Valeri P. Frolov,

Univ. of Alberta, Edmonton

International Conference on Geometry, Integrability and Quantization,

Varna, June 3 - 8, 2019

In total, since making history with the first-ever direct detection of gravitational waves in 2015, the LIGO-Virgo network has spotted evidence for two neutron star mergers, 13 black hole mergers, and one possible black hole-neutron star merger.

“Photo” of a black hole in M87 by EHT. [6 infrared telescopes form a giant interferometer]

Kerr metric besides evident explicit symmetries

possesses also (unexpected) hidden symmetry.

Complete integrability of geodesic equations

[Carter, 1968];

Complete separability of massless field equ

•

•

Main problems:

Higher dimensional generalizations of these results;

Separability

ati

of

ons

[Carter, 1968; T

Proca (massive vector

eukolsky,

) field e

q

1972]

uat

.

ions

•

•

“Black Holes, Hidden Symmetry and Complete Integrability”

V.F., Pavel Krtous and David Kubiznak

Living Reviews in Relativity, 20 (2017) no.1, 6; arXiv:1705.05482 (2017)

"Separation of variables in Maxwell equations in Plebanski-Demianski spacetime ",

V. F, P. Krtous and D. Kubiznák, Phys.Rev. D97 (2018) no.10, 101701;

"Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes" , P. Krtous, V.F.

and D. Kubiznák, Nucl.Phys. B934 (2018) 7-38;

"Massive Vector Fields in Kerr-NUT-(A)dS Spacetimes: Separability and Quasinormal Modes" ,

V.P., P. Krtous, D. Kubiznák, and J. E. Santo, Phys.Rev.Lett. 120 (2018) 231103;

"Duality and µ separability of Maxwell equations in Kerr-NUT-(A)dS spacetimes",

V. F. and P. Krtous, Phys.Rev. D99 (2019) no.4, 044044.

= + + − +

+

=

Let be - form on the Riemannian manifold.

1 1

If is a conformal KY ten

( ) ( ) 1

sor .

1[ (...)]

(...) 0

X

p

X Xp D p

Killing-Yano family

= + + −

=

+

If is a conformal KY ten

1 1( )

(...) 0,

( )

s

1

o

1

r.

X X Xp D p

= +

=

+ − +

=

= +

If (...) 0,

1 1( ) ( )

1 1

is a KY

tensor:

1( )

1

X

X

X Xp D p

Xp

= + + − +

= = =

= − +

If (...) 0, is a CCKY tenso

1 1( ) ( )

r

1 1

:

1 ( )

1

X

X

db

X Xp D p

XD p

= +

= + + − +

= −

+ − +

Then is a

Let be conformal KY - f

lso a conformal KY tensor

1 1( ) ( ) ( ),

1

orm

1 1( ) ( )

1 1

1

.

X

X

p

X Xp D

X Xp D p

D

p

p p

Hodge duality

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)

...

( ... )

A symmetric tensor which obeys an equation

0 is called a Killing tensor

.

Killing tensor

ab c

d ab c

k

k =

1 1

1 1

(1) (2)

(1) ... (2)

...

Let and are two KY tensors of rank . Then

= is a rank-2 Killing tensor.s

s

ac cab b

c c

f f s

k f f−

−

1 1 1 1 1... ... ( ...

If and are two symmetric tensors of the order

and , respectively, then ( ) is again a

symmetric tensor of the order 1:

Schouten-Nijenhuis bracket

r s

SN

a a cb b e a

r s

r s

c ra− −

=

+ −

=

a b

c a,b

1 1 1 11 1 1... ) ( ... ... )

SN bracket is invariant under a change of to

for any torsion-free corariant d

;

( ) is Schouten Nijenhuis bracket.

erivative.

s sr rcb b e b ba ca a

e e

SN

b sb a− −− −

−

= −c a,b

If and are two Killing tensors of the rank

and , respectively, then ( ) is again a

Killing tensor of the rank 1.

SNr s

r s

=

+ −

a b

c a,b

1-1

A special case ( ): a non-

degenerate closed conformal rank 2 KY tensor

1( ) ;

1

- , .

is a (primary) Killin

principal tensor

Principal tensor

= −

= =

X

b

c ab ca b cb a a baD

a

h

h X hD

h g g h

g vector. This is easy to

show in an Einstein ST, when R .= ab abg

Killing-Yano Tower

−

=

=

=

=

=

1(0) 1

( ) (0)

:

KY tensors:

Killing tensors:

Primary Killing vector:

...

*

Secondary Killing ve ct ors:

j

j times

jj

j j j

ba baD

j j

h h h h h

k h

K k k

K

C

h

CKY

A total number of conserved quantities:

+ + − +

= +

= + =

2

These Killing vectors and tensors

(i) are independent;

(ii) mut

( ) ( 1) 1

ually SN com t .

2

mu e

KV KT g

n D

D n

n n

Relativistic Particle

= =

= = =

12

Canonical coordinates in the phase space:

( , ). Hamiltonian ( , ) .

Symplectic fo

A relativistic particle in a curv

r

ed ST:

( )

m

.

=

, 0,

.

a b aba ab a b

aa

a

aa a a b a

a

dxx x

x p g x H p x g p p

dx

xd

dp

x x

The Euler-Lagrange equations 0

are equivalent to the Poisson equations

, .

a b

a

a

a a

a

x x

H Hx p

p x

=

= = −

=

=

=

1 2

1 2

... ...Let k be a Killing tensor then ...

Poisson-commutes with the Hamiltonian [ , ] 0.

Henc

Two integrals of motion are in

e, is an integral of m

involution, [ , ] 0,

iff ( )

otion.

0SN

a b a ba bK k

K

p p

K H

K

K

k k,

=

.

Liouville theorem: Dynamical equations in 2N dimensional phase space are completely integrable if there exist N independent

commuting integrals of motion.

This means that a solution of equations of motionof such a system can be obtained

by quadratures, i.e. by a finite number of algebraicoperations and integrations.

Geodesic equations in a ST admitting the

principal tensor are completely integrable.

Canonical coordinates

= +

+ +

= =

= =

2

0

2

0

Darboux frame:

ˆ2 , h= x

, Q ,

,

ˆ ˆ

ˆ ˆ ˆ, Q

ˆ ˆg=

,

( )

cab a bcQ h h e x e

h e

D n

x

e e

e e e e

e

e

e e

e

x

/

(i) It has different 2-eigenplanes;

(ii) are different and functionally independent in a domain near

given point ;

Principal tensor is no

(iii) can be u

n-denera

s

ed as coordin

t

e

:

t

e

a

n

x U

p n

h

x

s in this domain.

( ) ( )

( )

,

( )

( ) prime and secondary Killing vectors are

commuting. Moreover one has 0 0

According to Frobeneus theorem, there exist local

foliation such that are tangent to -dimen

.

s

j j

j

a

i

a

n

n

h

l

L x

+

= =

i

i

( )

ional

surfaces , and one can introduce coordinates

such that .a

i a

x const

l

=

=

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

2, 2

2ab abR g D n

D= = +

−

, mass, ( 1 ) rotation parameters,

( 1 ) `NUT' parameters

kM a n

N n

− − − +

− − −

# 2 1Total of parameters is n −

The metric coefficients of the Kerr-NUT-(A)dS

metric written in the canonical coordinates are

polynomials of one variable and the parameters

of these polynomials are constants that specify

a solution. If one substitutes these polynomials

by arbitrary functions of the same one variable

we call the corresponding metric off-shell.

All Kerr-NUT-AdS metrics and their off-shell extensions 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

If the metric pos

The off-shell ver

sesses a principa

sion of the Kerr-NUT-(A)dS metric

p

l tensor it is an

off-shell version

ossesses the prin

of the Kerr-NUT-

cipal te

(A)dS me

nsor.

tric.

Separability of the Klein–Gordon equation

2( ) 0− =m,

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

0

[Sergeev and Krtous (2008); Kolar and Krtous (2015)]

( ); ( ); .

[ , ] [ , ] [ ] 0;

ab ab aa b j a j b j j a

j k j k j k

K g K k L i

K K K L L L

= = − = −

= = =

1

0

2.

, ;

exp( );

( ') ' ' 0

− +

=

= =

=

+ + =

j j j j

n

k k

k

K

R i

X YX R R R

x X

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

Separability of HD Maxwell equations

[S.A. Teukolsky, Phys.Rev.Lett., 29, 1114-1118 (1972)]

Maxwell equations in 4D type D ST :

(i) can be decoupled, and (ii) the decoupled

equations allow complete separation of variables.

Method: Newman-Penrose equations;

Special choice of null frames;

( ) - fieldF i F anti self dual= −F

( )

" ’ - ",

1712 2017 138.

He proposed a specia

Remarkable progress b

l ansatz for em potential

in canonical coordinates and demonstrated that

Max

y Oleg Lunin:

w

Maxwell s equations in the Myers Perry geometry JHEP

ell equations in Kerr-deSitter spacetime are

separable.

Our recent results in:

"Separation of variables in Maxwell equations in Plebanski-Demianski spacetime",

V. F, P. Krtous and D. Kubiznák, Phys.Rev. D97 (2018) no.10, 101701;

"Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes", P. Krtous, V.F.

and D. Kubiznák, Nucl.Phys. B934 (2018) 7-38.

(i) Covariant description;

(ii) Ansatz for the field in terms of principal tensor ;

(iii) Analitical proof of separability;

(iv) Results are valid for any metric which admits

the principal tensor Fo

only

r off-shell Kerr-NUT-(A)dS STs.

1

0

1Polarization tensor: ,

1

( )

, exp ).

,

(

bc cab ab

n

kk

a

kZ Z

i

g h

R

i B

i

− +

=

= =

=+

+ =

h

A

B

B

(i) Lorenz condition: 0 second order

partial differential equation for , 0,

which allows the separation of variables;

(ii) Maxwell equations 0 in the Lorenz

gauge can be writt

aa

abb

A

Z DZ

F

=

=

=

en in the form 0.

This equation is separable. The separated

equations are the same as for the Lorenz

equation, with an additional condition that

one of sepation constantsis pu

abbB DZ =

t to zero.

Hodge duality:

0, , 0, 0

1,

1

( ) (

) e

,

ˆˆ ˆ ˆ ˆ,

x

p[ ].

0, , 0.

,

Maxwell equations in 4D

d d d

d d

i

Z R Y y i i

Z

r m

=+

=

= = = =

= = = =

− +

=

F F A F F

F F F

Bh

A B

F A F

( )[V. F. and Pavel Krtouš, Phys.Rev. D99 2019 044044.]

1ˆ ˆ ˆ ˆ ˆ, , ,ˆ

1

ˆ ˆ ˆ( ) ( ) exp[ ],

'ˆ ,

ˆ

-duality

r

r r

y

y y

R rR R

q qm

Y yY

Zi m

Z R r Y y

Yq qm

i im

=

= =

− + −

= − −

= −+

= − +

−

A B Bh

Massive vector field in

Kerr-NUT-(A)dS spacetime

2Proca (1936) equation: 0

implies Lorenz equation 0

ab ab

aa

F m A

A

+ =

=

( )

"Constraining the mass of dark photons and axion-like

particles through black-hole superradiance",V. Cardoso

Ó. Dias,G. Hartnett,M. Middleton,P. Pani,and J.Santos,

JCAP 1803 2018 043,1475-7516,arXiv:1801.01420

=

In our recent paper we prove separability of Proca

equations in off-shell Kerr-NUT-(A)dS spacetime

in any number of dimensions. We use the same

ansatz , to solve the Lorenz equation as earlier,

an

ZA B

2

d to show that the Proca equation is separable. The

only difference is that a separation constant, that

for Maxwell eqns vanishes, in the Proca case takes

the value m .

Ultralight bosons in the dark sector can have potentially

observable consequencies for astrophysical black holes.

(i) Gaps in the black hole mass-spin plane, to be revealed

by black hole surveys;

(ii) Gravitational wave 'sirens';

(iii) Significant transfers of mass-energy from the black hole

into surrounding bosonic 'cloud'.

•

•

•

Spacetimes with a principal tensor possess remarkable properties;

Complete integrability of geodesic equation

s;

Complete separability of

important fi

Brief summary

•

•

•

eld equations;

PT allows one to construct canonical coordinates, in which the

ST properties are more profound;

PT off-shell vesion of the Kerr-NUT-(A)dS spacetime;

Arbitrary number of ST d

•

•

imensions;

Interesting astrophysical applications (quasi-normal modes

of Proca field in the Kerr BH.

Future work: Separability of HD metric perturbation equations???