valeri p. frolov- black holes and hidden symmetries
TRANSCRIPT
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Peyresq, June 20-26, 2009
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Higher Dimensions:Motivations for Study
Higher Dimensions:
Kaluza-Klein models and Unification;
String Theory;
Brane worlds;
`From above view on Einstein gravity.
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Gravity in Higher Dim. Spacetime
(4 ) ( ) 3 ( ) k k kka G M x
F
( )
2
k
k
G MF
r
No bounded orbits
Gravity at small scales is stronger than in 4D
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`Running coupling constant
( )
( ) , ( ), ( ) k k
k k
G lG r G G l G r G
r r
1/2 22
*
*2 2 2
* *
( )
k
p pG r m Gme r lr r e
2
36
210
pGm
e
20
*
0.01 2 10 For l cm and k r cm
`Hierarchy problem
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1( , )
p m mv
*( ,0)
fp E
2( , )
p m mv
1/ 2
1 2 * */ , 2 , (2 ) E p p m E m E Em
2,
2 2
*
*
2
*,
*
(1 ) 2
2
/ 1
mE E
For E r E T
E
eV
m v m E
New fundamental scale
2*
*
( / ) ,
2 0.01 1
k
kPl Pl
M M l l
For k and l cm M TeV
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4D Newton law is confirmed for r>l.
Q.: How to make gravity strong (HD) atsmall scales without modifying it at largescales?
A.: Compactification of extra dimensions
`Our space dimensions
Extradims
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Gravity in ST with Compact Dims
Example: (4) 4 3 10, R SM
**
2 2 2 2
1 sinh( / )cosh( / )
( ) cosh ( / ) cos ( / )
n
G M r l r lG M
r z nl lr r l z l
**( , 0) coth( / ), /
G Mr z r l G G l
lr 24 * 2 *1
02 2W=- dx [ +(m ) ]nn
n
L
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Brane World Paradigm
Fundamental scale of order of TeV. Large extra
dimensions generate Planckian scales in 4D space
Gravity is not localized and `lives in (4+k)-Dbulk space
Bosons, fermions and gauge fields are localizedwithin the 4D brane
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Black Holes as Probes of Extra Dims
21 2610 10g M g
We consider first black holes in the mass range
Mini BHs creation in colliders
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BH formation
Bolding Phase
Thermal (Hawking) decay
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Black Objects
Black Holes[Horizon Topology S^(D-2)]
Exist in any # of dims
Generic solution is Kerr-NUT-(A)dS
Principal Killing-Yano existence istheir characteristic property
Hidden Symmetries, Integrability
properties
Black Rings
Black Saturns
Black Strings, ets
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Higher Dimensional Kerr-NUT-(A)dSBlack Holes
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Based on:V. F., D.Kubiznak, Phys.Rev.Lett. 98, 011101 (2007); gr-qc/0605058
D. Kubiznak, V. F., Class.Quant.Grav.24:F1-F6 (2007); gr-qc/0610144
P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP 0702:004 (2007); hep-th/0612029
V. F., P. Krtous , D. Kubiznak , JHEP 0702:005 (2007); hep-th/0611245
D. Kubiznak, V. F., JHEP 0802:007 (2008); arXiv:0711.2300
V. F., Prog. Theor. Phys. Suppl. 172, 210 (2008); arXiv:0712.4157
V.F., David Kubiznak, CQG, 25, 154005 (2008); arXiv:0802.0322
P. Connell, V. F., D. Kubiznak, PRD, 78, 024042 (2008); arXiv:0803.3259
P. Krtous, V. F., D. Kubiznak, PRD 78, 064022 (2008); arXiv:0804.4705
D. Kubiznak, V. F., P. Connell, P. Krtous ,PRD, 79, 024018 (2009); arXiv:0811.0012D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous, Phys.Rev.Lett. (2007); hep-th/0611083
P. Krtous, D. Kubiznak, D. N. Page, M. Vasudevan, PRD76:084034 (2007); arXiv:0707.0001
`Alberta Separatists
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Recent Reviews on HD BHs
Panagiota Kanti, Black holes in theories with large extra dimensions:
A Review, Int.J.Mod.Phys.A19:4899-4951 (2004).
Panagiota Kanti Black Holes at the LHC, Lectures given at 4thAegean Summer School: Black Holes, Mytilene, Island of Lesvos,Greece, ( 2007).
Roberto Emparan, Harvey S. Reall, Black Holes in HigherDimensions ( 2008) 76pp, Living Rev.Rel. arXiv:0801.3471
V.F. and David Kubiznak, Higher-Dimensional Black Holes: Hidden
Symmetries and Separation of Variables, CQG, Peyresq-Physics 12workshop, Special Issue (2008) ; arXiv:0802.0322
V.F, Hidden Symmetries and Black Holes, NEB-13, Greece, (2008);arXiv:0901.1472
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Hidden Symmetries of 4D BHs
Hidden symmetries play an important role in study 4D rotating blackholes. They are responsible for separation of variables in the Hamilton-Jacobi, Klein-Gordon and higher spin equations.
Separation of variables allows one to reduce a physical problem to asimpler one in which physical quantities depend on less number ofvariables. In case of complete separability original partial differentialequations reduce to a set of ordinary differential equations
Separation of variables in the Kerr metric is used for study:(1) Black hole stability(2) Particle and field propagation(3) Quasinormal modes(4) Hawking radiation
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Brief History of 4D BHs
1968: Forth integral of motion, separability of the Hamilton-Jacobi andKlein-Gordon equations in the Kerr ST, Carters family of solutions[Carter, 1968 a, b,c]
1970: Walker and Penrose pointed out that quadratic in momentumCarters constant is connected with the symmetric rank 2 Killing tensor
1972: Decoupling and separation of variables in EM and GP equations[Teukolsky]. Massless neutrino case [Teukolsky (1973), Unruh (1973)].Massive Dirac case [Chandrasekhar (1976), Page (1976)]
1973: Killing tensor is a `square of antisymmetric rank 2 Killing-Yanotensor [Penrose and Floyd (1973)]
1974: Integrability condition for a non-degenerate Killing-Yano tensor implythat the ST is of Petrov type D [Collinson (1974)]
1975: Killing-Yano tensor generates both symmetries of the Kerr ST[Hughston and Sommers (1975)]
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Type-D (withoutacceleration)
Killing tensor
Killing-Yanotensor
Separability
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Main Results (HD BHs)
Rotating black holes in higher dimensions, described by theKerr-NUT-(A)dS metric, in many aspects are very similar tothe 4D Kerr black holes.
They admit a principal conformal Killing-Yano tensor.This tensor generates a tower of Killing tensors and Killing
vectors, which are responsible for hidden and `explicitsymmetries.The corresponding integrals of motion are sufficient for acomplete integrability of geodesic equations.These tensors imply separation of variables in Hamilton-
Jacoby, Klein-Gordon, and Dirac equations.Any solution of the Einstein equations which admits a non-degenerate a principal conformal Killing-Yano tensor is aKerr-NUT-(A)dS spacetime.
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Spacetime Symmetries
L g g
0L g
( ; ) 0 (Killing equation)
( ; ) (conformal Killing eq) g
1
;
D D is # of ST dimensions
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Hidden Symmetries
( ; ) g
Killing vector
Killing tensor Killing-Yano tensor
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Symmetric generalization
CK=Conformal Killing tensor
1
1 2 1 2 2 1 2... ( ... ) ... ...,n n n nK K K K
1 2 1 2 3( ... ; ) ( ... )n nK g K
Integral of motion
1 2 1 2 1
1 2 1 2 1... ; ...( ... ) ...n n
n nu K u u u K u u u
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Antisymmetric generalization
CY=Conformal Killing-Yano tensor
1
1 2 1 2 2 1 2... [ ... ] ... ...,
n n n n k k k k
1 2) 3 1 1 2 3 1 3 1 2) 1( ... ... [ ( ... ]( 1)
n n nk g k n g k
If the rhs vanishes f=k is a Killing-Yano tensor
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2
2
...
... is the Killing tensorn
nK f f
1
1 1 1
...
... ...
.
n
n
n n
If f is a KY tensor then for a geodesic
motion the tensor p f u is
parallelly propagated along a geodesic
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Principal conformal Killing-Yano tensor
1[ ] 1
, (
0
*)
, na bc a na
c ab b c a
D
ca b
h h
h g g
1, (*)
, 2
1X
h
h db
X h
D
D
n
PCKY tensor is a closed non-degenerate(matrix rank 2n) 2-form obeying (*)
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Properties of CKY tensor
Hodge dual of CKY tensor is CKY tensor
Hodge dual of closed CKY tensor is KYtensor
External product of two closed CKY tensors
is a closed CKY tensor
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Darboux Basis
1 1
( ) ,n nab a b a b a b
ab a b
g e e e e e e
h x e e
1
2
( )m e ie
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Canonical Coordinates
h m ix m
A non-degenerate 2-form h has n independent
eigenvalues
essential coordinates x and Killing
coordinates are used as canonical coordinatesj
n n
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Principal conformal KY tensor
Non-degeneracy:
(1) Eigen-spaces of are 2-dimensional
(2) are functionally independent in some domain
(they can be used as essential coordinates)
h
x
(1) is proved by Houri, Oota and Yasui e-print arXiv:0805.3877
(2) Case when some of eigenvalues are constant studied in
Houri, Oota and Yasui Phys.Lett.B666:391-394,2008.e-Print: arXiv:0805.0838
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Killing-Yano Tower
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Killing-Yano Tower: Killing Tensors
2
2
1 2
1 21 1 2 2
2 4 2 2
... ... ..
2
.
* * *
4 2 2
*
j n
j times n times
j n
j n
j n
j j n n
j n
h h h h h h h h h h h h
k h k h k h k h
K k k K k k K k k K
D D D j
k
n
k
D
Set of (n-1) nontrivial rank 2 Killing tensors
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2 1 2 1
1 ( )
n j j j n n
ab a b a b a bA e e e e A e e
K
1 1
1 1
1 1
2 2 2 2,i i
i i
i i
j
i i
A x x A x x
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Killing-Yano Tower: Killing Vectors
11
is a primary Killing vectorna naD
h
1( ) ( )2 (*)
n
a b n a bD R h
( ; )On-shell ( ) (*) implies 0ab ab a bR g
Off-shell it is also true but the proof is much complicated
(see Krtous, V.F., Kubiznak (2008))
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1 1 2 2 1 1 j j n nK K K K
Total number of the Killing vectors is n
Total number of conserved quantities
( ) ( 1) 1 2n n n D
KV KT g
2
2 2
1
( )
, , ( )i
n i
i i
xd e
Q
x
U
U x
Reconstruction of metric
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Principal Conformal Killing-Yano Tensor
Reconstruction of metric
Coordinates
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0 1 1
1
: , ,..., ,
: ,...,
nn
n
Killing coordinates
Essential coordinates x x
2-planesof rotation
Off-Shell Results
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Off-Shell Results
A metric of a spacetime which admits a (non-
degenerate) principal CKY tensor can bewritten in the canonical form.
1
0
1 ,
n
ii
i
e dx e Q A d Q
, ( )X
Q X X x
U
Houri, Oota, and Yasui [PLB (2007); JP A41 (2008)] proved this result under additional assumptions:
0 and 0. Recently Krtous, V.F., Kubiznak [arXiv:0804.4705 (2008)] and
Houri, Oota, and Yasui
L g L h
[arXiv:0805.3877 (2008)] proved this without additional assumptions.
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On-Shell Result
A solution of the vacuum Einstein equationswith the cosmological constant which admits a
(non-degenerate) principal CKY tensor
coincides with the Kerr-NUT-(A)dS spacetime.
2
0
nk
k
k
X b x c x
Kerr-NUT-(A)dS spacetime is the most general BHsolution obtained by Chen, Lu, and Pope [CQG (2006)];See also Oota and Yasui [PL B659 (2008)]
"G l K NUT AdS t i i ll di i Ch
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"General Kerr-NUT-AdS metrics in all dimensions, Chen,L and Pope, Class. Quant. Grav. 23 , 5323 (2006).
/ 2 , 2n D D n
( 1) R D g
, , ( 1 ) ,
( 1 ) ` '
k M mass a n rotation parameters
M n NUT parameters
#Total of parameters is D
Ill i
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Illustrations
2 2 2 2 2
2 2 2 2 212
Let us consider a 4D flat spacetime with the metric
and let be the following 1-form
[ ( )],
dS dT dX dY dZ
b
b R dT a Y dX X dY R X Y Z
One has
( )*
h db dT XdX YdY ZdZ adY dX f h X dZ dY Z dY dX Y dX dZ a dZ dT
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1
[ ] 1
, (
0
*)
,n
a bc a na
c ab b c a
D
ca b
h h
h g g
0
0 00 0
0 0 0
na
X Y Z
X ahY a
Z
(0) , (0)13
( 1,0,0,0),n aa na a T h
(0)
, .1 ,TX X XX T h g etc
20a aY aX
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2 2
2 2
2 2 2
0
0
ab
a aY aX
aY Y Z XY XZ K
aX XY X Z YZ XY YZ X Y a
( ) (0) 2 2[ ]abb a T X Y T
K a a Y X a a
2 2 2 22 ( )a bab Z Z K p p L aEL a E p
1
(0) ( )
, / ,, ;
,
T t t
t
t T a aa a
D b di t
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Darboux coordinates
2
2 2
2 2
2
, ;
0
0
a ac a a a
b bc b b b H h h H
R aY aX
aY a X XY XZ
aX XY a Y YZ
XZ YZ Z
2 2 2 2 2
2 2 2 2 2 2 212
2 2
det( ) 0 ( ) 0,
[ ( ) 4 ],
, are essential coordinates
R a a Z
a R R a a Z
r y
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1 2 2 2 2
1 2
2 22 2 2 2 2 2 2 2
2 2 2
2 2
1
2 2 2 2
( )( ) cos ,
( )( ) sin ,
( )[ ] ( ) )
,
(dr dy
d
X a r a a y
Y a r a a y
Z a r
S dT r y a r a a y d r a a
y
y
2 ,T t a a
2 22 2 2 2 2 2 2
2 2
2 2 2 2
1(*) [ ( ) ( ) ] ( )[ ],
,
dr dydS R dt y d Y dt r d r y
r y R YR r a Y a y
This is a flat ST metric in the Darboux coordinatesassociated with the potential b.
2 2 2 2 21 [( ) ]b y r a dt r y d
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2[( ) ]b y r a dt r y d
Three important results can be proved:
(i) For arbitrary ( ) and ( ) is a potential generating a closed
conformal Killing-Yano tensor for the metric (*);
(ii) (*) with arbitrary functions ( ) a
R r Y y b
R r nd ( ) is the most general metric,
which admits a closed conformal Killing-Yano tensor;
(iii) (*) with arbitrary functions ( ) and ( ) belongs to Petrov type D
Y y
R r Y y
2 22 2
2 2
The Einstein equations, -3 , for these metrics are
satisfied iff the following equation is valid
12 ( ).
The most general solution of this equation can be written in the form
(
R g
d R d Y r ydr dy
R r
2 2 2 2 2 2)(1 ) 2 , ( )(1 ) 2a r Mr Y a y y Ny
Remark: the tranformation , makes ther ix M iM
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Remark: the tranformation , makes the
expressions for the metric (*), potential , and the solutions
symmetric with respect to the map:
r ix M iM
b
x y
Principal CKY tensor in Kerr-NUT-(A)dS
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V. F., D.Kubiznak, Phys.Rev.Lett. 98: 011101, 2007; gr-qc/0605058; D. Kubiznak, V. F., Class.Quant.Grav. 24:F1, 2007; gr-qc/0610144.
Principal CKY tensor in Kerr NUT (A)dS
1
( 1)12
0
,n
k
kk
h db b A d
(The same as in a flat ST in the Carter-type coordinates)
Complete integrability of geodesic motion in
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D. N. Page, D. Kubiznak, M. Vasudevan, P. Krtous,Phys.Rev.Lett. 98 :061102, 2007; hep-th/0611083
Complete integrability of geodesic motion ingeneral Kerr-NUT-AdS spacetimes
Vanishing Poisson brackets for integrals of
motion
P. Krtous, D. Kubiznak, D. N. Page, V. F., JHEP
0702: 004, 2007; hep-th/0612029
Separability of Hamilton-Jacobi and Klein-
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V. F., P. Krtous , D. Kubiznak , JHEP (2007); hep-th/0611245;Oota and Yasui, PL B659 (2008); Sergeev and Krtous, PRD 77 (2008).
Klein-Gordon equation
Multiplicative separation
2
0b m
Gordon equations in Kerr-NUT-(A)dS ST
Notes on Parallel Transport
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Notes on Parallel Transport
Case 1: Parallel transport along timelike geodesics
Let be a vector of velocity and be a PCKYT.
is a projector to the plane orthogona
Lemma (Page): is parallel p
l to .
Denote
ropagated along a ge
a
ab
b b b a
a a a
c d c cab a b cd ab a cb ac b
ab
u h
P u u u
F P P h h u u h h
F
u u
odesic:
0
Proof: We use the definition of the PCKYT
=
u ab
u ab a b a b
F
h u u
Suppose is a non degenerate then for a generic geodesich
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Suppose is a non-degenerate, then for a generic geodesic
eigen spaces of with non-vanishing eigen values are two
dimensional. These 2D eigen spaces are parallel propagated.
Thus a problem reduces
ab
ab
h
F
to finding a parallel propagated basis in 2D
spaces. They can be obtained from initially chosen basis by 2D
rotations. The ODE for the angle of rotation can be solved by
a separation of variables.
[Connell, V.F., Kubiznak, PRD 78, 024042 (2008)]
Case 2: Parallel transport along null geodesics
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p g g
Let be a tangent vector to a null geodesic and be a
parallel propagated vector obeying the condition 0.
Then the vector is parallel propagated,
provided .
This procedure
a a
aa
a ba a
b
a
a
l k
l k
w k h l
k
allows one to construct 2 more parallelpropagated vectors and , starting with .a a am n l
al
am
an
We introduce a projector 2 and c dP g l n F P P h
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( )
[ ]
We introduce a projector 2 , and .
One has: 2 0.
ab ab a b ab a b cd
c d c d
l ab a b l cd a b c d
P g l n F P P h
F P P h P P l
Thus is parallel propaged along a null geodesic. We use rotations
in its 2D eigen spaces to
[Kubiznak,V.F
construct
.,Krtous,
a parallel propagat
Connell, PRD 79, 024
ed basis.
018 (2009)]
abF
More Recent Developments
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Stationary string equations in the Kerr-NUT-(A)dSspacetime are completely integrable.[D. Kubiznak, V. F., JHEP 0802:007,2008; arXiv:0711.2300]
Solving equations of the parallel transport alonggeodesics [P. Connell, V. F., D. Kubiznak, PRD,78, 024042 (2008);arXiv:0803.3259; D. Kubiznak, V. F., P. Connell, arXiv:0811.0012 (2008)]
Separability of the massive Dirac equation in the
Kerr-NUT-(A)dS spacetime [Oota and Yasui, Phys. Lett.B 659, 688 (2008)]
Einstein spaces with degenerate closed
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Separability of Gravitational Perturbation inGeneralized Kerr-NUT-de Sitter Spacetime[Oota, Yasui, arXiv:0812.1623]
On the supersymmetric limit of Kerr-NUT-AdS
metrics [Kubiznak, arXiv:0902.1999]
p gconformal KY tensor [Houri, Oota and YasuiPhys.Lett.B666:391-394,2008. e-Print: arXiv:0805.0838]
GENERALIZED KILLING-YANO TENSORS
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1 12 3 3
1
3
1 1
3 2
Minimally gauged supergravity (5D EM with Chern-Simo
[Kubiznak, Kunduri, and Yasui, 0905.0722 (2009)]
ns term):*( ) * ,
0, * 0,
(ab ab ac
L R F F F F A
dF d F F F
R g F
1
3
1[ ] [ ]3
212
21
6 )
Appli
Torsion:
cation: Chong, Cvetic, Lu, Pope [PRL, 95,161301,2
005]
Note: This is type I metric
*
(* ) 2 ,
(
.
* ) (* )
T d
c ab c ab cd a b c a b
cd c
ab acd b ac b ab
c
b ab
T F
h h F h g
K h h h
F g
h g h
F
Summary
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8/3/2019 Valeri P. Frolov- Black Holes and Hidden Symmetries
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Summary
The most general spacetime admitting the PCKY tensor is described by thecanonical metric. It has the following properties:
It is of the algebraic type D
It allows a separation of variables for the Hamilton-Jacoby, Klein-Gordon,Dirac and stationary string equations
The geodesic motion in such a spacetime is completely integrable. Theproblem of finding parallel-propagated frames reduces to a set of the firstorder ODE
When the Einstein equations with the cosmological constant are imposedthe canonical metric becomes the Kerr-NUT-(A)dS spacetime