arxiv:gr-qc/9805052v1 14 May 1998 DUALITY AND STRUCTURAL EQUATIONS FOR KILLING TENSORS OF ORDER TWO Dumitru Baleanu 1 2 JINR, Bogoliubov Laboratory of Theoretical Physics 141980 Dubna,Moscow Region, Russia Abstract The structural equations for the Killing-Yano tensors of order two was investigated.the physical signification of the tensors L αβγ and M αβδγ was obtained using the dual transformation. The structural equations in the dual space was investigated. 1 Permanent address:institute for Space Sciences,P.O.BOX, MG-36, R 76900 Magurele- Bucharest, Romania 2 e-mail:baleanu@thsun1.jinr.ru 1
1 Introduction In a geometrical setting, symmetries are connected with isometries associated with Killing vectors, and more generally, Killing tensors on the configuration space of the system.an example is the motion of a point particle in a space with isometries [1], which is a physicist s way of studying the geodesic structure of a manifold. Any symmetric tensor K αβ which satisfies the condition K (αβ;γ) = 0 (1) is called a tensor of order two.here the parenthesis denotes a full symmetrization with all indices and coma denotes a covariant derivative.k αβ will be called redundant if it is equal to some linear combination with constants coefficients of the metric tensor g αβ and of the form S (α B β) where A α and B β are Killing vectors. For any Killing vector K α we have[2] K β;α = ω αβ = ω βα (2) ω αβ;γ = R αβγδ K δ (3) equations (2) and (3) may be regarded as a system of linear homogeneous first-order equations in the components K αβ,ω αβ. In [2] equations analogous to the above ones for a Killing vector are derived for a Killing vector are derived for a Killing tensor K αβ. Recently Holten [3] have presented a theorem concerning the reciprocal relation between two local geometries described by metrics which are Killing tensors with respect to one another.in [4] the non-generic symmetries was investigated for the Taub-NUT space-time. The main aim of this paper is to investigate the structural equations in the case of dual space and to give a natural physical interpretation for L αβδ and M αβδγ. 2 Geometric Duality Let a space with metric g µν admits a Killig tensor field K µν. A Killing-Yano tensor is a symmetric tensor which satisfies the following relation: K ( µν; λ) = 0 (4) The semicolon denotes a Riemannian covariant derivative, and the parentheses denote complete full symmetrization over the component indices.the equation of motion of a particle on a geodesic is derived from the action S = dτ( 1 2 g µν x µ ẋ ν ) (5) 2
The Hamiltonian is constructed in terms of the inverse metric H = 1 2 gµν p µ p ν The Poisson brackets are {x µ, p ν } = δµ ν (6) The equation of motion for a phase space functionf(x, p) can be computed from the Poisson brackets with the Hamiltonian F = {F, H} (7) From the contravariant components K µν of the Killing tensor we can construct a constant of motion K, We can easy verified that K = 1 2 Kµν p µ p ν (8) {H, K} = 0 (9) We have two different way to investigate the symmetries of the manifold. First, we consider the metric g µν and the Killing tensor K µν and the second we consider K µν like a metric and the Killing tensor g µν. If we denote D to be the covariant derivative in respect with g µν and ˆD the covariant derivative with respect with K µν we have the following properties: D λ g µν = 0 D λ K µν + D µ K νλ + D ν K λµ = 0 (10) ˆD λ K µν = 0 ˆD λ g µν + D µ g νλ + D ν g λµ = 0 (11) 3 The structural equations The following two vectors play roles analogous to that of a bivector ω αβ L αβγ = K βγ;α K αγ;β (12) M αβγδ = 1 2 (L αβ[γ;δ] + L γδ[α;β] ) (13) The properties of the tensors L αβγ and M αβγδ are derived in [2].M αβγδ has the same symmetries as the Riemannian tensor and the covariant derivatives of K αβ and L αβγ satisfy relations reminiscent of those satisfied by Killing vectors. From (4) and (5) we express M αβγδ in terms of second covariant derivative of K αβ, M αβγδ = 1 2 (K βγ;(αδ) + K αδ;(βγ) K αγ;(βδ) K βδ;(αγ) ) (14) 3
From (14) we obtain the identity M αβγδ + M γαβδ + M βγαδ = 0 (15) At this moment we are interested to investigate the structural equations when we work in the dual space. A very important relation is the relation between connections of initial and dual spaces. ˆΓ µ νλ = Γµ νλ Kµδ D δ K νλ (16) If we denote we obtain a very interesting result H µ νλ = ˆΓ µ νλ Γµ νλ (17) H δ νλ K δγ = D γ K νλ (18) By construction H µ νλ is a tensor. We are interested to make a connection between H µ νλ and the tensors L αβγ and M αβδγ. From (4) and (18) we can deduce immediately that Then L αβγ looks like an angular momentum. Using (14 and (18) the expression of M αβγδ is M αβγδ + + + L αβγ = H δ βγ K δβ H δ βγ K δα (19) = K σα 2 [ D δhβγ σ + D γ Hβδ σ + HθγH σ δβ θ HγβH θ θδ] σ K σβ 2 [D δh σ αγ D γh σ αδ + Hθ δα Hσ θγ Hθ γα Hσ θδ ] K σγ 2 [ D βhδα σ + D αhδβ σ Hθ βδ Hσ θα + Hθ αδ Hσ θβ ] K σδ 2 [ D αhγβ σ + D βhγα σ + Hθ βγ Hσ θα Hθ αγ Hσ θβ ] (20) The expression of a curvature tensor [5] is R β νρσ = Γ β νσ,ρ Γ β νρ,σ + Γ α νσγ β αρ Γ α νργ β ασ (21) When the initial space is a flat space the general expression for K αβ is K βγ = s βγ + 2 3 B α(βγ)x α + 1 3 A αβγδx α x δ (22) where s βγ, B αβγ and A αβγδ are arbitrary uniform fields having the same symmetries as K βγ,l αβγ and M αβγδ. respectively. From (20) and (21) the expressions of M αβγδ becomes M αβδγ = 1 2 (K σαr σ βγδ + K σβ R σ αδγ + K σγ R σ δαβ + K σδ R σ γβα ) (23) 4
where R σ νρσ has the same form like R σ νρσ from (21) with Hσ αδ instead of Γσ αδ. This important result hold if when we work on a curve in a initial manifold described by g µν [6]. Next step is to investigate the form of M αβγδ when the initial manifold g µν has a curvature. After tedious calculations the final expression for M αβγδ are where M αβγδ = K σα R σ βγδ + K σβ R σ αδγ + K σγ R σ δαβ + K σδ R σ γβα K σχ ( H χ βδ Gσ αγ + Hχ αδ Gσ βγ + Hχ βγ Gσ αδ Hχ αγ Gσ βδ ) + HβδK σ αγ,σ + HαγK σ βδ,σ HβγK σ αδ,χ HαδK σ βγ,σ (24) G µ νλ = Hµ νλ + Γµ νλ (25) In the presence of curvature the expression of M αβγδ is more complicated than the same expression in the flat space. Because g µν is dual to K µν from (1) and ( 2) we found immediately that ˆL µνλ = ˆD α g βγ ˆD β g αγ = H δ βγg δβ + H δ βγg δα (26) and ˆM αβγδ = 1 2 (ˆL αβ[γ;δ] + ˆL γδ[α;β] ) (27) Here the semicolon denotes the dual covariant derivative. In the dual space we have the following expression for ˆM αβγδ. ˆM αβγδ = g σα R σ βγδ g σβ R σ αδγ g σγ R σ δαβ g σδ R σ γβα g σχ (H χ βδĝσ αγ H χ αδĝσ βγ + H χ βγĝσ αδ + H αγĝσ χ βδ) Hβδ σ g αγ,σ Hαγ σ g βδ,σ + Hβγ σ g αδ,χ + Hαδ σ g βγ,σ (28) where Ĝσ αδ = H σ αδ+ˆγ σ αδ. The tensors ˆL αβγ and ˆM αβγ have the same symmetries like L µνλ and M αβγδ. From the consistency conditions (17) we have new identity satisfies by K αβ K σ β K νλ;σ + K σ λ K βν;σ + K σ ν K βλ;σ = 0 (29) In the dual space we have the following relations for g µν g σ β g νλ;σ + g σ λ g βν;σ + g σ nu g βλ;σ = 0 (30) In the last relation semicolon denotes the covariant derivative in the dual space. 4 Acknowledgments I would like to thank S.Manoff for many valuable discussions. 5
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