Feb 25, 2013 - space, for Aab a CKY bivector satisfying. Eq.(30), (1/3)âbAab is a Killing vector. It is well known21 that the Kerr metric admits a CKY bivector ...
Killing-Yano tensors in spaces admitting a hypersurface orthogonal Killing vector David Garfinkle1, 2 and E.N. Glass2 1)
Physics Department, Oakland University, Rochester, MI, 48309,
USA 2)
Michigan Center for Theoretical Physics, Randall Laboratory of Physics,
arXiv:1302.6207v1 [gr-qc] 25 Feb 2013
University of Michigan, Ann Arbor, MI 48109-1120, USA Methods are presented for finding Killing-Yano tensors, conformal Killing-Yano tensors, and conformal Killing vectors in spacetimes with a hypersurface orthogonal Killing vector. These methods are similar to a method developed by the authors for finding Killing tensors. In all cases one decomposes both the tensor and the equation it satisfies into pieces along the Killing vector and pieces orthogonal to the Killing vector. Solving the separate equations that result from this decomposition requires less computing than integrating the original equation. In each case, examples are given to illustrate the method. PACS numbers: 04.20.Cv, 04.20.Jb I.
tor, Killing-Yano, conformal Killing-Yano,
INTRODUCTION
etc.) it is natural to ask whether the approach of1 could be used on any of these
1
Recently Garfinkle and Glass presented a method for finding Killing tensors in spaces with a hypersurface orthogonal Killing vector.
The method involves a 3+1 (or
more generally (n-1)+1) decomposition of the Killing tensor equation using the foliation orthogonal to the Killing vector. The 1
approach of has been considered by Mirshekari and Will2 in showing that the BachWeyl metric does not admit a non-trivial Killing tensor. Since the Killing tensor equation is one of a class of similar tensor equations (Killing vector, conformal Killing vec1
other equations.
In fact, the use of 3+1
decomposition to study the equations for a Killing vector has a long history in general relativity begining with the work of Moncrief3 and Coll4 and continued e.g. by Beig and Chru´sciel5 . More recently G´omez-Lobo and Valiente-Kroon6 considered this 3+1 decomposition in spinor formalism, and have also studied Killing spinor initial data sets. The main difference between these earlier works and the method of1 is the assumption of a hypersurface orthogonal Killing vector. This as-
sumption greatly restricts the cases to which II.
THE KILLING-YANO TENSOR
the method applies; however it also provides METHOD a great simplification to the equations and thus makes them more tractable. A similar approach due to Bona and Coll7 treats the
The Killing-Yano (KY) equation for antisymmetric tensor Aab can be written as Aa(b;c) = 0.
conformal Killing equation in static space-
(1)
times, but then adds the further condition This generalizes Killing’s equation to antithat the conformal Killing field is Lie derived symmetric tensors. There are at most 10 inby the static Killing field. dependent solutions of the KY equation on manifold M. The maximum of 10 occurs if, and only if, M has constant curvature. There is an extensive literature covering KY tensors. In an early paper Collinson9 discussed the relationship between Killing vectors and This paper generalizes the technique of1 KY tensors. He pointed out that all type D by producing analogous methods for the vacuum solutions which admit a Killing tenKilling-Yano, conformal Killing, and confor- sor also admit a KY tensor. Two works by mal Killing-Yano equations. In each case the Dietz and R¨ udiger10,11 discuss the character spacetime is assumed to posess a hypersur- of spacetimes admitting KY tensors. More face orthogonal Killing vector, and the equa- recently, Ferrando and S´aez12 gave Rainich tions are decomposed with respect to the fo- conditions for systems to admit KY tensors. liation orthogonal to the Killing vector. As Hall13 studied the existence of KY tensors a simple illustration of these techniques, we in General Relativity, and Ibohal14 has used find the Killing-Yano tensors of the Bertotti- the Newman-Penrose formalism to integrate Robinson (BR) spacetime, and the conformal the KY equations and has found a number Killing-Yano tensors of a particular cylindri- of spacetimes which contain KY tensors, incal vacuum metric due to Linet.8
cluding FRW, Kerr-Newman, and Bertotti-
Notation: Lower case Latin indices, B a , Robinson15 .
Taxiarchis16 has proved that
range over n-dimensions. Greek indices, B µ , the only spacetimes which admit KY tensors range over n–1 dimensions. For Killing vec- have Petrov type D, N, or O. tor ξ a an overdot will denote a Lie derivative, Suppose that a spacetime has a hypersurA˙ := Lξ A. face orthogonal Killing vector ξ a . Define V 2
the Killing vector. Choose a coordinate sys-
such that ξ a ξa = ǫV 2
(2) tem (y, xµ) such that xµ are coordinates on
where ǫ = ±1. Then the metric in directions the surface orthogonal to the Killing vector and Lξ is simply a partial derivative with reorthogonal to ξ a is given by spect to y. Use ∂µ or a comma to denote hab = gab − ǫV −2 ξa ξb
(3)
a derivative with respect to the xµ coordi-
One can use ha b as a projection operator to nates. The Latin indices in this section are nproject any tensor in directions orthogonal dimensional, and the method below projects to ξ a . In particular, the KY tensor can be objects and equations down to n-1 dimensions with Greek indices.
decomposed as Aab = 2V −1 S [a ξb] + Qab
(4)
Equations (7-8) become Q˙ µν = ǫV ∂[µ Sν] + 2ǫS[µ ∂ν] V
where Sa and antisymmetric Qab are orthog-
S˙ µ = −Qµα hαν ∂ν V.
onal to ξ a . Projecting the KY equation using all combinations of ha b and ξ a yields the following Da Qbc + D b Qac = 0, D(a Sb) = 0, Lξ Qab = ǫV 3 D[a V −2 Sb] , Lξ S a = −Qab D b V.
(9) (10)
Thus the method for finding Killing-Yano tensors on the n-dimensional space consists
(5) of two steps: (6) (1) find all Killing-Yano tensors and all Killing vectors on the n−1 dimensional space (7) (2) subject those Killing-Yano tensors and (8) Killing vectors to the conditions of Eq.(9) and
Here Lξ denotes the Lie derivative with re- Eq.(10)
spect to Killing vector ξ a and Da denotes the
derivative operator on the space orthogonal III. KILLING-YANO TENSORS OF to ξ a . THE BERTOTTI-ROBINSON The first two equations say that Qab and METRIC S a are respectively a Killing-Yano tensor and a Killing vector on the space orthogonal to
The BR spacetime (up to an overall scale)
ξ a . The last two equations are additional has line element conditions that these tensors must satisfy. These last two equations are most easily implemented in a coordinate system adapted to 3
ds2 = (
1 )(−dt2 + dr 2 r2
+r 2 dϑ2 + r 2 sin2 ϑdϕ2 )
(11)
This spacetime is the direct product of the it has the three Killing vectors of that space. 2-sphere and 2-dimensional anti de-Sitter Thus we have spacetime, i.e. S 2 ⊗ AdS2 . Defining coordi-
Qµν = k1 2∂[µ w ∂ν] ϑ
nate w := − ln r allows the BR line element
Sµ = k2 ∂µ w + k3 ∂µ ϑ
to be transformed to the form 2
2w
2
2
2
2
+ k4 (ϑ∂µ w − w∂µ ϑ)
2
ds = −e dt + dw + dϑ + sin ϑdϕ . (12) For the convenience of the reader in following this section, additional properties of the BR spacetime are collected in Appendix A.
(14)
(15)
Using the ϕ Killing vector of metric (12) and recalling that the Killing vector norm is ǫV 2 , yields V = sin ϑ and ǫ = 1. Imposing Eq.(10) we find
The method of the previous section will be used to work out the KY tensors of the
k˙ 2 ∂µ w + k˙ 3 ∂µ ϑ + k˙ 4 (ϑ∂µ w − w∂µ ϑ)
BR metric. First the KY tensors of the 2-
= −k1 cos ϑ(∂µ w)
dimensional wϑ surface will be found, then these will be used to find the KY tensors of the 3-dimensional wϑϕ surface, and finally find the KY tensors of 4-dimensional BR spacetime.
(16)
It then follows that the quantities k1 , k˙ 2 , k˙ 3 and k˙ 4 all vanish. Thus we have Qµν = 0 and Sµ = c2 ∂µ w + c3 ∂µ ϑ+ c4 (ϑ∂µ w −w∂µ ϑ) (17)
will denote constants, and Now, using Eq.(9) we find
c1 , c2 etc.
k1 , k2 etc. will denote quantities that depend
0 = (c4 sin ϑ + c2 cos ϑ + c4 ϑ cos ϑ)
only on the coordinate associated with the
×(∂µ w ∂ν ϑ − ∂ν w ∂µ ϑ).
Killing vector.
(18)
This implies that c2 and c4 vanish. It follows wϑ and wϑϕ surfaces
that Sµ = c3 ∂µ ϑ. Use of Eq.(4) results in Aµν = c3 sin ϑ 2∂[µ ϑ ∂ν] ϕ
The 2-dimensional wϑ space has line ele-
(19)
ment ds2 = dw 2 + dϑ2 .
(13) the wϑϕ surface and the BR spacetime
For any 2-dimensional space, the unique solution (up to an overall scale) of Eq.(5) is
The 3-dimensional wϑϕ space has line element
the volume element. Since the wϑ space is just ordinary 2-dimensional Euclidean space, 4
ds2 = dw 2 + dϑ2 + sin2 ϑdϕ2
(20)
Here the Killing vectors are (∂/∂w)a and the on the right hand side and the term on the three Killing vectors of the 2-sphere, which left hand side. It then follows that this term will be denoted by ξ 1a , ξ 2a , ξ 3a . We therefore must vanish. Therefore ℓµ = 0, and the entire right hand side of this equation vanishes.
have Sµ = k2 ∂µ w + k3 ξµ1 + k4 ξµ2 + k5 ξµ3
(21)
The left hand side must therefore also vanish, and so k˙ 1 = 0. Thus k1 = c1 . Finally we have
From the results of the previous subsection it
Qµν = c1 sin ϑ 2∂[µ ϑ ∂ν] ϕ,
follows that
Sµ = c2 ∂µ w. Qµν = k1 sin ϑ 2∂[µ ϑ ∂ν] ϕ
(27) (28)
(22) Applying this result in Eq.(4) we find that the
The t Killing vector of metric (12) provides general Killing-Yano tensor of the BR spaceǫ = −1 and V = ew . Using Eq.(10) we have time is 0 = k˙ 2 ∂µ w + k˙ 3 ξµ1 + k˙ 4 ξµ2 + k˙ 5 ξµ3
(23)
Aµν = (c1 sin ϑ) 2∂[µ ϑ ∂ν] ϕ +(c2 ew ) 2∂[µ t ∂ν] w
Since the terms on the right hand side are
(29)
linearly independent, the coefficient of each Since the BR spacetime is the direct product term vanishes. Thus k˙ 2 = k˙ 3 = k˙ 4 = k˙ 5 = 0. S 2 ⊗ AdS2 , this result has a simple geometriTherefore one has k2 = c2 , k3 = c3 , k4 = c4 , cal interpretation. The Killing-Yano tensors k5 = c5 . It then follows that Sµ takes the of the BR spacetime are the volume elements form of S 2 and AdS2 . Sµ = c2 ∂µ w + ℓµ (24) where ℓµ is the sum of 2-sphere Killing vecIV.
tors, defined as
CONFORMAL
KILLING-YANO TENSORS ℓµ : = c3 ξµ1 + c4 ξµ2 + c5 ξµ3 .
(25) The tensor version of the conformally
Upon using Eq.(10) we find
covariant generalization of the KY equation, the CKY equation, was discovered by
k˙ 1 sin ϑ 2∂[µ ϑ ∂ν] ϕ w
w
= −e ∂[µ ℓν] − e 2ℓ[µ ∂ν] w
(26)
Tachibana18 . It can be written in the form
The last term on the right hand side is lin-
∇a Abc + ∇b Aac
early independent of both the first curl term
= 2Wc gab − Wa gbc − Wb gac
5
(30)
for some Wa . Abc is given in Eq.(4). It follows from Eq.(30) that
and the spacetime has a Lorentz signature. A
1 Wa = ∇b Aba . n−1 18
In
We now specialize to the case where n = 4 vector T a exists such that
(31) Qab = ǫabc T c
Tachibana showed that in a Ricci-flat
space, for Aab a CKY bivector satisfying Eq.(30), (1/3)∇bAab is a Killing vector. It is well known21 that the Kerr metric admits a CKY bivector, and indeed all type D vacuum solutions and their charged counterparts have a CKY bivector.19 In a manner just like the Killing-Yano case, we can decompose Aab as in Eq.(4).
(37)
where ǫabc is the volume element of the 3dimensional space orthogonal to the Killing vector. Then equations (33), (35), and (36) become D(a Tb) = ψhab 1 Lξ T a = − V 3 ǫabc Db (V −2 Sc ) 2 1 Lξ S a = V 3 ǫabc Db (V −2 Tc ) 2
(38) (39) (40)
Thus T a is a conformal Killing vector of the
Similarly, Wa can be decomposed as
3-dimensional space. The additional condiWa = γV
(32)
tions that T a and S a must satisfy are given by
where Xa is orthogonal to ξ a . Taking all pro-
equations (39) and (40) respectively. In the
−1
ξ a + Xa
adapted coordinate system these additional
jections of Eq.(30) we find the following:
conditions take the form Da Qbc + D b Qac = 2Xc hab − Xa hbc − Xb hac , D(a Sb) = γhab ,
(33) (34)
1 T˙ µ = − V 3 ǫµνα ∂ν (V −3 Sα ) 2 1 S˙ µ = V 3 ǫµνα ∂ν (V −3 Tα ) 2
(41) (42)
(35) Thus to find the CKY tensors of the 4dimensional spacetime, one does the followLξ S a = −Qab D b V − V Xa . (36) ing:
Lξ Qab = ǫV 3 D[a V −2 Sb] ,
As in the Killing-Yano case, the first two (1) find all the conformal Killing fields S a and equations have a simple geometrical inter- T a of the 3-dimensional surface orthogonal to pretation. On the n − 1 dimensional sub- the Killing vector. space orthogonal to the Killing vector Qab is (2) subject those conformal Killing fields to a CKY tensor and S a is a conformal Killing the conditions of equations (41) and (42). vector. The last two equations provide con- (3) use Eq.(37) to find Qab and then Eq.(4) to find Aab .
ditions that those tensors must satisfy. 6
V.
Subtracting Da of Eq.(46) from Lξ of Eq.(47)
CONFORMAL KILLING
provides
VECTORS Since, as shown in the previous section, one step in finding CKY tensors involves find-
Lξ Lξ Ba +
ing conformal Killing vectors, we now apply
ǫV 2 Da [V n−1 Db (V 1−n B b )] = 0. n−1
(49)
the general method of this paper to finding In the adapted coordinate system, the addia conformal Killing vectors. Recall that a con- tional conditions for B become formal Killing vector K a on an n-dimensional space is one for which ∇ a K b + ∇b K a =
2 (∇c K c )gab . n
(43)
∂[µ (V −2 B˙ ν] ) = 0 (50) 2 n−1 √ ¨µ + ǫV ∂µ V√ ∂ν ( hV 1−n B ν ) B n−1 h = 0.
(51)
In a manner similar to the Killing-Yano case, The equations for A are
we can decompose Ka as Ka = Aξa + Ba
(44)
where Ba is orthogonal to ξ a . Taking all projections of Eq.(43) it follows that 2 (Dc B c )hab n−1 V n−1 Lξ A = Da (V 1−n B a ) n−1
Da Bb + Db Ba =
A˙ =
√ V n−1 √ ∂ν ( hV 1−n B ν ) (n − 1) h
∂µ A = −ǫV −2 B˙ µ
(52) (53)
Thus the method for finding conformal (45)
Killing vectors on the n-dimensional space consists of three steps:
(46) (1) find all conformal Killing vectors on the
(47) n − 1 dimensional space (2) subject those conformal Killing vectors to As in the Killing-Yano case, Eq.(45) has a the conditions of Eq.(50) and Eq.(51) simple geometrical interpretation. On the (3) solve Eq.(52) and Eq.(53) for A. n − 1 dimensional subspace orthogonal to Da A = −ǫV −2 Lξ B a
Killing vector ξ a , B a is a conformal Killing vector. However, this conformal Killing vec-
VI.
CKY TENSOR OF LINET’S
VACUUM METRIC
tor is also subject to additional conditions, the integrability conditions for A. Taking the
element found by Linet8 is written as
curl of Eq.(47) we obtain D[a (V −2 Lξ B b] ) = 0.
The Petrov type D cylindrical vacuum line
(48) 7
ds2 = r 4 (−dt2 + dr 2 + dz 2 ) + r −2 dϕ2 . (54)
This static metric has Killing vectors ∂t , ∂z , then follows that ∂ϕ . We will use the method of the previous two sections to find the CKY tensors of this spacetime.
∂[µ (V −2 B˙ ν] ) = 21 r 6 [−∂u α˙ + ∂v β˙ ˙ ∂[µ u ∂ν] v + 3r −1(α˙ + β)]
(58)
and therefore from Eq.(50) ˙ = 0. − ∂u α˙ + ∂v β˙ + 3r −1 (α˙ + β)
tr and trϕ surfaces
(59)
Taking ∂u ∂v of this equation, and using the We begin by finding all the conformal Killing fields of the tr surface and then using those to find all the conformal Killing fields of the trϕ surface. The 2-dimensional tr space
pendent of u, we find ˙ = 0. − ∂u α˙ + ∂v β˙ − r −1 (α˙ + β)
(60)
Subtracting Eq.(60) from Eq.(59) yields
has line element ds2 = r 4 (−dt2 + dr 2 ).
fact that α is independent of v and β is inde-
(55)
α˙ + β˙ = 0.
(61)
However, since α is independent of v and β Like all 2-dimensional metrics, this metric is is independent of u, there exists a function conformally flat and the conformal Killing k (ϕ) such that α˙ = −k and β˙ = k . It then 1 1 1 fields are therefore those of the underlying follows from Eq.(57) that flat spacetime. For our purposes, it will be convenient to use null coordinates u = t − r
B˙ a = −k1 (∂u )a + k1 (∂v )a = k1 (∂r )a .
(62)
¨ and v = t+r. The line element then becomes Now taking ∂ϕ of Eq.(51) for Bµ , and using Eq.(62) we find n−1 √ 1−n ν ds2 = −r 4 dudv (56) ... ǫV 2 V ˙ 0 = Bµ + B ) ∂µ √ ∂ν ( hV n−1 h where r = (v − u)/2. It follows from metric = (r 4 k¨1 − 3r −4 k1 )∂µ r (63) (56) that the conformal Killing field takes the form B a = α(∂u )a + β(∂v )a
It then follows that k1 = 0. Therefore B˙ µ = 0 and so α and β are independent of ϕ. Thus
(57)
α is a function of u, and β is a function of v.
where α is independent of v, and β is inde- It then follows from Eq.(51) that there is a pendent of u. We now use this conformal constant c1 such that V n−1 √ Killing field to work out the general confor8c1 = √ ∂ν ( hV 1−n B ν ) h mal Killing field of the trϕ surface. We have = ∂u α + ∂v β + 3r −1 (β − α). gϕϕ = r −2 , therefore ǫ = 1 and V = r −1 . It 8
(64)
the trϕ surface and the Linet
Differentiating Eq.(64) by ∂u ∂v yields
spacetime 0 = ∂u α + ∂v β − r (β − α). −1
(65) The vector fields T a and S a are confor-
Subtracting Eq.(65) from Eq.(64) provides 8c1 = 4r −1 (β − α)
mal Killing fields on the trϕ surface, with z
(66) dependent coefficients and therefore take the form
from which it follows that β − α = 2c1 r = c1 (v − u).
T a = k1 [t(∂t )a + r(∂r )a + 4ϕ(∂ϕ )a ] (67)
But α depends only on u and β depends only on v and so there exists a constant c2 such that
+ k2 (∂t )a + k3 (∂ϕ )a
(73)
S a = k4 [t(∂t )a + r(∂r )a + 4ϕ(∂ϕ )a ] + k5 (∂t )a + k6 (∂ϕ )a
(74)
Since gzz = r 4 it follows that V = r 2 and α = c1 u + c2
(68) ǫ = 1. We then find
β = c1 v + c2
(69)
ǫµνλ ∂ν (V −2 Tλ ) = − 6r −7 ǫµrϕ (4k1 ϕ + k3 )
We therefore have
(75)
ǫµνλ ∂ν (V −2 Sλ ) = B a = c1 [u(∂u )a + v(∂v )a ] a
− 6r −7 ǫµrϕ (4k4 ϕ + k6 )
a
+ c2 [(∂u ) + (∂v ) ] = c1 [t(∂t )a + r(∂r )a ] + c2 (∂t )a
(70)
(76)
From the t component of Eq.(41) and Eq.(42) it follows that
It then follows from Eq.(52) and Eq.(53) that A˙ = 4c1 and ∂µ A = 0. We therefore have A = 4c1 ϕ + c3 .
k˙ 1 t + k˙ 2 = −3r −4 (4k4 ϕ + k6 ).
(77)
k˙ 4 t + k˙ 5 = 3r −4 (4k1 ϕ + k3 ).
(78)
(71) Therefore k1 , k3 , k4 , and k6 vanish, and k2
Finally, using Eq.(44) we find that the and k5 are constants. Thus the conformal general conformal Killing vector of the 3- Killing fields subject to restrictions (41) and dimensional trϕ surface is
(42) take the form
K a = c1 [t(∂t )a + r(∂r )a + 4ϕ(∂ϕ )a ] + c2 (∂t )a + c3 (∂ϕ )a
(72) 9
T a = c1 (∂t )a
(79)
S a = c2 (∂t )a
(80)
Finally, using Eq.(37) and Eq.(4) we find that λ2 characterizes the electromagnetic energy the general CKY tensor of the spacetime is
density. Since the Weyl tensor vanishes, the
Aµν = 2c1 r 3 ∂[µ r ∂ν] ϕ + 2c2 r 6 ∂[µ z ∂ν] t (81) Petrov type is 0. The BR spacetime has a diagonal trace-free Ricci tensor (with rows and columns along t, x, y, z) 1 In this work a method is developed which −1 α BR 2 [R β ] = λ decomposes the Killing-Yano tensor into sep −1 arate terms based on the surface geometry VII.
SUMMARY
1
of metrics with a hypersurface orthogonal Killing vector, and which thereby simplifies the solution of the Killing-Yano equation.
.
(A2)
The BR manifold is non-singular with Kretschmann scalar
Using this method, we have shown that the
Rαβµν Rαβµν = 8λ4 .
Bertotti-Robinson spacetime has a general The BR metric is spanned by the null
KY tensor which is the sum of volume bivectors. An enhancement of this method has
tetrad √ lα dxα = (1/ 2)[(1 + λ2 z 2 )1/2 dt
also been applied to the conformal KillingYano equation. The general CKY tensor has
+ (1 + λ2 z 2 )−1/2 dz] (A3) √ nα dxα = (1/ 2)[(1 + λ2 z 2 )1/2 dt
been constructed for Linet’s cylindrical vacuum metric. Acknowledgement
− (1 + λ2 z 2 )−1/2 dz] (A4) √ mα dxα = (1/ 2)[(1 − λ2 y 2)1/2 dx
We would like to
thank Jean Krisch for helpful discussions.
− i(1 − λ2 y 2 )−1/2 dy] (A5) √ m ¯ α dxα = (1/ 2)[(1 − λ2 y 2)1/2 dx
DG was supported by NSF Grants PHY0855532 and PHY-1205202 to Oakland Uni-
+ i(1 + λ2 y 2 )−1/2 dy]
versity.
(A6)
Eight Newman-Penrose spin coefficients van˜ = ν = ρ = µ = τ = π. The ish, κ = σ = λ
Appendix A: Bertotti-Robinson
The static Bertotti-Robinson (BR) metric remaining four are is BR dxµ dxν gµν
2 2
2
2 2 −1
2
= [(1 + λ z )dt − (1 + λ z ) dz ]
− [(1 − λ2 y 2)dx2 + (1 − λ2 y 2 )−1 dy 2]. (A1) 10
λ2 z 1 ǫ=γ=− √ √ 2 2 1 + λ2 z 2 λ2 y i α=β=− √ p 2 2 1 − λ2 y 2
(A7) (A8)
6
The null vectors are all geodesic
A.G.P. G´omez-Lobo and J.A.V. Kroon, J. Geom. Phys. 58, 1186 (2008). Killing
lα;β = 2γlα lβ + 2γlα nβ
(A9)
nα;β = −2ǫnα nβ − 2γnα lβ
(A10)
mα;β = 2αm ¯ αm ¯ β − 2αmα mβ
(A11)
spinor initial data sets 7
C. Bona and B. Coll, Gen. Rel. and Gravit, 23, 99 (1991). Invariant Conformal Vectors in Static Spacetimes
The BR manifold admits antisymmetric ten-
8
sor Aαβ as covariant constant bivectors Aαβ = k0 l[α nβ] + k1 m[α m ¯ β] Aαβ;ν = 0
B. Linet, Gen. Rel. Gravit. 17, 1109 (1985). The Static Metric with Cylindrical Symme-
(A12)
try Describing a Model of Cosmic Strings 9
(A13)
311 (1976). On the Relationship between Killing Tensors and Killing-Yano Tensor
Aαβ is therefore a KY solution. Note that l[α nβ] ∼ dt ∧ dz and m[α m ¯ β] ∼ dx ∧ dy. These
C.D. Collinson, Int. Jour. Theor. Phys. 15,
10
W. Dietz and R. R¨ udiger, Proc. Roy. Soc. London A 375, 360 (1981). Space-times ad-
are the volume elements of the BR manifold.
mitting Killing-Yano tensors. I 11
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