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10 ds 2 = N 2 (t))dt 2 + g αβ σ α i (x)σ β j (x)dxi dx j L = 1 2N G αβ(q) q α q β N. G αβ Q i = κ i

11 ϕ V (ϕ)

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15 3 + 1 k = 0 k 0

16 P h

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21 ds 2 = N 2 (t))dt 2 + g αβ σ α i (x)σ β j (x)dxi dx j L = 1 2N G αβ(q) q α q β NV (q). G αβ t

22 3 + 1 (M, γ µν ) M 4 γ µν ( + ++) M R σ σ 3 3 σ Σ t t n a γ µν g ab Σ t g ab = γ ab + n a n b

23 t a t a a t = 1 Σ t (t, x a ) (t + dt, x a + dx a ) t a Σ 0 t = 0 Σ t g ab (0) g ab (t) M t a Σ t N N a N = t a n a = (n a a t) 1, N a = g a b tb N τ t N a Σ t n µ = (1/N, N i /N) N µ = (0, N i ) n µ = ( N, 0, 0, 0) ( N 2 + N i N i γ µν = N i γ ij = g ij ( ) γ µν = 1 1 N i N 2 N i N 2 g ij N i N j 3+1 N i g ij (4) R = (4) R ij ij + 2 (4) R µ0 µ0 )

24 (4) R abcd = (3) R abcd + K ac K bd K ad K bc, (4) R 0bcd = K bd c K bc d K ij K ij = n i;j = n i,j + Γ λ ijn λ = NΓ 0 ij 3 g ij K ij = 1 ( ) Ni j + N 2N j i g ij,0 (4) R ij ij = (3) R + K 2 K ij K ij (4) R µ0 µ0 = ( n ν n µ ;µ n µ n ν ;µ) ;ν K2 + K ij K ij L = γ (4) R = N ( g (3) R + K ij K ij K 2) F,µ, µ F µ = 2 γ ( n µ n ν ;ν n ν n µ ) ;ν g ij N µ 3 3 g ij N, N i π ij = L ġ ij = N g ( K ij + Kg ij), p µ = L Ṅ µ = 0 ϕ µ = p µ 0,

25 ( ) 1 H c = p µ Ṅ µ + π ij ġ ij L = 2π ij N i j Ng 1/2 2 π2 π ij π ij + Rg H c = d 3 x ( NH + N i ) H i H = 1 g ( π ij π ij 1 2 π2 ) gr, j H i = 2π i j H p = H c + d 3 xv µ p µ {p µ, H p } = 0 H 0, H i 0 p µ, H µ ġ ij = δh p δπ ij, πij = δh p δg ij, Ṅ µ = δh p δp µ = ν µ, ṗ µ = δh p δn µ = H µ ν µ N µ Σ n a n b G ab = G 00 = 0, n a G ai = G 0i = 0 {g ij (x), π kl (y)} x 0 =y 0 = 1 ( ) δi k δj l + δ l 2 iδj k δ( x, y), {N µ (x), p ν (y)} x 0 =y 0 = δµ ν δ( x, y) {H(x), H(y)} = ( g ij H j (x) + g ij H j (y) ) i δ(x, y),

26 {H i (x), H(y)} = H(x) i δ(x, y), {H i (x), H j (y} (H j (x) i + H i (y) j ) δ(x, y) Diff(M) Diff(M) H(x) H i = 0 Γ Γ R Γ 2(N M M ) N Γ 2M M Γ R N, N i (6 + 4) 3 2 3

27 N µ R Σ F A (x, g(x), p(x)] (q, p) (Q, P, σ, π) (Q, P ) σ σ α = f α π Q, P, f α Σ F : R Σ M X A (F(x, t)) (T (x; g(t), p(t)], Z a (x; g(t), p(t)]), g(t) p(t) 4 F t (Σ) M (g ab (x), p cd (x)) (X A (x), P B (x), ϕ r (x), p s (x)) X A P B A, B = 0, 1, 2, 3 8 3

28 Z a T ϕ r, p s, s = 1, 2 {X A (x), P B (x)} = δ AB δ(x, y), {X A (x), X B (x)} = 0 = {P A (x), P B (x)}, {ϕ r (x), π s (y)} = δ r sδ(x, y), {ϕ r (x), ϕ s (y)} = 0 = {π r (x), π s (y)}, {X A, ϕ r } = {X A, π s } = {P A, ϕ r } = {P A, π s } = 0 S = dt d 3 x(p a X A + p ϕr r NH N a H a ) S = Σ dt d 3 x(p r ϕr h A (x; Xt B, ϕ r, p s ] X t A (x)) Σ X A Σ H = 0, H a = 0 P A (x) P A (x) + h A (x; X B, ϕ r, p s ] S = dt d 3 x(p r ϕr h A (x; Xt B, ϕ r, p s ] X t A (x)) Σ X A t (x) X A (x; g(x), p(x)] = χ A t (x)

29 H true (t) = Σ d 3 xh A (x; Xt B, ϕ r, p s ] X t A (x) ϕ r, p s

30 3 g ij Σ

31 Ψ[ϕ r (x)] Σ (Σ) iħ δψ[ϕr (x)] δx A (x) = h A (x; X B, ˆϕ r, ˆp s ]Ψ[ϕ r (x)] X A X A (x) ˆϕ r, ˆp s

32 Diff(M) Diff(Σ) V 3 = {V 1, V 2 } ˆV 3 = i ħ [ ˆV 1, ˆV 2 ] 3 g ab (x), p cd (x) [ĝ ab (x), ˆp cd (y)] = iħδ(a c δd b) δ(x, y) δ b a δ(x, y) ĝ ab

33 F (x; g, p] [Ĉa, Ĉb]ψ = 0 [Ĉa, Ĉb]ψ = Cab c (ĝ, ˆp)Ĉcψ Cab c (ĝ, ˆp) ħ Ψ[g] Σ ĝ ab Ψ[g ab (x)] = g ab (x)ψ[g ab (x)] ˆp cd δ Ψ[g ab (x)] = iħ δg cd (x) Ψ[g ab(x)] 3 Riem(Σ) Ψ[g] F F

34 Riem(Σ) g ab (x) x Σ H a 0, H 0 Ĥ a Ψ 2(D b g ac ) ħ δψ = 0, i δg bc ĤΨ ( 16πGħ 2 δ 2 ) g G abcd δg ab δg cd 16πG ((3) R 2Λ) Ψ = 0 D a 3 g ab Σ F o F phys F phys F o F. {A, H[f, f]} 0 H a = 0, H = 0 [Â, Ĥ] = 0 ĤΨ = 0, ĤaΨ = 0 F F o F phys F

35 Σ H i 0 Diff(Σ) x i x i = x i + δn i (x), i = 1, 2, 3 g ab ḡ ab = g ab D a δn b (x) D b δg a (x) Ψ[g ab ] Ψ[g ab ] 2 d 3 x δψ δg ab D a δn b (x) δn b (x) 0 δn a (x) D a δψ δg ab = 0 Ψ 3 Diff(Σ) Diff(Σ) Riem(Σ) Riem(Σ)/Diff(Σ) g ab (x) p cd g ab ħ 2 κ 2 δ 2 Ψ[g] g(x) G abcd (x, g] δg ab (x)δg cd (x) κ 2 R(x, g]ψ[g] = 0 G abcd G µνρσ = 1 2 (gµρ g νσ + g µσ g νρ 2g µν g ρσ )

36 δ(0) 0 Ψ[g] Ψ[g] = A[g]e is[g]/ħκ2 S[g] A[g]

37 ħκ 2 δa[g] δg ab A[g] δs[g] δg ab ħκ 2 lp 2, l Għ P S c 3 G abcd (x, g] δs[g] δs[g] δg ab (x) δg cd (x) g 1/2 (x)r(x, g] = 0, A δ ( G abcd (x, g] A 2 [g] δs[g] ) = 0 δg ab (x) δg cd (x) S i

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39 2 a(t) k ( ) dr ds 2 = dt a(t) 1 kr 2 + r2 dθ 2 + r 2 2 θdϕ 2 a(t) 3 3

40 L = 1 2N G αβ(q) q α q β NV (q), L ξ γ αβ = ϕγ αβ ϕ = 0 ϕ = const [ξ a, ξ b ] = Cab c ξ c, Cab c = Cc ba Cab c {ξ a} G G (M, γ αβ ) Σ t p, q Σ t g : M M g(p) = q G Σ t dimg = dimσ t = G Σ t g g(p) Σ t G Σ t G Σ t Σ t G m = 3 1 σi α dσ α = C α βγ σβ σ γ σ α i,j σ α j,i = 2C α βγ σγ i σβ j

41 Cβγ α t, x i ds 2 = (N α (t)n α (t) N 2 (t))dt 2 + 2N α (t)σ α i (x)dx i dt + g αβ σ α i (x)σ β j (x)dxi dx j ds 2 = N 2 (t)dt 2 + a 2 (t)dr 2 + b 2 (t) ( dθ 2 + f 2 (θ)dϕ 2) θ k = 1, f(θ) = θ k = 0, θ k = 1. k k = 0, 1 k = 1 G 3 S 2 R 2 2 t = const, r = const 3 C23 3 θ (θ) = 2 E α β K α β = gαρ K ρβ E 0. = K α β K β α K 2 + R = 0 E α. = K µ α C ϵ µϵ K µ ϵ C ϵ αµ = 0. = K α β NKKα β + NRα β + 2N ρ (K α ν C ν βρ Kν β Cα νρ) K αβ = 1 2N (ġ αβ + 2g αν C ν βρ N ρ + 2g βν C ν αρn ρ ) R αβ = C κ στ C λ µνg ακ g βλ g σν g τµ + 2C λ ακc κ βλ + 2Cµ ακc ν βλ g µνg κλ + 2C λ βκ Cµ µνg αλ g κν + 2C λ ακc µ µνg βλ g κν g αβ K αβ N, N α

42 g αβ g αβ t t = g(t) t = f( t). (ds 2 = d s 2 ) g αβ (t) g αβ (f( t)) ḡ αβ ( t) N(t) ±N(f( t)) df( t) d t N( t) N α (t) N α (f( t)) df( t) d t N α ( t) Kβ α df( t)/d t t = t t = t x i = g i (x j, t) x i = f i ( x j, t) ḡ αβ = Λ µ αλ ν β g µν N α = Λ β α(n β + P ρ γ ρβ ) N α = S α β (N β + P β ) N = N S = Λ 1 Λ α β, P α t Λ α µc µ βγ = Λρ β Λσ γc α ρσ P µ C α µνλ ν β = 1 2 Λ α β

43 (Λ 3 ) α β = (Λ 1) α ϱ (Λ 2 ) ϱ β (P 3 ) a = (Λ 1 ) α β (P 2) β + (P 1 ) a (Λ 1, P 1 ) (Λ 2, P 2 ) Λ α β (t) Cβγ α Λ α β (t) P α (t) Λ α β (t) = Λα β, P α (t) = 0 N, N a, g ab N, N a, ḡ ab N a = 0 g ab g ab g αβ g αβ ġ αβ X I = λ ρ Iα γ ρβ γ αβ λ α β λ α Iρ C ρ βγ = λρ Iβ Cα ργ + λ ρ Iγ Cα βρ. g αβ

44 t t + α g αβ λg αβ Y 1 = t, Y 2 = g 11 + g 12 + g 13 + g 22 + g 23 + g 33 g 11 g 12 g 13 g 22 g 23 g 33 X (I) [X I, Y α ] = 0 {I = 1,..., dim(aut(g)) α = 1, 2} S = 1 4κ 2 dtd 3 xn ( ) g K ij K ij K 2 + (3) R 2Λ + S matter S matter = d 4 x ( γ 1 ) 2 γµν µ ν ϕ U(ϕ) U(ϕ) N i = 0 K ij = 1 g ij 2N t G ijkl = 4N 2 (g ik g jl + g il g jk 2g ij g kl ) K ij K ij K 2 = 1 4N 2 g Gijkl ġ ij ġ kl

45 S = = = dtd 3 x g ( ( N g 16N 2 gκ 2 Gijkl ġ ij ġ kl + N g (3) R 4κ 2 ( dtd 3 1 x 16κ 2 N Gijkl ġ ij ġ kl + ( ) 1 dt 2N G µν q µ q ν NV (q) g 2N ϕ 2 N g N g2λ + N ) g 2N ϕ 2 2 (3) R 4κ 2 + 2Λ U(ϕ) )) G µν q µ q ν = 1 8κ 2 Gijkl ġ ij ġ kl + g ϕ V (q) = (3) R + 2Λ + U(ϕ) V (q) 4κ 2 N, q E 0 := 1 2N 2 G αβ q α q β + V = 0, E µ := q µ + Γ µ νλ qν q λ Ṅ N qµ + N 2 G µκ V,κ = 0 Γ µ νλ 2n 2 S = L(q(x), q(x))dx x t r L = 1 2N G αβ(q) q α q β NV (q). N(x) (x, q, N)

46 (x, q, N) X = X 1 + X 2 X 1 = ξ α (q) q α + τ(q)n N, X 2 = χ(x) x χ,x(x)n N L ξ G µν = τ(q)g µν, L ξ V = τ(q)v χ(x) x ξ G µν V X = χ(x) x + ξα (x) q α + N (τ(q) + c χ,x) N c = 0 χ(x) ξ α (q), τ L ξ G µν = τ(q)g µν, L ξ V = (τ + 2c)(q)V ξ α, τ X 1 L ξ G µν = τ(q)g µν, L ξ V = τ(q)v Y X 2

47 X 1 = ξ α (q) q α + τ(q)n N, X 2 = χ(x) x χ,x(x)n N, Y = q α q α N N x = f( x) N = NV q α L N(x) N( x) = N(f( x))f ( x), q α (x) q α ( x) = q α (f( x)). L = 1 2 N Ḡµν q µ q ν N Ḡµν = V G µν V = 1 ξ = ξ α q α X 1 L ξḡµν = 0, L ξ V = 0 ξ Ḡ µν, V Y L Y Ḡ µν = n 2 Ḡµν Ḡµν = V G µν X 2 n(n+1) H = q α p α L p α = L q α p α = 1 N Ḡ αβ q β

48 p a = 1 N G ab q b, p N = 0 N p N 0 N ( ) 1 H = N 2 Gαβ p α p β + 1 NH p N ṗ N = 0 {p N, H} = NH = 0 H (q, p) {q a, p b } = δb a Ḣ = {H, H} = 0 {p N, H} = 0 q α = {q α, H} = NG αµ p µ, ṗ α = {p α, H} = N 2 Ḡκλ,α p κ p λ Q = ξ α 1 N G κα q κ χ(x)e 0 E 0 Q = ξ α p α χ(x)nh ξ α p α Q H 0 ξ α G αβ L ξ G αβ = ω(q)g αβ L ξ G αβ = ω(q)g αβ

49 Q = ξ α p α dq dt = {Q, H} = Nω(q) ω(q) ξ ω(q) = 0 ξ α G αβ Q i Q i = ξ α p α = κ i ω(q) = c ξ α G αβ Q Q = ξ α p α = c dtn(t) + c c ω(q) 0 ξ α G αβ ξ α i ω(q) = 0 [ξ i, ξ j ] = c γ ij ξ k Q i c k ij ξ i {Q i, Q j } = c m ij Q m. ω(q) = const Q i = κ i, Q h = κ h dtn

50 p N, H ˆp N Ψ(q, N) i Ψ(q, N) = 0 Ψ Ψ(q), N ( ĤΨ(q) 1 ) 2 2 c + 1 Ψ(q) = 0. c 2 c 2 + d 2 R, 4(d 1) R (q, N) ( q, N) ˆX 1 Ψ = 0, ĤΨ = 0 X 1 := p N 0, H := 1 2 Gµν p µ p ν + V (q) 0

51 µ ˆX 1 = 1 2µ (µˆp N + ˆp N µ), Ĥ = 1 2µ ˆp µg µν ˆp ν + B + V (q) B q G µν (q) (q, N) ( q, N) µ( q, N)d Nd q = µ(q, N)dNdq µ( q, N) = µ(q, N) (q, N) ( q, N) ˆ X1 ˆΨ = ˆ H ˆΨ = 0 (q, N) ˆX 1 Ψ = ĤΨ = 0 Ψ(q, N) = Ψ( q, N) q q q(q) N N N = cn Ψ Ψ( q, N) = e W (q,n) Ψ(q, N) W, B ˆX 1 Ĥ = 1 2 G µg µν ν d 2 R + V (q) 8(d 1) N N d n qµψ 1 ψ 2 µ µ = G αβ ˆQ i Ψ(q) i 2µ (µξα i α + α µξ α i )Ψ(q) = κ i Ψ(q).

52 κ i {Q i, Q j } = c k ijq k, [ ˆQ i, ˆQ j ]Ψ = (κ i κ j κ j κ i )Ψ = 0 c k ijκ k = 0,

53 Ψ(x) = A(x)e is(x), A(x) S(x) ( A 2 ) S t A2 + = 0 m S t + ( S)2 m ħ2 2 A + V (x) 2m A = 0 ħ 0 S(x) ħ2 2 A 2m A 0 p i = i S p i = L q i 1 2 Gαβ α S β S 1 A 2 A d 2 R + 1 = 0. 4(d 1) L = S q i q i Q(q) 1 2A A = 1 2µ α(µg αβ β )A.

54 S(q) G αβ α S β A + A S = 0. 2 Q = 0 3

55 3 k = 0, k = 1 ν S = d 4 x g (R + ϵ µ ϕ µ ϕ + 2 U(ϕ)) R g g µν ϵ

56 g µν R µν 1 2 Rg µν = T µν T µν = ϵ ϕ,µ ϕ,ν 1 2 (ϵ ρϕ ρ ϕ 2 U(ϕ)) g µν ϵ ϕ U (ϕ) = 0, ϕ ( ) ds 2 = N(t) 2 dt 2 + a(t) k r 2 dr2 + r 2 dθ 2 + r 2 2 θdφ 2 N(t) a(t) N(t) (0i) T 0i = 0 L = 2a2 n ( a 2 U(ϕ) 3k ) ( 6ȧ 2 + ϵ a 2 ϕ2 ) n, n N N = n 2 a (a 2 U(ϕ) 3k). G µν = 4 a 2 ( a 2 U(ϕ) 3k ) ( ) ϵ a 2. k = 0 k = ±1

57 k = 0 ξ = a 6 U(ϕ) L ξ G µν = G µν a Q = a 6 p a + n(t)dt = a L 6 ȧ + n(t)dt = 4 a5 ȧu(ϕ) + n(t)dt. n Q = κ, κ ϕ(t) = t h(t) n(t) A(t) U(t) n(t) = ḣ(t), U(t) = (h(t) κ) ḣ(t) 4A(t). a(t) 5 ȧ(t) A(t) = 0, a(t) = ±6 1/6 (A(t) + c 1 ) 1/6. a(t) 2(A(t) + c 1 ) A(t)ḣ(t) + (κ h(t)) ( A(t) 2 6 ϵ (A(t) + c 1 ) 2) = 0, h(t) (ḣ2 ) 1/2 A(t) = µ4 6 ḣ ± + ϵ (κ h) 2 dt c 1, κ h µ k = 0 ϕ = t h(t) U(t) c 1

58 κ h(t) = κ+ ( ω 2 3 ϵ ω dt) h(t) N(t) ω > 0 N(t) = 1 3 µ2 ω e 3 ϵ ω dt, a(t) = µ 2/3 e ω/6, U(t) = 3 ( ω 2 6 ϵ ) e 6 ϵ ω dt 4 µ 4 ω 2 N(t) = 2 ϵ µ2 e ω 2 ω, a(t) = µ 2/3 e ϵ ω dt, U(t) = eω ( ω 2 6 ϵ ) 8 ϵ µ 4 ω < 0 ω = 6 ϵ ġ dt ω g(t) r rµ 2/3 µ = 3 m ds 2 = m 4 ω 2 e 6 (ϵ/ ω)dt dt 2 + e ω/3 ( dr 2 + r 2 dθ 2 + r 2 2 θdφ 2), ( ω 2 6 ϵ ) e 6 (ϵ/ ω)dt U(t) = 12m 4 ω 2. ω t ω ω(t) F (ω) t = ϵ F (ω) dω, 6 F (ω) ds 2 = e F (ω) dω 2 + e ω/3 ( dr 2 + r 2 dθ 2 + r 2 2 θdφ 2) U(ω) = 1 12 e F (ω) ( 1 F (ω) ). ϕ(t) = t ϕ(ω) = ϵ F (ω) dω. 6 ω F (ω)

59 ρ ϕ (t) = T µν u µ u ν P ϕ (t) = 1 3 T µν h µν u µ = ϕ,µ g κλ ϕ,κϕ,λ 4 h µν = g µν + u µ u ν 3 u µ ρ ϕ (ω) = 1 (ω) e F 12 P ϕ (ω) = 1 12 e F (ω) ( 2F (ω) 1 ) P ϕ = (2F (ω) 1)ρ ϕ. γ ϕ = P ϕ ρ ϕ F (ω) k 0 ξ = ϕ L ξ G µν = a2 U (ϕ) a 2 U(ϕ) 3k G µν a(t) 2 n(t)u (ϕ(t)) Q = p ϕ + a(t) 2 U(ϕ(t)) 3k dt = 4 ϵ ( a4 ϕ a 2 U(ϕ) 3k ) a(t) 2 n(t)u (ϕ(t)) + n a(t) 2 U(ϕ(t)) 3k dt ϕ(t) = t w(t) h(t) n(t) = 2ḣ ( a 2 U 3k ) a 2 U ẇ U(t) = a 6 dt,

60 h(t) = ϵ ẇ + 2h κ = 0 ḣ ( κ ± ) 4 c 1 + κ 2 8 ϵ w. w(t) w(t) = a v ȧ + 1 ( 4 c1 + κ 2) 6v, 8 ϵ v(t) L n = 0 6 ȧ v ϵ a ( ( 6ȧ v(t) = a ϵ a ) ) ( ka 5 ( ( 6ȧ dt ȧ 2ȧ + 36 v ȧ2 ϵ a k a 4 6 v = 0, a ϵ a ȧ ) ) ) dt dt + c 2, c 2 a(t) = e ω/6 ds 2 = ( ( 36 2e ω 6 (ϵ/ ω)dt e ω ω 2 c 2 + 3k (6 (ϵ/ ω)dt ω 3 ) ω ) + e ω/3 ( 1 1 kr 2 dr2 + r 2 dθ 2 + r 2 2 θdφ 2 ) )dt 2 dt ke 2ω 3 k = 0 c 2 = 1 72 m 4 ω ( ( 6 e ω ω 2 6 ϵ ) e ω 6 ( (ϵ/ ω)dt c k (6 (ϵ/ ω) ω dt) ) ) 3 ω dt + 3 k e 2ω 3 U(t) = ω 2, k = 0 c 2 = 1 72 m 4 [ ( 1 S (ω) t = ± 6 ϵ S (ω) + 1 )] 1/2 dω 3 S(ω) k 0 S(ω) = ( 12 k ) e F (ω) ω/3 dω 6 c 2 k

61 c 2 ( ) 1 ds 2 = e F (ω) dω 2 + e ω/3 1 kr 2 dr2 + r 2 dθ 2 + r 2 2 θdφ 2. U(ω) = 1 12 e F (ω) ( 1 F (ω) ) + 2 k e ω/3 ω [ 1 ϕ(ω) = ± (F (ω) + 12 k e F (ω) ω/3)] 1/2 dω, 6 ϵ ϕ(t) = t + k 0 F (ω) k = 0 k 0 F (ω) ρ ϕ (ω) = 1 12 e F (ω) + 3 k e ω/3 P ϕ (ω) = 1 12 e F (ω) ( 2F (ω) 1 ) k e ω/3, ( ) 2 e ω/3 (3F (ω) 1) P ϕ = 3 ( 36 k e F (ω) + e ω/3) 1 ρ ϕ. 3 k T µν = (ρ + P )ū µ ū ν + P g µν, ū µ = (1/N(t), 0, 0, 0) 4 ρ P P = γρ γ L m ρ g µν T µν ;ν = 0

62 T µν ρ P a ρ = m a 3(1+γ) m L m = 2 gρ = 2N m a 3γ N = n 2 (m a 3γ 3 k a + a 3 U(ϕ)) n L = 2a n ( m a 3γ + a 3 U(ϕ) 3 a k ) ( 6 ȧ 2 + ϵ a 2 ϕ2 ) n. R µν 1 2 g µνr = T µν + T µν G µν = 4 a ( m a 3γ + a 3 U(ϕ) 3 a k ) ( ) ϵ a 2. k = 0 ξ = ϕ L ξ G µν = Q = p ϕ + n a3(γ+1) U (ϕ) a 3(γ+1) U(ϕ) + m dt = 4 ϵ a3(1 γ) ϕ ( a 3(γ+1) U(ϕ) + m ) n a3(γ+1) U (ϕ) a 3(γ+1) U(ϕ) + m G µν, + n a3(γ+1) U (ϕ) a 3(γ+1) U(ϕ) + m dt. Q = const. L n = 0 k = 0 ϕ(t) = t n Q = const. k 0

63 ϕ = t a(t) =e ω/6 ( 6 n(t) = ω ω ) ( 1 e ( 1 (γ 1)ω+I) ) 1/2 ϵ 2 c 1e I 2 ϵ m e I 2 dt + 1 ω 3 m e 1 2 (γ 1)ω [ ( U(t) = 3 e 1 2 (γ+2)ω e ( 1 (γ 1)ω I) ) 2 (6 2 ω 2 c ϵ m dt ϵ ω 2 ) e ( 1 γω+i) 2 4 ϵ m e ω 2 ω ] c 1 ω(t) I I = ω 6 ϵ ω dt. S(ω) ( S(ω) = 6(γ + 1)m ( ) 1 γ + 1 ϕ(t) = t = ± + S (ω) 6 ϵ 2 S dω (ω) ) e F (ω) 1 2 (γ+1)ω dω 3 c 1 (γ + 1)m, U(ω) = 1 12 e F (ω) ( 1 F (ω) ) (γ 1) m e 1 2 (γ+1)ω ϕ(ω) [ 1 ϕ(ω) = ± (F (ω) 6(γ + 1)m e F (ω) 1 (γ+1)ω)] 1/2 2 dω. 6 ϵ F (ω) ρ ϕ = 1 12 e F (ω) m e 1 2 (γ+1)ω P ϕ = 1 12 e F (ω) ( 2F (ω) 1 ) γ m e 1 2 (γ+1)ω. ρ = m e 1 2 (γ+1)ω,

64 P = γρ ū µ u µ = ϕ,µ g κλ ϕ,κϕ,λ = e F (ω)/2 ρ = ρ ϕ + ρ = 1 (ω) e F 12 P = P ϕ + P = 1 12 e F (ω) ( 2F (ω) 1 ), T µν + T µν k = 0 k 0 k 0 ξ = ϕ L ξ G µν = ξ Q =p ϕ + a 3(γ+1) U (ϕ) n(t) 3ka 3γ+1 + a 3(γ+1) U(ϕ) + m dt = 4 ϵ a3(1 γ) ϕ ( 3ka 3γ+1 + a 3(γ+1) U(ϕ) + m ) n a 3γ+3 U (ϕ) 3ka 3γ+1 + a 3γ+3 U(ϕ) + m G µν + na 3(γ+1) U (ϕ) 3ka 3γ+1 + a 3(γ+1) U(ϕ) + m dt Q = const. L n = 0 ϕ = t a(t) =e ω/6 n(t) =( 6 ω ω ) ϵ (2 m e 1 2 (γ 1)ω 3 c 1 e I 6 ( 1 e I 2 (γ 1)ω m 3ke 1 (3γ+1)ω) ) 6 1/2 12 ϵ e I dt 6ke 2ω 3 ω [ U(t) = 3e 1 2 (γ+2)ω ( e I 1 2 (γ 1)ω m 3ke 1 (3γ+1)ω) 6 c 2 ω ϵ dt ( ω 2 6 ϵ ) e I+ 1 2 γω ω

65 12ke 1 6 (3γ+4)ω + 4 m e ω 2 ] c 1 I S(ω) ( 18S (ω) + 3(3γ 1)S (ω) + (3γ + 1)S ) (ω) 1/2 ϕ(ω) = t(ω) = ± 18 ϵ ((3γ + 1)S (ω) 6S dω, (ω)) F (ω) S (ω) = 1 6 e (γ+ 7 6)ω ( 18 c 1 e (γ+ 5 6)ω+F (ω) +S (ω) ((3γ + 1)e (γ+ 7 6)ω + 72ke (γ+ 5 6)ω+F (ω) 36(γ + 1) m e 1 (3γ+4)ω+F (ω)) 6 ) 12 (3γ + 1) k S(ω)e (γ+ 5 6)ω+F (ω) F (ω) U(ω) = 1 12 e F (ω) ( 1 F (ω) ) + 2 k e ω/ (γ 1) m e 1 2 (γ+1)ω [ 1 ( ϕ(ω) = ± F (ω) + 12 k e ω/3+f (ω) 6 ϵ 6 (γ + 1) m e F (ω) 1 (γ+1)ω)] 1/2 2 dω F (ω) ω ρ ϕ = 1 12 e F (ω) + 3 k e ω 3 m e 1 2 (γ+1)ω, P ϕ = 1 12 e F (ω) ( 2F (ω) 1 ) k e ω/3 m γ e 1 2 (γ+1)ω P = γρ ρ P ρ = ρ ϕ + ρ = 1 12 e F (ω) + 3 k e ω 3 P = P ϕ + P = 1 12 e F (ω) ( 2F (ω) 1 ) k e ω/3 k 0 T µν + T µν k = 0 ν N

66 P i = γ i ρ i, i = 1,..., ν. R µν 1 2 g µνr = T µν + ν i=1 ϵ ϕ(ω) + 1 ϕ (ω) U (ω) = 0, T (i) µν T (i) µν i [ 1 ( ϕ(ω) = ± F ω/3+f (ω) (ω) + 12 k e 6 6 ϵ 1/2 ν (γ i + 1) m i e F (ω) 1 2 i+1)ω)] (γ dω i=1 U(ω) = 1 12 e F (ω) ( 1 F (ω) ) + 2 k e ω/ ν (γ i 1) m i e 1 2 (γi+1)ω. i=1 ρ i = m i e 1 2 (γ I+1)ω, m i ρ ϕ = 1 ν 12 e F (ω) + 3 k e ω 3 m i e 1 2 (γi+1)ω, i=1 P ϕ = 1 12 e F (ω) ( 2F (ω) 1 ) k e ω/3 ν m i γ i e 1 2 (γi+1)ω. i=1

67 4 4 4 ds 2 = N 2 (t)dt 2 + γ αβ (t)σ α i (x)σ β j (x)dxi dx j, i, j = 1, 2, 3 N(t) γ αβ (t) 3 3 σi α (x) σ α i,j σ α j,i = C α βγ σβ j σγ i. σ α S tot = S grav + S mat = d 4 x ( g R 1 ) 2 gµν µ ϕ ν ϕ, ϕ R µν 1 2 Rg µν = 1 4 g µνg κλ κ ϕ λ ϕ µϕ ν ϕ T µν, g µν µ ν ϕ = 0

68 R T µν (0i) T 0i = 0 x i S tot = dt L L L = 1 2N G αβ(q) q α q β N. t t = f( t) N(t) N( t) = N(f( t))f ( t), q α (t) q α ( t) = q α (f( t)), ˆQ i G µν = R µν 1 2 Rg µν G µν = T (imf) µν, T (imf) µν T (imf) ij = (ρ + p) u i u j + pg ij + 2q (i u j) + π ij,

69 ρ 4 u µ q µ p π µν Π mn = G ij h i mh j n = p h mn + π mn, π mn = Π mn 1 3 Π k k h mn = Π mn ph mn, ρ = G ij u i u j, p = 1 3 Π i i, q k = G ij u i h j k, h ij u i h ij = g ij + u i u j u i u i = 1 i u j = u i u j + ω ij + σ ij θh ij, u i = u j j u i, θ = i u i, σ ij = (i u j) + u (i u j) 1 3 θh ij, ω ij = [i u j] + u [i u j], k = 1 k = 1 k = 0 k 0 ( ) dr ds 2 = N 2 (t)dt 2 + a 2 2 (t) 1 kr 2 + r2 dθ 2 + r 2 2 θdφ 2,

70 N(t) a(t) N(t) L = 6Nka 6aȧ2 N + a3 ϕ2 2N, L = n 36ka2 ȧ 2 n + 3ka4 ϕ2, n n = 6kNa H = n( p2 a 72ka 2 + p2 ϕ 1) = nh 0 12ka4 H q i (t) G αβ = 6ka ( 12a 0 0 a 3 ). ξ 1 = eϕ/ 3 a a 2 3e ϕ/ 3 a 2 ϕ, ξ 2 = e ϕ/ a 3 a + 2 3e ϕ/ 3 a 2 ϕ, ξ 3 = ϕ, ξ h = a 4 a i = 1, 2, 3 h [ξ 1, ξ 3 ] = 1 3 ξ 1, [ξ 2, ξ 3 ] = 1 3 ξ 2. k = 0 c 1 31 = c2 23 = 3 4 Q i = ξi αp α 12e ϕ 3 ka (6ȧ + 3a ϕ ) Q 1 =, n 12e ϕ 3 ka ( 6ȧ + 3a ϕ ) Q 2 = n, Q 3 = 6ka4 ϕ n k = 0

71 Q h = 18ka3 ȧ n + dt n(t) Q i = κ i i = 1, 2, 3, h κ i a = 2 31/4 κ 3 e ϕ 2 3 κ 1 + κ 2 e 2ϕ 3, κ3 e ϕ 2 3 (κ1 + κ 2 e 2ϕ 3 ) ϕ ȧ = n = 3 1/4 ( κ 1 + κ 2 e 2ϕ 3 ) 3/2 288κ 3e 2ϕ 3 k ϕ,, (κ 1 κ 2 e 2ϕ 3 ) 2 3κ3 (κ 1 + κ 2 e 2ϕ 3 ) dt n(t) = κ h, 2(κ 1 κ 2 e 2ϕ 3 ) t dt n(t) = n κ 1 κ k = 0. k = 0 Q i = κ i κ 1, κ 2 Q i Q 1 Q 2 ϕ = t a = 2 31/4 κ 3 t 1 2 3, N = 8 33/4 κ 3 t κ 1 + κ 2 t 2/ 3 ( κ 1 + κ 2 t 2/ 3 ) ds 2 λ = 4 T (1 + kt ) dt 2 + λ ( ) T dr 2 3 (1 + kt ) 1 kr 2 + r2 dθ 2 + r 2 2 θdφ 2. λ T

72 t t 12 3 ( ) κ 2 1 T 3/2 k 3(kT + 1)3 R = 2T 3/2 λ, λ = κ 3 3k 3/2 T 0 T ds 2 = dt 2 + T 2/3 dr 2 + T 2/3 r 2 dθ 2 + T 2/3 r 2 2 θdφ 2. R = 2 3T 2, T 0 Q i ˆQ 1 Ψ = ieϕ/ 3 ( 6 ϕ Ψ + 3a a Ψ) 3a 2 ˆQ 2 Ψ = ie ϕ/ 3 (6 ϕ Ψ + 3a a Ψ) 3a 2 = κ 1 Ψ, = κ 2 Ψ, ˆQ 3 Ψ = i ϕ Ψ = κ 3 Ψ, ĤΨ = 144ka4 Ψ 12 ϕϕ Ψ + a( a Ψ + a aa Ψ) 144ka 4 = 0, µ = G αβ ˆQ i Ψ = κ i Ψ ( ˆQ 1, ˆQ 2 ) ˆQ 1, ˆQ 2, ˆQ 3 ˆQ 1, ˆQ 2 ( ˆQ 1, ˆQ 2 ) ) Ψ = A (i a2 4 (κ 1e ϕ 3 + κ2 e ϕ 3 ).

73 L q = S q 1 ( ) 2 a e ϕ/ 3 (κ 1 + e 2ϕ/ 3 κ 2 ) ( 3e ϕ/ 3 (κ 1 e 2ϕ/ 3 κ 2 )) a 2 12 = 144kaȧ, n = 72ka2 ϕ, n S = 1 4 a2 e ϕ 3 (κ1 + κ 2 e 2ϕ 3 ) k = 0 ˆQ 3 Ψ cl (a, ϕ) = e iϕκ 3 (A 1 I i 3κ3 (6a 2 ) + B 1 I i 3κ3 (6a 2 )), Ψ op (a, ϕ) = e iϕκ 3 (A 2 J i 3κ3 (6a 2 ) + B 2 J i 3κ3 (6a 2 )), A 1 = B 1, A 2 = B 2 Ψ sm c 1 e iκ 3ϕ a. A 1 = B 1, A 2 = B 2 Ψ cl la ea2 a eiκ 3ϕ, Q sm = 1 144ka 4, Q cl + 4a4 la = 1 144a 4 k, Ψ op la ( 6a 2) e iκ3ϕ. a Qop la = 144ka ka 4, S = κ 3 ϕ a = c, n = 6ka4 κ 3 ϕ.

74 N(t) 8 3 3/4 t ( ( 3/2 48kt2/ 3 κ 1 κ 1 3 κ 1 κ 3/2 2 )) kt2/ 3 1 κ 2F 2 1 2, 3 4 ; 7 4 ; 144kt 3 1 κ 2 1 ϕ(t) = c 3 (144kt 2/ 3 κ 1 + κ 3 1 ), 2 F 1 (a, b; c; d) 4 ds 2 λ = 4 T (1 + T ϵ) dt ϵr 2 dr2 + r 2 dθ 2 + r 2 2 θdφ 2, c 2 = λ2 16 λ R = 6k ( ( Ψ(a, ϕ) = e iκ 3ϕ A 3 2 ) ( 3κ 3 a + B 3 2 )) 3κ 3 a. A = A 3 ( 2 3κ 3 a ) +B 3 ( 2 3κ 3 a ) S = κ 3 ϕ ds 2 = dt 2 + dr 2 + r 2 dθ 2 + r 2 2 θdφ 2. a k = ±1 π ij q i p = k, ρ = 3k, w = 1 3

75 π ij π ij = r θ 3 q i = 0, p = 1 3r 2, ρ = 1 r 2, w = 1 3 σ = 0 1 x γ γ = diag(e 2a, e 2b, e 2c ) a, b, c 4 ds 2 = N 2 dt 2 + e 2c dx 2 + e 2a dy 2 2e 2a xdydz + (e 2a x 2 + e 2b )dz 2, L = 1 2N ea+b+c ( 4ḃċ + 4ȧḃ + 4ȧċ ϕ 2) N 2 e3a b c. N = 2n ( 3a + b + c) n L = 1 4n e4a ( 4ȧḃ + 4ȧċ + 4ḃċ ϕ 2) n.

76 G αβ G αβ = e 4a ξ i, i = 1,..., 6 ξ h ξ 1 = (a + b) b (a + c) c, ξ 2 = 1 2 ϕ c + (a + b) ϕ, ξ 3 = b, ξ 4 = ϕ b + 2(a + c) ϕ, ξ 5 = c ξ 6 = ϕ ξ h = 1 4 a. [ξ 1, ξ 2 ] = ξ 2, [ξ 1, ξ 3 ] = ξ 3, [ξ 1, ξ 4 ] = ξ 4, [ξ 1, ξ 5 ] = ξ 5, [ξ 2, ξ 3 ] = ξ 6, [ξ 2, ξ 4 ] = ξ 1, [ξ 2, ξ 6 ] = 1 2 ξ 5, [ξ 4, ξ 5 ] = 2ξ 6 [ξ 4, ξ 6 ] = ξ 3, [ξ 4, ξ 6 ] = ξ 6. Q i ξ i, i = 1,..., 6 Q 1 = e4a n Q 3 = e4a n Q 5 = e4a n ) ((ȧ + ḃ)c + (ḃ ċ)a (ȧ + ċ)b, Q 2 = e4a 2n (ȧ + ċ), Q 4 = e4a n ( ) ȧ + ḃ, Q 6 = e4a 2n ϕ, ( (ȧ + ċ)ϕ + (a + c) ϕ ( ) (ȧ + ḃ)ϕ + (a + b) ϕ, ξ h Q h = e4a ) (ḃ + ċ 4n Q i = κ i, i = 1,..., 6 dt n. ), n = e4a ( ) ȧ + κ ḃ, ϕ = 2κ 6 (a + b) + 2κ 2, c = κ 3 (a + b) a + κ 4κ 5 2κ 2 κ 3, 5 κ 5 κ 5 κ 5 2κ 5 κ 6 2κ 1 κ 6 + κ 4 κ 5 2κ 2 κ 3 = 0.

77 b Q h ( ) κ5 c 2 1 e4a b = c 2 a 1 2 c 1ϵ κ 5, ϵ = ±1 c 1, c 2 H = 1 1 ( 4n e4a 4ȧḃ + 4ȧċ + 4ḃċ ϕ 2) κ c2 1 (κ2 6 κ 3κ 5 ) = 0 a = 1 4 ( (κ 3 κ 5 κ 2 6 ) 2 t ) ds 2 = e (α+ν)t tdt 2 + e αt tdx 2 + tdy 2 2x t dy dz + ( e νt t + x 2 t ) dz 2. t ± α, ν R = 1 2 e(α+ν)t (αν 1) t. 3 ( ˆQ 3, ˆQ 5, ˆQ 6 ) ( ˆQ 1, ˆQ 6 ) ( ˆQ 3, ˆQ 4 ) ( ˆQ 2, ˆQ 5 ) ˆQ 1 Ψ = i((a + c) c (a + b) b )Ψ, ˆQ 2 Ψ = i 2 (2(a + b) ϕ + ϕ c )Ψ, ˆQ 3 Ψ = i b Ψ, ˆQ 4 Ψ = i(2(a + c) ϕ + ϕ b )Ψ, ˆQ 5 Ψ = i c Ψ, ˆQ 6 Ψ = i ϕ Ψ, ĤΨ = 1 4 e 4a ((4e 4a 8) + 4 ϕϕ 4 c + cc 4 b 2 bc + bb + 4 a 2 ac 2 ab + aa )Ψ, µ = e 8a

78 ( ˆQ 3, ˆQ 5, ˆQ 6 ) Ψ(a, b, c, ϕ) = ( A 1 J λ (e 2a ) + A 2 J λ (e 2a ) ) e is, S = κ 3 b + κ 5 c + (κ 3 + κ 5 )a + κ 6 ϕ, λ 2 = 3 κ 3 κ 5 +κ 2 6 A 1 = A 2 Ψ sm e is e 2a (2λa). Q sm = e 4a (2 κ 3 κ 5 +κ 2 6 ) L q i = S q i Ψ la e is e 3a ( e 2a + e 2a ) Q la = 1 3e 4a. 4 a, b, c, n ds 2 = λ 1 e T dt 2 + T 2 dx 2 + dy 2 2xdydz λ 1, λ 2, λ 3 ( 4x 2 + (λ 2 T + λ 3 ) 2) dz 2 2 R = T 2 (λ 2 T + λ 3 ) 2λ2 2 T 2λ λ 2λ 3 λ 1 e T (λ 2 T + λ 3 ) T 0 ( ˆQ 1, ˆQ 6 ) + 2λ 2 λ 3 λ 2 3 λ 1 e T T (λ 2 T + λ 3 ). ˆQ cas = 4 ˆQ 1 ˆQ6 2 ˆQ 2 ˆQ3 2 ˆQ 3 ˆQ2 + ˆQ 4 ˆQ5 + ˆQ 5 ˆQ4 + A( ˆQ 3 ˆQ5 ˆQ 2 6),

79 A κ cas ˆQ cas Ψ = A( ϕϕ bc )Ψ = κ cas Ψ. ) Ψ =e (I is iκ1 (2 (a + b)(a + c)κ 2 6 ) + K iκ 1 (2 (a + b)(a + c)κ 26 ) ( ) A 1 J 3 (e 2a ) + A 2 J 3 (e2a ) S = κ 1 (a + b) + κ 6 ϕ κ 2 κ 1 6 ( (a + b)(a + c)) 2 ( ( )) ( A sm = e 2a κ 1 2 (a + b)(a + c)κ ) 3a Q sm (8a2 +8bc+8ab+8ac+κ 2 1 ) 4e 4a (a+b)(a+c) A A la = e 3a+2 (a+b)(a+c)κ 2 e 2a + e 2a 6 (a + b)(a + c)κ 2 6 ( 1+2 (a+b)(a+c)κ 2 6 (a+b)(a+c)(3+4e4a 4κ 2 6 ) ) Q la 4e 4a (a+b)(a+c) ( ds 2 = λ2 2(c 1 (1 + T ) 2 e2c 1T c 2 ) c 1 (1+T ) dt 2 + T 2 dx 2 + dy 2 (c2 c 2 1 2xdydz + T )2 c 1 + c 1 T + x2 ) dz 2 T 0 T c 1 < 0 c 1 > 0 c 1 ( ˆQ 2, ˆQ 5 ) ( ˆQ 3, ˆQ 4 ) ( ˆQ 2, ˆQ 5 ) Ψ = ( ) c 1 J 3 (e 2a ) + c 2 J 3 (e2a ) e is,

80 S = 4κ2 5 a2 + 4κ 2 5 bc + 4κ2 5 (b + c) (κ 5ϕ 2κ 2 ) 2, 4κ 5 (a + b) ( a ) ( A sm = e 2a + b 2 ) 3a, Q sm 2e 4a A A la = e2a + e 2a e 3a, a + b Q la 1 3 4e 4a ds 2 = λ 2 2e λ 3T dt 2 + λ 2 1T 2 dx 2 + dy 2 2xdydz + ( x 2 + (1 + T ) 2) dz 2, T 0 T λ 3 ( ˆQ 3, ˆQ 4 ) κ 5 κ 3, κ 2 κ 4 2 ( ˆQ 3, ˆQ 5, ˆQ 6 ) π ij e t, t, x t w = λ 1e t 2λ 2 t 3 + λ 2 (λ 2 3λ 3 )t 2 + λ 3 (λ 2 λ 5 )t 3e t. λ 1 3λ 2 t(λ 2 t + λ 3 ) ( ˆQ 1, ˆQ 6 ) w ( ˆQ 2, ˆQ 5 ) ( ˆQ 3, ˆQ 4 ) w

81 1 σ = 0 e x e x 1 0 0, γ γ = diag(a 4, b 4, c 2 ) 4 ds 2 = N 2 dt 2 + c 2 dx 2 + e 2x a 4 dy 2 + e 2x b 4 dz 2. G 1 2 = T 2 1 c = ab L = 6abN 4ab3 ȧ 2 N 16a2 b 2 ȧḃ N 4a3 bḃ2 N + a3 b 3 ϕ2 2N, N = n 6ab L = n 96a3 b 3 ȧḃ n 24a2 b 4 ȧ 2 n 24a4 b 2 ḃ 2 n + 3a4 b 4 ϕ2. n (u, v, ϕ) a = e u+v 4, b = e 1 4 (u v), ϕ = ϕ H = e 2u p 2 u e 2u p 2 v e 2u p 2 ϕ 0, G αβ = 18e 2u e 2u e 2u. ξ 1 = v, ξ 2 = ϕ v v ϕ, ξ 3 = ϕ, ξ h = 1 2 u, [ξ 1, ξ 2 ] = ξ 3, [ξ 2, ξ 3 ] = ξ 1.

82 Q 1 = 6e2u v 6e (ϕ 2u v v ϕ ) n, Q 2 =, Q 3 = 6e2u ϕ n n, Q h = 9e2u u dt n(t), n Q i = κ i ( n = 6e2u v, u = 2 ( )) 2 c 2 1 c2 (c 3 κ 1 2v), κ 1 κ 1 ϕ = c 1 + κ 3v, κ 1 κ 2 = c 1 κ 1, c i κ κ 2 3 = 48c 2. v = 1 4 ( 2c 3κ 1 κ 1 1 ( 1 t ) c2 ), c2 c2 ds 2 = 2 3 T dt T dx2 + 2e 2x λt T dy2 + 2e 2x+λT T dz2, λ = κ 1 4 c 2 T R = (λ2 3) 3 T c2, λ 2 = 3 ( ˆQ 1, ˆQ 3 ) ˆQ 2 ˆQ 1 Ψ = i v Ψ = κ 1 Ψ, ˆQ 2 Ψ = i(v ϕ ϕ v )Ψ = κ 2 Ψ, ˆQ 3 Ψ = i ϕ Ψ = κ 3 Ψ, ĤΨ = e 2u 144 µ = 9 2e 3u ( ( e 2u ) 12 ϕϕ 12 vv + u + uu ) Ψ = 0,

83 ( ˆQ 1, ˆQ 3 ) Ψ = e ivκ 1+iκ 3 ϕ e u 2 (AJ λ (6e u ) + BJ λ (6e u )), λ = 3 κ 2 1 κ2 3 e 6u Ψ sm ce u/2 (λu)e i(κ 1v+κ 3 ϕ), Ψ la ce u (6e u )e κ 1v+κ 3 ϕ. Q sm = e 2u (1 + 12κ κ2 3 ) Q la = 1 A 13 = B 13 S 13 = κ 1 v + κ 3 ϕ u = c 1, ϕ = c 2 + κ 3v κ 1, n = 6e2u v κ 1. n = e 2u v = κ 1t 6 + c 3. ds 2 = e3c 1 36 dt 2 + e c 1 dx 2 + e 2x+ κ 1 T 6 dy 2 + e 2x κ 1 T 6 dz 2. R = 1 2e c 1 (κ2 1 12e 2c 1 ), π ij q i p = κ2 1 4λ, ρ = 3κ λ 2,

84 w = 1 κ2 1+3κ σ ij = λet κ x λe t κ x ˆQ 2 ˆQ 2 ( ˆQ 2, ˆQ 4 ) ˆQ 13 = 1 2 ( ˆQ ˆQ 2 3), ˆQ 13 Ψ = 1 2 ( ϕϕ vv )Ψ = κ 13 Ψ, µ Ψ = e iκ 2 1 ( v ϕ ) e u 2 (A2 J λ (6e u ) + B 2 J λ (6e u )) J κ2 ( 2κ 13 (v 2 + ϕ 2 )), λ = i 6κ 13 e u u 2 + ϕ 2 A 2 = B 2 ( ( ) v Ψ sm iκ 2 1 u ) ( 6κ 13 u ) (v 2 + ϕ 2 ) κ 2 2, ϕ 2 ( ( ) v Ψ la iκ 2 1 u ) ( e u + e u ϕ 2 (v 2 + ϕ 2 ) 1/4 S = κ 2 1 ( v ϕ ( 2κ13(v 2 + ϕ 2 ) 1 2 πκ 2 + π 4 )), ). n = e 2u u = c 1,

85 v = 2c 2 ϕ 2, ( ) 2c2 12c2 c ϕ = 3 κ 2 t ( 12c 2c 3 κ 2 t 12c 2 ) 12c 2 ds 2 c 2 e 3c 1 = κ 2 2 T (2c 2 T ) dt 2 + e c 1 dx 2 + e 2x+ T +c1 dy 2 + e 2x T +c 1 dz 2, R = κ2 2 T + 2c 2κ c2 2 e2c 1 8c 2. 2 e3c 1 T T < c 2 T π ij = λ e t 2x 0, λ 8 e t 2x λt + κ 16 λt + κ + 48 p = 16µ 2, ρ = 16µ 2, w = λt + κ α σ ij = µ λt + κe t 2 x µ λt + κe t 2 x

86 σ = 0 e x e x 1 0 0, γ γ = diag(a 2, b 2, c 2 ) ds 2 = N 2 dt 2 + b 2 dx 2 + e 2x a 4 dy 2 + e 2x b 4 dz 2. G 1 2 = T 2 1 c = b L = 2a2 N b 2bȧ2 N 4aȧḃ N + a2 b ϕ 2 N. N = bn 2a 2 n L = n 4a2 ȧ 2 n 8a3 ȧḃ bn + 2a4 ϕ2 n. H = 1 G αβ = b 8a 3 p ap b + b2 16a 4 p2 b + 1 8a 4 p2 ϕ 0, 8a 2 8a3 b 0 8a3 b a 4. ξ 1 = a a b ( a 2 b 4) b 2ϕ ϕ, ξ 2 = bϕ b + 2 a ϕ, ξ 3 = b b, ξ 4 = ϕ, ξ h = b( ( ab ) ) b + ϕ 2 ϕ.

87 [ξ 1, ξ 2 ] = 2(ξ 2 + ξ 4 ), [ξ 1, ξ 3 ] = 4ξ 3, [ξ 1, ξ 4 ] = 2ξ 4, [ξ 2, ξ 4 ] = ξ 3. Q 1 = 1 ) (( 8a a 3 a + 32a 3 b)ȧ 8a4 n b ḃ 8a4 ϕ ϕ, Q 2 = 8a3 ϕ n ȧ + 8a4 a ϕ, n Q 3 = 8a3 ȧ n, Q 4 = 4a4 n ϕ, Q h = 4a3 ( ab 2) ȧ + 2a4 ϕ n n ϕ n dt. Q i = κ i, i = 1, 2, 3, 4, h n = 1 N = b 2a 3, a = 1 2 1/4 (8c 1 κ 3 t) 1/4, ϕ = c 2 κ 4 (8c 1 κ 3 t), 2κ 3 ( b = 2 1/8 2κ1 κ 3 + κ 2 3 8c 1c 3 κ 5 3 4c 2κ 3 κ 4 + 2κ c 1c 3 κ 2 3 κ2 4 + c 3κ 4 3 (κ2 3 2κ2 4 )t ) (8c 1 κ 3 t) κ 2 4 4κ 2 3, 8 c 3 = κ 3 3 (κ2 3 2κ2 4 ). 8κ 2 3 c 3 ds 2 = e2t T 5 4 +α β dt 2 + e2t T 1 4 α dx 2 + e 2x T dy 2 + e 2x T dz 2, 2β α, β R = α β e 2T T 3 4 α, α > 3 4 T 0 ( ˆQ 2, ˆQ 3 ) ( ˆQ 3, ˆQ 4 )

88 ˆQ 1 ˆQ 1 Ψ = i ( 2ϕ ϕ b ( a 2 b 4) b + a a ) Ψ = κ1 Ψ, ˆQ 2 Ψ = i( ( a 2) ϕ + bϕ b )Ψ = κ 2 Ψ, ˆQ 3 Ψ = ib b Ψ = κ 3 Ψ, ˆQ 4 Ψ = i ϕ Ψ = κ 4 Ψ, ĤΨ = 1 16a 4 ( 16a 4 2 ϕ,ϕ + b( b b b,b + 2a a,b ) ) Ψ = 0, µ = 16a5 b ( ˆQ 2, ˆQ 3 ) ( ˆQ 3, ˆQ 4 ) ( ( c 1 Ψ = i κ a( a) 1/2 3 ( ab ) (κ 2 κ 3 ϕ) 2 8a 4 )) a, 4κ 3 a Ψ = c 1 (i(κ a 3 b + 2a4 + κ 3 κ 3 2 a + κ2 4 a ) ). κ 3 A = c 1 a( a) 1/2 A = c 1 a ˆQ 1 Ψ = a 1+iκ 1 2 ( 1 + ( a 4 b 8) ) 1 iκ 1 8 ( A 3 J λ (a (a 4 b 8 )) + A 4 J λ (a (a 4 b 8 )) + a 1+iκ 1 2 ( 1 + ( a 4 b 8) ) 1+iκ 1 8 ( A 1 ( 2 ) ( 2a 2 ϕ + A 2 2 )) ) 2a 2 ϕ. ( ( Ψ sm A 4 + A 1 2 ) ( 2a 2 ϕ + A 2 2 )) 2a 2 ϕ e iκ 1 a + A 3 a 1 ( 1 + ( a 4 b 8) ) 1 4 e i κ 1 4 ( 1+(a 4 b 8 )). A 3 = 0 S sm = κ 1 a ( Ψ la (A 1 2 ) ( 2a 2 ϕ + A 2 2 ) 2a 2 ϕ )e iκ 1 a.

89 n = 1 a = c 2, b = c 3 2c 2 e κ1t 8c 4 2, ϕ = c 1. 2 ds 2 = 4eT κ 2 dt 2 + e T dx 2 + e 2x dy 2 + e 2x dz 2, κ = κ 1 T c 2 2 R = 2e T π ij = e t 2x e t+2x p = e t 3, ρ = e t, w = 1 3 θ = κe t/2 / σ ij = κet/ κe t/2 2x κe t/2+2x ds 2 = N 2 (t)dt 2 + a 2 (t)dr 2 + b 2 (t)(dθ θdφ 2 ),

90 a(t), b(t) 3 C 3 23 = θ 2 σα i = diag(1, 1, θ) L = 2aN 4bȧḃ N 2aḃ2 N ab2 ϕ2 2N. L = n + 8abȧḃ n + 4a2 ḃ 2 n n = 2Na + a2 b 2 ϕ2, n H = 1 p2 a 16b 2 + p ap b 8ab + p2 ϕ 4a 2 b 2 0, G αβ = 0 8ab 0 8ab 8a a 2 b 2. ξ 1 = a a + b b, ξ 2 = aϕ a + bϕ b 4 a ϕ, ξ 3 = ϕ, ξ i, i = 1, 2, 3 ξ h [ξ 1, ξ 2 ] = 4ξ 3, [ξ 2, ξ 3 ] = ξ 1, Q 1 = 8ab2 ȧ = κ 1, n 8ab (ϕȧ 2 a ϕ ) a Q 2 = = κ 2, n Q 3 = 2a2 b 2 ϕ = κ 3,, n Q h = 4a2 bḃ = κ h dt n(t), n

91 κ i, i = 1, 2, 3, h (a, ϕ) ϕ = κ 2 + 4κ 3 a κ 1. L red = n + 16κ2 3 b2 ȧ 2 κ 2 1 n + 8abȧḃ n + 4a2 ḃ 2 n. L red H red = 1 κ2 1 p2 a 16λb 2 + κ2 1 p ap b 8λab κ2 3 p2 b 4λa 2 0, λ 2 = κ 2 1 4κ2 3 G αβ = ( 32b 2 κ 2 3 κ 2 1 8ab 8ab 8a 2 ), ζ 1 = a a + b b, κ 1 ( λ a ζ 2 = bλ κ 1 ( λ a ζ 3 = bλ ζ h = a 2 a, κ 1 ) κ 1 ) ) ( ( λ a κ 1 + κ 1 λ λ a a + ) a ( ( λ a κ 1 + κ 1 λ λ a a + a κ 1 ) κ 1 ) b, b, [ζ 1, ζ 2 ] = λ κ 1 ζ 3, [ζ 1, ζ 3 ] = λ κ 1 ζ 2, [ζ 2, ζ h ] = 1 2 ζ 2 λ 2κ 1 ζ 3, [ζ 3, ζ h ] = λ 2κ 1 ζ ζ 3. Q redi ζ α i p α = c i, i = 1, 2, 3, h b = c 1 κ ( 1 ), λa c 3 λ a κ 1 c 2 λ a κ 1

92 c 1 ( c 3 κ c 3κ c 2κ 1 λ + ( c 2 λ 2 c 3 κ 1 λ ) λ a κ 1 ḃ = 1 λ a κ 1 ) ȧ λ 2 a 2 (c 2 c 3 λ a κ 1 ) 2, 8c 1 ȧ n = ( ), a c 3 λ a κ 1 c 2 λ a κ 1 c 1 ( c 3 κ 1 + c 2 λ + (c 2 κ 1 c 3 λ) λ a dt n(t) = c h + ( ), 2λ c 2 c 3 λ a κ 1 κ 1 ) c 2 3 c2 2 = 16 c 2 = 4 κ a = e t c 1 N = 4e t 2 (κ λt κ 1 ), b = c 1 κ ( 1 4λe t κ λt κ 1 ), ϕ = κ 2 + 4κ 3 t κ 1. ds 2 = βeαt 4 T dt 2 + e αt dr 2 + eαt β 2 T dθ2 + eαt 2 T 2 θdφ. R = e αt (α 2 4) 4 T 2β. T α 2 = 4 ( ˆQ 2, ˆQ 3 ) ˆQ 1, ˆQ 2, ˆQ 3 ˆQ 1 Ψ = i( b b + a a )Ψ = κ 1 Ψ, ( b ˆQ 2 Ψ = i abλ 2 ( λ 2 λ a + κ 1 λ λ a κ 1 κ 1 ) b aκ 1 λ λ a ) a Ψ = κ 2 Ψ, κ 1

93 ˆQ 3 Ψ = i abλ 2 ĤΨ = ( ( b κ 1 λ λ a + λ 2 λ a ) κ 1 κ 1 ( 1 κ2 3 4a 2 bλ 2 b κ 3 4a 2 λ 2 bb κ2 1 16ab 2 λ 2 a + κ2 1 8abλ 2 ab κ2 1 16b 2 λ 2 µ = 8 2a 2 b 2 ( ˆQ 2, ˆQ 3 ) b aκ 1 λ λ a ) a Ψ = κ 3 Ψ, κ 1 ) Ψ = 0. ( Ψ(a, b) = A i ab λ a ( )) c c 2 λ a 2. κ 1 κ 1 b κ 1 ( ( λ 16 + c c 2κ 1 ) λ a + (λc c 2 2 κ κ 1) λ a ) = 32κ2 3 b2 ȧ 1 κ 1 κ 2 1 n + 8abḃ n, a λ a κ 1 ( ) c c 2 λ a 2 = 8abȧ κ 1 n + 8a2 ḃ n, a = e t ˆQ 1 ˆQ 1 Ψ = e ic 1 a ( A 1 J ic 1 κ 1 λ ) ( i4ab) + B 1 Y ic 1 κ 1 ( i4ab), λ J ν (z), Y ν (z) ( c1 κ ) 1 Ψ sm D 1 λ (4ab) e ic 1 a, 1 ( Ψ la D 2 4ab + c 1κ 1 π ) e ic 1 a. ab 2λ c 2 1 κ2 1 Q sm = 16a 2 b 2 (κ 2 1 4κ2 3 ) Q la = a 2 b 2 32κ 2 3 b2 ȧ κ 2 1 n + 8abḃ n = c 1 a,

94 8abȧ n + 8a2 ḃ = 0. n n = 1 ( a = d 2 c 1 κ 2 1 t 8d 2 1 (κ2 1 4κ2 3 ) ), b = d ( 1 c 1 c 1 κ 2 1 t 8d 2 1 (κ2 1 4κ2 3 ) ) d i ds 2 = α T dt T dr2 + T dθ 2 + T 2 θdφ 2, α = 4d2 1 λ4 T 0 c 2 1 κ4 1 α = 1 4 R = 1 4α 2αT π ij = t , θ 3 q i = 0 p = 3 + 4α 12α t, ρ = 4α 1 4α tr, w = 3+4α 3 12α σ ij αt σ ij = t. 3 α t 3 α 2 θ

95 5 (1 + 1) N α = 0 ds 2 = N(t) 2 dt 2 + γ µν (t)σ µ i (x)σν j (x)dx i dx j, σi,j α σα j,i = Cα βγ σβ j σγ i

96 L = 1 2N(t) G αβ(q(t)) q(t) α q(t) β NV (q(t)), α, β = 0,..., n 1. N q α g µν I = ( 1 dtn 2 G dq α dq β ) αβ Ndt Ndt V t = f( t), q α (t) = q α (f( t)) =: q α ( t), N(t)dt = N(f( t)) df( t) d t d t =: N( t)d t. N q α E 0 := 1 2N 2 G αβ q α q β + V = 0, E µ := q µ + Γ µ νλ qν q λ Ṅ N qµ + N 2 G µκ V,κ t = f( t) (df/d t) 2 f q µ t = f( t) N q α q α N 2 Ṅ N N 2 = G αβ q α q β 2V =: K 2V q µ + Γ µ νλ qν q λ 1 K 2 K qµ V,κ V qκ q µ K 2V Gµκ V,κ = 0

97 V = V,κ q κ n 1 2G µρ q ρ 0 = 0 t q α M = Ne 2ω L = 1 2M Ḡαβ q α q β M V Ḡ αβ = e 2ω G αβ, V = e 2ω V. q µ + Γ µ νλ qν q λ 1 2 K K qµ + 1 V,κ 2 V qκ q µ K 2 V Ḡµκ V,κ = 0. Γ µ νλ = Γµ µλ + δµ ν ω,λ + δ µ λ ω,ν G νλ G µρ ω,ρ. V V q µ N N = N V L = 1 2 N Ḡαβ(q) q α q β N, α, β = 0,..., n 1, Ḡ αβ = V G αβ

98 L = 1 2 N G αβ q α q β N, α, β = 0,... n 1. 2 N 2 + G αβ q α q β = 0 q µ + Γ µ αβ qα q β Ṅ N qµ = 0. N 2 q µ + Γ µ νλ qν q λ 1 K 2 K qµ = 0 K K = G αβ q α q β t = f( t) q 0 = q(t) f q f := q 1 q α q q 0 =: q(t) = q(f( t)) = q(q 1 ( t)) = t, q i := q i (t) = q i (q 1 ( t)) = q i q 1 ( t) = f i (q), i = 1,..., n 1, q(t) q(t) q i (t) = f i (q(t)) q(t) = d dq K [ K = G G 0i f i i j + G ij f f ] q 2 =: h[q, f i (q), f i (q)] q 2. h[q, f i (q) f i (q)] h[q] (q, q i ), i = 1,..., n 1 G qq = ε = ±1, G qi = 0

99 µ = 0 µ = i µ = 0 q Γ q jk = 1 2G qq G jk,q, Γ i qk = 1 2 Gij G jk,q Γ i jk q = 1, q = 0 K K = [ ε + G ij (q, f i ) f i (q) f ] j (q) N 2 = h[q] 2, µ = 0 h h = 1 ε G ij,q f i f j µ = i f i + G ik G jk,q f j + Γ i f kl k f l 1 h 2 h f i = 0. f i + G ik G jk,q f j + Γ i kl f k f l + 1 2ε G jk,q f j f k f i = 0. f i q 2G ir f r h 2G ir f r f i N, q α N, q f i (q) N q f i (q) N, q f i (q) σ α i q i q i (t) = f i (q(t)) q(t) L red = 1 2 N(t) h[q(t)] q(t)2 N(t), N q f i (q(t))

100 f i (q) E 0 red := 2 N 2 + h[q] q 2 = 0 E red := q + h 2 h q2 Ṅ N q = 0. N q(t) q(t) f i (q) n 1 f i (q) q f i (q) σ α i (x) p N := L red Ṅ p := L red q = 0, = h N q. p N 0 H T = q p L + u N p N ( ) p 2 = N 2 h u N p N = N H + u N p N, H = p h ṗ N = {p N, H T } H 0, H

101 {p N, H} 0 H 0 p2 h 2 p h = ± 2, Q = p h Q = A(q)p+B(q) A(q) B(q) N h Q 0 {Q, H T } = N F (q) H ) (2 A + A h p N B p = F N h h h p2 + F N p B(q) = F (q) = 0 2 A + A h h = 0, A = c 1 h Q = c 1 p h + c 2 c 1 c 2 Q = p h. f i (q) f i (q) Q ˆp N = i N ˆp = i q

102 ħ = 1 {, } i[, ], ˆp N Ψ = 0 Ψ = Ψ(q) Ψ µ(q) ˆQ = i ( µ d 2 µ h dq + d ( )) µ. dq h Q ˆQΨ = κψ κ = ± 2 Ĥ = 1 ( ) d µ d + 1, 2 µ dq h dq µ = h µ Ĥ ĤΨ = 0 Ψ 2 h + h µ µ h 2 h 2 Ψ Ψ = 0. µ Ψ h [q] = ( h[q]) 1 4 µ(q) e i κ h[q] dq, µ[q] = ϕ[q] 2 h[q] κ = ± 2 ϕ[q] h ϕ 2 h ϕ = 0 ϕ[q] = c 1 + c 2 h[q] dq. µ[q] = ( ) h[q] 2 h[q] c 1 + c 2 dq. c 1 = 1 c 2 = 0

103 [ ˆQ, Ĥ] = 0 Ψ µ(q) µ[q] Ψ[q] Ψ[q]dq = h[q]dq h P h = h[q] dq. Ψ h [q] [0, 1] f i q i (t) P h q i (t) κ ± 2

104 P h f i (q) f i h[q] δp h = δ dq = 0, h[q, f, f] ( = G ij (q) f i (q) f ) 1 j 2 (q) ε. P h P h δp h = 0 f m + G mk G rk,q f r + Γ m rn f r f n h 2h f m = 0. L = 1 2 N G αβ q α q β N V, α, β = 0,... n 1, P h = h dq, h := ( V G αβ q α q β) q=t t h(q, f, f) h[q] q f i (q) h[q]

105 V G αβ q α q β α, β 0 n 1 G αβ g S BSW = R k ij k ij k 2 d 4 x g = g ij k ij = 2 N K ij = g ij t N i;j N j;i K ij δ hdq = 0 Ψ e is S Ψ e is BSW δ 2 h = 2 h f i f δ f i δ f j h j f i f δ f i δf j + 2 h j f i f j δf i δf j. 2 2 h f i f δ f i δf j = d ( 2 ) h j dq f i f δf i δf j d j dq ( 2 ) h f δf i δf j, i f j δf i 2 h f i f j δ 2 P h = b a [ 2 h f i f j δ f i δ f j + ( 2 h f i f j d dq ( 2 )) ] h f δf i δf j dq. i f j W ij = 2 h f i f j A ij = 2 h f i f j d dq ( 2 h f (i f j) ).

106 U A + Ú = U W 1 U [a, b] δ 2 P h W δp h = 0 W W W ij = 2 h f i f = j f i = f i ( 1 ( 1 h G kj f k = G ik G jl f k f l ( h) ( h f k ε + G kl f f )) l ) j + G ij h = 1 h ( 1 h G ik G jl f k f l + G ij f i ), ds 2 = a(r) 2 dt 2 + ( ) N(r) 2 dr 2 + b(r) 2 (dθ θ dϕ 2 ) 2a(r) r dr 2 L = 1 ( 16 a b ȧ 2 N ḃ + 8 a2 ḃ 2) N. ( ds 2 = c 2 1 2M ) ( dt M ) 1 ḃ(r) 2 dr 2 + b(r) 2 ( dθ θ dϕ 2). b(r) b(r)

107 b(r) G αβ = 0 8 a b, 8 a b 8 a 2 h h S [b] = 8 ( 2 b a(b) a (b) + a(b) 2). a(b) L = 1 2N hs[b]ḃ2 N b N b (r) 2 (ba(b)) = 1 4 N(r)2. N(r)dr = 2dτ τ = ρf (ρ) f (ρ) b = f (ρ) a 2 = 1 ( ρ 2 f f (ρ) 2ρf (ρ) + 2f(ρ) ) (ρ) f(ρ) a(r) a(b) a f P hs = hs [b] db a(b) d db ( h a ) h = 0 (b a + 2 a ) a + b a 2 = 0 a a S (b) = c 1 2 M b, a S W A

108 f i (q) a(b) W (b) = 2 h a a = 2 2 b 2 a 2 (a (a + 2 b a )) 3 2 A(b) = 2 h a a d ( 2 ) h db a a = 2 2 b a ( b 2 a 3 b 2 a a a + a 2 (2 a + b a ) ) (a (a + 2 b a )) 5 2 a = a S, W a=as (b) = 2 2 b 2 (1 2 M b ), c A a=as (b) = 2 2 M 2 c b (b 2 M). A a=as (b) + U (b) = 1 W a=as (b) U(b)2 2 2 c (b 2 M) U (b) + c 2 U(b) 8 M 2 = 0 U(b) P hs U(b) = 2 2 M c ( 1 ( b 2 M b 4 ) ) 2 c M c 1, 2 c 1 R b = 0 b = ±2 M (x) b ± a S (b) b W a=as (b) (0, ) b > 2 M a = a S b < 2 M P hs

109 ( ) N(t) 2 ds 2 = dt 2 + b(t) 2 dx 2 + e 2x b(t) 2 dy 2 + a(t) 2 dz 2. 2 a(t) a(b) = c 2 1 c 1 b c 1 b > c 1 t b < c 1 z h I Schw = q 2 22/3 3 5/3 q 1 m 2/3 6 3q 5/2 1, I III = q 2 (6 3q 5/ /3 c 2/3 1 q 7/3 1 ) q 1 q 2 = g µν q 1,µ q 1,ν I Schw, I III h(q) Ψ GC = ( h[q]) 1 4 µ[q] e i κ h[q] dq Dh(q)Dδ(I(q)), I(q) h f i (q) h f i (q)

110

111 6 ϕ = const

112 ˆQ i

113 S S (λ 1 + λ 2 )ϕ λ i, i = 1, 2 κ 3 (n 1) f i (q) (n 1) f i (q) q

114 U W V (ϕ)

115 gr-qc/ gr-qc/

116 gr-qc/ gr-qc/ gr-qc/

117 gr-qc/ gr-qc/ gr-qc/ gr-qc/ gr-qc/

118 gr-qc/ gr-qc/ gr-qc/ gr-qc/ gr-qc/ gr-qc/ gr-qc/

119