Killing-Yano tensors of rank three and Lax pair tensors

KILLING-YANO TENSORS OF RANK THREE AND LAX PAIR TENSORS ANCA VISINESCUa , MIHAI VISINESCUb Department of Theoretical Physics, “Horia Hulubei” National Institute for Physics and Nuclear Engineering, Atomistilor 407, RO-077125, POB-MG6, M˘agurele-Bucharest, Romania, EU Email: a [email protected]; b [email protected] Received January 13, 2012

Higher order first integrals of Hamiltonian systems are investigated and the special role of the Killing tensors is pointed out. A geometrical interpretation of the Lax representation in connection with third rank Killing-Yano tensors is discussed. Some examples of spacetimes involved in recent studies of higher dimensional black holes admitting Killing-Yano tensors of rank three are presented. Key words: Hidden symmetries, Killing tensors, Lax pair tensors. PACS: 02.40.Hw, 04.20.-q, 04.50.-h.

1. INTRODUCTION

The evolution of a dynamical system is described in the entire phase space and from this point of view it is natural to go in search of conserved quantities to genuine symmetries of the complete phase space, not just the configuration one. Such symmetries are associated with higher rank symmetric St¨ackel-Killing (SK) tensors which generalize the Killing vectors. These higher order symmetries are known as hidden symmetries and the corresponding conserved quantities are quadratic, or, more general, polynomial in momenta. Another natural generalization of the Killing vectors is represented by the antisymmetric Killing-Yano (KY) tensors [1] which in many respects are more fundamental than the SK tensors. The conformal extension of the SK tensor equation determines the conformal St¨ackel-Killing (CSK) tensors which define first integrals of motion of the null geodesics. Investigations of the hidden symmetries of the higher dimensional spacetimes have pointed out the role of the conformal Killing-Yano (CKY) tensors to generate background metrics with black hole solutions. The Lax representation of dynamical systems is one of the main tool to produce equations of evolution possessing conserved quantities [2]. The Lax tensors introduced in [3–5] lead to possibilities to incorporate the higher Killing symmetries in the field equations. Lax tensors arise in a covariant formulation of the Lax pair equations for (pseudo)-Riemannian spaces. In a special choice of the third rank Lax tensors, the Lax pair equation is identical to the KY equation. However, whereas RJP 57(Nos. Rom. Journ. Phys., 5-6), Vol. 1002–1010 57, Nos. 5-6,(2012) P. 1002–1010, (c) 2012-2012 Bucharest, 2012

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Killing-Yano tensors of rank three and Lax pair tensors

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the KY tensors are totally antisymmetric objects, the Lax tensor framework is more general deserving further investigations. The plan of the paper is as follows: In Section 2 the (C)SK and (C)KY tensors are introduced. In the next Section we give a brief overview of the use of Lax pair tensors as a unifying framework to incorporate the higher Killing symmetries in the equations of evolution. In Section 4 some examples of spaces admitting KY tensors of rank three involved in modern studies of hidden symmetries in gravitation are presented. The last Section contains some concluding remarks. 2. KILLING TENSORS

Let (M, g) be a D-dimensional differentiable manifold equipped with a Riemannian or pseudo-Riemannian metric ds2 = gµν dxµ dxν ,

(1)

and denote by 1 H = g µν pµ pν , (2) 2 the Hamiltonian function describing the geodesic motion in M. Using the natural Poisson bracket on the cotangent bundle, the geodesic equations are given by x˙ µ = {xµ , H} = g µν pν ,

p˙µ = {pµ , H} = Γνλµ pν pλ ,

(3)

where Γ is the Levi-Civita connection with respect to gµν . A vector field K on M is said to be a Killing vector field if the Lie derivative with respect to K of the metric g vanishes: LK g = 0 .

(4)

In components this means that K(µ; ν) = 0 , (5) where a semicolon precedes an index µ of covariant differentiation ∇µ associated with the Levi-Civita connection and we used a round bracket to denote symmetrization over the indices within. A symmetric generalization of the Killing vectors is that of SK tensors. A SK tensor of rank m is a totally symmetric m-index tensor Kµ1 ···µm = K(µ1 ···µm ) which satisfies the equation K(µ1 ···µm ;ν) = 0 . (6) Since the Poisson bracket of a constant of motion with the Hamiltonian vanishes, from the generalized Killing equation (6) we get that for any geodesic γ with tangent vector x˙ µ = pµ QK = Kµ1 ···µm x˙ µ1 · · · x˙ µm , (7) RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

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is constant along γ. KY tensors are a different generalization of Killing vectors which can be defined on a manifold. They were introduced by Yano [1] from a purely mathematical perspective and later on it turned out they have many interesting properties relevant to physics [6–8]. A KY tensor is a p-form f (p ≤ D) which satisfies ∇X f =

1 X −| df , p+1

(8)

for any vector field X, where ’hook’ operator −| is dual to the wedge product. This definition is equivalent with the property that ∇ν fµ1 ···µp is totally antisymmetric or, in components, fµ1 ···µp−1 (µp ;ν) = 0 . (9) These two generalizations SK and KY of the Killing vectors could be related. Let fµ1 ···µp be a KY tensor, then the symmetric tensor field µ2 ···µp

Kµν = fµµ2 ···µp fν

,

(10)

is a SK tensor and it sometimes refers to this SK tensor as the associated tensor with the KY tensor fµ1 ···µp . Having in mind the special role of null geodesic for the motion of massless particles, it is convenient to look for conformal generalization of KY tensor. Let us mention also that recently a lot of interest focuses on higher dimensional black holes. It was demonstrated the remarkable role of the CKY tensors in the study of the properties of such black holes (see e. g. [9–11] and the cites contained therein). A CKY tensor of rank p is a p-form which satisfies 1 1 X −| df − X [ ∧ d∗ f , (11) ∇X f = p+1 n−p+1 where X [ denotes the 1-form dual with respect to the metric to the vector field X and d∗ is the exterior co-derivative. Let us recall that the Hodge dual maps the space of p-forms into the space of (D − p)-forms. Drawing a parallel between definitions (8) and (11) we remark that all KY tensors are co-closed but not necessarily closed. From this point of view CKY tensors represent a generalization more symmetric in the pair of notions. CKY equation (11) is invariant under Hodge duality that if a p-form f satisfies it, then so does the (D − p)-form ∗f . Moreover the dual of a CKY tensor is a KY tensor if and only if it is closed. There is also a conformal generalization of the SK tensors, namely a symmetric tensor Ki1 ···ip = K(i1 ···ip ) is called a CSK tensor if it obeys the equation ˜ i ···i ) , K(i1 ···ip ;j) = gj(i1 K p 2 RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

(12)

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˜ is determined by tracing the both sides of equation (12). There where the tensor K is also a similar relation between CKY and CSK tensors as in equation (10). Let us consider a 2-form kµν which is a closed CKY tensor. At least locally, there exist a one form potential b so that k = db ,

(13)

Such a form b is usually called a KY potential and a non-degenerate 2-form k is called a principal CKY tensor. 3. LAX PAIR TENSORS

The complete integrability of the geodesic system (3) can be connected with the existence of a Lax pair of matrices L and A, functions on the phase space of the system, satisfying the Lax pair equation [2] dL L˙ := = [L, A] , dt

(14)

where the time derivative is L˙ = {L, H} . It follows from from (14) that the quantities

(15)

1 Ik := T rLk , (16) k are all constant of motion. The system is integrable in the sense of Liouville if it possesses sufficiently many of the integrals Ik which are in involution (i. e. [Ik , Il ] = 0 for all l, k). The Lax representation is not unique since the evolution equation (14) is invariant under a transformation of the form [12] ˜ = U LU −1 , L

˜ = U AU −1 − U˙ U −1 . A

(17)

For a geometrical interpretation of the Lax representation [13] we remark that according to invariances (17) L transforms as a tensor while A transforms as a connection. We choose a pair of matrices linear and homogeneous functions of momenta [3–5] Lµν = Lµν λ pλ , Aµν = Aµν λ pλ . (18) Taking into account the invariance (17) the simplest choice is Aµν λ = Γµν λ .

(19)

With this identification the Lax pair equation (14) leads to the equation Lµν(λ;σ) = 0 . RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

(20)

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Furthermore, choosing L to be an antisymmetric tensor we get a geometric object which is of particular interest. Indeed, comparing (20) with (9) one obtains an interesting realization of the Lax pair tensors in terms of KY tensors of rank three [3–5]. Using this special construction of Lax tensors as third rank KY tensors we can proceed with the evaluation of the constants of motion (16). Taking into account the antisymmetry of the KY tensors, the first quantity I1 vanishes. But for the second one, on the strength of the relation between SK and KY tensors (10), we get the conserved constant of motion (7) I2 = K µν pµ pν .

(21)

Here K µν is the SK tensor associated with the third rank KY tensor Lµνλ . 4. KILLING-YANO TENSORS OF RANK THREE

In what follows we shall illustrate the above simple realization of the Lax pair tensors on some spacetimes admitting KY tensors of rank three which appear in some modern studies in gravitation. 4.1. KIMURA IIC METRIC

Kimura IIC metric [14] is of Petrov type D given by ds2 =

r2 2 1 dt − 2 2 dr2 − r2 (dθ2 + sin2 θdϕ2 ) , b r b

(22)

with b a constant. By straightforward calculation one obtains two independent sets of KY tensors of rank three [15, 16]: (1)

ftθϕ = r4 sin θ ,

(23)

and (2)

ftθϕ = btr4 sin θ ,

(2)

frθϕ = r sin θ ,

(24)

all other components being zero. In the first case (23) the conserved constant of motion I2 (21) is the SK tensor (10) with the following non-null components (11)

Ktt

= 2r4 ,

(11)

Kθθ = −2br4 ,

(11) Kϕϕ = −2br4 sin2 θ .

(25)

However this SK tensor proves to be reducible since it can be written as a linear combination of symmetrized product of Killing vectors of the metric (22). More interesting is the second case, KY tensor (24) generating an irreducible SK tensor RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

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with the following non-null components [15, 16] (22)

Ktt

= 2b2 t2 r4 ,

2 , r2 (22) Kϕϕ = 2b2 r2 (1 − bt2 r2 ) sin2 θ.

(22)

Ktr

(22)

Kθθ = 2b2 r2 (1 − bt2 r2 ),

= 2btr,

(22) Krr =

(26)

Moreover, the SK tensor constructed from both KY (23) and KY (24) is irreducible with the following non-null components (12)

Ktt

= 2btr4 ,

(12)

(12)

Ktr

Kθθ = −2b2 tr4 ,

= r,

(27)

(12) Kϕϕ = −2btr4 sin2 θ .

4.2. MYERS-PERRY METRICS

The Myers-Perry (MP) metrics [17] are the most general known vacuum solutions for the higher dimensional rotating black holes. MP metrics can be written compactly as [18, 19] !2 n X U dr2 2M 2 2 2 ds = − dt + + dt + ai µi dφi V − 2M U i=1 (28) n X 2 2 2 2 2 2 2 + (r + ai )(µi dφi + dµi ) + r dµn+ , i=1

where the metric functions V = r−2

n Y

n

(r2 + a2i ) ,

X a 2 µ2 U i i , = 1− V r2 + a2i

(29)

i

i=1

and the latitude coordinates obey the relation n X

µ2i + µ2n+ = 1 ,

n = [(D − 1)/2] .

(30)

i

Here [(D − 1)/2] stands for the integer part of (D − 1)/2 and = 1 for even and = 0 for odd spacetime dimensions D. In [18] it was shown that MP metrics admit the CKY potential b ! n n X X 2b = r2 + a2i µ2i dt + ai µ2i (r2 + a2i )dφi , (31) i=1

i=1

and the corresponding closed CKY tensor (13) is k=

n X

ai µi dµi ∧ [ai dt + (r

2

+ a2i )dφi ] + rdr ∧

i=1

RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

dt +

n X i=1

! ai µ2i dφi

.

(32)

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Looking for KY tensors of rank three we shall concentrate on the D = 5 MP spacetimes. For D = 5 there arep 2 rotation parameters a1 and a2 and taking into account the constraint (30) µ2 = 1 − µ21 . The Hodge dual f = − ∗ k of (32) is a KY tensor of rank three: −f =rdt ∧ dr ∧ [a2 µ21 dφ1 + a1 (1 − µ21 )dφ2 ] + µ1 dt ∧ dµ1 ∧ [a2 (r2 + a21 )dφ1 − a1 (r2 + a22 )dφ2 ] + µ1 (r2 + a21 )(r2 + a22 )dφ1 ∧ dφ2 ∧ dµ1

(33)

+ rµ21 (1 − µ21 )(a22 − a21 )dφ1 ∧ dφ2 ∧ dr . An explicit calculation gives the following expression for the SK tensor (10) [18] K

µν

=

2 X

a2i (µ2i − 1)g µν

+ a2i µ2i δtµ δtν

i=1

1 + 2 δφµi δφν i µi

(34)

+ δµµ1 δµν 1 − 2Z (µ Z ν) − 2ξ (µ ζ ν) , where ξ = ∂t ,

ζ=

2 X

a i ∂ φi ,

Z = µ1 ∂µ1 .

(35)

1

Again the last term in (34) constructed from the Killing vectors ξ and ζ can be excluded from the SK tensor K representing a reducible part of it. 4.3. OTHER CONSTRUCTIONS INVOLVING PRINCIPAL CKY TENSORS

The above construction of KY tensors of rank three in 5-dimensional MP metrics can be extended in higher dimensions for spaces admitting principal CKY tensors. First of all let us use the fact that the external product k = k (1) ∧ k (2) of two closed CKY tensors k (1) and k (2) is also a closed CKY tensor [20]. Using this fact, starting with a principal CKY tensor k it is possible to generate a ”tower” of new closed CKY tensors: k (2j) = k ∧j = |k ∧ .{z . . ∧ k} .

(36)

j factors

Each of 2j- form k (2j) determines a (D − 2j)-form of KY tensors f (D−2j) = ∗k (2j) ,

(37)

and accordingly a SK-tensor as in (10). For the special case D −2j = 3 we are placed in the Lax pair tensors framework involving KY tensors of rank three. RJP 57(Nos. 5-6), 1002–1010 (2012) (c) 2012-2012

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5. CONCLUSIONS

The (C)SK and (C)KY tensors are related to a multitude of different topics such as classical integrability of systems together with their quantization, supergravity, string theories, hidden symmetries in higher dimensional spacetimes, etc. On the other hand a characteristic feature in the study of integrable systems is the existence of a pair of matrices satisfying the Lax equation. The Lax tensors considered in [3–5] represent an interesting approach to investigate hidden symmetries generated by third rank KY tensors. To conclude let us discuss shortly some problems that deserve a further attention. The geometrical setting for the Lax representation presented in this paper could be extended in various ways. First of all here we considered only third rank KY tensors which are totally symmetric objects, but in general the matrices L, A could have as well as symmetric parts. For example it is possible to define a sort of f-symbols [21] for third rank Killing tensors having both symmetric and antisymmetric parts. On the other hand it will be interesting to extend the present geometrical setting for (C)KY tensors of arbitrary rank. Acknowledgments. Support from ANCS, Romania through NUCLEU program PN 09370102 is acknowledged.

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