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???