Separability of Dirac equation in higher dimensional Kerr NUT de Sitter spacetime

Transcription

1 Physics Letters B wwwelseviercom/locate/physletb Separability of Dirac equation in higher dimensional Kerr NUT de Sitter spacetime Takeshi Oota a Yukinori Yasui b a Osaka City University Advanced Mathematical Institute OCAMI Sugimoto Sumiyoshi Osaka Japan b Department of Mathematics and Physics Graduate School of Science Osaka City University Sugimoto Sumiyoshi Osaka Japan Received 2 November 2007; accepted 28 November 2007 Available online 3 December 2007 Editor: M Cvetič Abstract It is shown that the Dirac equations in general higher dimensional Kerr NUT de Sitter spacetimes are separated into ordinary differential equations 2007 Elsevier BV All rights reserved Recently the separability of Klein Gordon equations in higher dimensional Kerr NUT de Sitter spacetimes [] was shown by Frolov et al [2] This separation is deeply related to that of geodesic Hamilton Jacobi equations Indeed a geometrical object called conformal Killing Yano tensor plays an important role in the separability theory [2 9] However at present a similar separation of the variables of Dirac equations is lacking although the separability in the four-dimensional Kerr geometry was given by Chandrasekhar [0] In this Letter we shall show that Dirac equations can also be separated in general Kerr NUT de Sitter spacetimes The D-dimensional Kerr NUT de Sitter metrics are written as follows []: a D = 2n g 2n = n dx 2 n Q x + Q x n A k dψ k 2 b D = 2n + g 2n+ = n dx 2 n Q x + Q x n 2 n 2 A k dψ k + c A n A k dψ k 2 The functions Q = 2naregivenby Q x = X U U = ν x 2 xν 2 3 * Corresponding author addresses: toota@sciosaka-cuacjp T Oota yasui@sciosaka-cuacjp Y Yasui /$ see front matter 2007 Elsevier BV All rights reserved doi:006/jphysletb

2 T Oota Y Yasui / Physics Letters B where X is a function depending only on the coordinate x and A k and A k are the elementary symmetric functions of {x 2 ν } and {x 2 ν } ν respectively: t x 2 ν = A 0 t n A t n + + n A n ν t x 2 ν = A 0 tn A tn + + n A n The metrics are Einstein if X takes the form [] a D = 2n X = n b D = 2n + X = c 2k x 2k + b x n c 2k x 2k + b + n c x 2 where cc 2k and b are free parameters D = 2n For the metric we introduce the following orthonormal basis {e a }={e e n+ } = 2n: e = dx Q The dual vector fields are given by e = Q e n+ = x e n+ = n Q A k dψ k n The spin connection is calculated as [] k x 2n k Q U ψ k 8 9 ω ν = x ν Qν x 2 x2 ν e x Q x 2 e ν x2 ν ν x Qρ ω n+ = Q e n+ x 2 ρ ρ x2 e n+ρ no sum over ω n+ν = x Qν x 2 x2 ν ω n+n+ν = x Qν x 2 x2 ν e n+ x Q x 2 x2 ν e x ν x 2 x2 ν e n+ν Q e ν Then the Dirac equation is written in the form γ a D a + m Ψ = 0 where D a is a covariant differentiation D a = e a + 4 ω bce a γ b γ c ν ν 0 2 From 9 0 and 2 we obtain the explicit expression for the Dirac equation

3 690 T Oota Y Yasui / Physics Letters B n γ Q + n X + + x 2 X 2 n γ n+ Q n ν k x 2n k X x x 2 x2 ν ˆΨ ψ k + 2 n ν x ν x 2 x2 ν Let us use the following representation of γ -matrices: {γ a γ b }=2δ ab γ ν γ n+ν ˆΨ + m ˆΨ = 0 3 γ = σ 3 σ 3 σ }{{} 3 σ I I γ n+ = σ 3 σ 3 σ }{{} 3 σ 2 I I where I is the 2 2 identity matrix and σ i are the Pauli matrices In this representation we write the 2 n components of the spinor field as Ψ ɛ ɛ 2 ɛ n ɛ =± and it follows that 4 γ Ψ ɛ ɛ 2 ɛ n = ɛ ν Ψ ɛ ɛ ɛ ɛ + ɛ n γ n+ Ψ = iɛ ɛ ɛ 2 ɛ n ɛ ν Ψ ɛ ɛ ɛ ɛ + ɛ n By the isometry the spinor field takes the form n Ψ ɛ ɛ 2 ɛ n x ψ = ˆΨ ɛ ɛ 2 ɛ n x exp i N k ψ k with arbitrary constants N k Substituting 5 into 3 we obtain n Q ɛ ρ + X + ɛ Y + n Ψ ɛ ɛ x 2 X 2 X 2 x ɛ ɛ ν x ɛ ɛ + ɛ n + mψ ɛ ɛ 2 ɛ n = 0 ν where we have introduced the function n Y = k x 2n k N k ν which depends only on x Consider now the region x x ν > 0for<νand x + x ν > 0 Let us define Φ ɛ ɛ 2 ɛ n x = <ν n Then one can obtain an equality x + ɛ ɛ ν x ν Φ ɛ ɛ ɛ ɛ + ɛ n x = ɛ ɛ ρ Φ ɛ ɛ 2 ɛ n x n U x ɛ ɛ ν x ν ν Now we show that the Dirac equation allows a separation of variables by setting ˆΨ ɛ ɛ 2 ɛ n x = Φ ɛ ɛ 2 ɛ n x χ ɛ x It should be noticed that x log ˆΨ ɛ ɛ ɛ ɛ + ɛ n = d dx log χ ɛ 2 n ν x ɛ ɛ ν x ν

4 T Oota Y Yasui / Physics Letters B By using 20 and 22 the substitution of 2 into 7 leads to n where P ɛ Putting P ɛ P n ν ɛ x ɛ x ɛ ν x ν + m = 0 is a function of the coordinate x only = ɛ n d X + dx 2 n 2 Qy = my n + q j y j j=0 with arbitrary constants q j we find P ɛ x = Qɛ x Thus the functions χ ɛ d + X + ɛ Y χ dx 2 X X χ ɛ X satisfy the ordinary differential equations ɛ ɛ n Qɛ x X X + ɛ Y X χ ɛ = 0 χ ɛ D = 2n + For the metric 2 we introduce the orthonormal basis {ê a }={ê ê n+ ê 2n+ } = 2n: ê = e ê n+ = e n+ ê 2n+ = S n A k dψ k with S = c/a n The -forms e and e n+ are defined by 8 The dual vector fields are given by ê = e ê n+ = e n+ + n x 2 Q U with 9 The spin connection is calculated as [] ψ n ê 2n+ = SA n ψ n S ˆω ν = ω ν ˆω n+ν = ω n+ν + δ ν ê 2n+ ˆω n+n+ν = ω n+n+ν x S ˆω 2n+ = ê n+ Q S ê 2n+ ˆω n+2n+ = ê x x x A similar calculation to the even-dimensional case yields the following Dirac equation n γ Q + X + + n x x 2 X 2x 2 x 2 Ψ x2 ν + n + γ 2n+ S n γ n+ Q n k x 2n k X 2x γ γ n+ + c ν ψ k + n x 2 X ψ n ψ n + 2 Ψ + mψ = 0 We use the representation of γ -matrices given by 4 together with γ 2n+ = σ 3 σ 3 σ 3 n ν x ν x 2 x2 ν γ ν γ n+ν Ψ

5 692 T Oota Y Yasui / Physics Letters B Thus the spinor field ˆΨ ɛ ɛ 2 ɛ n defined by Ψ ɛ ɛ 2 ɛ n x ψ = ˆΨ ɛ ɛ 2 ɛ n x exp satisfies the equation where n Q + i S ɛ ρ Ŷ = i n N k ψ k ɛ ρ + X + x 2 X 2 n n k x 2n k N k ɛ Ŷ X + 2x + 2 n ν ɛ + N n + m ˆΨ ɛ ɛ 2x c 2 ɛ n = 0 We find that the Dirac equation above allows a separation of variables ˆΨ ɛ ɛ 2 ɛ n x = Φ ɛ ɛ 2 ɛ n x χ ɛ x x with Φ ɛ ɛ 2 ɛ n defined by 9 Indeed 34 becomes n P n ν ɛ x ɛ x ɛ ν x ν + i c n ɛ ρ x ρ with the help of 24 Let us introduce the function where ˆQy = n j= 2 q j y j n x ɛ ɛ ν x ν ɛ + N n + m = 0 2x c q n = m q = i 2 n c q 2 = i c n N n ˆΨ ɛ ɛ ɛ ɛ + ɛ n Using the identities n n y 2 ν y y ν = n n y ν y ν y y ν = ν n n y n y ν we can confirm that the functions χ ɛ satisfy the ordinary differential equations 27 by the replacements Y Ŷ and Qɛ x ˆQɛ x We have shown the separation of variables of Dirac equations in general Kerr NUT de Sitter spacetimes An interesting problem is to investigate the origin of separability In the case of geodesic Hamilton Jacobi equations and Klein Gordon equations we know that the existence of separable coordinates comes from that of a rank-2 closed conformal Killing Yano tensor We can also construct the first order differential operators from the closed conformal Killing Yano tensor which commute with Dirac operators [2 4] However we have no clear answer of the separability of Dirac equations As another problem we can study eigenvalues of Dirac operators on Sasaki Einstein manifolds Indeed as shown in [5 7] the BPS limit of odd-dimensional Kerr NUT de Sitter metrics leads to Sasaki Einstein metrics Especially the five-dimensional metrics are important from the point of view of AdS/CFT correspondence 40 4

6 T Oota Y Yasui / Physics Letters B Acknowledgements We thank GW Gibbons for attracting our attention to Refs [2 4] This work is supported by the 2 COE program Construction of wide-angle mathematical basis focused on knots The work of YY is supported by the Grant-in Aid for Scientific Research Nos and from Japan Ministry of Education The work of TO is supported by the Grant-in Aid for Scientific Research Nos and from Japan Ministry of Education References [] W Chen H Lü CN Pope Class Quantum Grav hep-th/ [2] VP Frolov P Krtouš D Kubizňák JHEP hep-th/06245 [3] VP Frolov D Kubizňák Phys Rev Lett gr-qc/ [4] D Kubizňák VP Frolov Class Quantum Grav F gr-qc/06044 [5] DN Page D Kubizňák M Vasudevan P Krtouš Phys Rev Lett hep-th/06083 [6] P Krtouš D Kubizňák DN Page VP Frolov JHEP hep-th/ [7] P Krtouš D Kubizňák DN Page M Vasudevan hep-th/ [8] T Houri T Oota Y Yasui hep-th/ [9] T Houri T Oota Y Yasui Phys Lett B hep-th/ [0] S Chandrasekhar Proc R Soc London A [] N Hamamoto T Houri T Oota Y Yasui J Phys A F77 hep-th/06285 [2] GW Gibbons RH Rietdijk JW van Holten Nucl Phys B hep-th/93032 [3] M Tanimoto Nucl Phys B gr-qc/ [4] M Cariglia Class Quantum Grav hep-th/ [5] Y Hashimoto M Sakaguchi Y Yasui Phys Lett B hep-th/04074 [6] M Cvetič H Lü DN Page CN Pope Phys Rev Lett hep-th/ [7] M Cvetič H Lü DN Page CN Pope hep-th/