Black holes: From GR to HS V.E. Didenko Lebedev Institute, Moscow - PowerPoint PPT Presentation

Black holes: From GR to HS V.E. Didenko Lebedev Institute, Moscow Vienna, April 19, 2012

Plan • Introduction Unfolded dynamics. • GR black holes in d = 4 , 5 Algebraic facts. Unfolding formulation. • Generalization to higher-spins Strategy. Exact solution in d = 4 . Symmetries. • Conclusion 2

Unfolding of pure gravity Einstein equations R ab = 0 ⇔ R ab,cd = C ab,cd Riemann=Weyl Cartan equations dω ab + ω ac ∧ ω cb = C ac , bd e c ∧ e d , De a ≡ de a + ω ab ∧ e b = 0 . Unfolding.. DC ab,cd = C ab,cd | f e f : × = + + Bianchi identities: e b ∧ e d DC ab,cd = 0 ⇒ + = 0 DC ab , cd = ( 2C abf , cd + C abc , df + C abd , cf ) e f 3

DC abc,de = C abc,de | f e f ⇒ C abc,de | f ∼ Bianchi identities Gravity unfolded module C ...,... : , , , , . . . Unfolded equations DC a 1 ...a k +2 ,b 1 b 2 = (( k +2) C a 1 ...a k +2 c,b 1 b 2 + C a 1 ...a k +2 b 1 ,b 2 c + C a 1 ...a k +2 b 2 ,b 1 c ) e c Second Bianchi identity [ D, D ] ∼ C ab,cd ⇒ nonlinear corrections DC a 1 a 2 a 3 ,b 1 b 2 = (3 C a 1 a 2 a 3 c,b 1 b 2 + C a 1 a 2 a 3 b 1 ,b 2 c + C a 1 a 2 a 3 b 2 ,b 1 c + ( Weyl ) 2 ) e c schematically DC = F a ( C ) e a solved up to O ( C 2 ) in d = 4 , M.A. Vasiliev, ’89 4

Black holes in GR. • At least two isometries (any d ) • d = 4 Weyl tensor is of Petrov type D C ± ab,cd ∼ (Φ ± Φ ± ) ab,cd , Φ ab = − Φ ba . • d = 3 BTZ black hole is completely determined by an AdS 3 single isometry ξ BTZ = AdS 3 / exp tξ Type of BH is classified by inequivalent ξ with respect to AdS 3 adjoint action ξ → MξM − 1 5

• Hidden symmetries (Killing-Yano tensor Φ ab ) D Φ ab = v a e b − v b e a , Φ ab = − Φ ba General decomposition: D Φ ab = Φ ab | c e c Φ ab | c → × = + + Absence of and is a manifestation of the hidden symmetry Φ ab entails (on-shell): v a – Killing vector, K ( ab ) = Φ ac Φ cb – Killing tensor, K ab v b –Killing vector. 6

Φ ab in AdS (Minkowski) space-time AdS : dω ab + ω ac ∧ ω cb = Λ e a ∧ e b , d e a + ω ab ∧ e b = 0 embedding - zero curvature representation √ dW AB + W AC ∧ W CB = 0 W AB = ( ω ab , Λ e a ) ⇒ Global symmetries δW AB = D 0 ξ AB ≡ dξ AB + W AC ξ CB + W BC ξ AC = 0 , ξ AB → ( Φ ab , v a ) Dv a = − ΛΦ ab e b D Φ ab = v a e b − v b e a ξ AB – AdS global symmetry parameter 7

AdS Black holes from AdS global symmetry parameter • d = 3 BTZ black hole from a single AdS 3 isometry factorization (M. Henneaux+BTZ, ’93) • d=4 AdS − Kerr from an AdS isometry µ -deformation (V.D, A.S. Matveev, M.A. Vasiliev) d=5 (V.D) Dv a = − ΛΦ ab e b + F ( µ, Φ ab , e a ) D Φ ab = v a e b − v b e a ∂ Integrating flow ∂µ links AdS to BH g BH mn = g AdS mn + f mn ( ξ AB ) BH mass m and angular momenta a i are encoded in Casimir invari- ants Tr ( ξ n AB ). ♯ ( m, a i ) =rank O(d-1,2) 8

d=4 Kerr-NUT Black hole 9

Spinor form for AdS 4 equations: α = 1 α Φ γα + 1 γ ¯ 2 e γ ˙ 2 e α ˙ DV α ˙ Φ ˙ α ˙ γ D Φ αα = λ 2 e α ˙ γ V α ˙ α = λ 2 e γ ˙ D ¯ γ , Φ ˙ α V γ ˙ α . α ˙ 1. AdS 4 covariant form   � �  λ − 1 Φ αβ Ω αβ − λ e α ˙ V α ˙ β β  , K AB = K BA = Ω AB = Ω BA = = λ − 1 ¯ ¯ − λ e β ˙ Ω ˙ V β ˙ Φ ˙ α α ˙ α α ˙ β β 0 ∼ R 0 AB = d Ω AB + 1 2Ω AC ∧ Ω CB = 0 . D 2 D 0 K AB = 0 , K AB – AdS 4 global symmetry parameter 2. Two first integrals related to two AdS 4 invariants (Casimir operators) C 2 = 1 C 4 = 1 4 K AB K AB = I 1 , 4 Tr K 4 = I 2 1 + λ 2 I 2 10

Deformation of AdS 4 → black hole unfolded system (Keep the same form of the unfolded equations) α = 1 α Φ γα + 1 γ ¯ 2 ρ e γ ˙ ρ e α ˙ D V α ˙ 2¯ Φ ˙ γ , α ˙ D Φ αα = e α ˙ γ V α ˙ γ , α = e γ ˙ D ¯ Φ ˙ α V γ ˙ α . α ˙ Unlike the AdS 4 case with ρ = − λ 2 we assume ρ to be arbitrary Bianchi identities: D 2 ∼ R , D R = 0 fix ρ uniquely in the form G ) = M G 3 − λ 2 − q ¯ GG 3 , 1 ρ ( G, ¯ √ det Φ αβ G = 11

Integrating flow and solution space ∂ The flow ∂χ , where χ = ( M , q ) [ d, ∂ ∂χ ] = 0 allows one to express BH fields in terms of AdS 4 global symmetry pa- rameter K AB . This identifies the solution space • Generic K AB , M -complex – Carter-Plebanski class of metrics. Parameters: Re M , Im M , C 2 , C 4 , q , Λ • Kerr-Newman, C 2 > 0 , M > 0 Parameters: M, a ( C 4 ) , q B = − δ AB – Schwarzschild ( V 2 < 0), Taub-NUT ( V 2 > 0) • K 2 A 12

d = 5 black holes metric (Hawking-Hunter) dt − a sin 2 θ dψ � 2 + ∆ θ sin 2 θ dφ − b cos 2 θ adt − r 2 + a 2 dφ � 2 + ∆ θ cos 2 θ bdt − r 2 + b 2 ds 2 = − ∆ � � � dψ � 2 + ρ 2 ρ 2 ρ 2 Ξ a Ξ b Ξ a Ξ b abdt − b ( r 2 + a 2 ) sin 2 θ dφ − a ( r 2 + b 2 ) cos 2 θ + ρ 2 ∆ dr 2 + ρ 2 dθ 2 + (1 − Λ r 2 ) � dψ � 2 r 2 ρ 2 ∆ θ Ξ a Ξ b where ∆ = 1 r 2 ( r 2 + a 2 )( r 2 + b 2 )(1 − Λ r 2 ) − 2 M , ∆ θ = 1 + Λ a 2 cos 2 θ + Λ b 2 sin 2 θ ρ 2 = r 2 + a 2 cos 2 θ + b 2 sin 2 θ , Ξ a = 1 + Λ a 2 , Ξ b = 1 + Λ b 2 Horizon: ∆( r + ) = 0 13

Black hole unfolded system AdS 5 unfolded equations Dv αβ = − Λ D Φ αβ = 1 D 2 ∼ R ads 8(Φ αγ e γβ − Φ βγ e γα ) , 2( v αγ e γβ + v βγ e γα ) consistency µ -deformation → Dv αβ = − Λ 8(Φ αγ e γβ − Φ βγ e γα ) + µ H ( Φ − 1 α γ e γβ − Φ − 1 γ e γα ) , β D Φ αβ = 1 � 2( v αγ e γβ + v βγ e γα ) , H = det Φ αβ D 2 ∼ R ads + C BH C αβγδ = − 32 µ H 3 ((Φ − 1 ) αβ (Φ − 1 ) γδ + (Φ − 1 ) αγ (Φ − 1 ) βδ + (Φ − 1 ) αδ (Φ − 1 ) βγ ) 14

Black holes Φ αβ = 1 v αβ = v 0 2( v αγ x γβ + v βγ x γα ) + Φ 0 Φ 0 αβ = const , αβ , αβ = const Lorentz generator Φ 0 P 2 I 1 I 2 Type Killing vector αβ a Γ xy b 2 + a 2 ∂ αβ + b Γ zu − 1 Kerr 2 ab ∂t αβ a Γ xy ∂t + ∂ ∂ a 2 αβ + b Γ zu light-like Kerr 0 2 ab ∂x αβ a Γ ty a 2 − b 2 ∂ αβ + b Γ zu tachyonic Kerr +1 2 ab ∂x αβ Classification of Kerr-Schild solutions on 5d Minkowski space according to its Poincare invariants g mn = η mn + 2 µ H k m k n 15

Projectors and Kerr-Schild vectors Projectors: αβ = 1 Π ± 2( ǫ αβ ± X αβ ) X αγ X γβ = δ αβ ⇒ X αβ = X βα , α γ Π ± γβ = Π ± α γ Π ∓ Π + αβ = − Π − Π ± Π ± αβ , γβ = 0 , βα . Light-like vectors: βδ v γδ ⇒ v + v + = v − v − = 0 βδ v γδ , v + αγ Π + v − αγ Π − αβ = Π + αβ = Π − Specify X αβ : X αβ = 1 r 2 = 1 ( Φ αβ + HΦ − 1 αβ ) , 2( H − Q ) 2 r 16

Kerr-Schild vectors : v + v − v + v − = 1 αβ αβ 4 v + αβ v − αβ k αβ = v + v − , n αβ = v + v − , k a v a = n a v a = 1 , k a k a = n a n a = 0 geodetic condition: k a D a k b = n a D a n b = 0 Kerr-Schild vectors and massless fields φ a 1 ...a s = 1 Hk a 1 . . . k a s b = − Λ � φ a 1 ...a s − sD b D ( a 1 φ a 2 ...a s ) 2( s − 1 )( s + 2 ) φ a 1 ...a s Black holes mn + 2 µ g mn = g 0 H k m k n 17

Towards higher-spin BH • d=4 – Static BPS HS black hole (V.D, M.A. Vasiliev, 2009) • d=4 – D-type class of solutions (C. Iazeolla, P. Sundell, 2011) • d=3 HS asymptotic symmetries (M. Henneaux, S-J. Rey, ⊕ A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen, 2010) • d=3 – Static sl (3) ⊕ sl (3) black hole (M. Gutperle, P. Kraus, 2011) sl ( N ) ⊕ sl ( N ), hs ( λ ) ⊕ hs ( λ ) – great deal of interest 18

GR SUGRA HS → → black holes black holes black holes ??? Obstacles: 1. HS does not have decoupled spin-2 sector → all higher spins involved in the equations of motion. 2. The interval ds 2 = g µν dx µ dx ν is not gauge invariant quantity in higher spin algebra. Perturbative analysis available HS 0-th order 1-st order free field 2-nd order → → → theory vacua AdS Fronsdal equations interactions → . . . Program for HS black holes vacuum black hole HS → → → . . . AdS 4 massless fields φ µ 1 ...µ s ( x ) corrections 19

Kerr-Schild fields from free HS theory 20

• Free HS equations y | x ) = � ∞ α ( m ) y α . . . y α ¯ α . . . ¯ 1 HS field strengths y ˙ y ˙ α C ( y, ¯ n ! m ! C α ( n ) , ˙ n,m =0 ∞ � α ( m ) y α . . . y α ¯ α . . . ¯ 1 HS potentials y ˙ y ˙ α w ( y, ¯ y | x ) = n ! m ! w α ( n ) , ˙ n,m =0 Equations of motion: ˜ D 0 C ≡ dC − w 0 ⋆ C + C ⋆ ˜ ← twisted-adjoint w 0 = 0 D 0 w ≡ dw − [ w 0 , w ] ⋆ = R 1 ( C ) ← adjoint ˜ f ( y, ¯ y ) = f ( − y, ¯ y ) ← twist operator w 0 ( y , ¯ y | x ) − AdS 4 vacuum connection matter fields: scalar s = 0 → C ( x ) , fermion s = 1 / 2 → C α ( x ) ⊕ ¯ C ˙ α ( x ) HS fields: potentials → ω α ( s − 1) , ˙ α ( s − 1) , strengths → C α (2 s ) ⊕ ¯ C ˙ α (2 s ) 21

• Star-product operation Let Y A = ( y α , ¯ y ˙ α ) be commuting variables. � f ( Y + U ) g ( Y + V ) e U A V A dUdV − ( f ⋆ g )( Y ) = → associative algebra with [ Y A , Y B ] ⋆ = − 2 ǫ AB 22

• AdS 4 vacuum Introduce 1-form w 0 ∈ o (3 , 2) ∼ sp (4) w 0 = − 1 8( ω αα y α y α + ¯ α − 2 λ e α ˙ y ˙ y ˙ y ˙ α ¯ α y α ¯ α ) , ω ˙ α ¯ dw 0 − w 0 ⋆ ∧ w 0 = 0 α ˙ Equiv. to 2 ω αγ ∧ ω γα = λ 2 dω αα + 1 γ ∧ e α ˙ γ → AdS 4 Riemann tensor 2 e α ˙ α + 1 α + 1 2 ω αγ e γ ˙ α ˙ γ h α ˙ → d e α ˙ 2¯ ω ˙ γ = 0 zero torsion 23