Hidden symmetries and test fields inrotating black hole spacetimes
part II
Separability
Pavel Krtous
Charles University, Prague, Czech Republic
http://utf.mff.cuni.cz/~krtous/
with
Valeri P. Frolov, David Kubiznak
Atlantic General Relativity Conference
Antigonish, June 6, 2018
Contents
β’ KerrβNUTβ(A)dS geometry
β’ Integrability of the geodesic motion
β’ Separability
β’ Separability of the scalar field equation
β’ Separability of the vector field equation
β’ Quasi-normal modes
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 1
Off-shell KerrβNUTβ(A)dS geometry
geometry given by the existence of hidden symmetries encoded in
the principal tensor
(non-degenerate closed conformal KillingβYano tensor of rank 2)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 2
Off-shell KerrβNUTβ(A)dS geometryfor simplicity
even dimensionD = 2N
g =
NβΒ΅=1
[UΒ΅XΒ΅
dx2Β΅ +
XΒ΅UΒ΅
(Nβ1βj=0
A(j)Β΅ dΟj
)2]
explicit polynomials in coordinates xΒ΅
A(k) =
NβΞ½1,...,Ξ½k=1Ξ½1<Β·Β·Β·<Ξ½k
x2Ξ½1. . . x2
Ξ½kA(j)Β΅ =
NβΞ½1,...,Ξ½j=1Ξ½1<Β·Β·Β·<Ξ½jΞ½i 6=Β΅
x2Ξ½1. . . x2
Ξ½jUΒ΅ =
NβΞ½=1Ξ½ 6=Β΅
(x2Ξ½ β x2
Β΅)
unspecified N metric functions of one variable
XΒ΅ = XΒ΅(xΒ΅)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 3
On-shell KerrβNUTβ(A)dS geometryfor simplicity
even dimensionD = 2N
g =
NβΒ΅=1
[UΒ΅XΒ΅dx2
Β΅ +XΒ΅
UΒ΅
( Nβ1βj=0
A(j)Β΅ dΟj
)2]
Einstein equations β
XΒ΅ = Ξ»
NβΞ½=1
(a2Ξ½ β x2
Β΅
)β 2bΒ΅ xΒ΅
Parameters:
Ξ» cosmological parameter related to the cosmological constant Ξ = (2N β 1)(N β 1)Ξ»
bΒ΅ mass and NUT parameters
aΒ΅ rotational parameters
freedom in scaling of coordinates β one parameter can be fixed by a gauge condition
exact interpretation of parameters depends on coordinate ranges, signature, and gauge choices
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 4
KerrβNUTβ(A)dS geometry β Lorentzian signature
g =
NβΒ΅=1
[UΒ΅XΒ΅dx2
Β΅ +XΒ΅
UΒ΅
( Nβ1βj=0
A(j)Β΅ dΟj
)2]
Wick rotation:
time: Ο = Ο0 radial coordinate: xN = ir mass: bN = im
Gauge condition:
a2N = β1
Ξ» (suitable for limit Ξ»β 0)
New Killing coordinates:
t = Ο +βk
A(k+1)ΟkΟΒ΅aΒ΅
= Ξ»Ο ββk
(A(k)Β΅ β Ξ»A(k+1)
Β΅
)Οk
barred quantities refer to ranges of indices given by N β N = N β 1
(i.e., Β΅ = 1, . . . , N and j = 0, . . . , N β 1, etc.)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 5
KerrβNUTβ(A)dS geometry β Lorentzian signature
g = ββr
Ξ£
(βΞ½
1 + Ξ»x2Ξ½
1 + Ξ»a2Ξ½
dtββΞ½
J(a2Ξ½)
aΞ½(1 + Ξ»a2Ξ½)UΞ½
dΟΞ½
)2
+Ξ£
βrdr2
+βΒ΅
(r2+x2Β΅)
βΒ΅/UΒ΅dx2
Β΅ +βΒ΅
βΒ΅/UΒ΅(r2+x2
Β΅)
(1βΞ»r2
1+Ξ»x2Β΅
βΞ½
1+Ξ»x2Ξ½
1+Ξ»a2Ξ½
dt +βΞ½
(r2+a2Ξ½)JΒ΅(a2
Ξ½)
aΞ½(1+Ξ»a2Ξ½) UΞ½
dΟΞ½
)2
βr = βXN =(1βΞ»r2
)βΞ½
(r2+a2
Ξ½
)β 2mr Ξ£ = UN =
βΞ½
(r2 + x2Ξ½)
βΒ΅ = βXΒ΅ =(1+Ξ»x2
Β΅
)J (x2
Β΅) + 2bΒ΅xΒ΅ UΒ΅ =βΞ½
Ξ½ 6=Β΅
(x2Ξ½ β x2
Β΅)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 6
KerrβNUTβ(A)dS geometry β Lorentzian signature
g = ββr
Ξ£
(βΞ½
1 + Ξ»x2Ξ½
1 + Ξ»a2Ξ½
dtββΞ½
J(a2Ξ½)
aΞ½(1 + Ξ»a2Ξ½)UΞ½
dΟΞ½
)2
+Ξ£
βrdr2
+βΒ΅
(r2+x2Β΅)
βΒ΅/UΒ΅dx2
Β΅ +βΒ΅
βΒ΅/UΒ΅(r2+x2
Β΅)
(1βΞ»r2
1+Ξ»x2Β΅
βΞ½
1+Ξ»x2Ξ½
1+Ξ»a2Ξ½
dt +βΞ½
(r2+a2Ξ½)JΒ΅(a2
Ξ½)
aΞ½(1+Ξ»a2Ξ½) UΞ½
dΟΞ½
)2
βr = βXN =(1βΞ»r2
)βΞ½
(r2+a2
Ξ½
)β 2mr Ξ£ = UN =
βΞ½
(r2 + x2Ξ½)
βΒ΅ = βXΒ΅ =(1+Ξ»x2
Β΅
)J (x2
Β΅) + 2bΒ΅xΒ΅ UΒ΅ =βΞ½
Ξ½ 6=Β΅
(x2Ξ½ β x2
Β΅)
Rotating black holes with NUTs and ΞPavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 6
Off-shell KerrβNUTβ(A)dS geometry
g =
NβΒ΅=1
[UΒ΅XΒ΅
dx2Β΅ +
XΒ΅UΒ΅
(Nβ1βj=0
A(j)Β΅ dΟj
)2]
explicit function polynomial in coordinates xΒ΅ (symmetric polynomials)
A(k) =Nβ
Ξ½1,...,Ξ½k=1Ξ½1<Β·Β·Β·<Ξ½k
x2Ξ½1. . . x2Ξ½k A(j)
Β΅ =Nβ
Ξ½1,...,Ξ½j=1Ξ½1<Β·Β·Β·<Ξ½jΞ½i 6=Β΅
x2Ξ½1. . . x2Ξ½j UΒ΅ =
NβΞ½=1Ξ½ 6=Β΅
(x2Ξ½ β x2Β΅)
unspecified N metric functions of one variable
XΒ΅ = XΒ΅(xΒ΅)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 7
Killing tower and hidden symmetries
g =
NβΒ΅=1
[UΒ΅XΒ΅dx2
Β΅ +XΒ΅
UΒ΅
( Nβ1βj=0
A(j)Β΅ dΟj
)2]
Principal tensor
h =βΒ΅
xΒ΅ eΒ΅ β§ eΒ΅
Primary Killing vector
ΞΎ = βΟ0
Hidden symmetries β Killing tensors
k(j) =βΒ΅
A(j)Β΅
(eΒ΅eΒ΅ + eΒ΅eΒ΅
) Explicit symmetries β Killing vectors
l(j) = βΟj
[k(i),k(j)
]NS
= 0[k(i), l(j)
]NS
= 0[l(i), l(j)
]NS
= 0
Darboux frame
forms:
eΒ΅ =(UΒ΅XΒ΅
)12dxΒ΅ eΒ΅ =
(XΒ΅
UΒ΅
)12
Nβ1βj=0
A(j)Β΅ dΟj
vectors:
eΒ΅ =(XΒ΅
UΒ΅
)12βxΒ΅ eΒ΅ =
(UΒ΅XΒ΅
)12
Nβ1βk=0
(βx2Β΅)Nβ1βk
UΒ΅βΟk
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 KerrβNUTβ(A)dS geometry 8
Integrability of the geodesic motion
Killing tower
hidden symmetries β Killing tensors of rank 2
k(j)
explicit symmetries β Killing vectors
l(j)
commutation in sense of NijenhuisβSchouten brackets[k(i),k(j)
]NS
= 0[k(i), l(j)
]NS
= 0[l(i), l(j)
]NS
= 0
Conserved quantities
observables quadratic in momentum
Kj = p Β· k(j) Β· pobservables linear in momentum
Lj = l(j) Β· p
Hamiltonian
k(0) = g β H =1
2K0 =
1
2p2
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Integrability 9
Integrability of the geodesic motion
Killing tower
hidden symmetries β Killing tensors of rank 2
k(j)
explicit symmetries β Killing vectors
l(j)
commutation in sense of NijenhuisβSchouten brackets[k(i),k(j)
]NS
= 0[k(i), l(j)
]NS
= 0[l(i), l(j)
]NS
= 0
Conserved quantities
observables quadratic in momentum
Kj = p Β· k(j) Β· pobservables linear in momentum
Lj = l(j) Β· p
Hamiltonian
k(0) = g β H =1
2K0 =
1
2p2
Integrability
D observables Kj and Lj which are in involution and commute with the Hamiltonian{Ki, Kj
}= 0
{Ki, Lj
}= 0
{Li, Lj
}= 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Integrability 9
Integrability of the geodesic motion
Killing tower
hidden symmetries β Killing tensors of rank 2
k(j)
explicit symmetries β Killing vectors
l(j)
commutation in sense of NijenhuisβSchouten brackets[k(i),k(j)
]NS
= 0[k(i), l(j)
]NS
= 0[l(i), l(j)
]NS
= 0
Conserved quantities
observables quadratic in momentum
Kj = p Β· k(j) Β· pobservables linear in momentum
Lj = l(j) Β· p
Hamiltonian
k(0) = g β H =1
2K0 =
1
2p2
Integrability
D observables Kj and Lj which are in involution and commute with the Hamiltonian{Ki, Kj
}= 0
{Ki, Lj
}= 0
{Li, Lj
}= 0
Separability of HamiltonβJacobi equation
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Integrability 9
Separability of field equations
The field equations on the off-shell KerrβNUTβ(A)dS background can be separated for:
β’ Scalar field [2007] VF, PK, DK, A. Seregyeyev
β’Dirac field [2011] T. Oota, Y. Yasui, M. Cariglia, PK, DK
β’Vector field [2017] O. Lunin, VF, PK, DK
(including the electromagnetic case)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability 10
Standard method of separation of variables
Equation for a sum of terms depending on different variables
NβΒ΅=1
fΒ΅(xΒ΅) = S
implies
fΒ΅(xΒ΅) = SΒ΅
where SΒ΅ are constants satisfyingNβΒ΅=1
SΒ΅ = S
Solution is given by N β 1 independent separation constants
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability 11
Standard method of separation of variables
Equation for a sum of terms depending on different variables
NβΒ΅=1
fΒ΅(xΒ΅) = S
implies
fΒ΅(xΒ΅) = SΒ΅
where SΒ΅ are constants satisfyingNβΒ΅=1
SΒ΅ = S
Solution is given by N β 1 independent separation constants
U-separation of variables
Equation for a composition of terms depending on different variables
NβΒ΅=1
1
UΒ΅fΒ΅(xΒ΅) = C0
implies
fΒ΅(xΒ΅) = CΒ΅
where CΒ΅ =Nβ1βj=0
Cj(βx2Β΅)Nβ1βj are polynomials of degree N β 1 with the same coefficients Cj
Solution is given by N β 1 independent separation constants Cj
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability 11
U-separation of variables
Equation for a composition of terms depending on different variables
NβΒ΅=1
1
UΒ΅fΒ΅(xΒ΅) = C0 UΒ΅ =
NβΞ½=1Ξ½ 6=Β΅
(x2Ξ½ β x2
Β΅)
implies
fΒ΅(xΒ΅) = CΒ΅ CΒ΅ =
Nβ1βj=0
Cj(βx2Β΅)Nβ1βj
CΒ΅ are polynomials in variable x2Β΅ of degree Nβ1 with the same coefficients Cj
Solution is given by N β 1 independent separation constants Cj, j = 1, . . . , Nβ1
Related to the fact that A(j)Β΅ and
(βx2Β΅)Nβ1βj
UΒ΅are inverse matrices
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability 12
Separability of the scalar field equation
Massive scalar field equation
οΏ½Οβm2Ο = 0 οΏ½ = β Β· g Β·β
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 13
Separability of the scalar field equation
Scalar field equation as the eigenfunction equation
K0 Ο = K0 Ο K0 = ββ Β· g Β·β K0 = βm2
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 13
Separability of the scalar field equation
Scalar field equation as the eigenfunction equation
K0 Ο = K0 Ο K0 = ββ Β· g Β·β
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅)
Eigenfunction equation in coordinates
K0 Ο = K0 Ο ββΞ½
1
UΞ½
1
RΞ½
((XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½
)= K0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 13
Separability of the scalar field equation
Scalar field equation as the eigenfunction equation
K0 Ο = K0 Ο K0 = ββ Β· g Β·β
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅)
Eigenfunction equation in coordinates
K0 Ο = K0 Ο ββΞ½
1
UΞ½
1
RΞ½
((XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½
)οΈΈ οΈ·οΈ· οΈΈ
= KΒ΅
= K0
Separated ODEs (XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ β KΞ½RΞ½ = 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 13
Separation constants as eigenvalues
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅ ; Kj , Lj )
Separated ODEs (XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ β KΞ½RΞ½ = 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 14
Separation constants as eigenvalues
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅ ; Kj , Lj )
Separated ODEs (XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ β KΞ½RΞ½ = 0
Symmetry operators corresponding to the Killing tower
Kj = ββa kab(j) βb Lj = βi la(j)βa
Commutativity of the symmetry operators[Kk,Kl
]= 0
[Kk,Ll
]= 0
[Lk,Ll
]= 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 14
Separation constants as eigenvalues
Multiplicative separation ansatz
Ο =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅ ; Kj , Lj )
Separated ODEs (XΞ½R
β²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ β KΞ½RΞ½ = 0
Symmetry operators corresponding to the Killing tower
Kj = ββa kab(j) βb Lj = βi la(j)βa
Commutativity of the symmetry operators[Kk,Kl
]= 0
[Kk,Ll
]= 0
[Lk,Ll
]= 0
Common eigenvalue problem with the eigenvalues given by the separation constants
KjΟ = KjΟ LjΟ = LjΟ
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Scalar field 14
Separability of the vector field equation
Proca field equation
β Β· F = m2 A F = dA
Vacuum Maxwell equations: m2 = 0
β Β· F = 0 F = dA
Lorenz condition
β Β· A = 0 β Proca equation
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 15
Luninβs ansatz
Covariant form of the ansatz
A = B Β·βZ
Polarization tensor B
B = (g β Ξ²h)β1 B β‘ B(Ξ²) depends on an auxiliary parameter Ξ²
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 16
Luninβs ansatz
Covariant form of the ansatz
A = B Β·βZ
Polarization tensor B
B = (g β Ξ²h)β1 B β‘ B(Ξ²) depends on an auxiliary parameter Ξ²
Multiplicative separation ansatz
Z =
NβΒ΅=1
RΒ΅
Nβ1βk=0
exp(iLkΟk
)RΒ΅ = RΒ΅(xΒ΅)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 16
Vector field equations
Proca equation follows from the Lorenz condition and:
[οΏ½ + 2Ξ² ΞΎ Β·B Β·β
]Z = m2Z
m
βΞ½
1
UΞ½
1
RΞ½
((1+Ξ²2x2
Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½
)= C0 β‘ m2
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 17
Vector field equations
Proca equation follows from the Lorenz condition and:
[οΏ½ + 2Ξ² ΞΎ Β·B Β·β
]Z = m2Z
m
βΞ½
1
UΞ½
1
RΞ½
((1+Ξ²2x2
Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½
)οΈΈ οΈ·οΈ· οΈΈ
= CΞ½
= C0 β‘ m2
Separated ODEs:
(1+Ξ²2x2Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½ β CΞ½RΞ½ = 0
CΞ½ =
Nβ1βj=0
Cj(βx2Ξ½)Nβ1βj separation constants Cj, j = 0, . . . , N β 1
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 17
Vector field equations
Lorenz condition:
β Β·B Β·βZ = 0
m
βΞ½
1
UΞ½
1
1+Ξ²2x2Ξ½
1
RΞ½
((1+Ξ²2x2
Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½
)= 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 18
Vector field equations
Lorenz condition:
β Β·B Β·βZ = 0
m
βΞ½
1
UΞ½
1
1+Ξ²2x2Ξ½
1
RΞ½
((1+Ξ²2x2
Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½
)οΈΈ οΈ·οΈ· οΈΈ
= CΞ½
= 0
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 18
Vector field equations
Lorenz condition:
β Β·B Β·βZ = 0
m
βΞ½
1
UΞ½
1
1+Ξ²2x2Ξ½
1
RΞ½
((1+Ξ²2x2
Ξ½)( XΞ½
1+Ξ²2x2Ξ½
Rβ²Ξ½
)β²β L2
Ξ½
XΞ½RΞ½ + iΞ²
1βΞ²2x2Ξ½
1+Ξ²2x2Ξ½
Ξ²2(1βN)LRΞ½
)οΈΈ οΈ·οΈ· οΈΈ
= CΞ½οΈΈ οΈ·οΈ· οΈΈβ C(Ξ²2)
= 0
Condition on Ξ²:
C(Ξ²2) β‘Nβ1βj=0
CjΞ²2j = 0
Ξ²2 must be a root of the polynomial with coefficients given by the separation constants
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 18
Polarizations (just guessing)
Condition on Ξ²:
C(Ξ²2) β‘Nβ1βj=0
CjΞ²2j = 0
Ξ²2 must be a root of the polynomial with coefficients given by the separation constants
Polarization modes
choice of the root β choice of the polarization
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 19
Polarizations (just guessing)
Condition on Ξ²:
C(Ξ²2) β‘Nβ1βj=0
CjΞ²2j = 0
Ξ²2 must be a root of the polynomial with coefficients given by the separation constants
Polarization modes
choice of the root β choice of the polarization
β’ N β 1 roots
β’ N β 1 complex polarizations
β’ 2N β 2 = D β 2 real polarizations
β’ 1 polarization missing!
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Separability: Vector field 19
Quasi-normal modes of Proca field in 4 dimensions
[ with J. E. Santos ]
β’ Kerr solution in 4 dimensions
β’ massive vector field
β’ numerical search for quasi-normal modes
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 20
Quasi-normal modes of Proca field in 4 dimensions
[ with J. E. Santos ]
β’ Kerr solution in 4 dimensions
β’ massive vector field
β’ numerical search for quasi-normal modes
β’ separability allows much more effective calculations
β’ agreement with previous results based on numeric solution of PDEs and analytic aproximations
β’ possibility to extend a βcomputableβ range of parameters
β’ recovering 2 polarizations (out of 3)
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β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β
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ββββββββββ
βββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
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βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ β β β
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β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β
β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β
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οΏ½
οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½-οΏ½οΏ½
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Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 20
Quasi-normal modes of Proca field in 4 dimensions
Sam R. Dolan: Instability of the Proca field on Kerr spacetime, arXiv 1806.01604 (yesterday)
All three polarizations recovered!
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οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½
FIG. 1. Growth rate of the fundamental (n = 0) corotating dipole (m = 1) modes of the Proca field,
for the three polarizations S = β1 [solid], S = 0 [dashed] and S = +1 [dotted], and for BH spins of
a/M β {0.6, 0.7, 0.8, 0.9, 0.95, 0.99, 0.995}. The vertical axis shows the growth rate Οβ1 = (GM/c3)Im(Ο) on
a logarithmic scale, and the horizontal axis shows MΒ΅.
S n ΟR ΟI Ξ½R Ξ½I
β1 0 0.35920565 2.36943Γ 10β4 β0.60028373 β3.336873Γ 10β4
0 0 0.38959049 2.15992Γ 10β6 1.84437950 β9.497225Γ 10β7
+1 0 0.39606600 2.565202Γ 10β10 0.21767208 β3.367574Γ 10β11
β1 1 0.38814529 5.996892Γ 10β5 β0.64326832 β9.391193Γ 10β5
β1 2 0.39509028 1.781747Γ 10β5 β0.65428679 β2.863793Γ 10β5
TABLE I. Parameters for the bound state modes shown in Fig. 3, with m = 1, a/M = 0.99 and MΒ΅ = 0.4.
overtones of the dominant S = β1 mode will grow more substantially more rapidly than the funda-
mental mode. In essence, this is because in Eq. (2) the coefficient depends on the overtone n and
polarization S, whereas the index depends only on S.
Figure 5 shows the growth rate of the higher modes of the S = β1 polarization with azimuthal
numbers m = 2, 3 and 4. The superradiant instability persists for higher modes at larger values of
MΒ΅, but the rate becomes insignificant for MΒ΅οΏ½ 1, due to the exponential fall off of MΟmaxI with
MΒ΅ seen in Fig. 5.
Figure 6 highlights the maximum growth rate for the dominant S = β1, m = 1 mode. For
a = 0.999M , we find a maximum growth rate of MΟI β 4.27 Γ 10β4 which occurs at MΒ΅ β 0.542.
This corresponds to a minimum e-folding time of Οmin β 2.34 Γ 103(GM/c3). For comparison, a
11
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 21
Quasi-normal modes of Proca field in 4 dimensions
So, how is it with those polarizations?
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Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 22
Summary
Hidden symmetries encoded in the principal tensor determine geometry
β’ Off-shell KerrβNUTβ(A)dS spacetimes
β includes generally rotating BH in higher dimensions
β includes charged BH in 4 dimensions
β includes conformally rescaled PlebanskiβDemianski metric
These spacetimes have nice properties:
β’ Integrability of the geodesic motion
β’ Separability of standard field equations
β scalar field
β Dirac field
β vector field (including EM)
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Summary 23
Summary
Hidden symmetries encoded in the principal tensor determine geometry
β’ Off-shell KerrβNUTβ(A)dS spacetimes
β includes generally rotating BH in higher dimensions
β includes charged BH in 4 dimensions
β includes conformally rescaled PlebanskiβDemianski metric
These spacetimes have nice properties:
β’ Integrability of the geodesic motion
β’ Separability of standard field equations
β scalar field
β Dirac field
β vector field (including EM)
suitable for calculation of quasi-normal modes and
study of an instability of the Proca field
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Summary 23
References
KerrβNUTβ(A)dS and Hidden symmetries
β’ Frolov V. P., Krtous P., Kubiznak D.: Black holes, hidden symmetries, and complete integrability,Living Rev. Relat. 20 (2017) 6, arXiv:1705.05482
Scalar field
β’ Frolov V. P., Krtous P., Kubiznak D.: Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes,JHEP02(2007)005, arXiv:hep-th/0611245
β’ Sergyeyev A., Krtous P.: Complete Set of Commuting Symmetry Operators for KleinβGordon Equation in . . . Spacetimes,Phys. Rev. D 77 (2008) 044033, arXiv:0711.4623
Dirac field
β’ Oota T., Yasui Y.: Separability of Dirac equation in higher dimensional KerrNUTde Sitter spacetime,Phys. Lett. B 659 (2008) 688, arXiv:0711.0078
β’ Cariglia M., Krtous P., Kubiznak D.: Dirac Equation in Kerr-NUT-(A)dS Spacetimes: Intrinsic Characterization of Separability . . . ,Phys. Rev. D 84 (2011) 024008, arXiv:1104.4123
β’ Cariglia M., Krtous P., Kubiznak D.: Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets,Phys. Rev. D 84 (2011) 024004, arXiv:1102.4501
Vector field
β’ Lunin O.: Maxwellβs equations in the Myers-Perry geometry,JHEP12(2017)138, arXiv:1708.06766
β’ Krtous P., Frolov V. P., Kubiznak D.: Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes,preprint (2018), arXiv:1803.02485
β’ Frolov V. P., Krtous P., Kubiznak D.: Separation of variables in Maxwell equations in Plebanski-Demianski spacetime,Phys. Rev. D 97 (2018) 101701(R), arXiv:1802.09491
β’ Frolov V. P., Krtous P., Kubiznak D., Santos J. E.: Massive vector fields in rotating black-hole spacetimes: Separability and quasinormal modes,Phys. Rev. Lett. 120 (2018) 231103, arXiv:1804.00030
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 References 24