Separability - utf.mff.cuni.cz
Transcript of Separability - utf.mff.cuni.cz
![Page 1: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/1.jpg)
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
![Page 2: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/2.jpg)
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
![Page 3: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/3.jpg)
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
![Page 4: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/4.jpg)
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
![Page 5: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/5.jpg)
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
![Page 6: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/6.jpg)
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
![Page 7: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/7.jpg)
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
![Page 8: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/8.jpg)
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
![Page 9: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/9.jpg)
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
![Page 10: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/10.jpg)
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
![Page 11: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/11.jpg)
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
![Page 12: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/12.jpg)
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
![Page 13: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/13.jpg)
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
![Page 14: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/14.jpg)
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
![Page 15: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/15.jpg)
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
![Page 16: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/16.jpg)
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
![Page 17: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/17.jpg)
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
![Page 18: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/18.jpg)
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
![Page 19: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/19.jpg)
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
![Page 20: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/20.jpg)
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
![Page 21: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/21.jpg)
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
![Page 22: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/22.jpg)
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
![Page 23: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/23.jpg)
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
![Page 24: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/24.jpg)
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
![Page 25: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/25.jpg)
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
![Page 26: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/26.jpg)
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
![Page 27: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/27.jpg)
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
![Page 28: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/28.jpg)
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
![Page 29: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/29.jpg)
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
![Page 30: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/30.jpg)
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
![Page 31: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/31.jpg)
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
![Page 32: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/32.jpg)
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
![Page 33: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/33.jpg)
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
![Page 34: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/34.jpg)
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
![Page 35: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/35.jpg)
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
![Page 36: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/36.jpg)
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)
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
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
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½-οΏ½
οΏ½οΏ½-οΏ½
οΏ½οΏ½-οΏ½
οΏ½οΏ½-οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
ββββββββββ
βββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ 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
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½
οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½-οΏ½οΏ½
οΏ½οΏ½-οΏ½οΏ½
οΏ½οΏ½-οΏ½οΏ½
οΏ½οΏ½-οΏ½οΏ½
οΏ½οΏ½-οΏ½
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 20
![Page 37: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/37.jpg)
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!
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½οΏ½
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
![Page 38: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/38.jpg)
Quasi-normal modes of Proca field in 4 dimensions
So, how is it with those polarizations?
-N
ew
York
er
Cart
oon
Post
er
Pri
nt
by
Sam
Gro
ssat
the
Conde
Nast
Collecti
on
Pavel Krtous, Charles University Hidden symmetries and test fields in BH spacetimes: Separability, Antigonish, June 6, 2018 Quasi-normal modes 22
![Page 39: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/39.jpg)
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
![Page 40: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/40.jpg)
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
![Page 41: Separability - utf.mff.cuni.cz](https://reader033.fdocuments.net/reader033/viewer/2022042818/62696191dfb3150d785ffcf9/html5/thumbnails/41.jpg)
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