Hidden Symmetries of Charged Kerr Black Hole Tsuyoshi Houri Osaka - PowerPoint PPT Presentation

高次元 BH 研究会 , 京都大学 , 2009 年 12 月 25 日 . Hidden Symmetries of Charged Kerr Black Hole Tsuyoshi Houri （ Osaka City University ） * collaboration with D.Kubizˇ n´ ak, C.M.Warnick (DAMTP) and Y.Yasui (OCU) * in preparation 1

Motivation • String theory implies the existence of extra dimensions and motivates us to study a gravity in a higher-dimensional framework. • There is gravity/gauge duality which is one of the most exciting ideas in particle physics. ( d + 1)-dim. gravitaional theory ⇔ d -dim. gauge theory • Understanding in higher-dimensional framework might give us further understanding in 4-dimension. Black hole solutions provide important and useful gravitational back- grounds for these purposes, since black holes possess properties such as entropy and a singularity that fundamental physics aims to address. 2

Black hole metrics in a vacuum • 4-dimensional black hole metric mass a.m. NUT Λ ○ Schwarzschild (1915) ○ ○ Kerr (1963) ○ ○ ○ Carter (1968) ○ ○ ○ ○ Plebanski (1975) • Higher-dimensional ( D ≥ 4) black hole metric mass a.m.s NUTs Λ ○ Tangherlini (1963) ○ ○ Myers, Perry (1986) ○ ○ ○ Gibbons, L¨ u, Page, Pope (2004) ○ ○ ○ ○ Chen, L¨ u, Pope (2006) 3

Kerr-NUT-AdS metric in D -dimension The most general known solution (Chen-L¨ u-Pope metric) is called Kerr- NUT-AdS metric, which is given by 2 2     dx 2 n − 1 n n n µ A ( k ) � � � � A ( k ) dψ k g = + Q µ µ dψ k + εS     Q µ µ =1 µ =1 k =0 k =0 in D = 2 n + ε dimension, where ε = 0 for even dimensions and ε = 1 for odd dimesions. Here the functions are n + ε ( − 1) k c Q µ = X µ � � c k x 2 k + b µ x 1 − ε ( x 2 µ − x 2 , U µ = ν ) , X µ = , µ x 2 U µ µ ν � = µ k = ε � � A ( k ) = = A (0) = 1 , A ( k ) x 2 ν 1 · · · x 2 x 2 ν 1 · · · x 2 A (0) = ν k , ν k , µ µ 1 ≤ ν 1 < ··· <ν k ≤ n 1 ≤ ν 1 < ··· <ν k ≤ n ν i � = µ c S = A ( n ) , c = const. . This metric satisfies R ab = − ( D − 1) c n g ab in all dimesions. 4

Kerr metric (4-dimension) � △ − a 2 sin 2 θ � ds 2 =Σ △ dr 2 + Σ dθ 2 − dt 2 Σ − 4 Mar sin 2 θ � ( r 2 + a 2 ) 2 − △ a 2 sin 2 θ � sin 2 θdφ 2 dtdφ + Σ Σ where Σ = r 2 + a 2 sin 2 θ , △ = r 2 + a 2 − 2 Mr 5

Kerr metric (4-dimension) ds 2 = x 2 − y 2 dx 2 + y 2 − x 2 dy 2 X Y X Y x 2 − y 2 ( dψ 0 + y 2 dψ 1 ) 2 + y 2 − x 2 ( dψ 0 + x 2 dψ 1 ) 2 + where X = x 2 − a 2 − 2 Mx , Y = y 2 − a 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kerr-NUT metric (4-dimension) ds 2 = x 2 − y 2 dx 2 + y 2 − x 2 dy 2 X Y X Y x 2 − y 2 ( dψ 0 + y 2 dψ 1 ) 2 + y 2 − x 2 ( dψ 0 + x 2 dψ 1 ) 2 + where X = x 2 − a 2 − 2 Mx , Y = y 2 − a 2 − 2 Ly 6

Ansatz metric (4-dimension) ds 2 = x 2 − y 2 X ( x ) dx 2 + y 2 − x 2 Y ( y ) dy 2 X ( x ) Y ( y ) x 2 − y 2 ( dψ 0 + y 2 dψ 1 ) 2 + y 2 − x 2 ( dψ 0 + x 2 dψ 1 ) 2 + We can determine the functions X and Y by imposing Einstein condition R ab = − 3 c g ab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kerr-NUT-AdS metric (4-dimension) ds 2 = x 2 − y 2 dx 2 + y 2 − x 2 dy 2 X Y X Y x 2 − y 2 ( dψ 0 + y 2 dψ 1 ) 2 + y 2 − x 2 ( dψ 0 + x 2 dψ 1 ) 2 + where X = cx 4 + x 2 − a 2 − 2 Mx , Y = cy 4 + y 2 − a 2 − 2 Ly 7

Kerr-NUT-AdS metric (5-dimension) ds 2 = x 2 − y 2 dx 2 + y 2 − x 2 dy 2 X Y X Y x 2 − y 2 ( dψ 0 + y 2 dψ 1 ) 2 + y 2 − x 2 ( dψ 0 + x 2 dψ 1 ) 2 + c x 2 y 2 ( dψ 0 + ( x 2 + y 2 ) dψ 1 + x 2 y 2 dψ 2 ) 2 + X = c 4 x 4 + c 2 x 2 + c 0 + b 1 + c Y = c 4 y 4 + c 2 y 2 + c 0 + b 2 + c x 2 , y 2 8

Kerr-NUT-AdS metric (6-dimension) ds 2 =( x 2 − y 2 )( x 2 − z 2 ) dx 2 + ( y 2 − x 2 )( y 2 − z 2 ) dy 2 + ( z 2 − x 2 )( z 2 − y 2 ) dz 2 X Y Z X ( x 2 − y 2 )( x 2 − z 2 )( dψ 0 + ( y 2 + z 2 ) dψ 1 + y 2 z 2 dψ 2 ) 2 + Y ( y 2 − x 2 )( y 2 − z 2 )( dψ 0 + ( z 2 + x 2 ) dψ 1 + z 2 x 2 dψ 2 ) 2 + Z ( z 2 − x 2 )( z 2 − y 2 )( dψ 0 + ( x 2 + y 2 ) dψ 1 + x 2 y 2 dψ 2 ) 2 + where X = c 6 x 6 + c 4 x 4 + c 2 x 2 + c 0 + b 1 x , Y = c 6 y 6 + c 4 y 4 + c 2 y 2 + c 0 + b 2 y , Z = c 6 z 6 + c 4 z 4 + c 2 z 2 + c 0 + b 3 z 9

Kerr-NUT-AdS metric (7-dimension) ds 2 =( x 2 − y 2 )( x 2 − z 2 ) dx 2 + ( y 2 − x 2 )( y 2 − z 2 ) dy 2 + ( z 2 − x 2 )( z 2 − y 2 ) dz 2 X Y Z X ( x 2 − y 2 )( x 2 − z 2 )( dψ 0 + ( y 2 + z 2 ) dψ 1 + y 2 z 2 dψ 2 ) 2 + Y ( y 2 − x 2 )( y 2 − z 2 )( dψ 0 + ( z 2 + x 2 ) dψ 1 + z 2 x 2 dψ 2 ) 2 + Z ( z 2 − x 2 )( z 2 − y 2 )( dψ 0 + ( x 2 + y 2 ) dψ 1 + x 2 y 2 dψ 2 ) 2 + c x 2 y 2 z 2 ( dψ 0 + ( x 2 + y 2 + z 2 ) dψ 1 + ( x 2 y 2 + y 2 z 2 + x 2 z 2 ) dψ 2 + x 2 y 2 z 2 dψ 3 ) 2 + where X = c 6 x 6 + c 4 x 4 + c 2 x 2 + c 0 + b 1 − c x 2 , Y = c 6 y 6 + c 4 y 4 + c 2 y 2 + c 0 + b 2 − c y 2 , Z = c 6 z 6 + c 4 z 4 + c 2 z 2 + c 0 + b 3 − c z 2 10

We can assume the ansatz metric � n − 1 � 2 � � 2 n n n dx 2 � � � � µ A ( k ) A ( k ) dψ k g = + Q µ µ dψ k + εS Q µ µ =1 µ =1 k =0 k =0 in D = 2 n + ε dimension, where ε = 0 for even dimensions and ε = 1 for odd dimesions. Here the functions are Q µ = X µ � ( x 2 µ − x 2 U µ = ν ) , X µ = X µ ( x µ ) , , U µ ν � = µ � � A ( k ) = = A (0) = 1 , A ( k ) x 2 ν 1 · · · x 2 x 2 ν 1 · · · x 2 A (0) = ν k , ν k , µ µ 1 ≤ ν 1 < ··· <ν k ≤ n 1 ≤ ν 1 < ··· <ν k ≤ n ν i � = µ c S = c = const. . A ( n ) , Imposing Einstein condition R ab = λg ab , we can determine the form of the functioin X µ n + ε ( − 1) k c � c k x 2 k + b µ x 1 − ε X µ = . µ x 2 µ k = ε 11

Separabilities of Kerr-NUT-AdS spacetime in higher-dimensions It is known that the separation of variables for various field equations on Kerr-NUT-AdS background. • Geodesic equation Frolov-Krtous-Kubiznak-Page(2006) • Klein-Gordon equation Kubiznak-Krtous-Kubiznak(2006) • Dirac equation Oota-Yasui(2008), Wu(2009) • gravitational perturbation equation (tensor modes) Kundri-Lucietti-Reall(2006), Oota-Yasui(2008) • Maxwell equation ? 12

Killing vector Def. A generator of isometry of spacetime ξ , i.e., ∇ ( a ξ b ) = 0 ( L ξ g = 0 ) , is called Killing vector . If the orbit of Killing vector is closed, it generates axial symmetry. If not, it gener- ates translation symmetry. Conformal Killing vector Def. A generator of conformal symmetry of spacetime ξ , i.e., ∇ ( a ξ b ) = φ g ab ( L ξ g = 2 φ g ) , is called conformal Killing vector . 13

Geodesic integrability For geodesic Hamiltonian H = 1 2 g ab p a p b , E.O.M. gives geodesic equation p b ∇ b p a = 0 x a + Γ abc ˙ x b ˙ x c = 0 ) . ( ¨ We assume that a C.O.M. is written as C = K a 1 ...a n p a 1 · · · p a n . Then the condition { C, H } P = 0 leads to the equation ∇ ( b K a 1 ...a n ) = 0 . This equation is called Killing equation and K is called Killing tensor of rank-n . When n = 1, K is a Killing vector. Since Killing tensor gives C.O.M. along geodesic, geodesic equation is integrable if there are the dimension number of Killing vectors and Killing tensors totally. 14

Contents motivation solution admitting a closed conformal Killing-Yano tensor solution admitting a generalized closed conformal Killing-Yano tensor summary and discussion Killing vector conformal Killing vector symmetric Killing tensor conformal Killing tensor anti-symmetric Killing-Yano tensor conformal Killing-Yano tensor 15

Geodesic integrability of Kerr spacetime in 4-dimension Carter (1968) . . . There exists an nontrivial Killing tensor K , so there are four constants of motion. ξ = ∂ t , η = ∂ φ , g , K Penrose and Floyd (1973) . . . Killing tensor K is written as the square of rank-2 Killing-Yano tensor f . ∃ f K ab = f ac f bc , f ba = − f ab , ∇ ( a f b ) c = 0 s.t. KY equation Hughston and Sommers (1987) . . . Two Killing vectors, ξ and η , are also constructed from the Killing-Yano tensor f . ξ a = ∇ b ( ∗ f ) ba , η a = K ab ξ b ⇒ KY tensor is more fundamental. 16

Killing tensor Def. When a rank- n symmetric tensor K satisfies the equation ∇ ( b K a 1 ...a n ) = 0 , K is called Killing tensor . Killing-Yano tensor Def. When a rank- n anti-symmetric tensor f satisfies the equation ∇ ( b f a 1 ) a 2 ...a n = 0 , f is called Killing-Yano (KY) tensor . 17

Geodesic integrability of Kerr-NUT-AdS spacetime in D -dimension Page, Frolov, Kubizˇ n´ ak, Krtous and Vasdevan (2006) There exist n − 1 nontrivial Killing tensors K ( j ) in D -dimension, so there are the dimension number of constants of motion, which are mutually commuting. η ( j ) = ∂ φ i , K ( j ) η ( n ) ξ = ∂ t , g , and ( j = 1 , . . . n − 1) # Dimension # Killing vector # Killing tensor D = 2 n n n D = 2 n + 1 n + 1 n As the 4-dimension, Killing vectors and tensors, ξ , η ( j ) and K ( j ) , are constructed from rank-( D − 2 j ) Killing-Yano tensors f ( j ) . K ( j ) ab = f ( j ) a ··· f ( j ) b ··· , ξ a = ∇ b ( ∗ f (1) ) ba , η ( j ) a = K ( j ) ab ξ b 18