Brief course of lectures at 18th APCTP Winter School on Fundamental Physics

Transcription

1 Brief course of lectures at 18th APCTP Winter School on Fundamental Physics Pohang, January January 28, 2014

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4 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

5 Higher dimensional gravity is stronger at small scales and weaker at large scales than the 4D one: F G M M / r ( D) ( D2) 1 2 If extra dimensions are compact with size L, then at the distance larger than L one has standard 4D gravity with G G / L ( D) ( D4). For L 0.1 mm interactions. higher dim. gravity may be as strong as other Properties of HD black holes with L are determined by D-dim Einstein equations, so that testing their properties in our 4D experiments one probes extra dimensions (e.g. micro BH at colliders?) r g

6 Gravity in Higher Dim. Spacetime (4 k) ( k) 3k k a G M x ( ) F ( k ) G M F, k D 4 2 k r No bounded orbits (prove!) Gravity at small scales is stronger than in 4D

7 4D Newton law is confirmed for r>l. Q.: How to make gravity strong (HD) at small scales without modifying it at large scales? A.: Compactification of extra dimensions `Our space dimensions Extra dims

8 Gravity in ST with Compact Dims Example: 0, M R S (4) * GM n * 1 G M sinh( r / l)cosh( r / l) r z nl lr r l z l ( ) cosh ( / ) cos ( / ) * GM * (, 0) coth( / ), / r z r l G G l lr

9 In a general case: # STdimensions is D, D G # non-compactified dimensionsis D4 : G. l D4 Gl M M GMM For r l F r l r r One can introduce a function Gr ( ) such that D ; for F D2 2 D4 Gl for r l G( r) and for r l G( r) G. D4 r G() r M1M 2 Then F Running gr.coupling `constant'. 2 r At small scales, that is at large energy, the running coupling `constant' becomes large and gravitational interaction becomes as strong as other physical interaction..

10 `Solution' of the hierarchy problem 2 Gm p 2 e e Gm * p r p 2 2 * 2 * * r r e G( r ) m 1/ k l 20 For l 0.01cm and D 4 k 2 r* 10 cm

11 Kaluza-Klein tower (x,y)= (x,y+2 L) (x,y)= n(x ) e W=- dx dy [ +m ] * 2 * 0 n W=- dx [ +(m ) ] * 2 * 2 n 0 2 n L n iny/l KK tower with the mass spectrum m m n L /

12 Brane World Models Bosons, fermions and gauge fields are localized within the 4D brane Gravity is not localized and `lives in D-dim bulk space Fundamental scale of order of TeV. Large extra dimensions generate Planckian scales in 4D space

13 Black Holes as Probes of Extra Dims We consider black holes in the mass range g M 10 g Mini BHs creation in colliders

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15 BH formation Bolding Phase Thermal (Hawking) decay

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17 Angular momentum in Higher Dimensions ( D2 n ) J J J Jn 1 2 i, Ji ji 0 j

18 5D vac. stationary black holes

19 (1) No Uniquness Theorem: For given M and J more than one BH solution (2) Complete integrability and separation of variables are generic properties of HD analogues of Kerr BH ( hidden symmetry generators V. F & D. K. 07) (3) Stability of HD BHs?

20 We focus on higher dimensional rotating black holes with the spherical topology of horizon in a ST which asympotically is either flat or (Anti)DeSitter: R = g

21 Main Results In the most general ( Kerr NUT ( A) ds) higher dimensional black hole spacetime : (1) Geodesic motion is completely integrable. (2) Hamilton Jacobi and Klein Gordon equa tions allow the complete separation of variables

22 Motivations Separation of variables allows one to reduce a physical problem to a simpler one in which physical quantities depend on less number of variables. In case of complete separability original partial differential equations reduce to a set of ordinary differential equations Separation of variables in the Kerr metric is used for study: (1) Black hole stability (2) Particle and field propagation (3) Quasinormal modes (4) Hawking radiation

23 Main lesson: Properties of higher dimensional rotating BHs and the 4D Kerr metric are very similar. HDBHs give a new wide class of completely integrable systems

24 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 with spherical topology of the horizon)

25 "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, M mass, a ( n 1 ) rotation parameters, k M ( n 1 ) ` NUT ' parameters Total # of parameters is D

26 Differential forms Antisymmetric tensor differential form B p 1 A[ 1... ] A A[ 1... ] dx... dx p p p q A[[... ] B[... ]] A B 1 p 1 A [..., ] 1 p p [... ] e... p1 D p1 D q da p p A B A [... ] 1 p p D p p

27 Generator of Symmetries PRINCIPLE CONFORMAL KY TENSOR 2-form [ ] with the following properties: (i) Non-degenerate (maximal matrix rank) (ii) Closed dh 0 (iii) Conformal KY tensor h g g, 1 b h c ab ca b cb a a D1 ba a h is a primary Killing vector

28 All Kerr-NUT-AdS metrics possess a PRINCIPLE CONFORMAL KY TENSOR (V.F.&Kubiznak 07)

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

30 4D Kerr metric example Principal CCKY tensor h db 1 b [ ( y 2 r 2 ) d r 2 y 2 d] 2 H h h ; det( H ) 0, y 2, r 2 Lesson: Essential coordinates ( r, y), Killing coordinates (, )

31 All of this is valid for arbitrary functions R( r) and Y( y) (`off shel') : 2 R( r) Y( y) ds ( d y d ) ( d r d ) r y r y dr dy ( r y )[ ]. R( r) Y( r) For Kerr metric: R r a, Y a y

32 BRIEF REMARKS ON COMPLETE INTEGRABILITY

33 Dynamical System 2m Phase space: { M,, H}; AB is a non-degenerate 2-form (symplectic structure); H is a scalar function on M called a Hamiltonian. 2m A BA, B of the Hamiltonian flow BC C ( ). is a generator A A Equation of motion is z =. Its solutions determine evolution of the system. AB Poisson brackets { F, G} F G. One has F { H, F}. AB H, A, B If { H, F} 0 F is called an integral of motion. One can always introduce such (canonical) coordinates A m 1 m 1 m i i i=1 in which z ( q,..., q, p,..., p ) and = dq dp. A

34 Liouville theorem: Dynamical equations in 2m dimensional phase space are completely integrable if there exist m independent commuting integrals of motion. In such a case a solution can be found by using algebraic relations and integrals. F( p, q) and Q( p, q) commute if their Poisson brackets vanish, { FQ, } 0 These integrals of motion can be used as coordinatesi on the phase space. Moreover, there exist canonical coordinates ( I ( F), ), called action-angle variables, in which the equations j i i of motion are F i trivial: I 0, H / I const i i i

35 Relativistic Particle as a Dynamical System Preferable coordinates in the phase 1 2 Monomial in momenta integrals of motion... K p... p imply that K... (... ; ) space: ( x, p g x ), H ( p, x) g p p. K= K 0 is 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 (vectors)

36 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 integrals of order n (n) (m) m. The corresponding Killing tensor is called reducible.

37 Denote D2 n. D-dimensional Kerr-NUT-AdS metric has n Killing vectors. For complete integrability of geodesic equations one needs n more integrals of motion. In 4D: n 2, 0, 2 Killing vectors, t

38 GENERAL SCHEME

39 (... 1 k Killing - Yano 2) 3 p1 0 (KY) tensor K 2... k k p... 2 p is a Killing tensor ( K k k ) K ( ; ) 0

40 CKY=Conformal Killing-Yano tensor,... [... ] h h h h 1 2 p 1 2 p 2 p 1 2 p h g h ( p 1) g h ( [ (... ] 1 2) 3 p p ) p1 h 1 D p h 2 p1 1 2 p Closed conformal Killing - Yano tensor (CCKY) This is a CKY tensor h with an additional property: dh 0.

41 Properties of CKY tensor Hodge dual of CKY tensor is CKY tensor Hodge dual of closed CKY tensor is KY tensor: *CCKY=KY and *KY=CCKY External product of two closed CKY tensors is a closed CKY tensor (Krtous,Kubiznak,Page&V.F. '07; V.F. '07)

42 CKY CCKY * KY R2-KT

43 Killing-Yano Tower

44 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 Killing vectors: j K j

45 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 Kerr-NUT -AdS ST are completely integrable.

46 Separation of variables in HD Black Holes Separation of variables in HJ and KG equations in 5D ST (V.F. Stojkovic 03) Separability depends on the choice of coordinates

47 In D 2 n dimensional ST the Principal CKY tensor (as operator) has n 2D eigenspaces with eigenvalues x a. They can be used as coordinates. D canonical coordinates: n essential coordinates x and n Killing coordinates j. a Complete separability takes place in these canonical coordinate s.

48 Hamilton-Jacobi ( S) WKB ~ exp( is) m Klein-Gordon ( m ) 0 "Dirac eqn= KG eqn " Dirac equation ( m) 0

49 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)

50 FURTHER DEVELOPMENTS

51 Separability of gravitational perturbations in Kerr-NUT-(A) ds Spacetime (Oota andyasui '10) Charged particle motion in a weakly charged HD black holes (Krtous&V. F. '11) Metrics admitting a principal Killing-Yano tensor with torsion (Houri, Kubiznak, Warnick and Yasui '12)

52 SUMMARY OF THIS PART

53 The most general spacetime admitting PCKY tensor is Kerr-NUT-(A)dS. It has the following properties: It is of the algebraic type D It allows a separation of variables for the Hamilton- Jacoby, Klein-Gordon, Dirac, tensorial gravitational perturbations. The geodesic motion in such a spacetime is completely integrable.

54 Possible generalizations to degenerate PCKY tensor and non-vacuum STs

55 Review Articles Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables (V.F. and Kubiznak 08) Hidden Symmetries and Integrability in Higher Dimensional Rotating Black Hole Spacetimes, (Cariglia, Krtous and Kubiznak 11) Hidden Symmetry and Exact Solutions in Einstein Gravity (Yasui and Houri 11)

56 A Problem BLACK HOLES HIDE THEIR SYMMETRIES. WHY AT ALL HAVE THEY SOMETHING TO HIDE?