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 2 2

Phase space: { , , } Symplectic form is a closed non-degenerate 2-form d =0 ( =d ) Hamiltonian is a scalar function on the symplectic manifold are coordinates on

m m A m

M H H M z M     

, , ,

Poisson bracket { , } is a generator of the Hamiltonian flow Equation of motion is = One has { , }

AB A B A BA B A A

F G F G H z F H F         

A 1 1 1

In the vicinity of any point it is always possible to choose canonical coordinates z ( , , , Darboux th , , ) in whi eo em: c r h

m m m i i i

p p q q dp dq



   



 

Integrability is linked to `existence of constants of motion' How many constants of motion Integrability means: reduci How precisely they ble to quadra are related tures 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

1

Liouville theorem (Bour, 1855; Liouville, 18 55): , { , } , 0, ,

i j m

If a Hamiltonian system with m degrees of freedom has F H F and m integrals of motion in involution F F functionally indepen

• n the

inte de r nt    ' "Miracle": Each of the integrals of motion " works tw , ic , e" .

i i

the solutions of the corresponding Hamilton s e section of the levels sets of the m functions F f quations can be found by quadratures 

The main idea behind Liouville's theorem is that the first integrals can be used as local coordinates. The involution condition implies that the vector fields, generated by commute with each othe

i i

F m F 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

Liouville Integrable System

,

(1) are tangent to ; (2) [ , ]

A AB i i B i f i j

X F X M X X   

2

On

• ne has:

{ , } , , ( , ) ( , ) ( , ) ( , ) , / , ,

A B f AB i j i j i i i i i i i q i i i q i i i i i i i i i i i i i i

M X X F F d p dq F p q f p p f q S F q p f q dq S F dS dF p dq d S dp dq d dF                           

      

, 1 1

To obtain 2 operations are require There exists a canonical transformati d: (1) Find The system in the ne ( , ), and (2) calculate some i w va

• n (

, ) ( ) ntegrals r

i i i i m m i i i i i i i i i

p p f p q F dp dq dF q d

 

         

 

1

iables takes the form { , } 0; { , } ; ,

i i i i i i i i

F H F H H F F const a bt                

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.

2 2 2 1 1 2 2 1 1

The Neumann model: ( ) ( 1)

N N k k k k k k

L x a x x

 

    

 



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

PARTICLE MOTION IN GR

1 , 2 1 2

Phase space in GR: anonical coordinates ( , ) Symplectic form = Hamiltonia Equations of motion: { , } , { , n ( } a ) re equivalent x H x g p p H p g p p C p x p dx H g x p p

                

       



 

;

to the geodesic equation 0. p p

   

1 1 1 1

( ; )

Consider a special monomial on the phase space . A condition that it is a first integral

• f motion implies:

=0, i.e. is a Killing tens Remark: is a trivial Kill

• r.

s s s s

K p p K g K

         

 K

  



1 2 1 2 1 2 1 2 1 2

The Poisson bracket { , } [ , ] . The first integrals of motion and are in involution when [ , ] If there exist non-degenerate functionally independent ing tensor o . f rank 2 Killi . K K K K K K m     K K K K  ng tensors in involution then the geodesic equations in dimensions are completely integrable. m

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

New physically interesting wide class

• f completely integrable systems

Separation of variables in HJ eqs

1 1

1 1 1 1 1 1 1 2 1 1

Suppose and enter t For the Ham his equation as ( , ). Then the variable can be separa iltonian ( , ), , , , , , , the Hamilton-Jacobi equation ted: is ( , ) ( , ( . ) '

q m q m P

H P Q P p p Q q q H S Q q S S q q S S q C S q           

1 2

1 1 1 1 2 1

, , ), ( , ) , ( ', , , ; )

m q q

q S q C H S q C        

1 1 1 2 2 1 2 1

The constants generate first integrals on the phase s Complete separation of variables: ( , ) ( , , ) ( ,

• pace. When these integrals are independent and in

involution th , , ). e sy

i m m m

S S q C S q C C S q C C C     stem is integrable in the Liouville sence.

KILLING-YANO TENSORS

Forms (=AStensor)

1 1

(1) External product: ( ) (2) Hodge dual: *( ) (* ) (3) External derivative: ( ) ( ) (4) Closed form: ( ) 0 (locally ( ))

q p q p q D q q q q q q

d d d d           

   

      

CKY=Conformal Killing-Yano tensor

           

   

1 1 2 1 2 2 1 2

... [ ... ] ... ...

,

p p p p

k k k k

                 

   

          

1 2) 3 1 1 2 3 1 3 1 2) 1 1 2 1 1 2

( ... ... [ ( ... ] ... ...

( 1) 1 1

p p p p p

k g k p g k k k D p

If vanishes f=k is a Killing-Yano tensor

2 2

... ...

is the Killing tensor

n n

K f f

     



k 

is an integral of geodesic motion K p p

  

is a parallel propagated vector f 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

CCKY KY CKY

*

R2-KT

Principal Killing-Yano tensor

1 [ ] 1

, ( *) ,

b a bc a ba c ab b c a D ca b

h h h g g   



      

PKY tensor is a closed non-degenerate (matrix rank 2n) 2-form obeying (*)

is a primary Killing vector (off-shell!!)

a



Killing-Yano Tower

Killing-Yano Tower

... *

j j times j j j j j

h h h h h k h K k k

 

       

j j

K    

Total number of conserved quantities

( ) ( 1) 1 2 n n n D KV KT g         

Existence of the Principal CCKY tensor in the most general known solution for higher dimensional rotational Kerr-NUT-(A)dS black hole metric was discovered in:

• 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

Constructed KY tower produces a set of D non-degenerate, functionally independent Killing integrals of motion in the involution

• P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP 0702:004 (2007)
• D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous, Phys.Rev.Lett. (2007)
• P. Krtous, D. Kubiznak, D. N. Page, M. Vasudevan, PRD76:084034 (2007);

Metrics which admit Principal CCKY tensor allow complete description

ˆ ˆ 1 1

( ) ,

n n ab a b a b a b

g e e e e e e

    



 

  



1 ˆ ( )

1 ,

n i i i

e dx e Q A d Q

     



 

 



2 2

, , ( ), ( ) X Q U x x X X x U

           

   



2 ( ) 1

(1 )

n n j j j

tx t A

   

  

 

1 2 1 2 ( ) 1

(1 ) (1 )

n n k k k

tx tx t A

       

   

 

Houri, Oota, and Yasui [PLB (2007); JP A41 (2008)] proved this result under additional assumptions: 0 and

• 0. More recently Krtous, V.F., Kubiznak [arXiv:0804.4705 (2008)] and

Houri, Oota, and Ya L g L h

 

  sui [arXiv:0805.3877 (2008)] proved this without additional assumptions.

Canonical Coordinates

ˆ

( ) h m ix m m e ie

        

   

A non-degenerate 2-form h has n independent eigenvalues. There exist n (mutually orthogonal) 2D eigenspaces

D=2n+ Canonical coordinates: essential coordinates x and Killing coordinates

j

n n



   

n  

Principal CCKY tensor

Non-degeneracy: (1) Eigen-spaces of are 2-dimensional (2) are functionally independent in some domain (they can be used as essential coordinates) h x

(1) is proved by Houri, Oota and Yasui e-print arXiv:0805.3877 (2) Case when some of eigenvalues are constant studied in Houri, Oota and Yasui Phys.Lett.B666:391-394,2008. e-Print: arXiv:0805.0838

On-Shell Result

A solution of the vacuum Einstein equations with the cosmological constant which admits a (non-degenerate) principal CKY tensor coincides with the Kerr-NUT-(A)dS spacetime.

2 n k k k

X b x c x

    

 

Kerr-NUT-(A)dS spacetime is the most general BH solution obtained by Chen, Lu, and Pope [CQG (2006)]; See also Oota and Yasui [PL B659 (2008)]

"General Kerr-NUT-AdS metrics in all dimensions“, Chen, Lü and Pope, Class. Quant. Grav. 23 , 5323 (2006).

 

/ 2 , 2 n D D n    

( 1) R D g

 

  

, , ( 1 ) , ( 1 ) ` '

k

M mass a n rotation parameters M n NUT parameters



         

# Total

• f parameters is D 



SEPARATION OF VARIABLES

1

( )

n m k k k

S w S x

  

 

 

    

 

2 2 1 2 2 1 2

1 1 ') ( ( ( ) ) ( )

m m n k n k k k k k

S x x c X X

          

     

  

Separability of the Hamilton–Jacobi equation

ab a b

S g S S       

• V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007); hep-th/0611245

Separability of the Klein–Gordon equation

2

( )     

1

( )

k k

n m i k

R x e

      

  

 

2 1 2 2 1

( ) ( ( ) ) ( )

m m n k n k k k k k

X R X R R x b x R x X

         



     

      

 

,

• V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007); hep-th/0611245

Weakly charged higher dimensional rotating black holes

1 Hamiltonian ( )( ) 2

ab a a b b

H g p qA p qA   

2

HJ equation ( )( )

ab a b a b

S S g qA qA x x               

2

Klein-Gordon equation ( )( )

ab a a b b

g iqA iqA 

     

      

b a a b ab a b

F A A

   

    

(in Ricci flat)

a b b a a

A Q  

  

 

2 2 2 2 2 2 (0)

[ ] 0, 2

ab a b

S S g M x x M M e e                 

For a primary Killing vector field one again has a complete separation of variables

[V.F. and P. Krtous, 2010 ]

OTHER CASES OF COMPLETE SEPARABILITY

Separability of the massive Dirac equation in the Kerr-NUT-(A)dS spacetime [Oota and Yasui,

• Phys. Lett. B 659, 688 (2008)]

Stationary string equations in the Kerr-NUT- (A)dS spacetime are completely integrable. [D. Kubiznak, V. F., JHEP 0802:007,2008] Separability of Gravitational Perturbation in Generalized Kerr-NUT-de Sitter Spacetime [Oota, Yasui, arXiv:0812.1623]

PARALLEL TRANSPORT

Parallel transport along timelike geodesics

Let be a vector of velocity and be a PCKYT. is a projector to the plane orthogona Lemma (Page): is parallel p l to . Denote ropagated along a ge

a ab b b b a a a a c d c c ab a b cd ab a cb ac b ab

u h P u u u F P P h h u u h h F u u       

• desic:

Proof: We use the definition of the PCKYT =

u ab u ab a b a b

F h u u      

Suppose is a non-degenerate, then for a generic geodesic eigen spaces of with non-vanishing eigen values are two

• dimensional. These 2D eigen spaces are parallel propagated.

Thus a problem reduces

ab ab

h F 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, 024042 (2008)]

Parallel transport along null geodesics

Let be a tangent vector to a null geodesic and be a parallel propagated vector obeying the condition 0. Then the vector is parallel propagated, provided . This procedure

a a a a a ba a b a a

l k l k w k h l k         allows one to construct 2 more parallel propagated vectors and , starting with .

a a a

m n l

a

l

a

m

a

n

( ) [ ]

We introduce a projector 2 , and . One has: 2 0.

c d ab ab a b ab a b cd c d c d l ab a b l cd a b c d

P g l n F P P h F P P h P P l         

Thus is parallel propaged along a null geodesic. We use rotations in its 2D eigen spaces to [Kubiznak,V.F construct .,Krtous, a parallel propagat Connell, PRD 79, 024 ed basis. 018 (2009)]

ab

F

FURTHER DEVELOPMENTS

On the supersymmetric limit of Kerr-NUT-AdS metrics [Kubiznak, arXiv:0902.1999] Einstein spaces with degenerate closed conformal KY tensor [Houri, Oota and Yasui, Phys.Lett.B666:391-394,2008]

1 1 2 3 3 1 3 1 1 3 2

Minimally gauged supergravity (5D EM with Chern-Simo GENERALIZED KILLING-YANO TENSORS [Kubiznak, Kunduri, and Yasui, 0905.0722 (2009)] ns term): *( ) * , 0, * 0, (

ab ab ac

L R F F F F A dF d F F F R g F               

1 3 1 [ ] [ ] 3 2 1 2 2 1 6

) Appli Torsion: cation: Chong, Cvetic, Lu, Pope [PRL, 95,161301,2 005] Note: This is type I metric * (* ) 2 , ( . * ) (* )

T d c ab c ab cd a b c a b cd c ab acd b ac b ab c b ab

T F h h F h g K h h h F g h g h F           

SUMMARY

The most general spacetime admitting PCKY tensor is described by the canonical metric. It has the following properties:

• It is of the algebraic type D
• It

allows a separation

• f

variables for the Hamilton-Jacoby, Klein-Gordon, Dirac, tensorial gravitational perturbations, and stationary string equations

• The geodesic motion in such a spacetime is

completely integrable.

• The

problem

• f

finding parallel-propagated frames reduces to a set of the first order ODE. This is a new interesting example of completely integrable system.

• When

the Einstein equations with the cosmological constant are imposed the canonical metric becomes the Kerr-NUT-(A)dS spacetime

• Possible generalizations to degenerate PCKY

tensor and non-vacuum STs

BIG PICTURE

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