Valeri P. Frolov- Black Holes and Hidden Symmetries

8/3/2019 Valeri P. Frolov- Black Holes and Hidden Symmetries

1/57

Peyresq, June 20-26, 2009


2/57


3/57

Higher Dimensions:Motivations for Study

Higher Dimensions:

Kaluza-Klein models and Unification;

String Theory;

Brane worlds;

`From above view on Einstein gravity.


4/57

Gravity in Higher Dim. Spacetime

(4 ) ( ) 3 ( ) k k kka G M x

F

( )

2

k

k

G MF

r

No bounded orbits

Gravity at small scales is stronger than in 4D


5/57

`Running coupling constant

( )

( ) , ( ), ( ) k k

k k

G lG r G G l G r G

r r

1/2 22

*

*2 2 2

* *

( )

k

p pG r m Gme r lr r e

2

36

210

pGm

e

20

*

0.01 2 10 For l cm and k r cm

`Hierarchy problem


6/57

1( , )

p m mv

*( ,0)

fp E

2( , )

p m mv

1/ 2

1 2 * */ , 2 , (2 ) E p p m E m E Em

2,

2 2

*

*

2

*,

*

(1 ) 2

2

/ 1

mE E

For E r E T

E

eV

m v m E

New fundamental scale

2*

*

( / ) ,

2 0.01 1

k

kPl Pl

M M l l

For k and l cm M TeV


7/57

4D Newton law is confirmed for r>l.

Q.: How to make gravity strong (HD) atsmall scales without modifying it at largescales?

A.: Compactification of extra dimensions

`Our space dimensions

Extradims


8/57

Gravity in ST with Compact Dims

Example: (4) 4 3 10, R SM

**

2 2 2 2

1 sinh( / )cosh( / )

( ) cosh ( / ) cos ( / )

n

G M r l r lG M

r z nl lr r l z l

**( , 0) coth( / ), /

G Mr z r l G G l

lr 24 * 2 *1

02 2W=- dx [ +(m ) ]nn

n

L


9/57

Brane World Paradigm

Fundamental scale of order of TeV. Large extra

dimensions generate Planckian scales in 4D space

Gravity is not localized and `lives in (4+k)-Dbulk space

Bosons, fermions and gauge fields are localizedwithin the 4D brane


10/57

Black Holes as Probes of Extra Dims

21 2610 10g M g

We consider first black holes in the mass range

Mini BHs creation in colliders


11/57

BH formation

Bolding Phase

Thermal (Hawking) decay


12/57

Black Objects

Black Holes[Horizon Topology S^(D-2)]

Exist in any # of dims

Generic solution is Kerr-NUT-(A)dS

Principal Killing-Yano existence istheir characteristic property

Hidden Symmetries, Integrability

properties

Black Rings

Black Saturns

Black Strings, ets


13/57

Higher Dimensional Kerr-NUT-(A)dSBlack Holes


14/57

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:0811.0012D. 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


15/57

Recent Reviews on HD BHs

Panagiota Kanti, Black holes in theories with large extra dimensions:

A Review, Int.J.Mod.Phys.A19:4899-4951 (2004).

Panagiota Kanti Black Holes at the LHC, Lectures given at 4thAegean Summer School: Black Holes, Mytilene, Island of Lesvos,Greece, ( 2007).

Roberto Emparan, Harvey S. Reall, Black Holes in HigherDimensions ( 2008) 76pp, Living Rev.Rel. arXiv:0801.3471

V.F. and David Kubiznak, Higher-Dimensional Black Holes: Hidden

Symmetries and Separation of Variables, CQG, Peyresq-Physics 12workshop, Special Issue (2008) ; arXiv:0802.0322

V.F, Hidden Symmetries and Black Holes, NEB-13, Greece, (2008);arXiv:0901.1472


16/57

Hidden Symmetries of 4D BHs

Hidden symmetries play an important role in study 4D rotating blackholes. They are responsible for separation of variables in the Hamilton-Jacobi, Klein-Gordon and higher spin equations.

Separation of variables allows one to reduce a physical problem to asimpler one in which physical quantities depend on less number ofvariables. In case of complete separability original partial differentialequations 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


17/57

Brief History of 4D BHs

1968: Forth integral of motion, separability of the Hamilton-Jacobi andKlein-Gordon equations in the Kerr ST, Carters family of solutions[Carter, 1968 a, b,c]

1970: Walker and Penrose pointed out that quadratic in momentumCarters constant is connected with the symmetric rank 2 Killing tensor

1972: Decoupling and separation of variables in EM and GP equations[Teukolsky]. Massless neutrino case [Teukolsky (1973), Unruh (1973)].Massive Dirac case [Chandrasekhar (1976), Page (1976)]

1973: Killing tensor is a `square of antisymmetric rank 2 Killing-Yanotensor [Penrose and Floyd (1973)]

1974: Integrability condition for a non-degenerate Killing-Yano tensor implythat the ST is of Petrov type D [Collinson (1974)]

1975: Killing-Yano tensor generates both symmetries of the Kerr ST[Hughston and Sommers (1975)]


18/57

Type-D (withoutacceleration)

Killing tensor

Killing-Yanotensor

Separability


19/57

Main Results (HD BHs)

Rotating black holes in higher dimensions, described by theKerr-NUT-(A)dS metric, in many aspects are very similar tothe 4D Kerr black holes.

They admit a principal conformal Killing-Yano tensor.This tensor generates a tower of Killing tensors and Killing

vectors, which are responsible for hidden and `explicitsymmetries.The corresponding integrals of motion are sufficient for acomplete integrability of geodesic equations.These tensors imply separation of variables in Hamilton-

Jacoby, Klein-Gordon, and Dirac equations.Any solution of the Einstein equations which admits a non-degenerate a principal conformal Killing-Yano tensor is aKerr-NUT-(A)dS spacetime.


20/57

Spacetime Symmetries

L g g

0L g

( ; ) 0 (Killing equation)

( ; ) (conformal Killing eq) g

1

;

D D is # of ST dimensions


21/57

Hidden Symmetries

( ; ) g

Killing vector

Killing tensor Killing-Yano tensor


22/57

Symmetric generalization

CK=Conformal Killing tensor

1

1 2 1 2 2 1 2... ( ... ) ... ...,n n n nK K K K

1 2 1 2 3( ... ; ) ( ... )n nK g K

Integral of motion

1 2 1 2 1

1 2 1 2 1... ; ...( ... ) ...n n

n nu K u u u K u u u


23/57

Antisymmetric generalization

CY=Conformal Killing-Yano tensor

1

1 2 1 2 2 1 2... [ ... ] ... ...,

n n n n k k k k

1 2) 3 1 1 2 3 1 3 1 2) 1( ... ... [ ( ... ]( 1)

n n nk g k n g k

If the rhs vanishes f=k is a Killing-Yano tensor


24/57

2

2

...

... is the Killing tensorn

nK f f

1

1 1 1

...

... ...

.

n

n

n n

If f is a KY tensor then for a geodesic

motion the tensor p f u is

parallelly propagated along a geodesic


25/57

Principal conformal Killing-Yano tensor

1[ ] 1

, (

0

*)

, na bc a na

c ab b c a

D

ca b

h h

h g g

1, (*)

, 2

1X

h

h db

X h

D

D

n

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


26/57

Properties of CKY tensor

Hodge dual of CKY tensor is CKY tensor

Hodge dual of closed CKY tensor is KYtensor

External product of two closed CKY tensors

is a closed CKY tensor


27/57

Darboux Basis

1 1

( ) ,n nab a b a b a b

ab a b

g e e e e e e

h x e e

1

2

( )m e ie


28/57

Canonical Coordinates

h m ix m

A non-degenerate 2-form h has n independent

eigenvalues

essential coordinates x and Killing

coordinates are used as canonical coordinatesj

n n


29/57

Principal conformal KY 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


30/57

Killing-Yano Tower


31/57

Killing-Yano Tower: Killing Tensors

2

2

1 2

1 21 1 2 2

2 4 2 2

... ... ..

2

.

* * *

4 2 2

*

j n

j times n times

j n

j n

j n

j j n n

j n

h h h h h h h h h h h h

k h k h k h k h

K k k K k k K k k K

D D D j

k

n

k

D

Set of (n-1) nontrivial rank 2 Killing tensors


32/57

2 1 2 1

1 ( )

n j j j n n

ab a b a b a bA e e e e A e e

K

1 1

1 1

1 1

2 2 2 2,i i

i i

i i

j

i i

A x x A x x


33/57

Killing-Yano Tower: Killing Vectors

11

is a primary Killing vectorna naD

h

1( ) ( )2 (*)

n

a b n a bD R h

( ; )On-shell ( ) (*) implies 0ab ab a bR g

Off-shell it is also true but the proof is much complicated

(see Krtous, V.F., Kubiznak (2008))


34/57

1 1 2 2 1 1 j j n nK K K K

Total number of the Killing vectors is n

Total number of conserved quantities

( ) ( 1) 1 2n n n D

KV KT g

2

2 2

1

( )

, , ( )i

n i

i i

xd e

Q

x

U

U x

Reconstruction of metric


35/57

Principal Conformal Killing-Yano Tensor

Reconstruction of metric

Coordinates


36/57

0 1 1

1

: , ,..., ,

: ,...,

nn

n

Killing coordinates

Essential coordinates x x

2-planesof rotation

Off-Shell Results


37/57

Off-Shell Results

A metric of a spacetime which admits a (non-

degenerate) principal CKY tensor can bewritten in the canonical form.

1

0

1 ,

n

ii

i

e dx e Q A d Q

, ( )X

Q X X x

U

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

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

Houri, Oota, and Yasui

L g L h

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


38/57

On-Shell Result

A solution of the vacuum Einstein equationswith the cosmological constant which admits a

(non-degenerate) principal CKY tensor

coincides with the Kerr-NUT-(A)dS spacetime.

2

0

nk

k

k

X b x c x

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

"G l K NUT AdS t i i ll di i Ch


39/57

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

/ 2 , 2n D D n

( 1) R D g

, , ( 1 ) ,

( 1 ) ` '

k M mass a n rotation parameters

M n NUT parameters

#Total of parameters is D

Ill i


40/57

Illustrations

2 2 2 2 2

2 2 2 2 212

Let us consider a 4D flat spacetime with the metric

and let be the following 1-form

[ ( )],

dS dT dX dY dZ

b

b R dT a Y dX X dY R X Y Z

One has

( )*

h db dT XdX YdY ZdZ adY dX f h X dZ dY Z dY dX Y dX dZ a dZ dT


41/57

1

[ ] 1

, (

0

*)

,n

a bc a na

c ab b c a

D

ca b

h h

h g g

0

0 00 0

0 0 0

na

X Y Z

X ahY a

Z

(0) , (0)13

( 1,0,0,0),n aa na a T h

(0)

, .1 ,TX X XX T h g etc

20a aY aX


42/57

2 2

2 2

2 2 2

0

0

ab

a aY aX

aY Y Z XY XZ K

aX XY X Z YZ XY YZ X Y a

( ) (0) 2 2[ ]abb a T X Y T

K a a Y X a a

2 2 2 22 ( )a bab Z Z K p p L aEL a E p

1

(0) ( )

, / ,, ;

,

T t t

t

t T a aa a

D b di t


43/57

Darboux coordinates

2

2 2

2 2

2

, ;

0

0

a ac a a a

b bc b b b H h h H

R aY aX

aY a X XY XZ

aX XY a Y YZ

XZ YZ Z

2 2 2 2 2

2 2 2 2 2 2 212

2 2

det( ) 0 ( ) 0,

[ ( ) 4 ],

, are essential coordinates

R a a Z

a R R a a Z

r y


44/57

1 2 2 2 2

1 2

2 22 2 2 2 2 2 2 2

2 2 2

2 2

1

2 2 2 2

( )( ) cos ,

( )( ) sin ,

( )[ ] ( ) )

,

(dr dy

d

X a r a a y

Y a r a a y

Z a r

S dT r y a r a a y d r a a

y

y

2 ,T t a a

2 22 2 2 2 2 2 2

2 2

2 2 2 2

1(*) [ ( ) ( ) ] ( )[ ],

,

dr dydS R dt y d Y dt r d r y

r y R YR r a Y a y

This is a flat ST metric in the Darboux coordinatesassociated with the potential b.

2 2 2 2 21 [( ) ]b y r a dt r y d


45/57

2[( ) ]b y r a dt r y d

Three important results can be proved:

(i) For arbitrary ( ) and ( ) is a potential generating a closed

conformal Killing-Yano tensor for the metric (*);

(ii) (*) with arbitrary functions ( ) a

R r Y y b

R r nd ( ) is the most general metric,

which admits a closed conformal Killing-Yano tensor;

(iii) (*) with arbitrary functions ( ) and ( ) belongs to Petrov type D

Y y

R r Y y

2 22 2

2 2

The Einstein equations, -3 , for these metrics are

satisfied iff the following equation is valid

12 ( ).

The most general solution of this equation can be written in the form

(

R g

d R d Y r ydr dy

R r

2 2 2 2 2 2)(1 ) 2 , ( )(1 ) 2a r Mr Y a y y Ny

Remark: the tranformation , makes ther ix M iM


46/57

Remark: the tranformation , makes the

expressions for the metric (*), potential , and the solutions

symmetric with respect to the map:

r ix M iM

b

x y

Principal CKY tensor in Kerr-NUT-(A)dS


47/57

V. F., D.Kubiznak, Phys.Rev.Lett. 98: 011101, 2007; gr-qc/0605058; D. Kubiznak, V. F., Class.Quant.Grav. 24:F1, 2007; gr-qc/0610144.

Principal CKY tensor in Kerr NUT (A)dS

1

( 1)12

0

,n

k

kk

h db b A d

(The same as in a flat ST in the Carter-type coordinates)

Complete integrability of geodesic motion in


48/57

D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous,Phys.Rev.Lett. 98 :061102, 2007; hep-th/0611083

Complete integrability of geodesic motion ingeneral Kerr-NUT-AdS spacetimes

Vanishing Poisson brackets for integrals of

motion

P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP

0702: 004, 2007; hep-th/0612029

Separability of Hamilton-Jacobi and Klein-


49/57

V. F., P. Krtous , D. Kubiznak , JHEP (2007); hep-th/0611245;Oota and Yasui, PL B659 (2008); Sergeev and Krtous, PRD 77 (2008).

Klein-Gordon equation

Multiplicative separation

2

0b m

Gordon equations in Kerr-NUT-(A)dS ST

Notes on Parallel Transport


50/57

Notes on Parallel Transport

Case 1: 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 cab 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

odesic:

0

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 geodesich


51/57

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)]

Case 2: Parallel transport along null geodesics


52/57

p g g

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

aa

a ba a

b

a

a

l k

l k

w k h l

k

allows one to construct 2 more parallelpropagated vectors and , starting with .a a am n l

al

am

an

We introduce a projector 2 and c dP g l n F P P h


53/57

( )

[ ]

We introduce a projector 2 , and .

One has: 2 0.

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)]

abF

More Recent Developments


54/57

Stationary string equations in the Kerr-NUT-(A)dSspacetime are completely integrable.[D. Kubiznak, V. F., JHEP 0802:007,2008; arXiv:0711.2300]

Solving equations of the parallel transport alonggeodesics [P. Connell, V. F., D. Kubiznak, PRD,78, 024042 (2008);arXiv:0803.3259; D. Kubiznak, V. F., P. Connell, arXiv:0811.0012 (2008)]

Separability of the massive Dirac equation in the

Kerr-NUT-(A)dS spacetime [Oota and Yasui, Phys. Lett.B 659, 688 (2008)]

Einstein spaces with degenerate closed


55/57

Separability of Gravitational Perturbation inGeneralized Kerr-NUT-de Sitter Spacetime[Oota, Yasui, arXiv:0812.1623]

On the supersymmetric limit of Kerr-NUT-AdS

metrics [Kubiznak, arXiv:0902.1999]

p gconformal KY tensor [Houri, Oota and YasuiPhys.Lett.B666:391-394,2008. e-Print: arXiv:0805.0838]

GENERALIZED KILLING-YANO TENSORS


56/57

1 12 3 3

1

3

1 1

3 2

Minimally gauged supergravity (5D EM with Chern-Simo

[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

212

21

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


57/57

Summary

The most general spacetime admitting the PCKY tensor is described by thecanonical metric. 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 and stationary string equations

The geodesic motion in such a spacetime is completely integrable. Theproblem of finding parallel-propagated frames reduces to a set of the firstorder ODE

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

Documents

Valeri P. Frolov- Black Holes and Hidden Symmetries