Kaluza-Klein

Transcription

1 Kaluza-Klein

2 (Hawking ) Hawking Planck Planck Hawking 4 Planck Planck ( TeV) Kaluza-Klein Kaluza-Klein Kaluza-Klein Kaluza-Klein Planck 4 Kaluza-Klein Kaluza-Klein (Gregory-Laamme ) Hawking Hawking Hawking Hawking Kaluza-Klein Hawking Hawking Kaluza-Klein Gregory-Laamme 1

3 surface gravity horizon constraint dilaton gravity dilaton gravity horizon constraint Hawking Hawking Schwarzschild gravitational anomaly Hawking gravitational anomaly Kerr Hawking Myers-Perry Hawking Hawking CGHS RST horizon constraint RST

4 5 Kaluza-Klein Kaluza-Klein Gregory-Laamme instability Kaluza-Klein Einstein-Hilbert dimensional reduction dilaton gravity radion A Wald 79 B Cardy 86 C 89 C.1 Weyl C.2 (dilaton ) C.3 Louville action C.4 (dilaton ) D (2+1) 101 D.1 (2+1) D.2 (2+1) D.3 (2+1)

5 1 (Hawking ) ( M Pl ) Hawking Planck 4 Planck Planck Planck 1 object ( 1.1) Planck ( TeV) Kaluza-Klein Kaluza-Klein ( 1.1) Kaluza-Klein Kaluza-Klein Planck 4 Kaluza-Klein Kaluza-Klein Hawking Hawking Hawking Hawking Kaluza-Klein 4

6 1.1: Kaluza-Klein 4 Hawking Kaluza-Klein Hawkin Hawking 2 Hawking Hawking 3 Hawking 3 Hawking Hawking eective action Hawking Hawking Hawking 4 6 gravitational anomaly Hawking Hawking Hawking cancellation 5

7 gravitational anomaly Kerr Myers-Perry 4 Hawking 4 Hawking Hawking Kaluza-Klein 2 5 Kaluza-Klein, Kaluza-klein 6 Kaluza-Klein Hawking 7 6

8 2 Bekenstein-Hawking S = A 4 G [1] A ( ) [2] c = G = 1 timelike Killing vector k = / t k k 2 = 1 Cauchy surface V V ( ) M = 1 ds µν µ k ν (2.1.1) 8π V Komar ( ADM ) 7

9 Gauss Killing ξ M = 1 ds µ ν µ k ν (2.1.2) 4π V ρ µ ξ ν = R ν µρσξ σ (2.1.3) M = 1 ds µ R µ νk ν (2.1.4) 4π V T µν (T 00 T 0i T ij ) Einstein R µ νk ν = 8π(T µ νk ν 1 2 T kµ ) 8π(T µ T 00k µ ) (2.1.5) V t = const M = 2 d 3 x(t T 00) = V V d 3 x T 00 (2.1.6) (2.1.1) Schwarzschild Schwarzschild ( ds 2 = 1 2m r ) ( dt m r ) 1 dr 2 + r 2 dω 2 (2.1.7) timelike Killing vector k = / t V t V r M = 1 ds µν µ k ν 8π V = 1 1 dx ρ dx σ ɛ ρσµν µ k ν (2.1.8) 8π 2! V = 1 16π 4 dθ dϕ ɛ θϕtr t k r ( Killing ) V t k r = g tt Γ r tt = m r 2 (2.1.9) M = 1 4π = 1 4π V dθ dϕ g m r 2 r 2 dω m r 2 = m (2.1.10) m 8

10 Kerr Kerr ds 2 = a2 sin 2 θ Σ + (r2 + a 2 ) 2 a 2 sin 2 θ Σ determinant dt 2 2a sin 2 θ r2 + a 2 dtdϕ Σ sin 2 θdϕ 2 + Σ (2.1.11) dr2 + Σdθ 2 Σ = r 2 + a 2 cos 2 θ, = r 2 2mr + a 2 (2.1.12) g = Σ sin θ (2.1.13) (t, φ) g tt = (r2 + a 2 ) 2 a 2 sin 2 θ Σ g φφ = a2 sin 2 θ Σ sin 2 θ g tφ = a(r2 + a 2 ) Σ,, (2.1.14) Schwarzschild M = 1 dθdϕ Σ sin θ t k r (2.1.15) 4π Σ t k r = m(r2 + a 2 )( r 2 + a 2 cos 2 θ) Σ 2 m (r ) (2.1.16) M = m 4π dθdϕ sin θ = m (2.1.17) rotation Killing vector m = / ϕ m m 2 = 1 J = 1 ds µν µ m ν (2.1.18) 16π V 9

11 J = V t J = J = V V ds µ (T µ νm ν 1 2 T mµ ) (2.1.19) V d 3 x T 0 νm ν (2.1.20) m x 1 x 2 x2 (2.1.21) x 1 d 3 x (T 0 2x 1 T 0 1x 2 ) = ɛ 3jk V d 3 x x j T k0 (2.1.22) Kerr Kerr asymptotic rotation Killing vector m = / ϕ Schwarzchild V t V r J = 1 ds µν µ m ν 16π V = 1 dθ dϕ ɛ θϕtr t m r (2.1.23) 8π V = 1 dθdϕσ sin θ t m r 8π Σ t m r = Ma sin2 θ(3r 4 + a 2 r 2 + a 2 r 2 cos 2 θ a 4 cos 2 θ) Σ 2 3Ma sin 2 θ (r ) (2.1.24) J = 3 4 Ma dθ sin 3 θ = Ma (2.1.25) Ma a surface gravity Killing horizon N Killing ξ N surface gravity κ ξ ν ν ξ µ = κξ µ (2.1.26) 10

12 surface gravity timelike Killing 4 u µ = ξ µ ( ξ 2 ) 1/2 (2.1.27) timelike Killing a µ = ξ ν ν u µ = ξν ν ξ µ ( ξ 2 ) 1/2 + ξµ ξ ν ξ ρ ν ξ ρ ( ξ 2 ) 3/2 = ξν ν ξ µ (2.1.28) ( ξ 2 ) 1/2 Killing Killing horizon a µ = κξµ ( ξ 2 ) 1/2 (2.1.29) a a µ a µ = κ (2.1.30) ξ ξ 2 = 1 (ξ ) surface gravity event horizon Killing horizon event horizon surface gravity Kerr event horizon = 0 r + = M + M 2 a 2 event horizon Killing ξ µ = k µ + a r a2 mµ (2.1.31) regular Kerr dv = dt + r2 +a 2 dr dχ = dϕ + a dr dr = dr ds 2 = a2 sin 2 θ Σ v = t χ = φ r = r r2 +a 2 t a φ dv 2 + 2dvdr 2a sin 2 θ r2 + a 2 dvdχ Σ 2a sin 2 θdχdr + (r2 + a 2 ) 2 a 2 sin 2 θ Σ = 0 regular (2.1.32) sin 2 θdχ 2 + Σdθ 2 (2.1.33) g vv = a2 sin 2 θ Σ g vχ = 1 Σ g vr = r2 +a 2 Σ g χχ = 1 Σ sin 2 θ g χr = a Σ g r r = Σ (2.1.34) 11

13 r = r + l = g µν ( ν r ) µ r+ = g r µ µ r+ = r2 + + a 2 ( ) Σ(r + ) v + a r+ 2 + = r2 + + a 2 a2 χ Σ(r + ) ξ (2.1.35) l 2 = g r r = 0 r = r + null surface l ξ event horizon r = r + Killing ξ ξ ν ν ξ µ = r + r 2(r a2 ) ξµ (r ± M ± M 2 a 2 ) (2.1.36) Kerr surface gravity κ = r + r 2(r a2 ) (2.1.37) Schwarzschild ds 2 = f(r)dt 2 + f(r) 1 dr 2 + r 2 dω 2 (2.1.38) surface gravity f(r) f(r H ) = 0, f(r) 1 (r ) (2.1.38) r = r H regular dr v = t + (2.1.39) f(r) (2.1.38) ds 2 = f(r)dv 2 + 2dvdr + r 2 dω 2 (2.1.40) r = r H regular Killing k = / t = / v surface gravity k ν ν k µ = 1 2 f (r H )k µ (2.1.41) κ = 1 2 f (r H ) (2.1.42) 12

14 2.1.4 future event horizon Killing Ω H Kerr (2.1.31) ξ ξ µ = k µ + Ω H m µ (2.1.43) Ω H = a r a2 (2.1.44) ξ µ µ (ϕ Ω H t) = 0 (2.1.45) ξ orbit ϕ = Ω H t+const null genarator k( time translation) Ω H induced metrc h µν A = d 2 x h (2.1.46) H Kerr r = r + (2.1.33) induced metric h µν dx µ dx ν = (r2 + + a 2 ) 2 sin 2 θdχ 2 + Σdθ 2 (2.1.47) Σ A = dω(r a 2 ) = 4π(r a 2 ) (2.1.48) 2.2 0, 1, 2 Kerr Kerr [3] 0 future event horizon surface gravity κ 13

15 1 M J future event horizon suface gravity κ, Ω H, A horizon M, J dm = κ 8π da + Ω HdJ (2.2.1) 2 future event horizon (Hawking ) 0 (2.1.37) suface gravity 1 Kerr κ, A, Ω H 1 (2.2.1) A A surface gravity κ 0 3 Hawking Hawking T = κ/2π S = A/4 4 4 Einstein Kaluza-Klein Wald [4, 5] n S = 2π d n 2 x ( 1 h g H δi δr µναβ ) ɛ µν ɛ αβ (2.2.2) H : horizon spatial section h : H induced metric determinant ɛ µν : H binormal H volume form I : φ Killing ξ L ξ φ = 0 A 14

16 dm = T ds + Ω (µ) H dj (µ) (2.2.3) T Hawking T = κ/2π M, J (µ), Ω (µ) H A Einstein Einstein-Hilbert Wald (2.2.2) S = 2π = 1 4G D = A H 4G D 1 I = 16πG D H H d D 2 x h gµα g νβ 16πG D ɛ µν ɛ αβ d D x gr (2.2.4) d D 2 x h ( g µα g νβ ɛ µν ɛ αβ = 2) (2.2.5) A H Einstein dilaton gravity I = M D 2 D d D x g f(φ)(r + 2a( φ) 2 + V (φ)) (2.2.6) Wald (2.2.2) S = 2πM D 2 D d D 2 x h f(φ)g µα g νβ ɛ µν ɛ αβ H = 4πM D 2 D d D 2 x h f(φ) H = 4πM D 2 D f(φ H)A H (2.2.7) φ H φ L ξ φ = 0 dm = T ds (2.2.8) M A Einstein ADM [4, 5] 4 Einstein Schwarzschild ( ds 2 = 1 C ) ( dt C ) 1 dr 2 + r 2 dω 2 (2.2.9) r r 15

17 (2.1.42) Hawking T = κ 2π = 1 4πC (2.2.10) S = A 4 = πc2 (2.2.11) dm = 1 4πC d ( πc 2) = 1 2 dc M = C 2 (2.2.12) Schwarzschild 2.3 horizon constraint Hawking gravitational anomaly 3.3 Hawking horizon constraint [2] dilaton gravity dilaton gravity I[A, g] = d 2 x g(ar + V (A)) (2.3.1) A c = = 16πG = 1 dilaton gravity Einstein I = d 4 x gr (2.3.2) ds 2 = g (2) ab (xa ) + e 2φ(xa) dω 2 2 (2.3.3) (2.3.2) I = 4π d 2 x g (2) e 2φ (R (2) + 2( φ) 2 + 2e 2φ ) (2.3.4) 16

18 A 4πe 2φ, g ab 2e φ g (2) ab I = d 2 x g(a R + 8π 3/2 A 1/2 ) (2.3.5) (2.3.1) 4 Einstein A = 4πe 2φ 4 r (2.3.1) ds 2 = (J(x) C)dt 2 + (J(x) C)dx 2 A = x (2.3.6) [6] J(A) = V (A )da (2.3.7) J(x) = C (2.3.1) Wald dilaton gravity S = 2π 1 g δi ɛ µν ɛ αβ δr µναβ H = 2πAg µα g νβ ɛ µν ɛ αβ = 4πA H (2.3.8) = A H 4 G ( H ) dilaton gravity null dyad {l a, n a } l n = 1, l 2 = n 2 = 0, g ab = l a n b l b n a (2.3.9) κ, κ l, n a l a = κn a l b κl a l b a n b = κn a n b + κl a n b (2.3.10) l = σdu + αdv n = βdu + τdv (2.3.11) 17

19 Lagrangian L = 1 στ αβ [2(τ A βa )( α σ ) + 2(α A σa )( τ β )] + (στ αβ)v (A) (2.3.12) / u, / v κ, κ 1 κ = στ αβ ( α σ ) 1 κ = στ αβ ( τ β ) (2.3.13) Lagrangian(2.3.12) π α δi δα = 2 στ αβ (τ A βa ) π τ δi δτ = 2 στ αβ (α A σa ) π A δi δa = 2 στ αβ [τ( α σ ) + α( τ β )] π σ δi δσ = 0 π β δi δβ = 0 (2.3.14) H = π α α + π τ τ + π A A L = σc + β τ (C αc ) (2.3.15) C Hamiltonian constraint C momentum constraint C = π α π απ A τv (A) C = π A A απ α τπ τ (2.3.16) (2.3.14) C π = τπ τ απ α + 2A = 0 (2.3.17) constraint constraint dyad l f(x)l, n f(x) 1 n, (f ) constraint generator {A, C [ξ]} = ξa (C [ξ] dvξc ) (2.3.18) 18

20 2.1: stretched horizon C v generator {C [ξ], C [η]} = 0 {C [ξ], C [η]} = C [ξη ] {C [ξ], C [η]} = C [ξη ηξ ] {C [ξ], C π [η]} = C [ξη] {C [ξ], C π [η]} = C π [ξη ] {C π [ξ], C π [η]} = 0 (2.3.19) horizon constraint dilaton gravity input u = 0 null surface du / l = σdu + αdv α = ɛ 1 1 (2.3.20) α = stretch (stretched horizon) ɛ 1 0 expantion 2 expantion θ = l a a A/A (2.3.5) A r l a a A κa = ɛ 2 1 (2.3.21) 19

21 l l al κ aκ κ (2.3.21) A 1 2 ɛ 2A H π A = 0 (2.3.22) A H A constraint K 1 = α ɛ 1 K 2 = A 1 2 ɛ 2A H π A + a 2 C π (2.3.23) constraint C π = τπ τ απ α + 2A = 0 A (τπ τ απ α )/2 K 2 (a/2)c π a K 1 K 2 second class constraint ( ) {K i (u, v), K j (u, v 0 a )} = 2 αδ(v v ) a 2 αδ(v v ) (1 + a)ɛ 2 A H v δ(v (2.3.24) v ) P P P + c 1 K 1 + c 2 K 2 (2.3.25) c 1, c 2 {P (u, v), K i (u, v )} {P (u, v), K i (u, v )} = 0 (2.3.26) {, } Dirac Dirac K 1 = K 2 = 0 Poisson Dirac constraint K 1, K 2 C C π C = C 4(1 + a) K 1 + a 2 ɛ 2 A H 2 ɛ 1 a K 2 C π = C π + 2 a (1 + 2 a )ɛ 2A H K 1 ɛ 1 2 a K 2 C C π {C [ξ], C [η]} = C [ξη ηξ 2(1 + a) ] a 2 ɛ 2 A H dv(ξ η η ξ ) {C [ξ], C π [η]} = C π [ξη ] + 2 a (2 a + 1)ɛ 2A H dvξ η {C π [ξ], C π [η]} = 2 a 2 ɛ 2A H dv(ξη ηξ ) (2.3.27) (2.3.28) {C [ξ], C π [η]} anomalous a = 2 c 48π 1 2 ɛ 2A H (2.3.29) 20

22 c {C [ξ], C [η]} = C [ξη ηξ ] c 48π dv(ξ η η ξ ) (2.3.30) Virasoro c central charge ({, } i[, ]) [C [ξ], C [η]] = ic [ξη ηξ ] ic dv(ξ η η ξ ) (2.3.31) 48π C Cardy L n C [ξ n ] (ξ n A 2πA e2πina/a H ) (2.3.32) ln ρ( ) 2π c 6 (2.3.33) L 0 ρ( ) B c (2.3.29) L 0 = 0 stretched horizon (stretched horizon ) v = v H C boundary term C [ξ] C [ξ] ξπ A A vh (2.3.34) δπ A = δk 1 = δk 2 = 0 boundary term C [ξ] ξπ A A vh = ξ 0 π A A vh = A2 H π A 2πA = A H (2.3.35) vh πɛ 2, 16πG (2.3.33) c = 3ɛ 2A H 2 G = A (2.3.36) H 16π 2 Gɛ 2 1 S = 2π 6 3ɛ 2 A H 2 G A H 16π 2 Gɛ 2 = A H 4 G (2.3.37) 21

23 2.3.5 c = 0 L n phys = 0 (2.3.38) L 0 phys = 0 (2.3.33) c 0 (2.3.38) 0 = [L m, L n ] phys = ((m n)l m+n + c 12 m3 δ m+n ) phys = c 12 m3 δ m+n phys (2.3.39) phys L n phys = 0 (2.3.40) L 0 phys = 0 (2.3.33) c = 0 c 0 22

24 3 Hawking A 1 dm = κ 8π da T κ T κ Hawking Hawking Kaluza-Klein Hawking Hawking Hawking Hawking [7, 3] Hawking Hawking [8, 9] Hawking gravitational anomaly [10, 11, 12] gravitational anomaly Kerr Myers-Perry 3.1 Hawking Schwarzschild Hawking Schwarzschild Schwarzschild ( ds 2 = 1 2M r ) ( dt M r ) 1 dr 2 + r 2 dω 2 (3.1.1) (c = G = 1) singular r = 2M, r = 0 r = 0 r = 2M R µνρσ R µνρσ = 48M 2 /r 6 r = 2M ( ) r 2M r = r + 2M ln 2M 23 (3.1.2)

25 3.1: Kruskal u = t r v = t + r (3.1.3) ( ds 2 = 1 2M r ) dudv + r 2 dω 2 (3.1.4) r = 2M singular U = e u/4m, V = e v/4m (3.1.5) ds 2 = 32M 3 e r/2m dudv + r 2 dω 2 (3.1.6) r (U, V ) Kruskal r = 2M r = r(u, V ) UV = r 2M 2M er/2m (3.1.7) r (U, V ) singularity r = 0 (U, V ) UV = U = tan Ũ V = tan Ṽ (3.1.8) 24

26 3.2: Schwarzschild Penrose 3.2 Penrose Schwarzshild Penrose Hawking S = d 4 x g[ 1 2 ( φ)2 1 2 m2 φ 2 ] (3.1.9) Klein-Gordon ( m 2 )φ = 0 (3.1.10) S {φ n } S sympectic φ α φ β = Σ ds µ φ α µ φ β (3.1.11) Σ Cauchy surface f µ g f µ g g µ f Σ Gauss (φ α φ β ) Σ (φ α φ β ) Σ = d 4 x µ (φ α µ φ β ) (3.1.12) 25 S

27 3.3: Penrose (S Σ Σ 4 ) µ (φ α µ φ β ) = φ α φ β φ β φ α = m 2 φ α φ β m 2 φ β φ α = 0 (3.1.13) φ α φ β α, β (φ α φ β ) =... (3.1.14) φ 1 φ 2 = φ 3 φ 4 = φ 5 φ 6 = = 1 (3.1.15) ψ n (φ 2n 1 iφ 2n )/ 2 (ψ i, ψ j ) = i ds µ ψ i µ ψ j (3.1.16) Σ {ψ i } ( ) (ψ i, ψ j ) = δ ij (ψ i, ψj ) = 0 (ψ i, ψj ) = 0 (ψ i, ψ j ) = δ ij (3.1.17) ( ψ i 2 < 0 ) φ = i (a i ψ i + a i ψ i ) (3.1.18) 26

28 φ = i (a i ψ i + a i ψ i ) (3.1.19) [a i, a j ] = 0, [a i, a j ] = δ ij (3.1.20) a i vac = 0 ( i), vac vac = 1 (3.1.21) (3.1.17) (3.1.17) ψ i = j (A ij ψ j + B ij ψ j ) (3.1.22) ψ i = j ((A ) ij ψ j (B T ) ij ψ j) (3.1.23) {ψ i } {ψ j } (3.1.17) A, B AA BB = 1 AB T BA T = 0 A A B T B = 1 A B B T A = 0 (3.1.24) {ψ i } time like Killing k µ k µ µ u i = iω i u i (3.1.25) k µ µ S well dined k µ µ S S k µ µ k = / t t t φ S ( m 2 )k µ 1 µ φ = ( µ gg µν ν m 2 ) t φ g 1 = t ( µ gg µν ν m 2 )φ = 0 g (3.1.26) 27

29 3.4: k µ µ φ S Σ ( t f, g) + (f, t g) = i d 3 x g (3) (f 2 t g g 2 t f ) (3.1.27) g (3) determinant Klein-Gordon ( i 3 ) 2 t = i i m 2 (3.1.28) ( t f, g) + (f, t g) = i d 3 x g (3) (f i i g g i i f ) = i d 3 x g (3) i (f i g g i f ) = 0 (3.1.29) (f, k µ µ g) = (k µ µ f, g) (3.1.30) k µ µ k µ µ t < t 1 M t 1 < t < t 2 M 0 t > t 2 M + ( 3.4) M +, M φ = i φ = i (a i u i + a i u i ) in M (a iu i + a i u i ) in M + (3.1.31) 28

30 u i, u i M, M + timelike Killing u i u i S u i = (A ij u j + B ij u j) (3.1.32) j M + φ = i = i = i (a iu i + a i u i ) [a i j (A ij u j + B ij u j) + a i (a i u i + a i u i ) (A iju j + Biju j )] j (3.1.33) a j = i u i = j (a ia ij + a i B ij) (3.1.34) (A iju j + B iju j ) (3.1.35) a j = i (a i A ij + a i B ij) (3.1.36) Bogoliubov A, B Bogoliubov M M + M M + M vac M + vac N i vac = vac a i a i vac = j,k vac (a k B ki )(a j B ji) vac = (B B ) (3.1.37) ii ( 3.5) M + I +, M I H + Klein-Gordon u ω = e iωu (3.1.38) Schwarzschild timelike Killing / t 3.5 H + ane distance ɛ 29

31 3.5: γ ɛ = 0 γ H U ane distance γ U = ɛ γ (3.1.38) γ u ω u ω = e iωu = exp ( ) iω κ ln( U) ( ) iω = exp κ ln ɛ (3.1.39) ɛ 0 u ω γ I I γ γ H ane distence ɛ cɛ ( ɛ f(ɛ) f(0) = 0 ɛ f(ɛ) cɛ ) I u ω ( ) iω u ω = exp κ ln(cɛ) (3.1.40) I ds 2 = dudv + r 2 dω 2 γ γ H ane distance v v ɛ (u, v) u ω I ( ) iω u ω = exp κ ln( c v) (3.1.41) v < 0 v > 0 null I u ω = 0 u 0 (v > 0) ω = exp ( iω κ ln( c v) ) (3.1.42) (v < 0) 30

32 3.6: I u ω = e iωv u ω u ω Bogoliubov dω u ω = 2π C ωω u ω (3.1.43) C ωω = = 0 dv e iω v u ω (v) C ωω ( dv exp iω v + iω ) (3.1.44) κ ln( c v) C ω, ω = e πω κ Cωω (for ω > 0). (3.1.45) v ln ω > 0 C ωω i +i0 v = ix ( C ωω = i dx exp ω x + iω ) 0 κ ln(c xe iπ/2 ) = ic iω/κ πω exp( 2κ ) dx exp ( ω x + iωκ ) (3.1.46) ln(x) 0 C ω, ω i i0 v = ix ( C ω, ω = i dx exp ω x + iω ) 0 κ ln(c xe iπ/2 ) = ic iω/κ πω exp( 2κ ) dx exp ( ω x + iωκ ) (3.1.47) ln(x) 0 31

33 (3.1.45) (3.1.32) u ω = A, B C 0 dω 2π (A ωω u ω + B ωω u ω ) (3.1.48) A ωω = C ωω B ωω = C ω, ω = e πω κ Cωω (3.1.49) B ij = e πω i/κ A ij (3.1.50) (3.1.24) δ ij = (AA BB ) ij = k (A ik A jk B ikb jk ) (3.1.51) = (e π(ω i+ω j )/κ 1)(BB ) ij i = j (BB ) ii = 1 e 2πω i/κ 1 (3.1.23) Bogoliubov Bogoliubov (3.1.52) I + (3.1.37) B = B T (3.1.53) N i I + = (B B ) ii = (B B T ) ii = (BB ) ii (3.1.54) (3.1.52) N i I + = 1 e 2πω i/κ 1 (3.1.55) Planck T H = κ 2π (3.1.56) Hawking object 32

34 3.2 eective action ( Hawking ) [8, 9] 2 2 Hawking Hawking 2 Schwarzschild ( ds 2 = 1 2M r ) ( dt M r ) 1 dr 2 (3.2.1) massless S = 1 d 2 x gg µν µ φ ν φ (3.2.2) 2 Weyl g µν e 2ω(x) g µν, φ φ Weyl Weyl 0 T µ µ = R 24π. (3.2.3) eective action W [g] = 1 96π d 2 x gr 1 R (3.2.4) Liouville action ( Liouville action C ) Liouville action (3.2.4) nonlocal 1 χ W [χ, g] = 1 96π d 2 x g( χ χ + 2χR) (3.2.5) χ χ = R (3.2.5) (3.2.4) T µν = 2 g δw [χ, g] δg µν = 1 48π [ µχ ν χ 2 µ ν χ + g µν {2R 1 2 ( χ)2 }] (3.2.6) χ ( 1 2M r ) 1 2 t χ + r ( 1 2M r ) r χ = 4M r 3 (3.2.7) 33

35 χ = at log ( 1 2M r ) + A(r + 2M log(r 2M)) + B (3.2.8) a, A, B χ T tt = 1 ) 2Mr 7M 2 ( + 12π 3 2r A 2 + a 2, 48π 2 T rr = 1 ( 1 2M ) 2 ( 2M 2 48π r r 4 A2 + a 2 ), (3.2.9) 2 T rt = 1 Aa 48π 1 2M. r a, A Unruh vacuum condition [13] ingoing ux r = r = 2M regular T vv = 0 (at r = ), (3.2.10) T uu = 0 (at r = 2M), u = t r 2M log ( ) r 2M 2M v = t + r + 2M log ( ) (3.2.11) r 2M A = a = 1/4M Hawking T tt = 1 12π T rr = 1 48π T rt = 2Mr 7M 2 ( + 3 2r 4 ( 1 2M r πM 2 1 2M r ) + 2M 1 768πM 2 ) 2 ( 2M 2 r M 2. ) (3.2.12) T uu 1/196πM 2 ux T tr 1/768πM 2 Hawking ux Φ = 1/768πM 2 Hawking ux 2 ux Φ = πt 2 /12 T = 1/8πM = κ/2π Hawking 3.3 gravitational anomaly Hawking 2.3 [2] 34

36 Hawking Hawking (1+1) Schwarzschild Hawking Hawkig (2+1) Robinson and Wilczek gravitational anomaly Schwarzschild Hawking [10] Hawking gravitational anomaly compensating ux Schwarzschild gravitational anomaly Schwarzschild Iso et al. Reissner-Nordstrom Hawking gravitational anomaly U(1) [11] Robinson and Wilczek [10] Kerr Myers-Perry [12] Robinson and Wilczek [10] Kerr Myers-Perry Hawking gravitational anomaly gravitational anomaly [10, 11] ds 2 = f(r)dt f(r) dr2 + r 2 dω 2 D 2. (3.3.1) r = r H f(r H ) = 0 surface gravity (2.1.42) κ = f (r H )/2 2 S[ϕ] = 1 d D x g ϕ 2 ϕ 2 = 1 d D x r D 2 ( γϕ 1 2 f 2 t + 1 r D 2 rr D 2 f r + 1 ) (3.3.2) r 2 Ω ϕ, γ dω 2 D 2 determinant Ω 2 35

37 r r H dominant term S[ϕ] = r H D 2 d D x γ ϕ ( 1f ) 2 2t + r f r ϕ = r D 2 ( H dtdr ϕ n 1 ) (3.3.3) 2 f 2 t + r f r ϕ n n 2 ϕ (D 2) ds 2 = f(r)dt f(r) dr2. (3.3.4) D 2 2 ingoing mode ingoing mode r H r r H + ɛ r H + ɛ r ɛ 0 2 gravitational anomaly [14, 15, 16] µ T µ 1 ν = 96π g ɛβδ δ α Γ α νβ, (3.3.5) ɛ 01 = +1 A ν N µ ν µ T µ ν A ν 1 g µ N µ ν. (3.3.6) r H + ɛ r A ν = N µ ν = 0 r H r r H + ɛ N t t = N r r = 0, N r t = 1 192π (f 2 + f f), N t r = 1 192πf 2 (f 2 f f), A t = 1 192π (f 2 + f f), A r = 0, (3.3.7) (3.3.8) r eective action ( ) W [g µν ] = i ln Dφ e is[φ, gµν] (3.3.9) S[φ, g µν ] x µ x µ λ µ (3.3.10) 36

38 eective action δ λ W = d 2 x g λ ν µ {T (H) µν H(r) + T (o) µ ν Θ +(r)} = d 2 x λ t { r (N r th) + (T (o) rt T (H) r t + N r t)δ(r r H ɛ)} + d 2 x λ r (T (o) rr T (H) r r )δ(r r H ɛ), (3.3.11) Θ + (r) = Θ(r r H ɛ), H(r) = 1 Θ + (r) H o r H r r H + ɛ, r H + ɛ r T (o) µν T (H) µ ν (3.3.5) T (H) µ ν T (o) µν (3.3.6) T t t = K + Q f T r r = K + Q f B(r) f + B(r) f I(r) f + I(r) f T r t = K + C(r) = f 2 T t r,, + T α α(r), (3.3.12) [17] T T (H) T (o) C(r) B(r) I(r) 1 2 r r H A t (r )dr, r r H f(r )A r (r )dr, r r H T α α(r )f (r )dr, (3.3.13) K Q r H r r H + ɛ K H, Q H r H + ɛ r K o, Q o (3.3.8) B(r) = 0 r r H, C(r) 0, I(r)/f 1 2 T α α(r H ) (3.3.12) T t t = K + Q + 1 f 2 T α α(r), T r r = K + Q + 1 f 2 T α α(r), T r t = K = f 2 T t r. (3.3.14) 4 K H, K o, Q H, Q o gravitational anomaly (3.3.14) (3.3.11) ɛ 0 δ λ W = d 2 x λ t { r (N r th) + ( K H + K o + N r t)δ(r r H )} + d 2 x λ r K H + Q H K o Q o δ(r r H ). f (3.3.15) 37

39 1 δ ingoing mode K o = K H Φ, Q o = Q H + Φ, (3.3.16) Φ f 2 = κ2 192π 48π. (3.3.17) r=rh Hawking ux K H [11] covariant energy momentum tensor T µν covariant anomaly equation µ T µ ν = 1 96π g ɛ µν µ R. (3.3.18) dieomorphism invariant covariant anomalous energy µν momentum tensor T (H) K H = 2Φ T (H) r t = T (H) r t 1 192π (ff 2f 2 ) = 0. (3.3.19) T (o) rt = Φ. (3.3.20) Φ Hawking ux 2 ux Φ = π 12 T 2 (3.3.17) Hawking T = κ 2π (3.3.21) Kerr Hawking Kerr Hawking gravitational anomaly cancellation Kerr 2 Kerr Boyer-Linquist Kerr ds 2 = a2 sin 2 θ Σ + (r2 + a 2 ) 2 a 2 sin 2 θ Σ dt 2 2a sin 2 θ r2 + a 2 dtdφ Σ sin 2 θdφ 2 + Σ (3.3.22) dr2 + Σdθ 2 38

40 Σ = r 2 + a 2 cos 2 θ, = r 2 2Mr + a 2 = (r r + )(r r ). (3.3.23) outer horizon inner horizon r = r +, r determinant g = Σ sin θ (3.3.24) (t, φ) g tt = (r2 + a 2 ) 2 a 2 sin 2 θ Σ g φφ = a2 sin 2 θ Σ sin 2 θ g tφ = a(r2 + a 2 ) Σ,., (3.3.25) Kerr S[ϕ] = 1 d 4 x g ϕ 2 ϕ 2 = 1 d 4 x g ϕ 1 [ ( (r 2 + a 2 ) 2 2 Σ ) a 2 sin 2 θ ( ) 1 + sin 2 θ a2 φ 2 + r r + 1 sin θ θ sin θ θ t 2 2a(r2 + a 2 ) t φ ] ϕ (3.3.26) r r + dominant term S[ϕ] = 1 2 d 4 x sin θ ϕ [ (r2 + + a 2 ) 2 t 2 2a(r2 + + a 2 ] ) t φ a2 2 φ + r r ϕ. (3.3.27) ψ = φ Ω H t, ξ = t, (3.3.28) Ω H (ξ, r, θ, ψ) (3.3.27) S[ϕ] = a d 4 x sin θ ϕ 2Ω H a r a2 (3.3.29) ( 1 f(r) 2 ξ + rf(r) r ) ϕ, (3.3.30) f(r) Ω H a 39 (3.3.31)

41 ϕ(x) = l,m ϕ l m(ξ, r)y l m (θ, ψ) 2 S[ϕ] = a 1 Ω H 2 l,m eective 2 ( dξdrϕ l m 1 ) f(r) 2 ξ + rf(r) r ϕ l m. (3.3.32) ds 2 = f(r)dξ f(r) dr2 (3.3.33) geometry r + > r Rindler extremal r + = r AdS 2 [18] consistent Kerr Hawking 3.3 anomaly cancellation Hawking (3.3.21) T = 1 4π rf r+ = r 2 + a 2 4πr + (r a2 ) = M 2 a 2 4πM(M + M 2 a 2 ). (3.3.34) Kerr Hawking Planck 4 (ξ, ψ) (exp(ω/t ) 1) 1 ω m ϕ exp(iωξ + imψ) (t, φ) ϕ exp(i(ω mω H )t + imφ)) (t, φ) 1 exp((ω mω H )/T ) 1 (3.3.35) Kerr ux Q, P Kerr- Newman Hawking a 2 a 2 +Q 2 +P Myers-Perry 1 Myers-Perry D Myers- Perry [19, 20] ds 2 = dt 2 + Udr2 V 2M + 2M n U (dt + a i µ 2 i dφ i ) 2 + i=1 n (r 2 + a 2 i )(µ 2 i dφ 2 i + dµ 2 i ) + ɛr 2 dµ 2 n+ɛ i=1 (3.3.36) 40

42 V = r ɛ 2 n i=1 U = V (1 (r 2 + a 2 i ), n a 2 i µ2 i r 2 + a 2 ), i i=1 (3.3.37) n (D 1)/2 ɛ = 1(D:even), 0(D:odd) µ i n µ 2 i + ɛµ 2 n+ɛ = 1. (3.3.38) i=1 1 Myers-Perry a 1 = a, a i = 0 (for i 1), µ 1 = µ, φ 1 = φ µ = cos θ, µ 2 = sin θ cos θ 2, µ 3 = sin θ sin θ 2 cos θ 3, (3.3.39). ( ds 2 = 1 + 2M ) dt 2 + µ (r a 2 + 2Ma2 µ 2 ) dφ 2 U U (3.3.40) + 4Maµ2 dtdφ + Udr2 U V 2M + (r2 + a 2 )U dθ 2 + r 2 dγ 2 V dγ 2 sin 2 θdω 2 n+ɛ 2 + (µ 2 2dφ µ 2 ndφ 2 n) (3.3.41) dω 2 n+ɛ 2 Sn+ɛ 2 (θ 2,, θ n+ɛ 1 ) (t, φ) g tt = V {(r2 + a 2 )U + 2a 2 µ 2 M} (r 2 + a 2 )(V 2M)U g φφ (U 2M)V = µ 2 (r 2 + a 2 )(V 2M)U, g tφ 2aMV = (r 2 + a 2 )(V 2M)U,, (3.3.42) determinant µ(r 2 + a 2 )U g = r D 4 γ. (3.3.43) V V (r = r + ) = 2M r = r + (t, r, φ) (θ, θ 2,, θ n+ɛ 1, φ 2,, φ n ) (θ, θ 2,, 41

43 θ n+ɛ 1, φ 2,, φ n ) S[ϕ] = 1 d D x g ϕ 2 ϕ 2 = 1 [ d D x r+ D 4 γ µ(r a 2 )ϕ 2M ( ) ] a 2 t V 2M r+ 2 + V 2M a2 φ r 2M r ϕ ψ = φ + ξ = t. S[ϕ] = (r2 + + a 2 )r+ D 4 2 = (r2 + + a 2 )r+ D 4 2 n a r 2 + +a2 t, d D x γµ ϕ( 1 f 2 ξ + rf r )ϕ (3.3.44) (3.3.45) dξdr ϕ n ( 1 f 2 ξ + rf r )ϕ n (3.3.46) f(r) = V 2M (3.3.47) 2M γµ (θ, θ 2,..., θ n+ɛ 1, ψ, φ 2,..., φ n ) ϕ (3.3.46) 2 ds 2 = f(r)dξ f(r) dr2 (3.3.48) 3.3 Myers-Perry Hawking T = V (r + ) 8πM = (D 3)r2 + + (D 5)a 2 4πr + (r a2 ) (3.3.49) Kerr, Myers-Perry Hawking gravitational anomaly gravitational anomaly Kerr [21] Myers-Perry 2 [22, 23, 24] [22, 23, 24, 25, 26, 27, 28] future work [2, 17, 29, 30] Hawking Hawking gravitational anomaly

44 4 Hawking Hawking Planck Hawking Kaluza-Klein Hawking 4 Hawking Hawking Kaluza-Klein Hawking Hawking T = κ/2π Stephan-Boltzmann dm dt = σat 4 H (σ = π2 60 ) (4.1.1) A A = M 2 /MPl 4, T H MPl 2 /M dm dt M 4 Pl M 2 (4.1.2) τ M 3 M 4 Pl (4.1.3) (M M Pl ) Hawking (3.1.56) Planck Planck Hawking Plank 43

45 + S = 1 16πG d 4 x gr + d 4 x g[ 1 2 ( f)2 ] (4.1.4) eective action ( W [g] = i ln Df exp(i d 4 x g[ 1 ) 2 ( f)2 ]) (4.1.5) S = 1 16πG d 4 x gr + W [g] (4.1.6) W [g] Hawking Einstein G µν = 8πG T µν, T µν 2 g δw δg µν (4.1.7) Hawking 4 eective action 4 2 eective action (3.2.4) Einstein 2 eective action (3.2.4) CGHS 2 Einstein-Hilbert d 2 x gr Einstein-Hilbert 2 Einstein 0 g µν dilaton φ S CGHS = 1 2π d 2 x g[e 2φ (R + 4( φ) 2 + 4λ 2 ) 1 2 N ( f i ) 2 ] (4.2.1) f i λ mass Callan, Gidding, Harvey, Strominger CGHS [31] dilaton gravity Einstein-Hilbert dimensional reduction f i Hawking f i i=1 44

46 2e 2φ ( µ ν φ g µν 2 φ + g µν ( φ) 2 λ 2 g µν ) g µν( f i ) 2 1 (4.2.2) 2 µf i ν f i = 0, R 4( φ) φ + 4λ 2 = 0, (4.2.3) 2 f i = 0 (4.2.4) f i 2 i ds 2 = e 2ρ dx + dx (4.2.5) (conformal gauge) x + x + (x + ), x x (x ) ds 2 2ρ dx+ = e = exp dx dx dx+ dx (2ρ + ln dx+ dx dx + + ln dx dx + ) dx + dx (4.2.6) conformal gauge ρ ρ + (x + ) + (x ) (4.2.7) conformal gauge (4.2.10),(4.2.11) e 2φ (4 + ρ + φ 2 2 +φ) f i + f i = 0, (4.2.8) e 2φ (4 ρ φ 2 2 φ) f i f i = 0, (4.2.9) e 2φ (2 + φ 4 + φ φ λ 2 e 2ρ ) = 0, (4.2.10) 4 + φ φ φ ρ + λ 2 e 2ρ = 0, (4.2.11) + f i = 0 (4.2.12) + (ρ φ) = 0 (4.2.13) ρ = φ + (x + ) + (x ) ρ = φ (4.2.14) x ± (4.2.14) (4.2.10) + e 2φ = λ 2 e 2φ = λ 2 x + x + A(x + ) + B(x ) (4.2.15) 45

47 f i ingoing shock wave + f i + f i = 2aδ(x + x + 0 ) f i = 0 (4.2.16) (4.2.12) consistent (4.2.8) +e 2 2φ = aδ(x + x + 0 ) A (x + ) = aδ(x + x + 0 ) ( (4.2.15)) A(x + ) = a(x + x + 0 )θ(x+ x + 0 ) + c 1x + + c 3, (4.2.17) (4.2.9) 2 e 2φ = 0 B (x ) = 0 ( (4.2.15)) B(x ) = c 2 x + c 4, (4.2.18) e 2φ = a(x + x + 0 )θ(x+ x + 0 ) λ2 x + x + c 1 x + + c 2 x + c (4.2.19) x ± c 1 = c 2 = 0 ( x + 0 ) e 2φ = e 2ρ = a(x + x + 0 )θ(x+ x + 0 ) λ2 x + x + c (4.2.20) c = 0 shock wave x + < x + 0 ds 2 = dx+ dx λ 2 x + x = dσ+ dσ λx + (4.2.21) = e λσ+ λx = e λσ shock wave c = 0 ds 2 = dx + dx a(x + x + 0 )θ(x+ x + 0 ) λ2 x + x (4.2.22) x + > x + 0 R = 8e 2ρ + ρ = 4λ 2 ax + (4.2.23) 0 a(x + x + 0 ) + λ2 x + x x + (x + a λ 2 ) = ax+ 0 λ 2 (4.2.24) curvature singularity 4.1 conformal Penrose

48 4.1: shock wave (Hawking ) 4.2: CGHS Penrose 47

49 4.3 RST CGHS f i f i + δf i (4.3.1) classical part f i quantum part δf i CGHS S CGHS = 1 d 2 x g[e 2φ (R + 4( φ) 2 + 4λ 2 ) 1 2π 2 ( f i) ( δf i) 2 ] (4.3.2) (δf i 1 ) quantum part δf i eective action S RST = i ln N i=1 = S CGHS κ 8π Dδf i exp(is CGHS [g, φ, f i + δf i ]) d 2 x g(r 1 R + 2φR). (4.3.3) κ N/12 C Russo, Susskind, Thorlacius RST [32] R 1 R Liouville action C local counter term 2φR Hawking solvable solvable local counter term [33, 34] RST (4.3.3) nonlocal χ (4.3.3) S RST = S CGHS κ 8π d 2 x g[( χ) 2 + 2(χ + φ)r]. (4.3.4) χ 2 χ = R (4.3.4) (4.3.3) 2(e 2φ + κ 4 )( µ ν φ g µν 2 φ) +2e 2φ g µν {( φ) 2 λ 2 } g µν( f i ) µf i ν f i (4.3.5) κ 4 ( µχ ν χ 2 µ ν χ 1 2 g µν( χ) 2 + 2g µν 2 χ) = 0, (1 + κ 4 e2φ )R 4( φ) φ + 4λ 2 = 0, (4.3.6) 2 f i = 0, (4.3.7) 2 χ = R. (4.3.8) CGHS conformal gauge ds 2 = e 2ρ dx + dx (4.3.9) 48

50 (4.3.8) + χ = 2 + ρ χ = 2(ρ + A + (x + ) + A (x )) (4.3.5) (+ ) (4.3.6) (4.3.10) 2(e 2φ + κ 4 ) + φ + 4e 2φ + φ φ + λ 2 e 2(ρ φ) + κ + ρ = 0 (4.3.11) 2(e 2φ + κ 4 ) + ρ + 4e 2φ + φ φ 4e 2φ + φ + λ 2 e 2(ρ φ) = 0 (4.3.12) 2 + (ρ φ) = 0 (4.3.13) CGHS ρ = φ (4.3.11) 2(e 2φ κ 4 ) + φ + 4e 2φ + φ φ + λ 2 κ κ + ( 2 φ + e 2φ ) + λ 2 = 0 κ κ Ω 2 φ + e 2φ = λ2 x + x + B + (x + ) + B (x ) κ κ (4.3.14) κ B + (x + ) + B (x ) = 4 ln( λ2 x + x ) (4.3.15) (4.3.5) (±±) 2(e 2φ + κ 4 )( 2 ±φ 2( ± φ) 2 ) 1 2 ±f i ± f i + κ(( ± φ) 2 ±φ 2 ( ± A ± ) 2 ±A 2 ± ) = 0 (4.3.16) κt ± (x ± ) = κ ±Ω ±f i ± f i t ± (x ± ) ( ± A ± ) 2 2 ±A ± t ± f i = 0 f i = 0 (4.3.15) t ± = 1 ] 2 κ ± [ λ2 κ x + x κ 4 ln( λ2 x + x ) (4.3.17) 1 = 4(x ± ) 2 + Ω = λ2 (4.3.18) κ κ ±Ω 2 = 4(x ± ) κ ±f i ± f i (4.3.19) [ ] κ Ω 2 φ + e 2φ κ 49

51 ingoing shock wave + f i + f i = 2aδ(x + x + 0 ) f i = 0 (4.3.20) (4.3.18), (4.3.19) Ω = λ2 κ x + x κ 4 ln( λ2 x + x ) a (x + x + κ 0 )θ(x+ x + 0 ) (4.3.21) CGHS x + < x + 0 R = 8e 2ρ + ρ (4.3.22) + ρ = 1 Ω ( + Ω Ω Ω 2 +Ω Ω ), [ d dφ ] (4.3.23) Ω = 0 curvature singularity x + > x + 0 e 2φ = κ/4 singular (4.3.21) singularity 1 ln κ 4 = 4λ2 κ x+ x ln( λ 2 x + x ) 4a κ (x+ x + 0 ) (4.3.24) apparent horizon 2 apparent horizon e 2φ r 2 + φ = 0 (4.3.25) apparent horizon + φ = 0 + Ω = 0 x + (x + a λ 2 ) = κ 4λ 2 (4.3.26) apprent horizon 4.3 singularity (4.3.24) apparent horizon (4.3.26) x + = κ 4a (e4ax+ 0 /κ 1) x = a λ 2 (1 e 4ax+ 0 /κ ) 1 (4.3.27) singularity apparent horizon Hawking singularity space like time like 4.3 conformal Penrose

52 4.3: shock wave (Hawking ) 4.4: RST Penrose 51

53 4.4 horizon constraint RST Hawking 2.3, 3.3 Hawking Hawking 2.3 Hawking ( 4.5) 4.5: Hawking RST RST ( Hawking ) ( (4.3.3)) Hawking horizon constraint constraint constraint Virasoro [L[ξ], L[η]] = il[ξη ηξ ] i dv c(g, φ, f, χ)(ξ η η ξ ) (4.4.1) 48π central charge Hawking ux central charge central charge Hawking 4.5 [35, 36, 37, 38] horizon constraint 52

54 5 Kaluza-Klein 4 Kaluza-Klein Kaluza-Klein Kaluza-Klein Kaluza-Klein Kaluza-Klein ( ) Kaluza-Klein (Gregory-Laamme instability[39]) [40] dimensional reduction ds 2 = g µν (x µ )dx µ dx ν + dy 2 (5.1.1) y L masless S = 1 d 5 x g 2 (5) ( (5) φ) 2 = 1 2 d 4 xdy g[( φ) 2 + ( y φ) 2 ] (5.1.2) g (5) 5 (5) 5 φ y Fourier y y + L φ φ = n φ n (x µ )e 2πin L y (5.1.3) 53

55 φ φ n = φ n S = 1 L d 4 x [ ( ) ] 2πn 2 g µ φ n µ φ n + φ n φ n 2 L n = 1 d 4 x g( φ 0 ) 2 1 d 4 x [ ( ) ] 2πn 2 (5.1.4) g µ φ n µ φ n + φ n φ n 2 2 L n 0 2 canonical L φ n massless + massive massive L 1 L 1 ( TeV) massive 4 Einstein-Hilbert Einstein-Hilbert (5.1.1) S = M5 3 d 5 x g (5) R (5) = M5 3 L d 4 x gr (5.1.5) M 5 5 Planck + S = M5 3 d 5 x g (5) R (5) 1 d 5 x g 2 (5) ( (5) φ) 2 M5 3 L d 4 x gr 1 d 4 x (5.1.6) g( φ 0 ) 2 2 canonical L 4 eective M Pl = M5 3L (5.1.7) 4 L D Einstein-Hilbert action S = M D 2 D d D x gr (5.2.1) M D D Planck R µν = 0 ( ( ds 2 rh = 1 r ) ) ( D 3 ( dt 2 rh + 1 r ) D 3 ) 1 dr 2 + r 2 dω 2 D 2. (5.2.2) 54

56 5.1: dω 2 D 2 (D 2) D Schwarzschld Kerr Myers-Perry S 2 S D 2 S n 2 R D n ds 2 = V dt 2 + V 1 dr 2 + r 2 dω 2 n 2 + dy i dy i ( rh ) n 3 (5.2.3) V 1 r (D n = 1 ) n Schwarzschild (D n) Euclid Schwarzschild ( 5.1) n = 4 (y 1,, y D n ) 4 Schwarzschild 5.3 Kaluza-Klein Schwarzschild (5.2.2) (5.2.3) 2 Einstein (5.2.1) S = 4πM D 2 D A H (5.3.1) A H Schwarzschild 55

57 S BH = 4πM D 2 D Ω D 2rBH D 2 S BB = 4πM D 2 D Ω (5.3.2) n 2rBS n 2 LD n BH Schwarzschild BB r BH, r BS L Ω N N Ω N 2π(N+1)/2 Γ((N + 1)/2) (5.3.3) Hawking T BH = D 3 4πr BH T BB = n 3 4πr BB (5.3.4) S Hawking T dm = T ds M BH = (D 2)M D 2 D Ω D 2rBH D 3, M BB = (n 2)M D 2 D Ω n 2rBB n 3 LD n. (5.3.5) S BH (M) > S BB (M) (5.3.6) (5.3.6) Ω D 2 r D 2 BH (D 2)Ω D 2 r D 3 BH 2 r BH L > ( ) D 2 D 2 n 2 > Ω n 2r n 2 BS LD n (5.3.7) = (n 2)Ω n 2r n 3 BB LD n (5.3.8) D n ( Ω D 2 Ω n 2 ) 1 D n rbb (5.3.9) k = 2π/L ( ) D 2 ( ) 1 n 2 D n Ω n 2 D n kr BB < 2π D 2 Ω D 2 (5.3.10) 56

58 5.3.2 Gregory-Laamme instability [39, 41, 42] dimensional reduction c = = M D = 1 Einstein-Hilbert (5.2.1) (n 2) ds 2 = g (D n+2) MN (x M )dx M dx N + e 2φ(xM ) dω 2 n 2 (5.3.11) s-wave S =Ω n 2 d D n+2 x g (D n+2) e (n 2)φ [R (D n+2) + (n 2)(n 3)( φ) 2 + (n 2)(n 3)e 2φ ] (5.3.12) dilaton gravity R (D n+2) g (D n+2) MN Ω N (5.3.3) N (D n) (D n 1) g (D n+2) D n 1 MN (x M )dx M dx N = g µν (3) (x µ )dx µ dx ν + dy i dy i (5.3.13) (D n 1) zero mode L (5.3.12) S = Ω n 2 L D n 1 d 3 x g (3) e (n 2)φ [R (3) + (n 2)(n 3)( φ) 2 + (n 2)(n 3)e 2φ ] (5.3.14) R (3) g µν (3) (3) R (n 2)(n 3)( φ) 2 + 2(n 3) 2 φ + (n 4)(n 3)e 2φ = 0, (5.3.15) i=1 G µν + (n 2) µ ν φ (n 2) µ φ ν φ + (n 2)g µν { 2 φ + n 1 ( φ) 2 n 3 e 2φ } = (5.3.16) ds 2 = V dt 2 + V 1 dr 2 + dy 2 e 2φ = r 2 (5.3.17) ( rh ) n 3] [V 1 r 3 g µν g µν + h µν φ φ + δφ (5.3.18) 57

59 back ground (5.3.17) perturb (5.3.16) δg µν (n 2)δΓ ρ µν ρ φ + (n 2)h µν { 2 φ + n 1 ( φ) 2 n 3 e 2φ } (n 2)g µν {h ρσ ρ σ φ + g ρσ δγ α ρσ α φ n 1 h ρσ ρ φ σ φ} 2 + (n 2) µ ν δφ 2(n 2) (µ φ ν) δφ + (n 2)g µν { 2 δφ + (n 1)g ρσ ρ φ σ δφ (n 3)e 2φ δφ} = 0 (5.3.19) δg µν = 1 2 [ ρ µ h νρ + ρ ν h µρ h µν µ ν h Rh µν g µν ( ρ σ h ρσ h R ρσ h ρσ )] (5.3.20) δγ ρ µν = 1 2 gρσ ( µ h νσ + ν h σµ σ h µν ) h µν δφ h µν (t, r, y) = h(r)e Ωt+iky δφ(t, r, y) = δφ(r)e Ωt+iky (5.3.21) e iωt e Ωt Ω > 0 (5.3.14) h µν δφ h µν h µν = µ ξ ν ν ξ µ δφ δφ = ξ µ µ φ (5.3.22) h tt h tt = 2Ωξ t + V V ξ r h tr h tr = ξ t + V h ty h ty = ikξ t Ωξ y h rr h rr = 2ξ r V V ξ t Ωξ r V ξ r (5.3.23) h ry h ry = ikξ r ξ y h yy h yy = 2ikξ y δφ δφ = 1 r V ξ r d/dr ξ µ (5.3.21) δφ = 0, h ty = 0, h yy = 0 (5.3.24) (5.3.23) ξ t, ξ r, ξ y (5.3.19) 58

60 tt (n 2){ (n 3) + (n 3)V + rv }h tt +{(n 2)(n 3)V + 2(n 2)rV + k 2 r 2 }V 2 h rr + (n 2)rV 3 h rr +ikrv 2 {rv + 2(n 2)}h ry + 2ikr 2 V 2 h ry = 0 (5.3.25) tr (n 2){rV + (n 3)V (n 3)}h tr +(n 2)ΩrV h rr + iωkr 2 h ry = 0 (5.3.26) ty ik{rv + (n 2)V }h tr ikrv h tr + iωkrv h rr (n 2)ΩV h ry ΩrV h ry = 0 (5.3.27) rr {k 2 r 2 + (n 2)rV }h tt (n 2)rV h tt + 2(n 2)ΩrV h tr (n 2)(n 3)V 2 h rr ikrv {rv + 2(n 2)V }h ry = 0 (5.3.28) ry ikr 2 V h tt 2ikr 2 V h tt + 2iΩkr 2 V h tr ikrv 2 {2(n 2)V + rv }h rr 2V {(n 2)(n 3)V 2 (n 2)(n 3)V + 2(n 2)rV V + Ω 2 r 2 + r 2 V }h ry = 0 (5.3.29) yy { 2(n 2)rV V 2r 2 V V + r 2 V 2 }h tt + {2(n 2)rV r 2 V }h tt + 2r 2 V 2 h tt 2ΩrV {2(n 2)V + rv }h tr 4Ωr 2 V 2 h tr +{2(n 2)(n 3)V 2 + 6(n 2)rV V + r 2 V 2 + 2r 2 V V + 2Ω 2 r 2 }V 2 h rr +{2(n 2)rV + r 2 V }V 3 h rr = 0 (5.3.30) tr, ty, rr, ry 4 h tr, h rr, h ry h tt P( r) d2 d r 2 h tt + Q( r) d d r h tt + R( r)h tt = 0. (5.3.31) P( r) 2(n 2)( k 2 + Ω 2 ) r 3n 9 + {(5n 9) k 2 + 4(n 2) Ω 2 } r 2n 6 2{(2n 3) k 2 + (n 2) Ω 2 } r n 3 + (n 1) k 2, Q( r) 1 r [ 2(n 2)2 ( k 2 + Ω 2 ) r 3n 9 + (n 1){(5n 12) k 2 + 2(n 2) Ω 2 } r 2n 6 {(5n 2 16n + 9) k 2 + 2(n 2) Ω 2 } r n 3 + (n 1)(2n 5) k 2 ], R( r) 2(n 2)( k 2 + Ω 2 ) 2 r 3n 9 (3n 5) k 2 ( k 2 + Ω 2 ) r 2n 6 + (n 1) k 4 r n 3 (5.3.32) (n 1)(n 3) 2 k2 r n 5 + (n 1)(n 3)2 k2 r 2, 59

61 r r/r H, k kr H, Ω Ωr H h tr, h rr, h ry h tt (5.3.31) (5.3.31) r 1 r 1 d 2 h tt d r 2 + n 2 r dh tt d r ( k 2 + Ω 2 )h tt = 0, (5.3.33) d 2 h tt d r dh tt Ω2 r 1 d r (n 3) 2 (r 1) 2 h tt = 0 (5.3.34) k2 A e + Ω 2 r + B e k2 + Ω 2 r ( r 1) h tt A H (r 1) Ω n 3 + BH (r 1) Ω n 3 ( r 1) (5.3.35) regular B = B H = 0 Runge- Kutta algorithm r 1 = r 2 = h tt e k2 + Ω 2 r h tt ( r 1 ) = 1.0, dh tt ( r 1 )/d r = k2 + Ω 2 k Ω (5.3.31) r = r 2 B H B H Ω k ( 5.2) Ω = 0 k = k crit 5.1 [39, 41, 42] k crit ( (5.3.10) D n = 1 ) 60

62 5.2: n = 4, 5, 6, 7, 8, 9 n k crit r H ( ) k crit r H ( ) : n = 4, 5, 6, 7, 8, 9 k crit 61

63 6 Kaluza-Klein Kaluza-Klein 3 4 Kaluza-Klein Kaluza-Klein Hawking Kaluza-Klein Kaluza-Klein 6.1 Hawking ( ) 6.1 r H L Hawking r H < L ( ) Hawking Hawking ( 6.2) Hawking 62

64 6.1: Hawking 6.2: Hawking 63

65 6.2 Einstein-Hilbert dimensional reduction D Einstein-Hilbert S g = M D 2 D d D x gr (6.2.1) 1 Schwarzchild n D = n + 1 Einstein-Hilbert ds 2 = g (3) µν (x µ )dx µ dx ν + l 2 e 4 n 2 φ(xµ) dω 2 n 2 (6.2.2) S g = Ω n 2 M D (M D l) n 2 d 3 x ge 2φ [R+4 n 3 (n 2)(n 3) n 2 ( φ)2 + l 2 e 4 n 2 φ ] (6.2.3) l (5.2.3) ds 2 = V dt 2 + V 1 dr 2 + dy 2 l 2 e 4 n 2 φ = r 2 [ ( rh V 1 r ) n 3 ] (6.2.4) Hawking D S m = d D x g [ 1 2 ( f)2 ] (6.2.5) dimensional reduction S m = d 3 x g (3) e 2φ [ 1 2 ( f)2 ] (6.2.6) f canonical D 3 dimensional reduction S = Ω n 2 M D (M D l) n 2 d 3 x ge 2φ [R + 4 n 3 (n 2)(n 3) n 2 ( φ)2 + l 2 e 4 n 2 φ ] + d 3 x g e 2φ [ 1 2 ( f)2 ] (6.2.7) 3 dilaton gravity context 64

66 6.3 3 dilaton gravity 3 dilaton gravity 3 dilaton gravity S g = M 3 d 3 x g e F (φ) [R + 2a( φ) 2 + U(φ)] (6.3.1) ds 2 = g (2) ab (xa )dx a dx b + dy 2 (6.3.2) g (2) ab (6.3.2) dimensional reduction (6.2.3) (6.3.1) φ g µν F (φ)[r 2a( φ) 2 + U(φ)] 4a 2 φ + U (φ) = 0, (6.3.3) G µν F (φ) µ ν φ + (2a F (φ) F (φ) 2 ) µ φ ν φ +g µν {F (φ) 2 φ + (F (φ) + F (φ) 2 a)( φ) U(φ)} = 0 (6.3.4) (6.3.4) (y, y) 1 2 R + F 2 φ + (F + F 2 a)( φ) U = 0 (6.3.5) (6.3.3) (6.3.5) R (2F 2 4a) 2 φ + F (2F + 2F 2 4a)( φ) 2 + U = 0 (6.3.6) (6.3.4) (a, b) (a, b y) F 2 φ + (F + F 2 )( φ) 2 U = 0 (6.3.7) (6.3.6) (6.3.7) φ 2 (6.3.6) (6.3.7) 2F 2 4a F = F (2F + 2F 2 4a) (F + F 2 ) af (φ) = 0 F (φ) = 0 or a = 0 (6.3.8) 2F 2 4a F = U U 65 (6.3.9)

67 F = 0 (6.3.9) F (φ) = bφ + c (6.3.10) U(φ) exp( 2b2 4a φ) (6.3.11) b bφ + c 2φ, 4a/b 2 a (6.3.1) S g = M 3 d 3 x g e 2φ [R + 2a( φ) 2 + λ 2 e 2(2 a)φ ] (6.3.12) λ a = 0 (6.3.9) U(φ) e 2F (φ) (6.3.13) F (φ) 2φ (6.3.1) S g = M 3 d 3 x g e 2φ [R + λ 2 e 4φ ] (6.3.14) (6.3.12) a = 0 (6.3.2) 3 dilaton gravity S g = M 3 d 3 x g e 2φ [R + 2a( φ) 2 + λ 2 e 2(2 a)φ ] (6.3.15) (6.3.15) a = 2 CGHS (4.2.1) CGHS 2 1 (6.3.15) M 3 = Ω n 2 M D (M D l) n 2 a = 2 n 3 n 2 (6.3.16) λ 2 (n 2)(n 3) = l 2 S g = Ω n 2 M D (M D l) n 2 d 3 x ge 2φ [R + 4 n 3 (n 2)(n 3) n 2 ( φ)2 + l 2 e 4 n 2 φ ] (6.3.17) dimensional reduction (6.2.3) (6.3.15) (6.2.3) n = 4, 5, 6, n R (6.3.15) a 0, 2 e (2 a)φ = λ(2 a) r 2a ds 2 = V dt 2 + V 1 dr 2 + dy 2 [ ( rh ) a ] 2 a V 1. r (6.3.18) 66

68 a = 2 φ = λ 2 r ds 2 = (1 Ce λr )dt 2 + (1 Ce λr ) 1 dr 2 + dy 2 (6.3.19) a = 0 e 2φ = λr ( r ds 2 = ln r H ) ( ( )) r 1 (6.3.20) dt 2 + ln dr 2 + dy 2 r H a = 2 CGHS [43, 44, 45, 46] [46] 2.3 horizon constraint a = 0 a = 0 (6.3.16) n = 3 (6.3.16) λ = 0 λ (6.3.20) D 6.4 (6.3.15) (6.2.3) (6.3.15) ds 2 = g (2) ab dxa dx b + e 2χ dy 2 (6.4.1) χ radiaon radion r H L 2 g (2) ab radion χ y (6.3.15) y S g = M 3 L d 2 x g (2) e 2φ χ [R (2) + 2a( φ) φ χ + U(φ)], (6.4.2) [U(φ) λ 2 e 2(2 a)φ ] g (2) ab R(2) R = R (2) χ 2( χ) 2 (6.4.3) 67

69 2 radion 1 S m = d 2 x g (2) e 2φ χ [ 1 2 ( f)2 ] (6.4.4) L f Hawking eective action ( ) W [g (2), φ, χ] i ln Df e ism[g (2),φ,χ,f] (6.4.5) (2) g ab M 3 Le 2φ χ [2 a b φ + a b χ 2(2 a) a φ b φ a χ b χ +g ab { 2 2 φ 2 χ + (4 a)( φ) φ χ + ( χ) U(φ)}] = 1 2 T ab, (6.4.6) φ 2M 3 Le 2φ χ [R + 2a 2 φ χ 2a( φ) 2 2a φ χ 2( χ) 2 + (a 1)U(φ)] = X, (6.4.7) χ M 3 Le 2φ χ [R φ 2(4 a)( φ) 2 + U(φ)] = Y (6.4.8) T ab 2 g δw δg ab X 1 δw g δφ Y 1 δw g δχ (6.4.9) (6.4.6) 2 2 φ 2 χ + 4( φ) φ χ + ( χ) 2 U(φ) = 1 2M 3 L e2φ+χ T a a (6.4.10) (6.4.7) (6.4.8) R 2(2 a) 2 φ χ + 4(2 a)( φ) 2 2a φ χ 2( χ) 2 (2 a)u(φ) = 1 2M 3 L e2φ+χ (X 2Y ) (6.4.11) (6.4.10) (6.4.11) 2 φ ( χ) 2 2 χ + 2 φ χ = 1 2(4 a)m 3 L e2φ+χ ((2 a)t a a X + 2Y ) (6.4.12) 68

70 (6.4.4) φ χ 2φ + χ eective action W φ χ 2φ + χ local counter term φ χ eective action X = 1 δw [g, 2φ + χ] = 2 1 δw [g, 2φ + χ] = 2Y (6.4.13) g δφ g δχ X 2Y = 0 (6.4.12) ( χ) 2 2 χ + 2 φ χ = 2 a 2(4 a)m 3 L e2φ+χ T a a (6.4.14) full non-linear Hawking T ab, X, Y perturb back ground χ = 0 (6.4.14) perturb δχ + 2 φ δχ = 2 a 2(4 a)m 3 L e2φ T a a (6.4.15) δχ δχ ( back ground χ = 0) (6.4.15) (6.4.15) eective action (6.4.4) Weyl Weyl Weyl radion induce C T a a = 1 24π [R ( f(φ))2 ] (6.4.16) f(φ) φ f(φ) (r H = 0) Hawking T µν = 0 r H = 0 0 f = 0 a = 2 CGHS (6.4.15) CGHS CGHS (6.4.15) radion 69

71 6.5 radion a = 2 a = 0 D 0 < a < 2 (6.3.16) (6.4.15) δχ + 2 φ δχ = 1 T a a 2(n 1)Ω n 2 M n 1 D L (6.5.1) (le 2 n 2 φ ) n 2 (t, r) δχ,tt V 2 δχ,rr + (2V φ,r V,r )V δχ,r = F (r) (6.5.2) ( rh V 1 r ) n 3 (6.5.3) F (r) radion F (r) 1 V T a a 2(n 1)Ω n 2 M n 1 D L (6.5.4) (le 2 n 2 φ ) n 2 (6.5.2) δχ δχ(t = 0, r) = δχ,t (t = 0, r) = 0 δχ δχ = δχ,t = 0 δχ,tt = F (r) δχ = 1 2 F (r)t2 (6.5.5) F (r) (6.2.4) le 2 n 2 φ = r (6.5.6) (6.4.16) f = 0 T a a = F (r) = R 24π = V,rr 24π (n 2)(n 3) 1 V = (6.5.7) 24π r 2 (n 2)(n 3) 48π(n 1)Ω n 2 M n 1 D L V (1 V ) r n (6.5.8) F (r) 6.3 radion 6.4 radion (6.5.5) (6.5.5) δχ,tt δχ,rr δχ F (r)t 2 t F/F F (r) r H t r H (6.5.5) F (r) 1/(M n 1 D L rn H ) δχ δχ t 2 M n 1 D L rn H 70 (t r H ) (6.5.9)

72 6.3: F (r) 6.4 radion non-linear radion Hawking singularity Hawking 6.4 radion Hawking ( 6.5) 6.6 radion radion (6.4.15) radion (6.4.15) r = dr V (6.6.1) δχ,tt δχ, + 2φ, δχ, = F (r) (6.6.2) 71

73 6.4: 6.5: 72

74 , / r 2φ, δχ, r r r (r ) r ( ) 1 n 3 r H ln r rh (6.6.3) r H (r r H ) r r (r ) r r H r H e (n 3)r /r H (r ) F (r) r F (r ) (n 2)(n 3) 1 ( rh 48π(n 1)Ω n 2 M n 1 D L r n r (6.6.4) ) n 3 (r ) (6.6.5) (n 2)(n 3) 2 F (r ) 48π(n 1)Ω n 2 M n 1 D L e (n 3)r /r H (r ) (6.6.6) rn H F (r) r : F (r) (r ) F (r ) r H F (r) (6.5.8) F (r) 1/(M n 1 D L rn H ) F (r ) Gaussian 1 F (r ) M n 1 D L e (r /r H) 2 (6.6.7) rn H [ 1 f(r + t) + f(r t) δχ M n 1 D L rn 1 2 H f(r ) Gaussian f(r ) r dy 73 y ] f(r ) (6.6.8) dx 1 e x 2 r H 2 (6.6.9) r H

75 f(r) 6.7 t δχ : f(r) 6.8: radion δχ t r H δχ δχ t M n 1 D L rn 1 H (t r H ) (6.6.10) (6.6.2) 2φ, δχ, φ, = V φ,r V r 1/r (r ) 1 (r ) r H e (n 3)r /r H (6.6.11) φ, (r ) r H F (r) r O(r H ) δχ, (6.6.8) φ,, δχ, : φ, δχ, δχ φ, 1/r H, δχ, δχ, /r H δχ, φ, δχ, δχ, 74

76 dominate r H δχ, φ, supress δχ, φ, δχ, 2φ, δχ, dominate 2φ, δχ, 6.7 Gregory-Laamme instability (6.5.9), (6.6.10) radion t 2 (t r M δχ n 1 H ) D Lrn H t (t r H ) M n 1 D Lrn 1 H (6.7.1) τ pinch δχ 1 M n 1 D τ pinch Lrn 2 H r H (τ pinch r H ) M n 1 D Lrn 2 H r H (τ pinch r H ) (6.7.2) M n 1 D Lrn 2 H r H (r H 1/(M n 1 D L) 1 n 2 ) M n 1 Lrn 2 H r H (r H 1/(M n 1 D L) 1 n 2 ) D (M n 1 D L) 1 n 2 (D 1) eective Planck r H 1/(M n 1 D L) 1 n 2 Planck dominant r H 1/(M n 1 D L) 1 n 2 M D 1 (M n 1 D L) 1 n 2 τ pinch M n 1 D Lrn 1 H M n 2 D 1 rn 1 H (6.7.3) 1 (D 1) Stephan-Boltzmann dm dt AT n (6.7.4) A (D 1) A r n 2 H Hawking T 1/r H M M n 2 D 1 rn 3 H dr H dt 1 M n 2 D 1 rn 2 H (6.7.5) τ lifetime M n 2 D 1 rn 1 H τ pinch (6.7.6) 75

77 Gregory-Laamme instability r H Gregory-Laamme instability L τ GL M n 1 D (rn H L n ) + L M n 1 D rn H τ lifetime (6.7.7) τ pinch τ GL τ lifetime (6.7.8) non-linear δχ(t = 0, r) = δχ,t (t = 0, r) = RST Hawking onset radion ( 6.4) Schwarzschild Kruskal spacelike Schwarzschild 76

78 7 Kaluza-Klein Hawking Kaluza-Klein Hawking Hawking Hawking non-linear Hawking Kaluza-Klein Gregory-Laamme instability Kaluza-Klein S 1 Hawking Kaluza-Klein Hawking 77

79 VIENDAYANTI, Kiki 78

80 A Wald n S = 2π d n 2 x ( ) 1 δi h ɛ ab ɛ cd g H δr abcd (A.1) H : horizon spatial section h : H induced metric determinant ɛ µν : H binormal H volume form I : [4, 5] φ null Killing ξ L ξ φ = 0 Wald n-form Lagrangian L = L(g ab, R abcd, a1 R abcd,, (a1... am)r abcd, ψ, a1 ψ,, (a1... al )ψ) (A.2) ψ matter eld Lagrangian a b ψ = (a b) ψ + [a b] ψ Lagrangian δl = E(φ)δφ + dθ(φ, δφ) (A.3) φ g ab (φ = (g ab, ψ)) Θ φ δφ symplectic potential form symplectic potential form Θ Θ + dy (φ, δφ) Lagrangian L L + dµ(φ) total Θ Θ + δµ(φ) + dy (φ, δφ) E = 0 Lemma Θ Θ = 2E bcd R d δg bc + Θ (A.4) m 1 Θ = S ab (φ)δg ab + T i (φ) abcda 1 a i δ (a1... ai )R abcd i=0 l 1 + U i (φ) a 1 a i δ (a1... ai )ψ i=0 (A.5) 79

81 δg δ (E bcd R ) b 2 b n = E abcd R ɛ ab 2 b n E abcd R = 1 g δ δr abcd ( )L = ɛl (ɛ volume form) ( L δl = ɛ δg ab + L L δr abcd + + δ g ab R abcd (a1... (a1... am)r abcd am)r abcd + L ψ δψ + + L (a1... al )ψ δ (a 1... al )ψ L ) gab δg ab L (A.6) (A.7) L ɛ δ a1 a2 R abcd (a1 a2 )R abcd (A.8) schematic δ R = δ( R + Γ R) = δ R + Γδ R + δγ R = δ R + ( δg ) (A.9) (A.8) L ɛ a1 δ a2 R abcd + ɛ( δg ) (a1 a2 )R abcd ] [ ] L L = a1 [ɛ δ a2 R abcd a1 ɛ δ a2 R abcd (a1 a2 )R abcd (a1 a2 )R abcd + a1 [ɛ(δg ) a 1 ] + ɛ(δg ) ] L = dv a1 [ɛ δ a2 R abcd + ɛ(δg ) (a1 a2 )R abcd [ ] L V b2 b n = ɛ a1 b 2 b n δ a2 R abcd + (δg ) a 1 (a1 a2 )R abcd (A.10) (A.11) δ R = δr + ( δg ) (A.12) 80

82 (A.10) ] L dv a1 [ɛ a2 δr abcd (a1 a2 )R abcd + ɛ(δg ) + ɛ( δg ) [ ) ] [ ] L L = dv a2 a1 (ɛ δr abcd + a2 a1 ɛ δr abcd (a1 a2 )R abcd (a1 a2 )R abcd + a2 [ɛ(δg ) a 2 ] + ɛ(δg ) ] = dv L + a2 a1 [ɛ δr abcd + ɛ(δg ) (a1 a2 )R abcd [ ( V b 2 b n = V b2 b n + ɛ a2 b 2 b n a1 L (a1 a2 )R abcd (A.7) (A.13) ) ] δr abcd + (δg ) a 2 (A.14) δl = ɛ(a ab g δg ab + E abcd R δr abcd + E ψ δψ) + d Θ (A.15) Θ (A.5) (n 1)-form ER abcd = L a1 R abcd = 1 g δ δr abcd L + + ( 1) m L a1 R (a1... am) abcd (a1... am)r abcd L (A.15) δr abcd δg (A.16) E abcd R δr abcd = 2E abcd a d δg bc + E abcd R e abc δg de R = 2 a d E abcd R R δg bc + E abcd R e abc δg de R + a (2ER abcd d δg bc ) d (2 a ER abcd δg bc ) (A.17) (A.15) δl = ɛ(ãbc g δg bc + 2 a d ER abcd δg bc + E ψ δψ) + d(2er bcd d δg bc 2 d ER bcd δg bc + Θ) (A.18) (E bcd R Ã bc g ) b2 b n = E abcd = A bc g R ɛ ab2 b n (A.18) symplectic potential form + E pqrb R R pqr c (A.19) Θ = 2E bcd R d δg bc + Θ (Θ 2 d E bcd R δg bc + Θ) (A.20) 81

83 δ diomorphism transformation ˆδφ = L ξ φ Lagrangian ˆδL = d(ξ L) (A.21) (ξ L) b2 b n = ξ a L ab1 b n (A.21) Λ Lie L ξ Λ = ξ dλ + d(ξ Λ) (A.22) (A.3) ˆδL = E(φ)L ξ φ + dθ(φ, L ξ φ) (A.23) (A.21) (A.23) d(θ(φ, L ξ φ) ξ L) = EL ξ φ (A.24) Noether current J = Θ(φ, L ξ φ) ξ L (A.25) dj = EL ξ φ EOM = 0 (A.26) dj = 0 Noether charge Q J = dq (A.27) J Q L L + dµ, Θ Θ + δµ + dy J J J + L ξ µ + dy (φ, L ξ φ) ξ dµ = J + d(ξ µ) + dy (φ, L ξ φ) L ξ µ = ξ dµ + d(ξ µ) (A.28) (A.29) Q Q Q + ξ µ + Y (φ, L ξ φ) + dz (A.30) Proposition Noether charge Q = W c (φ)ξ c + X cd [c ξ d] + Y (φ, L ξ φ) + dz(φ, ξ) (A.31) 82

84 W, X, Y, Z ( )Θ (A.4) (X cd ) c3 c n = E abcd R ɛ abc3 c n, Y = Z = 0 (A.32) J = 2E bcd R d ( b ξ c + c ξ b ) + Θ (φ, L ξ φ) ξ L (A.33) Θ (δg, δr, δ R, ) Θ (φ, L ξ φ) (ξ, ξ) (ξ ) ER bcd c d (A.33) 2Ebcd R d c ξ b = ER bcdr dcb e ξ e J ξ 2ER bcd d b ξ c Q Q = W c (φ)ξ c + X cd [c ξ d] ( ) (A.34) (X cd ) c3 c n ER abcd ɛ abc3 c n 2E R bcd d b ξ c Q Q Q = W c (φ)ξ c + X cd [c ξ d] + Y (φ, L ξ φ) + dz(φ, ξ) ξ diemorphism genaretor Cauchy surface C H = J + (boundary term) C (A.35) (A.36) (boundary term) H δh J δj = δθ(φ, L ξ φ) ξ δl = δθ(φ, L ξ φ) ξ (Eδφ + dθ(φ, δφ)) ( (A.3)) = δθ(φ, L ξ φ) L ξ Θ(φ, δφ) + d(ξ Θ(φ, δφ)) ( E = 0, (A.22)) H δh = (δθ(φ, L ξ φ) L ξ Θ(φ, δφ)) + C ξ Θ(φ, δφ) + δ(boundary term) (A.37) (A.38) δ(boundary term) = ξ Θ(φ, δφ) δh = δ J ξ Θ(φ, δφ) (A.39) C H (n 2)-form B δ ξ B = ξ Θ(φ, δφ) 83 (A.40)

85 H = = C J ξ B (Q ξ B) (A.41) t a asymptotic time transration vector( ) canonical energy E (Q[t] t B) (A.42) canonical energy Einstein ADM mass ϕ a asymptotic rotation vector canonical angular momentum J (Q[ϕ] ϕ B) (A.43) stationaly bifurcation (n 2)-surface Σ Σ Killing ξ a = t a + Ω (µ) H ϕa (µ) (A.44) δφ δe = 0 δj = dδq (A.45) L ξ φ = 0 δθ(φ, L ξ φ) = L ξ Θ(φ, δφ) (A.46) Θ(φ, L ξ φ) = φl ξ φ δθ(φ, L ξ φ) = δφ L ξ φ +φ δl ξ φ = L ξ Θ(φ, δφ) (A.47) }{{}}{{} =0 =L ξ δφ Θ (A.45), (A.46) (A.37) dδq = d(ξ Θ) (A.48) C Σ δq = (δq[ξ] ξ Θ) Σ = δe Ω (µ) H δj (µ) (A.49) 84

86 Theorem S S = 2π X cd ɛ cd. Σ ɛ Σ binormal Σ volume element ( ) (A.31) δq = Σ Σ κ δs = δe Ω(µ) H 2π δj (µ) (δw c (φ)ξ c + δx cd [c ξ d] + δy (φ, L ξ φ) + dδz(φ, ξ)) (A.50) (A.51) (A.52) 1 Σ ξ = 0 4 Σ 3 δy (φ, L ξ φ) = Y (φ, L ξ δφ) = L ξ Y (φ, δφ) = ξ dy + d(ξ Y ) Σ δq = δx cd [c ξ d] Σ Σ Σ c ξ d = κɛ cd (A.53) (A.54) (A.55) Σ ξ = 0 Σ t t c c ξ d = 0 Killing t c c ξ d = t c d ξ c = 0 (A.56) c ξ d Σ Σ α c ξ d = αɛ cd (A.57) 2 α α = κ (A.55) (A.55) κ 2π δs = δq = δe Ω (µ) H δj (µ) (A.58) Σ (A.51) S X cd ER abcd (A.32), (A.6) S = 2π = 2π Σ Σ d n 2 x her abcd ɛ ab ɛ cd d n 2 x ( 1 δi h g δr abcd ) ɛ ab ɛ cd (A.59) I = L Wald Σ bifurcation surface stationaly 85

87 B Cardy Cardy [47, 48, 49] Cardy ln ρ(, ) 2π ce c e (B.1), : L 0, L 0 ρ(, ) :, c e = c 24 0, c e = c 24 0 c, c : central charge 0 = min, 0 = min Z(τ, τ) = Tr e 2πiτ(L 0 0 ) e 2πi τ( L 0 0 ) =, ρ(, )e 2πiτ( 0) e 2πi τ( 0 ) (B.2) ρ(, ) Z 0 (τ, τ) = Tr e 2πiτ(L 0 c 24 ) e 2πi τ( L 0 c 24 ) (B.3) Z 0 CFT CFT H = L 0 + L 0 (c + c)/24 P = L 0 L 0 Z 0 Z 0 (τ, τ) = Tr e 2πiτ 1P 2πτ 2 H (B.4) τ = τ 1 + iτ 2 B.1 CFT CFT modular τ τ + 1, τ 1 τ (B.5) τ τ + 1 B.2 τ 1 τ B.3 Z 0 modular Z modular [ ( c ) ] [ ( ) ] c Z(τ, τ) = exp 2πi τ exp 2πi τ Z 0 ( 1/τ, 1/ τ) = exp [ 2πi 24 0 ( c 0) ( 24 τ + 1 τ )] exp [ 2πi ( c 24 0 ) ( τ + 1 τ )] Z( 1/τ, 1/ τ) (B.6)

88 B.1: B.2: τ τ + 1 B.3: τ 1 τ 87

89 q e 2πiτ ρ(, ) (B.2) ρ(, ) = 1 dq d q (2πi) 2 q Z(q, q) (B.7) 0+1 q 0 +1 ( ) ρ( ) = dτe 2πi( 0)τ exp [ ( c 2πi 0) ( 24 τ + 1 τ )] Z( 1/τ) (B.8) c, 0 ( d ( c 2πi( 0 )τ + 2πi 0) ( dτ 24 τ + 1 )) = 0 τ ce τ i 24 (B.9) τ i c e /24 i τ = iɛ Z( 1/τ) ɛ 1 Z(i/ɛ) = ρ( )e 2π( 0)/ɛ ρ( 0 ) (B.10) Z( 1/τ) const ( ) c e ρ( ) exp 2π Z(i ) (B.11) 6 τ ln ρ(, ) 2π ce c e (B.12) 88

90 C C.1 Weyl 2 2 S = 1 d 2 x g( X) 2, 2 S = 1 d 2 x ge 2φ ( X) 2 2 (C.1.1) (C.1.2) (C.1.2) Lagrangian dilaton e 2φ dimensional reduction Euclidean (C.1.1), (C.1.2) (C.1.1) (C.1.2) g µν e 2α(x) g µν, X X, (C.1.3) g µν e 2α(x) g µν, X X, φ φ (C.1.4) Weyl Weyl 0 = δs[φ, g] = 1 2 d 2 x g T µν δg µν = d 2 x g T µν g µν α(x) (C.1.5) T µ µ = 0 Weyl 0 Weyl (C.1.1) T µ µ = 1 24π R, (C.1.2) (C.1.6) T µ µ = 1 [ R ( f(φ)) 2 ] (C.1.7) 24π f(φ) φ (C.1.2) φ = 0 (C.1.1) (C.1.1) (C.1.2) 89

91 C.2 (dilaton ) (C.1.1) S = 1 d 2 x g( X) 2 2 (C.2.1) Fujikawa method [50, 51] ( Z[g] = DX exp 1 d 2 x ) g( X) 2 (C.2.2) 2 Weyl δz[g] = Z[e 2α g] Z[g] = DX e S[X,g ] Z[g] = DXJe S[X,g] Z[g] = DX(J 1)e S[X,g] DX ln Je S[X,g] (C.2.3) J 2 3 Weyl 0 J J (f, g) d 2 x gf g (C.2.4) δx 2 = (δx, δx) (C.2.5) {ϕ n } ((ϕ n, ϕ m ) = δ nm ) δx = n δc nϕ n δx 2 = δc n 2 (C.2.6) n DX = n dc n (C.2.7) Weyl (C.1.3) X DX Weyl X Weyl (C.2.4) Weyl {ϕ n } Weyl Weyl {ϕ n} δ nm = d 2 x g ϕ n ϕ m = d 2 x ge 2α ϕ n ϕ m (C.2.8) 90

92 δ nm = d 2 x g ϕ nϕ m (C.2.9) ϕ n ϕ n ϕ n = e α ϕ n = (1 + α)ϕ n (C.2.10) X ϕ n ϕ n X = n c n ϕ n = n c nϕ n = n = n c n(1 + α)ϕ n (c n + αc n )ϕ n ( c n = c n + O(α)) (C.2.11) n c nϕ n = n c n (1 α)ϕ n c n c n c n = m (ϕ m, (1 α)ϕ m )c m = m (δ nm α nm )c m (C.2.12) α nm (ϕ n, αϕ m ) DX = dc n = dc n det(δ nm α nm ) n n = DX exp( α nn ) n (C.2.13) det(1 + ε) exp(tr ε) (C.2.14) ε (C.2.13) J ln J = α nn = d 2 x g α(x)ϕ nϕ n (C.2.15) n n n ϕ nϕ n = δ(0) ϕ nϕ n n n ϕ n exp( /M 2 )ϕ n (C.2.16) M M ln J = d 2 x g α(x)ϕ n exp( /M 2 )ϕ n n = d 2 x (C.2.17) g α(x) x exp( /M 2 ) x 91

93 x exp( /M 2 ) x comformal gauge g µν = e 2σ δ µν = e 2σ 2 (C.2.18) (C.2.19) 2 = δ µν µ ν δ µν contraction ϕ k = e σ e ikx d 2 x g ϕ k ϕ k = d 2 x e 2σ e σ e ikx e σ e ikx = (2π) 2 δ(k k ) (C.2.20) (C.2.21) x exp( /M 2 ) x d 2 k = (2π) 2 e σ e ikx exp( /M 2 )e σ e ikx = e 2σ d 2 [ ] k 1 (2π) 2 exp M 2 eσ e ikx e 2σ 2 e σ e ikx ( B 1 e A B = e B 1AB (A, B : )) = e 2σ d 2 [ ] k 1 (2π) 2 exp M 2 e ikx e σ 2 e σ e ikx (C.2.22) 2, fg = ( f)g (C.2.23) fg = ( f)g + f( g) + fg (C.2.24) µ e f(x) = e f(x) ( µ + µ f(x)) (C.2.22) e 2σ d 2 [ k 1 (2π) 2 exp M 2 e σ ( ] µ + ik µ ) 2 e σ = e 2σ d 2 [ k 1 (2π) 2 exp M 2 e σ ( k 2 + 2ik µ µ + ] 2 )e σ = e 2σ M 2 d 2 [ k (2π) 2 exp e σ ( k 2 + 2i M k µ µ + 1 ] 2 M 2 )e σ (C.2.25) (C.2.26) 92

94 k/m k H 0 k 2 e 2σ ( 2i H I e σ M k µ µ + 1 ) 2 M 2 e σ x exp( /M 2 ) x = e 2σ M 2 { 1 e H 0+H I = e H dth I (t) + 0 (H I (t) e H 0t H I e H 0t ) 1 0 dth I (t) d 2 k (2π) 2 eh 0+H I t 0 } dt H I (t ) (C.2.27) (C.2.28) (C.2.29) e H 0+H I M 2 H I H I = 2i M e 2σ k µ ( µ µ σ) + 1 M 2 e 2σ ( 2 2 µ σ µ 2 σ + ( σ) 2 ) H I (t) H I (t) = 2i M e 2σ k µ ( µ µ σ + 2tk 2 µ σe 2σ ) + 1 M 2 e 2σ (( + 2tk 2 µ σe 2σ ) 2 2 µ σ( µ + 2tk 2 µ σe 2σ ) 2 σ + ( σ) 2 ) = 2i M e 2σ k µ ( µ µ σ + 2tk 2 µ σe 2σ ) + 1 M 2 e 2σ [ 2 + 4tk 2 µ σe 2σ µ 2 µ σ µ + 2tk 2 ( 2 σ 4( σ) 2 )e 2σ + 4t 2 (k 2 ) 2 ( σ) 2 e 4σ 2 σ + ( σ) 2 ] (C.2.30) (C.2.31) 1 0 dth I (t) k = 1 M 2 e 2σ [ 2 + 2k 2 µ σe 2σ µ 2 µ σ µ + k 2 ( 2 σ 4( σ) 2 )e 2σ (k2 ) 2 ( σ) 2 e 4σ 2 σ + ( σ) 2 ] (C.2.32) k (C.2.28) k = dth I (t) t 0 dt H I (t ) dt 4 M 2 k µk ν e 2σ t( µ µ σ + 2tk 2 µ σe 2σ ) e 2σ ( µ µ σ + tk 2 µ σe 2σ ) 93 (C.2.33)