 μ ln r 4 black hole 5 Λ parameter 5 Λ 5 parameter black hole 4 Λ 5 =0 Λ 5 < 0 4 [16] black hole black hole 4 4 brane Brane 5 (3+1) domain wall 5 bulk domain wall [16] 5 bulk μ lapse δ 2 Friedmann brane energy big bang

82 82 5 lapse δ brane energy energy 5 bulk μ brane domain wall 3 black hole 5 bulk SU(2) gauge brane lapse brane 5 bulk 5 5 bulk 5 bulk [15] brane 5.2 gauge gauge Einstein Yang-Mills M 0 Einstein Einstein [68] [69] black hole Einstein black hole [70] gauge field strength [71] gauge black hole [60] gauge black hole 1 gauge brane 1 [72]

83 COE ( )

84

85 85 A Riemann tensor Bianchi [29, 30] (M,g ab ) n A.1, Bianchi Riemann tensor R a bcd vector v a 2 R a bcdv b = 2 [c d] v a (A.1) Ricci tensor R ab, Ricci scalar R R a bcd Weyl tensor C abcd R abcd, R ab, R R ab = R c acb (A.2) R = R a a (A.3) C abcd = R abcd 2 n 2 (g 2 c[ar b]d g d[a R b]c )+ (n 1)(n 2) g c[ag b]d R (A.4) R abcd Bianchi R ab, R [e R ab]cd = 0 (A.5) [e R a]c = 1 2 d R eacd (A.6) e R = 2 c R ec (A.7) A.2 M M h ab h ab = g ab n a n b (A.8)

86 86 A n a M vector M timelike (n a spacelike vector) M spacelike (n a spacelike vector) + M K ab K ab = ( c n d )h a c h b d (A.9) Gauss (M,h ab ) Riemann tensor R a bcd R abcd, K ab R a bcd = R e fghh a eh f b h g c h h d ± K a ck bd K a dk bc (A.10) (M,h ab ) Ricci tensor R ab Ricci scalar R R abcd, R ab, R, K ab R ab = R cd h c a h d b R c defn c h d a n e h f b ± KK ab K c a K cb (A.11) R = R 2R ab n a n b ± K 2 K ab K ab (A.12) K = K a a Codacci K ab R ab D b K ab D a K = R cd n c q a d (A.13) D a (M,h ab )

87 87 B gauge gauge B.1 B SU(2) gauge [28, 38] B.1 gauge gauge [38] B.1.1 gauge Killing vector D Riemann M M N Lie S S Lie M compact M N Killing vector ξ n (n =1,,N) Killing vector ξ λ n (n =1,,N) ξ = λ nξn (B.1) Killing vector ξ M g ab ξ g ab = 0 (B.2) ξ ξ Lie Killng vector ξ vector Lie f mnp [ ξ m, ξ n ] a ξm b bξn a ξb n bξm a = f mnp ξp a (B.3) vector η tensor T ab... cd... Lie η T ab... cd... = 0 (B.4) vector η gauge A a gauge (3.6) A a η

88 88 B gauge gauge η Lie g M scalar W η A a = D a W = a W [A a,w] (B.5) gauge A a field strength η F ab = [F ab,w] (B.6) A a (3.6) W W = gwg 1 + η a ( a g)g 1 = W + η V [W, V ] (B.7) W (B.5) gauge gauge Killing vector ξ Lie g M scalar W (B.5) Lie g M 1 A a Killing vector N Killing vector ξ n λ n (B.1) (B.5) ξn A a = D a W n (B.8) W n Lie g M scalar W = λ n W n (B.9) Killing vector (B.3) W n ξm W n ξn W m [W m,w n ] f mnp W p = 0 (B.10) F ab F ab (B.5) ξn a F ab = D b Ψ m (B.11) Ψ n = ξna a a W n (B.12) ξm a ξb n F ab = f mnp Ψ p [Ψ m, Ψ n ] (B.13)

89 B.1. gauge 89 B.1.2 gauge M p M S X M N 1 M x a x a = (x α,y α ) (B.14) x α =const. X α,α α =1,,N, α =1,,D N Frobenius X Killing vector X Killing vector ξ ξ a = (0,ξ α ) (B.15) ξ y α x α N X X N S A α A α,w m y α Lie S Lie s s Killing vector Lie X Killing vector ξ n S vector ξ n R [ ξ R m, ξ R n ]ˆα = f mnp ξ Rˆα p (B.16) S vector ˆα 1 N vector ξ n R Killing vector ξ n S Lie J n s s S sj n = ξ Rs (B.17) n X 1 q X q R S S N N S/R X Rs S/R X y α R y ω (ω =1,,N N ) s 0 (y α ) Rs(y α ) 1 S y ˆα y ˆα = (y ω,y α ) (B.18) s(y ˆα ) = r(y ω )s 0 (y α ) (B.19) r R S vector ξ n R ξ Rˆα n = (ξn Rω,ξn Rα ) (B.20) 1 N N, D ξ Rα n = ξ α n (B.21)

90 90 B gauge W n (y α ) Lie g S scalar W n W n (y ˆα ) W n (y α ) (B.22) W n (B.10) W n ξ R m Wn ξ R n Wm [ W m, W n ] f mnp Wp = 0 (B.23) Lie g S vector Wˆα W n = ξ Rˆα n Wˆα (B.24) (B.7) Wˆα W n gauge W ˆα = Wˆα + ˆα V [ Wˆα,V] (B.25) (B.23) ˆα W ˆβ ˆβ Wˆα [ Wˆα, W ˆβ] = 0 (B.26) Wˆα S gauge 2 gauge Wˆα =0 W n =0 gauge gauge A α (y β ) Lie g S 1 Ā ˆα(y ˆβ) Ā ˆα (y ˆβ) (0,A α (y β )) (B.27) Ā ˆα A μ gauge gauge A μ (B.8) y α ξ RĀ ˆα n = D ˆα Wn (B.28) W n =0 gauge ξ RĀ ˆα n = 0 (B.29) S vector Ā ˆα B.1.3 tensor S vector ξ R tensor T ˆγˆδ... ˆα ˆβ... ξ RT ˆγˆδ... ˆα ˆβ... = 0 (B.30) 2 Field strength F μν 0 gauge gauge gauge gauge A μ =0