VARNA, JUNE 3-8, 2011 http://neondude.uw.hu/ esher_mosaic_ii.jpg - PowerPoint PPT Presentation

VARNA, JUNE 3-8, 2011 http://neondude.uw.hu/ esher_mosaic_ii.jpg

Based on: V. F., D.Kubiznak, Phys.Rev.Lett. 98, 011101 (2007); gr-qc/0605058 D. Kubiznak, V. F., Class.Quant.Grav.24:F1-F6 (2007); gr-qc/0610144 P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP 0702:004 (2007); hep-th/0612029 V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007); hep-th/0611245 D. Kubiznak, V. F., JHEP 0802:007 (2008); arXiv:0711.2300 V. F., Prog. Theor. Phys. Suppl. 172, 210 (2008); arXiv:0712.4157 V.F., David Kubiznak, CQG, 25, 154005 (2008); arXiv:0802.0322 P. Connell, V. F., D. Kubiznak, PRD, 78, 024042 (2008); arXiv:0803.3259 P. Krtous, V. F., D. Kubiznak, PRD 78, 064022 (2008); arXiv:0804.4705 D. Kubiznak, V. F., P. Connell, P. Krtous , PRD, 79, 024018 (2009); arXiv:0811.0012 D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous, Phys.Rev.Lett. (2007); hep-th/0611083 P. Krtous, D. Kubiznak, D. N. Page, M. Vasudevan, PRD76:084034 (2007); arXiv:0707.0001 `Alberta Separatists’

Main focus Higher dimensional rotating black holes and their properties Spherical topology of the horizon No black rings, black branes ets ST is either asymptotically flat (vacuum) or (A)deSitter (with cosmological constant) Particle motion (mainly geodesics) Field propagation

Key words Hidden symmetries Complete integrability Separation of variables

COMPLETE INTEGRABILITY

Phase Space  2 m Phase space: { M , , H }  Symplectic form is a closed non-degenerate 2-form    d =0 ( =d ) Hamiltonian H is a scalar function on the symplectic 2 m manifold M 2 A m z are coordinates on M

  AB Poisson bracket { , } F G F G , A , B    A BA is a generator of the Hamiltonian flow H , B   A A Equation of motion is z =   One has F { H F , }

Darboux th eo r em: In the vicinity of any point it is always possible to choose canonical coordinates m        A z ( p , , p , q , , q ) in whi c h dp dq 1 m 1 m i i  i 1

Integrability means: reduci ble to quadra tures Integrability is linked to `existence of constants of motion' How many constants of motion How precisely they are related How the phase space is foliated by their level sets A system of differential equations is said to be integrable by quadratures if its solution can be found after a finite number of steps involving algebraic operations and integrations

Liouville theorem (Bour, 1855; Liouville, 18 55): If a Hamiltonian system with m degrees of freedom has   m integrals of motion F H , , F in involution , 1 m  { , F F } 0, and functionally indepen de nt on the i j  inte r section of the levels sets of the m functions F , f , i i ' the solutions of the corresponding Hamilton s e quations can be found by quadratures . "Miracle": Each of the integrals of motion " works tw ic e"

Liouville Integrable System The main idea behind Liouville's theorem is that the first integrals can be used as local coordinates. The involution F i condition implies that the vector fields, generated by m F i commute with each othe r and provide a choice of canonical coordinates. In these coordinates, the Hamiltonian is reduced to a sum of decoupled Hamiltonians that can be integrated m

  A AB X F , i i B (1) are tangent to ; X M i f  (2) [ X , X ] 0 i j

   A B On one has: { , } 0 M X X F F f AB i j i j       d , p dq , i i i    F p q ( , ) f p p ( , ) f q i i i i q    S F q ( , ) p ( , ) f q dq , i i i 0 q          S / F , dS dF p dq , i i i i i i i i           2 d S 0 dp dq d dF i i i i i i

  There exists a canonical transformati on ( p q , ) ( F ) i i i , i m m         dp dq dF d i i i i   i 1 i 1  To obtain 2 operations are require d: i  (1) Find p p ( , ), and (2) calculate some i f q ntegrals i i The system in the ne w va r iables takes the form    F { H F , } 0; i i  H        1 { H , } ;  i i i F i     F const , a bt i i

Integrability and chaotic motion are at the two ends of ` properties’ of a dynamical system. Integrability is exceptional, chaoticity is generic. In all cases, integrability seems to be deeply related with some symmetry, which might be partially hidden: the existence of constants of motion reflects the symmetry.

Important known examples of integrable mechanical systems include: (1) Motion in Euclidean space under central potential (2) Motion in the two Newtonian fixed centers (3) Geodesics on an ellipsoid (Jacobi, 1838) (4) Motion of a rigid body about a fixed point (several cases; Euler, Lagrange, Kowalevski) (5) Neumann model The Neumann model: N N         2 2 2 1 1 L ( x a x ) ( x 1) k k k k 2 2   k 1 k 1

PARTICLE MOTION IN GR

Phase space in GR:  C anonical coordinates ( p , x )      Symplectic form = p dx    1 Hamiltonia n H g ( x p p )   2 Equations of motion:       { , } , x H x g p       1 p { H p , } g p p      2 , a re equivalent to the geodesic equation     p p 0. ;

Consider a special monomial on the phase space     K K p p . A condition that it is a first integral s 1    1 s of motion implies: K =0, i.e. K is a Killing tens or.        ( ; ) 1 s 1 s Remark: g is a trivial Kill ing tensor o f rank 2 .      The Poisson bracket { K K , } [ K K , ] K K . 1 2 1 2 1 2 The first integrals of motion and are in involution K K 1 2  when [ K K , ] 0 . 1 2 If there exist non-degenerate functionally independent m Killi ng tensors in involution then the geodesic equations in dimensions are completely integrable. m

New physically interesting wide class of completely integrable systems Geodesic motion in the gravitational field of 4 and higher dimensional rotating black holes with spherical topology of the horizon (with `NUT’ parameters) in the asymptotically flat or (A)dS

Separation of variables in HJ eqs For the Ham iltonian     H P Q ( , ), P p , , p , Q q , , q , 1 1 m m the Hamilton-Jacobi equation is   H ( S Q , ) 0 . P    Suppose q and S enter t his equation as ( S q , ). 1 q 1 q 1 1 1 Then the variable q can be separa ted: 1    S S q C ( , ) S q ' ( , , q ), 1 1 1 2 m    ( , ) , S q C 1 q 1 1 1     H ( S ', , q , ; C ) 0 1 q 2 1 2

Complete separation of variables:      S S q C ( , ) S q C C ( , , ) S ( q , C , , C ). 1 1 1 2 2 1 2 m m 1 m The constants C generate first integrals on the phase i s pace. When these integrals are independent and in involution th e sy stem is integrable in the Liouville sence.

KILLING-YANO TENSORS

Forms (=AStensor)        (1) External product: ( )  q p q p    (2) Hodge dual: *( ) (* )  q D q    (3) External derivative: ( d ) ( d )  q q 1      (4) Closed form: ( ) 0 (locally ( )) d d  q q q 1

CKY=Conformal Killing-Yano tensor      k k , k k 1            ... [ ... ] ... ... 1 2 1 2 2 1 2 p p p p       k g k ( p 1) g k             ( ... ... [ ( ... ]    1 2) 3 p 1 1 2 3 p 1 3 1 2) p 1 1     k k 1        ... ...  D p 1 2 p 1 1 2 p

 k If vanishes f=k is a Killing-Yano tensor p  f is a parallel propagated vector     ... K f f is the Killing tensor 2 n     ... 2 n  is an integral of geodesic motion K p p  

Properties of CKY tensor Hodge dual of CKY tensor is CKY tensor Hodge dual of closed CKY tensor is KY tensor External product of two closed CKY tensors is a closed CKY tensor

CKY * CCKY KY R2-KT

Principal Killing-Yano tensor      h g g , ( *) c ab ca b c b a      b 1 h 0 , h  [ a bc ] a ba D 1 PKY tensor is a closed non-degenerate (matrix rank 2n) 2-form obeying (*)  a is a primary Killing vector (off-shell!!)

Killing-Yano Tower

Killing-Yano Tower       j h h h h ... h j times     j j * k h K k k j j j     K j j Total number of conserved quantities          ( ) ( 1) 1 2 n n n D KV KT g