Black holes: From GR to HS

Transcript

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

2 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

3 Unfolding of pure gravity Einstein equations R ab = 0 R ab,cd = C ab,cd Riemann=Weyl Cartan equations dω ab + ω c a ω cb = C ac,bd e c e d, Unfolding.. De a de a + ω a b e b = 0. 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

4 DC abc,de = C abc,de f e f Bianchi identities C abc,de f Gravity unfolded module C...,... :,,,,... Unfolded equations DC a1...a k+2,b 1 b 2 = ((k+2)c a1...a k+2 c,b 1 b 2 +C a1...a k+2 b 1,b 2 c+c a1...a k+2 b 2,b 1 c)e c Second Bianchi identity [D, D] C ab,cd nonlinear corrections DC a1 a 2 a 3,b 1 b 2 = (3C a1 a 2 a 3 c,b 1 b 2 + C a1 a 2 a 3 b 1,b 2 c + C a1 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

5 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 BT Z = AdS 3 / exp tξ Type of BH is classified by inequivalent ξ with respect to AdS 3 adjoint action ξ MξM 1

6 Hidden symmetries (Killing-Yano tensor Φ ab ) DΦ ab = v a e b v b e a, Φ ab = Φ ba General decomposition: DΦ 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 a c Φ cb Killing tensor, K ab v b Killing vector.

7 Φ ab in AdS (Minkowski) space-time AdS : dω ab + ω a c ω cb = Λe a e b, de a + ω b a e b = 0 embedding - zero curvature representation W AB = (ω ab, Λe a ) dw AB + W C A W CB = 0 Global symmetries δw AB = D 0 ξ AB dξ AB + W C A ξ CB + W C B ξ AC = 0, ξ AB (Φ ab, v a ) Dv a = ΛΦ ab e b DΦ ab = v a e b v b e a ξ AB AdS global symmetry parameter

8 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 ) DΦ 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 invariants T r(ξ n AB ). (m, a i)=rank O(d-1,2)

9 d=4 Kerr-NUT Black hole

10 Spinor form for AdS 4 equations: DV α α = 1 2 eγ αφ γα e α γ Φ α γ DΦ αα = λ 2 e α γ V α γ, D Φ α α = λ 2 e γ αv γ α. 1. AdS 4 covariant form K AB = K BA = λ 1 Φ αβ V β α V α β λ 1 Φ α β, Ω AB = Ω BA = ( Ωαβ λe α β λe β α Ω α β ) = D 0 K AB = 0, D 2 0 R 0AB = dω AB Ω A C Ω CB = 0. K AB AdS 4 global symmetry parameter 2. Two first integrals related to two AdS 4 invariants (Casimir operators) C 2 = 1 4 K ABK AB = I 1, C 4 = 1 4 TrK4 = I λ2 I 2

11 Deformation of AdS 4 black hole unfolded system (Keep the same form of the unfolded equations) DV α α = 1 2 ρ eγ αφ γα ρ e α γ Φ α γ, DΦ αα = e α γ V α γ, D Φ α α = e γ αv γ α. Unlike the AdS 4 case with ρ = λ 2 we assume ρ to be arbitrary Bianchi identities: D 2 R, DR = 0 fix ρ uniquely in the form ρ(g, Ḡ) = MG 3 λ 2 q ḠG 3, G = 1 det Φαβ

12 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 parameter K AB. This identifies the solution space Generic K AB, M-complex Carter-Plebanski class of metrics. Parameters: ReM, ImM, C 2, C 4, q, Λ Kerr-Newman, C 2 > 0, M > 0 Parameters: M, a(c 4 ), q K 2 A B = δ A B Schwarzschild (V 2 < 0), Taub-NUT (V 2 > 0)

13 metric (Hawking-Hunter) ds 2 = ρ 2 ( dt a sin 2 θ Ξ a d = 5 black holes dφ b cos2 θ ) 2 θ sin 2 θ dψ + Ξ b + ρ2 dr2 + ρ2 dθ 2 + (1 Λr2 ) θ r 2 ρ 2 ρ 2 ( adt r 2 + a 2 Ξ a ( abdt b(r 2 + a 2 ) sin 2 θ Ξ a dφ ) 2 + θ cos 2 θ ρ 2 ( bdt r 2 + b 2 Ξ b dψ ) 2 + dφ a(r2 + b 2 ) cos 2 θ Ξ b dψ ) 2 where = 1 r 2(r2 + a 2 )(r 2 + b 2 )(1 Λr 2 ) 2M, θ = 1 + Λa 2 cos 2 θ + Λb 2 sin 2 θ ρ 2 = r 2 + a 2 cos 2 θ + b 2 sin 2 θ, Ξ a = 1 + Λa 2, Ξ b = 1 + Λb 2 Horizon: (r + ) = 0

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

15 Black holes v αβ = v 0 αβ = const, Φ αβ = 1 2 (v α γ x γβ + v β γ x γα ) + Φ 0 αβ, Φ0 αβ = const Type Killing vector Lorentz generator Φ 0 αβ P 2 I 1 I 2 Kerr t aγ xy αβ + bγzu αβ 1 b 2 + a 2 2ab light-like Kerr t + x aγ xy αβ + bγzu αβ 0 a 2 2ab tachyonic Kerr aγ ty αβ +1 a 2 b 2 2ab x αβ + bγzu Classification of Kerr-Schild solutions on 5d Minkowski space according to its Poincare invariants g mn = η mn + 2µ H k mk n

16 Projectors and Kerr-Schild vectors Projectors: Π ± αβ = 1 2 (ɛ αβ ± X αβ ) X αβ = X βα, X α γ X γ β = δ α β Light-like vectors: Π ± α γ Π ± γβ = Π± αβ, Π± α γ Π γβ = 0, Π+ αβ = Π βα. v + αβ = Π+ αγπ + βδ vγδ, v αβ = Π αγπ βδ vγδ v + v + = v v = 0 Specify X αβ : X 1 αβ = 2r (Φ αβ + HΦ 1 αβ ), r2 = 1 (H Q) 2

17 Kerr-Schild vectors : k αβ = v+ αβ v + v, n αβ = v αβ v + v, v+ v = 1 4 v+ αβ v αβ 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 a1...a s = 1 H k a 1... k as φ a1...a s sd b D (a1 φ a2...a s ) b = Λ 2 (s 1)(s + 2)φ a 1...a s Black holes g mn = g 0 mn + 2µ H k mk n

18 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

19 GR black holes SUGRA black holes HS 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 theory 0-th order vacua AdS 1-st order free field Fronsdal equations... 2-nd order interactions Program for HS black holes vacuum AdS 4 black hole massless fields φ µ1...µ s (x) HS corrections...

20 Kerr-Schild fields from free HS theory

21 Free HS equations HS field strengths HS potentials w(y, ȳ x) = C(y, ȳ x) = n,m=0 n,m=0 1 n!m! C α(n), α(m)y α... y α ȳ α... ȳ α 1 n!m! w α(n), α(m)y α... y α ȳ α... ȳ α Equations of motion: D 0 C dc w 0 C + C w 0 = 0 twisted-adjoint D 0 w dw [w 0, w] = R 1 (C) adjoint f(y, ȳ) = f( y, ȳ) twist operator w 0 (y, ȳ 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(2s) C α(2s)

22 Star-product operation Let Y A = (y α, ȳ α ) be commuting variables. (f g)(y) = associative algebra with f(y + U)g(Y + V)e U AV A dudv [Y A, Y B ] = 2ɛ AB

23 AdS 4 vacuum Introduce 1-form w 0 o(3, 2) sp(4) w 0 = 1 8 (ω ααy α y α + ω α α ȳ α ȳ α 2λe α α y α ȳ α ), dw 0 w 0 w 0 = 0 Equiv. to dω αα ω α γ ω γα = λ2 2 e α γ e α γ AdS 4 Riemann tensor de α α ω α γ e γ α ω α γ h α γ = 0 zero torsion

24 Kerr-Schild HS solution Fix AdS 4 global symmetry parameter K AB = K BA D 0 K AB = 0 D 0 K AB Y A Y B dk AB Y A Y B [w 0, K AB Y A Y B ] = 0 any function f(k AB Y A Y B ) is a HS global symmetry parameter D 0 f(k AB Y A Y B ) = 0 Solving for linearized C(y, ȳ x) in the curvature sector dc w 0 C + C w 0 = 0, C = 2πf( 1 2 K ABY A Y B ) δ (2) (y) K AB = K BA = λ 1 Φ αβ V β α V α β λ 1 Φ α β C(y, ȳ x) = d 2 uf( 1 2 Φ αβu α u β + V α α u α ȳ α Φ α βȳ α ȳ β ) exp(iuα y α )

25 Reality condition (C(y, ȳ x)) = C( y, ȳ x) Generic f( 1 2 K ABY A Y B ) does not meet reality condition for C Choose f in the form f = M exp 1 2 K ABY A Y B C = C K A C K C B = δa B V α α time-like, M = M Schwarzschild V α α space-like, M = M Taub-NUT

26 Higher-spin curvatures: C = M r exp (1 2 Φ 1 ααy α y α Φ 2 α αȳ α ȳ α iφ 1 αγ v γ αy α ȳ α ) C α(2n) = M r (Φ 1 αα) n, C α(2n) = M r ( Φ 1 α α )n Connections: φ µ1...µ n = M r k µ 1... k µn BH Fronsdal fields

27 Nonlinear HS equations w(y, ȳ x) W (y, ȳ, z, z x), C(y, ȳ x) B(y, ȳ, z, z x) Pure gauge compensator 1-form S(Z, Y x) = S α dz α + S α dz α Introducing A = d + W + S, HS nonlinear equations take the form A A = R(B, v, v), [B, A] = 0 Component form (bosonic eqs.) dw W W = 0, db W B + B W = 0, ds α [W, S α ] = 0, d S α [W, S α ] = 0, S α S α = 2(1 + B v), S α S α = 2(1 + B v), [S α, S α ] = 0, B S α + S α B = 0, B S α + S α B = 0, Dynamical potentials and field strengths: W (Y, Z x) Z=0, B(Y, Z x) Z=0

28 New ingredients (Y, Z) star-product: Let Y A = (y α, ȳ α ) and Z A = (z α, z α ) be commuting variables. (f g)(y, Z) = f(y + s, Z + s)g(y + t, Z t)e s At A dsdt associative algebra with [Z A, Z B ] = [Y A, Y B ] = 2ɛ AB, [Y A, Z B ] = 0 Klein operators v = exp (z α y α ), v = exp ( z α ȳ α ) v v = v v = 1, v f(y, z) = f( y, z) v, v f(ȳ, z) = f( ȳ, z)

29 Solving nonlinear HS equations Main idea: Function F K = exp ( 1 2 K ABY A Y B ) generates invariant subspace in the star-product algebra and provides suitable ansatz for solving nonlinear HS equations Properties of F K 1. F K F K = F K, Y A F K = F K Y +A = 0, Y ±A = Π ±A C Y C = 1 2 (δ A B ± K A B )Y B Fock vacuum projector 2. D 0 F K = 0 by definition 3. Generates subalgebra of the form F K φ(a x), where a A = Z A +K B A Y B (F K φ 1 (a x)) (F K φ 2 (a z)) = F K (φ 1 (a x) φ 2 (a x)) * - is Fock induced associative star-product operation on the space of a A - oscillators

30 * - properties 1. associativity (φ 1 φ 2 ) φ 3 = φ 1 (φ 2 φ 3 ) [a A, a B ] = 2ɛ AB 2. Admits Klein operators of the form K = 1 r exp (1 2 Φ 1 ααa α a α ), K = 1 r exp (1 2 Φ 1 α αā α ā α ) K K = K K = 1, {K, a α } = { K, ā α } = 0 3. Differential Q = ˆd 1 2 dkab 2 a A a B Q(f(a x) g(a x)) = Qf(a x) g(a x) + f(a x) Qg(a x), Q 2 = 0, Qa A = 0, QK = 0

31 The Ansatz B = MF K δ(y), S α = z α + F K σ α (a x), S α = z α + F K σ α (ā x), W = w 0 (y, ȳ x) + F K (ω(a x) + ω(ā x)), w 0 is the AdS 4 connection HS equations reduce to 3d massive equations : [s α, s β ] = 2ɛ αβ (1 + M K), Qs α [ω, s α ] = 0, Qω ω ω = 0, where s α a α + σ α (a x) - the so called deformed oscillators (Wigner). Note: 3d HS equations around the vacuum B 0 = ν = const were considered by Prokushkin and Vasiliev and were shown to provide massive field dynamics with the mass scale ν

32 Exact solution S α = z α + MF K a + α r 1 0 dt exp ( t 2 κ 1 ββ aβ a β ), S α = z α + MF K ā + α r 1 0 dt exp ( t 2 κ 1 β β ā βā β ), B = M r exp (1 2 κ 1 αβ yα y β κ 1 α β ȳ α ȳ β κ 1 αγ v γ αy α ȳ α ), W = W 0 +( M 8r F Kdτ αβ π + β α a α a α 1 0 dt(1 t) exp ( t 2 κ 1 ββ aβ a β )+F K f 0 +c.c.),

33 Symmetries Let ɛ(z, Y x) be a global symmetry parameter B ɛ ɛ B = 0, [S, ɛ] = 0, dɛ [W, ɛ] = 0 ɛ(y x) = m,n=1 f 0A(m),B(n) (x) Y A +... Y A + }{{} m [F K, ɛ] = 0 Y B... Y B +c }{{} 0 (x) =: f(y, Y + ) : +c 0, D 0 f = 0 n Max. finite dimensional subalgebra: T AB = Y (A + Y B), T = Y AY A + su(2) u(1) More (Supersymmetry)! Global SUSY is a quarter of the N = 2 SUSY with two supergenerators Q α A of the AdS 4 vacuum. Vacuum AdS 4 symmetry algebra osp(2, 4) is broken giving BPS HS black hole

34 Conclusion It is demonstrated that unfolded formulation allows to describe GR black holes in terms of AdS global symmetry parameter in coordinate invariant way. The construction admits natural generalization to higher-spins, resulting in HS Schwarzschild and Taub-NUT exact solutions. Open problems What is black about HS black hole? (Horizons, singularities, entropy, temperature) Black rings, are they within reach of the AdS global symmetry parameter?