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

“Everything should be made as simple as possible, but not simpler.” — Albert Einstein

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

2

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 . 5D exact solutions

for stationary black rin

DS

gs (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)

5D vac. stationary black holes

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.

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 2 dimensions

= g

ab ab

D n

R

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.

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)

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

/ 2 , 2n D D n

( 1) ab abR D g

, mass, ( 1 ) rotation parameters,

( 1 ) `NUT' parameters

kM a n

M n

#Total of parameters is D

1

1

00

0

00

0

n

n

J

J

J

J

tStationary - Killing vector ξ ;

Axisymmetric - (n - 1 + ε) Killing vectors ξ ;

When cosmological constant λ and NUT parameters

vanish one has Myers - Perry metric (1986)

Generator of Symmetries

Principal Closed Conformal KY Tensor

[ ]

1-1

2

0

- ,

,

c ab ca b cb a

b

a abaD

h

n

dh

h g g

h

2 - form with the following properties :

(i) Non - degenerate (maximal matrix rank, )

(ii) Closed

is

(iii) Conformal KY tensor :

a primary Killing ve

ctor

For briefness, we call this object a "Principal Tensor"

All Kerr-NUT-AdS metrics in any number of ST dimensions possess a PRINCIPAL TENSOR

(V.F.&Kubiznak ’07)

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;)

Uniqueness Theorem

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.

BRIEF REMARKS ON COMPLETE INTEGRABILITY

2

2

-1

Phase space: Differential manifold with

a symplectic form (non-denerate rank 2 closed, 0).

Observables are scalar functions on .

= is a Hamiltonian vect

Hamiltonian system

D

D

F

M

d

M

X dF

-1

or field associated

with an observable .

The dynamical equations for the Hamiltonian are .

Poisson bracket { , } [ , ].

Integral of motion { , } 0.

Integrals of motion , are in inv

H

F Q

F

dzH X

dt

F Q dF dQ X X

F F H

F Q olution if { , } 0F Q

1 1

Darboux's theorem: Suppose that is a simplectic 2 - form on

an 2 dimensional manifold . Then in a neighrhood of

each point on , there is a coordinate chart in which

= = ... .

is

m m

n m M

q M U

d dq dp dq dp

U

called a Darboux chart. The manifold can be covered by

such charts (Darboux atlas).

Liouville theorem: Dynamical equations in 2D dimensional phase space are completely integrable if

there exist D independent commuting integrals of motion.

General idea: D commuting independent integrals of motion

can be used as coordinates on the phase space.

Denote by the level set for { }. Gradients are linearly

independent is D dimens

i

f i i

f

F

M F dF

M 2

ij

ional submanifold of .

The involution condition implies that | 0,

hence is a Lagrangian submanifold. The vector fields are

tangent to it and mutually commute.

F i j

i

D

L F F

f F

M

X X

M X

0

Denote = and consider the function ( , ) | .

is a point of the phase space, and are its coordinates.

Since 0 on , the integral does not depend on a choice

of the path. ( , ) can be

f

m

M

m

f

d W q P

m q

d M

W q P

used as a generating function, which

produces a canonical transformation from the original ( , ) to

( , )new canonical coordinates ( , ) : . In these

new coordinates the Hamilton's equat

i ii i

i

q p

W q PQ P F Q

P

ions become trivial:

{ , } 0, { , } .i ii i

i

HP P H Q Q H const

P

0 0( , )q p

( , )q p

0q

q

Lagrandgiansubmanifold

Relativistic Particle as a Dynamical System

......

( ...

1

; )

2

Monomial in momenta integ

Preferable co

rals of motion

... imply tha

ordinates in the phase

t is a Ki

space:

( , ),

lling tensor:

( , ) .

a ba b a b

a

a b

b c

aba ab a bx p g x H p x

K

g

K p

K

p p

p

K =

1 1 2{ } 0 [ , ] 0.

Motion of particle in D-dimensional ST is completely

integrable if there exist D independent commuting

Killing tensors (and vecto

0

rs)

K K

2K ,K

If and are 2 monomial integrals

of order and , then is a monomial

integral of order . The corresponding

Killing tensor is called reducible.

n m

n m

(n) (m)

(n) (m)

K K

K K

Metric is a best known example of rank 2 Killing tensor

( ; ) ( 0)

Walker & Penrose '70

ab cKT K( ; )( 0)

Penrose & Floyd '73

a b cKY KT f

( ) ( )

;13

( * , )

a b c ab c c a b

b

a ba

PCCKY h f h db

h g g

h

( )

Hughston & Sommers '75

a is primary KV

is a SKV

t

a ab

bK

( )Einstein space

which admits PCCKY is either

Kerr - NUT - AdS or C - metric

ab abR g

Properties of 4D black holes

Carter ’68 H-J separation 4th integral of

motion

( ; )

; ; ;

Parallel transport : Consider a geodesic, and let be a tangent vector.

Let be a Killing-Yano tensor: 0. Then a vector is

parallel propagated. Really,

0.

a

bab a b c a ab

c b c b ca c ab c ab c

p

k k q k p

p q k p p k p p

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).

«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. 680-689, (1994)

Tidal disruption of a white dwarf by a massive black hole

r

b

M

m

1/3

3 2

Condition of a static disruption:

GMb ( / )

r

Gmr M m b

b

2 .

Denote

D - dimensional AdS - Kerr - NUT metric

has Killing vectors. For complete

integrability of geodesic equations one

needs more integrals of motion.

D n

n

n

Higher dimensions

GENERAL SCHEME

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

Let be - form on the Riemannian manifold.

Then its Hodge dual * is ( - )- form.

Two operations: deri

(...) ( * *)

If

vative

(...

and coder

) 0 is

i

a conformal KY tensor;

If (..

vative .

p

D p

d

d g d

.) 0 is a KY tensor;

If d (...) 0 is a closed conformal

KY tensor.

Killing-Yano family

2

2

...

...is the Killing tensorn

nK f f

is an integral of geodesic motionK p p

1 2 ... parallel propagated

along a

g

i

eo

s a

desic form;

n

nf p

1 2 1 2... ...( ; ) 0,n n

f f Let be a Killing - Yano tensor then

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

Integrability conditions of the KY equation:

1

2Maximal number of independent KY tensor

(D+1)!of rank is

(D-p)!(p+1)!

Maximal number of independent CCKY tensor

of rank

p p

ea b c c c ea bc c c

pf R f

p

(D+1)! is

(D-p+1)!p!p

(a;b) ; ;Example : Killing vector 0,

Maximum number of independent Killing vectors :

1 1[ ] ( 1) ( 1)

2 2

fa b c abc f

D

R

N D D D D D

Flat sacetime (as well as (anti)deSitter space) has the maximum number of

Killing-Yano and closed conformal Killing-Yano tensors. The Killing-Yano

-forms in the flat spacetime allow quite simple desn

1 1 2

1 1

,..., }

1 1

cription. Consider a

set of integer numbers

,..., }, 1 ...

Translationally invariant Killing-Yano -forms

...

Consider another set of 1 integer numbers

,..., }

n n

n n

a a a a a

n

n

a a a a D

n

k dx dx dx

n

a a

1 1 1 2

1 1

,..., } [ 1]

, 1 ...

Rotational Killing-Yano -forms

ˆ ...

Closed conformal Killing-Yano tensors are dual of the above objects.

n n

n n

a a a a a

a a a D

n

k x dx dx

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)

CCKY

KY

CKY

*

R2-KT

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

The Principal CCKY tensor (or simply the Principal Tensor) is a no

matrix rank (2 )

n-denerate

2 r

.

-fo

n

m which is CCKY object. (Some additional requirements will be added later.)

Killing-Yano Tower

Killing-Yano Tower

11

:

KY tensors:

Killing tensors:

...

*

Primary Killing vector:

Secondary Killin

g vectors:

j

j times

jj

jj j

ba baD

j j

h h h h h

k h

K k k

h

CCKY

K

Total number of conserved quantities:

The integrals of motion are functionally

independent and in involution. The

geodesic equations in the AdS-Kerr-NUT

ST ar

( ) ( 1)

e completely integra

1 2

ble.

n n

KV KT

n

g

D

Canonical Coordinates

ˆ( )h m ix m m e ie

We include in the definition of the Principal Tensor

the following requirement:

The 2-form in the ST with 2 dimensions

has non-equal independent eigen-values , so

that there exists n (mutu

h

h D n

n x

ally orthogonal) two-planes.

Canonical coordinates: essential coordinates x

and Killing coordinates . Total number of

such "canonical" coordinates is 2 .

j

n

n

D n

ˆ ˆ ˆ1 1( ) ,

n ng e e e e e e h x e e

The components of Killing tower objects in such

a basis are polynomials in .x

“Off-shell” metrics, which admit the Principal Tensor, allow complete description

ˆ ˆ 1 1( ) ,n n

ab a b a b a bg e e e e e e

1ˆ ( )

0

1,

ni

i

i

e dx e Q A dQ

2 2, , ( ), ( )X

Q U x x X X xU

2 ( )

01

(1 )n n

j j

j

tx t A

12 1 2 ( )

01

(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. Later Krtous, V.F.,

Kubiznak [arXiv:0804.4705 (2008)] and Houri, Oota, and Yasui

[ar

L g L h

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

Arbitrary functions ( ) after substitution into Einstein equations

become polynomials. Their coefficients are just papameters of the metric.

X x

Solutions of the Einstein equations with the cosmological constant (“On-shell” metrics)

2

0

for even.

A similar expression for odd.

nk

k

X b x c x D

D

This is nothing but the Kerr-NUT-(A)dS metric,

written in special (canonical) coordinates.

1. The Principal Tensor (if it exists) generates the Killing tower, which

roughly contains a half of the Ki

lling vectors and a half of the Killing

t so

en

Intermediate Summary

rs. 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.

2 2

2

Hamilton-Jacobi ( )

Klein-Gordo

WKB ~ exp( )

"Dirac eqn= KG

n ( ) 0

Dirac equat

eqn

ion ( )

"

0

S m

m

m

iS

Separation of variables

This is a very special property, which depends both on the

type of the equaion and special choice of the coordinates.

Separation of variables in HJ eqs

1 11 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

( , ) 0

( , (

.

) '

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

, , ),

( , ) ,

( ', , , ; ) 0

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 mS S q C S q C C S q C

C

C

stem is integrable in the Liouville sence.

1 0

( )n m

k k

k

S w S x

2 2 1 2 2 1

20 0

1 1')( ( ( ) ) ( )

m mn k n k

kk

k k

S x xcX X

Separability of the Hamilton–Jacobi equation in canonical coordinates

0ab

a b

Sg S S

V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007)

Separation of variables in HJ and KG equations in 5D ST (V.F. and Stojkovic ’03)

Separability of the Klein–Gordon equation

2( ) 0 1 0

( ) k k

n mi

k

R x e

2 1 2 2 1

0 0

( ) ( ( ) ) ( ) 0m m

n k n k

k k

k k

X RX R R x b x R

x X

,

V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007)

Weakly charged higher dimensional rotating black holes

1Hamiltonian ( )( )

2

ab

a a b bH g p qA p qA

2HJ equation ( )( )ab

a ba b

S Sg qA qA

x x

2

Klein-Gordon equation

( )( ) 0ab

a a b bg iqA iqA

0 0 0b a a b

ab a bF A A

0 (in Ricci flat)a b

b a aA Q

2

2

2 2 2 2

0 (0)

0

[ ] 0,

2

ab

a b

S Sg M

x x

M

M e e

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

[V.F. and Krtous, 2011]

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)

Metrics adm

Separabilit

itting a pr

y of gravitational per

incipal Killing-Yano

te

turbations in

Kerr-NUT-(A)dS Spacetime (Oota

nsor with torsion (Houri, Kubiznak,

Warnick

and

and Y

Yasui '

asui

10)

'12)

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

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 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)]

Possible generalizations to degenerate PCKY tensor and non-vacuum STs

Infinite set of new interesting completely integrable dynamical systems.

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

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

BIG PICTURE

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

TO HIDE?

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’