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3 : :
4 Outline 4
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6 G MN T t g x mu... g µν = G MN X M x µ X N x ν X G MN c.f., : X 5, X 6,... g µν
7 : : g µν, A µ : R µν 1 2 Rg µν = 8πGT µν : G MN : R MN 1 2 RG MN = 0 ( ) Kaluza-Klein G MN = A 1 g µν. A 4 A 1 A 4 ϕ 2 G 55 = ϕ 2, G µ5 = A µ ( ϕ A µ )
8 Kaluza & Klein macroscopic large 3 - dim. space macroscopic L large 3 - dim. space & extra - compact space E c L L: X 5 ( )
9 : (Akama 82 Rubakov-Shaposhnikov 83) L 3 - dim. brane (Kaluza-Klein )
10 : Arkani-Hamed, Dimopoulos & Dvali 98) m 2 pl = Ln M n+2 D D = 4 + n, n: n 2 M D TeV L 0.1mm ( ) 3 - dim. brane L n : vol. of extra-dim TeV LHC ( )
11 : Randall & Sundram dim. brane L curved extra-dim. ( L)
12 : Kaluza-Klein
13
14 Black Hole
15 = Minkowski Observer's world-line Null Infinity "Time-like Infinity" Null Infinity " Star " An isolated object "STAR" Observer (null infinity)
16 = ( I + : null infinity ) singularity black hole event horizon null infinity observer's world-line =
17 ( H = Time
18 : (Hawking 71) A : A 0 Remark: 2 : S : S 0 (Bekenstein 73) Question: ( ) ()
19 : t
20 (Schwarzschild 16) () M Q : (Reissner 16 Nordstrom 18) (Regge & Wheeler 57 Kay & Wald 87) κ = 1/4M
21 ds 2 = (1 r H /r)dt 2 + Singularity Horizon t Null infinity dr 2 1 r H /r + r2 dω 2 : r H = 2M t a = ( / t) a t c c t a = κ(r)( / r) a r r H : t c c t a = κ(r H )t a κ = κ(r H ) = 1/4M:
22 (Kerr 63) M J : J < M 2 J 0 Q : (Newman 65) (Teukolsky 72 Whiting 89)
23 (Bardeen, Carter & Hawking 73) M J M + M J + J κ = const., M = κ 8π A + Ω H J A: Ω H :
24 κ = const., M = 1 8π κ A + Ω H J 0 1 T = const., E = T S P V : T = κ 2π (Hawking 75) : S = A 4
25 : (Israel-Carter-Robinson-Mazur-Bunting-Chrusciel) M Q J Black Hole Kerr
26 SIZE A B
27 : M J Q Black Holes Have No Hair Black Hole Kerr
28 10 20
29
30
31
32 Kerr
33 Kerr
34 In my entire scientific life... the most shattering experience has been the realization that an exact solution of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe Chandrasekhar ( ) Truth and Beauty (1987)
35 : 4 M J Q (Q = 0 Kerr ) 4?
36 ...!
37
38 macroscopic large 3 - dim. space macroscopic L large 3 - dim. space & extra - compact space
39 3 - dim. space L R horizon 3 - dim. space L R horizon
40 : (5) G MN dx M dx N = (4) g µν dx µ dx ν + dz dim. space L R horizon Z 5th - dim. L R 3 - dim. space 3 - dim. space Z =
41 (Gregory & Laflamme 93) R 5th- dim. L 3 - dim. space (Horowitz & Maeda 01) (Choptuik et al 03)
42 C.f. M ( ) : f ϕ = GM r n 2 + L2 /2M r 2 f n :
43 : (Tangherlini 63) (D 2)- (AI & Kodama 03) S p S D-2-p (D 2)- (Gibbons & Hartnoll 02)
44 (Myers & Perry 82) (D 2)- : J 1 J 2 etc
45 : Z- } X 2 {{ + Y } 2 +Z 2 = 1
46 : X } 2 {{ + Y } 2 +Z 2 = 1 : } X 2 {{ + Y } 2 + Z } 2 + {{ W } 2 =
47 D 6 ( C.f. J M 2 ) Ultra-spinning hole 0 = 1 + (J/M)2 r 2 GM r D 3
48 Horizon Area ( M fixed ) Horizon Area Horizon Area Extremal limit 4D Kerr hole J 2 ( M fixed ) ( M fixed ) Extremal limit (singular) 5D Myers-Perry hole J 2 6D Myers-Perry hole J 2
49
50 ! (Emparan & Reall 02) S 1 S 2 ( ) M J
51 Area ( same M ) Rotating Hole Thin Ring (M, J 1, J 2 = 0) Fat Ring 3 Black Objects J 2
52 Partial uniquness in D > 4 Uniquness results under additional assumptions Static holes: (Gibbons, Ida & Shiromizu 02) 5D rotating holes w/ Isom U(1) U(1): for rotating holes w/ S 3 topology (Morisawa-Ida 04) for rings w/ S 2 S 1 (Hollands & Yazadjiev 07 Morisawa-Tomizawa & Yasui 07) for holes in 5D minimal supergravity (Tomizawa-Yasui & AI 09)
53 !
54 Exact vacuum solutions in D = 5 Solutions akin to Emparan-Reall s ring (M, J 1 0, J 2 = 0) Black-ring w/ two angular momenta (M, J 1 0, J 2 0) (Pomeransky & Sen kov 06) Multi-black objects in vacuum gravity: Black di-rings ( ring + ring ) (Iguchi & Mishima 07) Black-Saturn ( hole + ring ) (Elvang & Figueras 07) Orthogonal-di-/Bicycling-Rings ( ring + ring ) (Izumi 07 Elvang & Rodriguez 07)
55 Introduction GR & Dimensions BH in 4D GR BHs in D > 4 石橋 明浩 Topology Stability Symmetry Summary
56 M = i ( ) 1 8π κ i A i + Ω i J i κ 1 κ 2
57
58
59 Black Hole D > 4 : (Galloway & Schoen 05 Galloway 07) Σ Σ Remarks: S D 2 ( ) S 1 S D 3 ( ) S m S n D = 5 S 1 S 2 S 3 S 3 /Γ
60
61 D 4 : g ab exp(+ωt) (Ω > 0) : g ab exp( iωt Ωt) (ω R) g ab g ab + g ab 4 (Regge & Wheeler 57 - Kay & Wald 87)
62 D 4 : ω 2 Φ = AΦ := ( 2 r 2 ) + U(r) Φ, Φ exp ( iωt) A 0 ω 2 0: ω U(r) > 0 A > 0 U(r)
63 A (AI & Kodama 03)
64 Ultra-spinning black hole in D 6 : Gregory-Laflamme? (Emparan & Myers 03) (Dias-et al 09) c.f. (Murata & Soda 08)
65 Gregory Laflamme 5th - dim. L R 3 - dim. space 3 - dim. space R 5th- dim. L 3 - dim. space
66 -
67 D 6 Area = Entropy Static-hole Rotating-ring Ultra-spinning Stable (AI-Kodama) J (probably) unstable (Emparan-Myers)
68
69 : t Killing t a i.e., a t b + b t a = 0
70 Killing Horizon : N Killing : Killing vector K a N K a N κ : N a (K b K b ) = 2κK a ( ) Remarks: Eq. ( ) K b b K a = κk a Event horizon () H
71 Killing horizon vs Event Horizon Multi extreme charged black holes in de Sitter space (Kastor-Traschen 93) ds 2 = 1 U 2 dt2 + e 2Ht U 2 dx 2, U := i ( ) M i 1 + e Ht x x i The event horizon of two extremal black holes in de Sitter space non-stationary is not a Killing horizon
72 : (Hawking 73) Remarks: Einstein
73 : (1) Killing (2) Black Hole Black Hole
74 D = 4 Hawking Σ 2 :
75 H Killing field K a K Σ t S K a K a = 0 and t K a = 0 on H K g ab = 0 on H α = const. (K c c K a = αk a ) on H Σ α Σ K a t a : Killing field : α Σ
76 K a Σ S a D a Ψ(x) = J(x) S a t a S S a Σ
77 4 Hawking 73 fixed point 4, Σ 2 S a Σ S a κ κ(x) = 1 P P 0 κ = const. α[ϕ s (x)]ds
78 D > 4 S a Σ S a 5D Myers-Perry 2-rotations Ω (1), Ω (2) : Σ S 3, t a = K a + S a S a = Ω (1) φ a (1) + Ω (2)φ a (2) Killing vector φ a Ω (1) /Ω (2) S a
79 : (Hollands, AI & Wald 07 09) S a Ergodic theorem guarantees the limit κ(x) exists κ 1 T (x) := lim α(x)ds T T 0 Einstein s equations yields κ (x) is a constant, κ so that 1 T lim α(x)ds = κ = T T 0 1 α(x)dσ Area(Σ) Σ time-average space-average
80 D 4 Remarks: U(1) U(1) N U(1) N
81 : : Kerr : ( )
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26 th July 2016 / 1991(M1)-1995(D3), 2005( ) 26 th July 2016 / 1. 2. 3. 4. . ( ) 1960-70 1963 Kerr 1965 BH Penrose 1967 Hawking BH Israel 1971 (Carter)-75(Robinson) BH 1972 BH theorem(,, ) Hawk 1975 Hawking
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