XVIII Int. Conference Geometry, Integrability and Quantization, Varna, June 3-8, 2016

Transcription

1 XVIII Int. Conference Geometry, Integrability and Quantization, Varna, June 3-8, 2016

2 Everything should be made as simple as possible, but not simpler. Albert Einstein

3 Motivations : (1) Extra - dimensions and string theory (2) Brane - world models (3) Black holes as probes of extra dimensions (4) Micro BHs production in colliders? (5) Generic and non - generic properties of BHs

4 Two big "surprises" in study of HD black holes: 1. The topology of the horizon can be more complicated than the topology of the sphere D2 S. 5D exact solutions for stationary black rings (Emparan and Reall, 2002) plus many later publications. Stability of such solutions? 2. Properties of HD stationary black holes with spherical topology of the horizon are quite similar to the properties of their 4D "cousins" (Kerr metric): geodesic equations are completely inegrable and wave equations are completely separable (V.F. and Kubiznak, 2007, plus Alberta separatists' later publications)

5 5D vac. stationary black holes

6 In this talk I shall focus on the "second big surprize": Hidden Symmetries and Complete Integrability in HD BHs. 1. Brief history; 2. Liouville theory of the complete integrability; 3. Origin and properties of the hidden symmetries; 4. Killing tensors and Killing towers; 5. Applications and results; 6. Recent developments.

7 Remark: Whenever I tell a black hole, it means that I discuss an isolated higher dimensional stationary rotating black hole with the spherical topology of horizon in a ST, which asympotically is either flat or (Anti)DeSitter. Its metric is a solution of the Einstein equations in D 2 n dimensions R ab = g ab

8 Main Message: Properties of higher dimensional rotating BHS and the 4D Kerr metric are very similar. HDBHs give a new wide (infinite) class of completely integrable dynamicalsystems.

9 Higher Dimensional Black Holes Tangherlini '63 metric (HD Schw.analogue)... Myers &Perry '86 metric (HD Kerr analogue)... Kerr - NUT - AdS '06 (Chen, Lu, and Pope; The most general HD BH solution)

10 "General Kerr-NUT-AdS metrics in all dimensions, Chen, Lü and Pope, Class. Quant. Grav. 23, 5323 (2006). n D / 2, D 2n R ( D 1) g ab, M mass, a ( n 1 ) rotation parameters, M k ( n1 ) `NUT' parameters ab Total # of parameters is D

11 0 J1 0 J J n 0 J n 0

12 Stationary - Killing vector ξ t; Axisymmetric - (n ε) Killing vectors ξ ; When cosmological constant λ and NUT parameters vanish one has Myers - Perry metric (1986)

13 Generator of Symmetries Principal Closed Conformal KY Tensor 2 - form h with the following properties : 1 D-1 [ ] (i) Non - degenerate (maximal matrix rank, 2n) (ii) Closed dh 0 (iii) Conformal KY tensor : a b h ba, a h g - g, c ab ca b cb a is a primary Killing vector For briefness, we call this object a "Principal Tensor"

14 All Kerr-NUT-AdS metrics in any number of ST dimensions possess a PRINCIPAL TENSOR (V.F.&Kubiznak 07)

15 Uniqueness Theorem A solution of Einstein equations with the cosmological constant, which possesses a PRINCIPAL TENSOR is a Kerr-NUT-AdS metric (Houri,Oota&Yasui 07 09; Krtous, V.F..&Kubiznak 08;)

16 Geodesic equations in ST with a PRINCIPAL TENSOR are completely integrable in Liouville sense. Important field equations allow a complete separation of variables. ( Alberta separatists ) In the rest of the talk I shall explain why it happens.

17 BRIEF REMARKS ON COMPLETE INTEGRABILITY

18 Hamiltonian system Phase space: Differential manifold a symplectic form (non-denerate rank 2 closed, d 0). 2D Observables are scalar functions on M. X F -1 = df is a Hamiltonian vector field associated with an observable F. M 2D with dz The dynamical equations for the Hamiltonian H are XH. dt -1 Poisson bracket { F, Q} df dq [ X, X ]. Integral of motion F { F, H} 0. Integrals of motion FQ, are in involution if { FQ, } 0 F Q

19 Darboux's theorem: Suppose that is a simplectic 2 - form on an n 2 m dimensional manifold M. Then in a neighrhood of each point q on M, there is a coordinate chart U in which = d = dq dp... dq dp. 1 1 m U is called a Darboux chart. The manifold can be covered by such charts (Darboux atlas). m

20 Liouville theorem: Dynamical equations in 2D dimensional phase space are completely integrable if there exist D independent commuting integrals of motion.

21 General idea: D commuting independent integrals of motion Fi can be used as coordinates on the phase space. Denote by M the level set for { F }. Gradients df are linearly independent f i i M f 2D is D dimensional submanifold of M. The involution condition implies that X X 0, hence M is a Lagrangian submanifold. The vector fields X are f tangent to it and mutually commute. ij L F F F i j F i

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23 Denote = d and consider the function W( q, P). mis a point of the phase space, and qare its coordinates. Since d0on M, the integral does not depend on a choice f of the path. W( q, P) can be used as a generating function, which produces a canonical transformation from the original ( qp, ) to i i W( q, P) new canonical coordinates ( Q, Pi Fi): Q. In these P new coordinates the Hamilton's equations become trivial: i i H Pi { Pi, H} 0, Q { Q, H} const. P i i m m 0 M f

24 Lagrandgian submanifold ( q, p) ( q, p ) 0 0 q q 0

25 Relativistic Particle as a Dynamical System Preferable coordinates in the phase a ( x, p g x ), H( p, x) g pp. b 1 ab a ab 2 a b Monomial in momenta integrals of motion K= K a... b p... p imply that a b a... b K ( a... bc ; ) K space: is a Killing tensor: { K 1,K 2 } 0 [ K1, K2] 0. Motion of particle in D-dimensional ST is completely integrable if there exist D independent commuting Killing tensors (and vectors) 0

26 Metric is a best known example of rank 2 Killing tensor If K and K are 2 monomial integrals (n) (m) of order n and m, then K K is a monomial integral of order n m. The corresponding Killing tensor is called reducible. (n) (m)

27 Properties of 4D black holes KY KT ( f a b c 0) KT ( K ( ab; c) 0) ( ; ) Walker & Penrose '70 Penrose & Floyd '73 Carter 68 H-J separation 4 th integral of motion PCCKY ( h * f, h db) a h g g ( a b) c ab c c( a b) 1 3 h ; b ba a () t K a is primary KV ab is a SKV Hughston & Sommers '75 b Einstein space ( R g ) ab ab which admits PCCKY is either Kerr - NUT - AdS or C - metric

28 Parallel transport : Consider a geodesic, and let p b Let k be a Killing-Yano tensor: k 0. Then a vector q k p is ab a( b; c) a ab parallel propagated. Really, c b c b c p q k p p k p p 0. a; c ab; c ab ; c a be a tangent vector. A 2-plane, determined by a 2-form =*(p q) is also parallel propagated along a geodesic. In order to find 2 parallel propagated vectors, that span it it is sufficient to solve a simple ODE (Marck, 1983).

29 Tidal disruption of a white dwarf by a massive black hole «Relativistic tidal interaction of a white dwarf with a massive black hole» Frolov, V. P., Khokhlov, A. M., Novikov, I. D., & Pethick, C. J. Astrophysical Journal, vol. 432, p , (1994)

30 M m r b Condition of a static disruption: GMb r Gm b 3 2 1/3 r ( M / m) b

31 Higher dimensions Denote D 2 n. D - dimensional AdS - Kerr - NUT metric has n Killing vectors. For complete integrability of geodesic equations one needs n more integrals of motion.

32 GENERAL SCHEME

33 Forms (=AStensor) (1) External product: ( ) (2) Hodge dual: *( ) (* ) q p q p (3) External derivative: d( ) ( d) q Dq q1 (4) Closed form: d( ) 0 (locally d( )) q q q q1

34 Killing-Yano family Let be p - form on the Riemannian manifold. Then its Hodge dual * is ( D- p)- form. Two operations: derivative d and coderivative. d g (...) ( * d*) If (...) 0 is a conformal KY tensor; If (...) 0 is a KY tensor; If d (...) 0 is a closed conformal KY tensor.

35 Let f be a Killing - Yano tensor then... f...( ; ) 0, 1 2 n 1 2 n f n is a parallel propagated n p along a geodesic form; K f f... 2 n is the Killing tensor 2... n K p p is an integral of geodesic motion

36 Integrability conditions of the KY equation: p 1 e ab fc 1c2... c R p ea[ bc f 1 c2... cp] 2 Maximal number of independent KY tensor (D+1)! of rank p is (D-p)!(p+1)! Maximal number of independent CCKY tensor of rank p (D+1)! is (D-p+1)!p!

37 Example: Killing vector 0, R f (a;b) a; b; c abc f Maximum number of independent Killing vectors : 1 1 ND[ ] D D( D 1) D( D 1) 2 2

38 Flat sacetime (as well as (anti)desitter space) has the maximum number of Killing-Yano and closed conformal Killing-Yano tensors. The Killing-Yano n-forms in the flat spacetime allow quite simple description. Consider a set of n integer numbers a,..., a }, 1 a... a D 1 n 1 Translationally invariant Killing-Yano n-forms 1,..., n } 1 2 k dx dx... dx a a a a a Consider another set of n 1 integer numbers a 1 an 1,..., }, 1 a... a a D ˆ 1 n n 1 n n1 Rotational Killing-Yano n-forms k x dx dx a,..., a } [ a a a 1] n n Closed conformal Killing-Yano tensors are dual of the above objects.

39 Properties of CKY tensor Hodge dual of CKY tensor is CKY tensor Hodge dual of CCKY tensor is KY tensor; Hodge dual of KY tensor is CCKY tensor; External product of two CCKY tensors is a CCKY tensor (Krtous,Kubiznak,Page &V.F. '07; V.F. '07)

40 CKY CCKY * KY R2-KT

41 Principal Tensor External powers of CCKY tensor give new CCKY. That is why they are "better" as symmetry generators than KY tensors. For a 1-form one has =0. For higher than rank 1 forms their external powers are, generally, non-vanishing. The lower rank of the CCKY form, the more non-trivial new CCKY tensors it can generate. The rank 2 CCKY form is in this sence the most promicing. In order to generate the largest number of non-vanishing CCKY forms, a 2-form of the CCKY object must have the highest possible matrix rank (2 n). The Principal CCKY tensor (or simply the Principal Tensor) is a non-denerate 2-form which is CCKY object. (Some additional requirements will be added later.)

42 Killing-Yano Tower

43 Killing-Yano Tower CCKY : j h h h h... h j times KY tensors: k j * h j Killing tensors: j K k k j j Primary Killing vector: a 1 D1 b h ba Secondary Killin g vectors: j K j

44 Total number of conserved quantities: ( n ) ( n1) 1 2n D KV KT g The integrals of motion are functionally independent and in involution. The geodesic equations in the AdS-Kerr-NUT ST are completely integrable.

45 Canonical Coordinates ˆ h ( m ) ix m m e ie We include in the definition of the Principal Tensor h the following requirement: The 2-form h in the ST with D 2 n dimensions has n non-equal independent eigen-values x, so that there exists n (mutually orthogonal) two-planes.

46 Canonical coordinates: n essential coordinates x and n Killing coordinates. Total number of such "canonical" coordinates is D2 n. j g ( e e e e ) e e, h x e e ˆ ˆ n1 n1 ˆ The components of Killing tower objects in such a basis are polynomials in x.

47 Off-shell metrics, which admit the Principal Tensor, allow complete description ˆ ˆ n 1 n 1 ab a b a b a b g ( e e e e ) e e, n1, ˆ () i i0 1 e dx e Q A d Q i X Q U x x X X x 2 2,, ( ), ( ) U

48 n 1 (1 tx ) n 2 j ( j) t A j0 2 1 n 2 n1 k ( k) 1 k0 (1 tx ) (1 tx ) t A Arbitrary functions X ( x ) after substitution into Einstein equations become polynomials. Their coefficients are just papameters of the metric. Houri, Oota, and Yasui [PLB (2007); JP A41 (2008)] proved this result under additional assumptions: L g 0 and L h 0. Later Krtous, V.F., Kubiznak [arxiv: (2008)] and Houri, Oota, and Yasui [arxiv: (2008)] proved this without additional assumptions.

49 Solutions of the Einstein equations with the cosmological constant ( On-shell metrics) n 2k k0 X b x c x for D A similar expression for D even. odd. This is nothing but the Kerr-NUT-(A)dS metric, written in special (canonical) coordinates.

50 Intermediate Summary 1. The Principal Tensor (if it exists) generates the Killing tower, which roughly contains a half of the Killing vectors and a half of the Killing tensors. The Killing vectors are responcible for explicite symmetry of the spacetime, while the Killing tensors describe its hidden symmetries. 2. Eigen-values of the Principal Tensor together with the Killing parameters, determined by the Killing vectors provide one with special coordinates, in which the metric and the PT has "simple" canonical form. 3. Equations for the Principal Tensor and their integrability conditions, written in the canonical coordinates, allows one to find the (off-shell) metric. 4. After imposing the Einstein equations, this metric becomes Kerr-NUT-(A)dS solution.

51 Separation of variables This is a very special property, which depends both on the type of the equaion and special choice of the coordinates. Hamilton-Jacobi ( S) WKB ~ exp( is) m Klein-Gordon ( m ) 0 "Dirac eqn= KG eqn " Dirac equation ( m) 0

52 Separation of variables in HJ eqs For the Hamiltonian H ( P, Q), P p,, p, Q q,, q, 1 m 1 the Hamilton-Jacobi equation is H ( S, Q) 0. P m Suppose q and S enter this equation as ( S, q ). Then the variable S S ( q, C ) S '( q 1 q 1 q ( S, q ) C, 1 q q 1 q can be separated:,, q ), H ( S ',, q, ; C ) 0 m

53 Complete separation of variables: S S ( q, C ) S ( q, C, C ) S ( q, C,, C ) m m 1 m The constants C i generate first integrals on the phase space. When these integrals are independent and in involution the system is integrable in the Liouville sence.

54 Separation of variables in HJ and KG equations in 5D ST (V.F. and Stojkovic 03) Separability of the Hamilton Jacobi equation in canonical coordinates S ab g as bs 0 S w S ( x ) n m 1 k0 k k 1 1 ( S ( ( x ) ) ( x ) m m 2 2 n1 k 2 2 n1 k ') 2 k c k X k 0 X k0 V. F., P. Krtous, D. Kubiznak, JHEP 0702:005 (2007)

55 Separability of the Klein Gordon equation, 2 ik k R ( x ) e ( ) 0 1 k0 n m X R m m 2 n1 k 2 2 n1 k ( X R ) R ( ( x ) k) bk( x ) R 0 x X k0 k0 V. F., P. Krtous, D. Kubiznak, JHEP 0702:005 (2007)

56 Weakly charged higher dimensional rotating black holes 1 ab Hamiltonian H g ( pa qaa )( pb qab ) 2 S x S x 2 ab HJ equation g ( qaa)( qab) a b Klein-Gordon equation g iqa iqa ab 2 ( a a )( b b) 0

57 b a ab ab a b F 0 A 0 A 0 ab 0 A Q (in Ricci flat) b a a g ab S S 2 M 0 a b x x 2 [ M ] 0, M 2e e (0) For a primary Killing vector field one again has a complete separation of variables [V.F. and Krtous, 2011]

58 Complete separation of variables in KG and HJ eqns in Kerr-NUT-AdS ST (V.F.,Krtous&Kubiznak 07) Separation constants KG and HJ eqns and integrals of motion (Sergyeyev&Krtous 08) Separation of variables in Dirac eqns in Kerr-NUT- AdS metric (Oota&Yasui 08, Cariglia, Krtous&Kubiznak 11)

59 Separability of gravitational perturbations in Kerr-NUT-(A)dS Spacetime (Oota andyasui ' 10) Metrics admitting a principal Killing-Yano tensor with torsion (Houri, Kubiznak, Warnick and Yasui '12)

60 Parallel transport along timelike geodesics a Let u be a vector of velocity and h be a PCKYT. b b b a P u u is a projector to the plane orthogonal to u. a a a Denote F P P h h u u h h c d c c ab a b cd ab a cb ac b ab uu Lemma (Page): F ab u is parallel propagated along a geodesic: F ab 0 Proof: We use the definition of the PCKYT h = u u u ab a b a b

61 Suppose h ab eigen spaces of is a non-degenerate, then for a generic geodesic F ab with non-vanishing eigen values are two dimensional. These 2D eigen spaces are parallel propagated. Thus a problem reduces to finding a parallel propagated basis in 2D spaces. They can be obtained from initially chosen basis by 2D rotations. The ODE for the angle of rotation can be solved by a separation of variables. [Connell, V.F., Kubiznak, PRD 78, (2008)]

62 Possible generalizations to degenerate PCKY tensor and non-vacuum STs Infinite set of new interesting completely integrable dynamical systems.

63 Lax-pairs for integrable geodesics in Kerr-NUT-(A)dS, (Cariglia, V.F., Krtous, Kubiznak, '13) Deformed and twisted black holes with NUTs, (Krtous, Kubiznak, V.F., Kolar, '16) Review in Living Review in Relativity (V.F., Krtous, Kubiznak), under preparation

64 A higher dimensional black hole demonstrates approximately half of its symmetries as explicit ST symmetries, while the other half is hidden.

65 BIG PICTURE BLACK HOLES HIDE THEIR SYMMETRIES. WHY AT ALL HAVE THEY SOMETHING TO HIDE?

66 Based on: V. F., D.Kubiznak, Phys.Rev.Lett. 98, (2007); gr-qc/ D. Kubiznak, V. F., Class.Quant.Grav.24:F1-F6 (2007); gr-qc/ P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP 0702:004 (2007); hep-th/ V. F., P. Krtous, D. Kubiznak, JHEP 0702:005 (2007); hep-th/ D. Kubiznak, V. F., JHEP 0802:007 (2008); arxiv: V. F., Prog. Theor. Phys. Suppl. 172, 210 (2008); arxiv: V.F., David Kubiznak, CQG, 25, (2008); arxiv: P. Connell, V. F., D. Kubiznak, PRD, 78, (2008); arxiv: P. Krtous, V. F., D. Kubiznak, PRD 78, (2008); arxiv: D. Kubiznak, V. F., P. Connell, P. Krtous,PRD, 79, (2009); arxiv: D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous, Phys.Rev.Lett. (2007); hep-th/ P. Krtous, D. Kubiznak, D. N. Page, M. Vasudevan, PRD76: (2007); arxiv: `Alberta Separatists