November, 2007 OCU-PHYS 280
arXiv:0711.0078v2 [hep-th] 16 Jan 2008
Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime
Takeshi Ootaa1 and Yukinori Yasuib2
a Osaka
City University Advanced Mathematical Institute (OCAMI) 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN
b Department
of Mathematics and Physics, Graduate School of Science, Osaka City University 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN
Abstract It is shown that the Dirac equations in general higher dimensional Kerr-NUT-de Sitter spacetimes are separated into ordinary differential equations.
1 2
[email protected] [email protected]
1
Recently, the separability of Klein-Gordon equations in higher dimensional Kerr-NUT-de Sitter spacetimes [1] was shown by Frolov, Krtouˇs and Kubizˇ na´k [2]. This separation is deeply related to that of geodesic Hamilton-Jacobi equations. Indeed, a geometrical object called conformal KillingYano tensor plays an important role in the separability theory [3, 4, 5, 2, 6, 7, 8, 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 [10]. In this paper 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 [1]: (a) D = 2n !2 n−1 n n X X X dx2µ (k) (2n) (1) Aµ dψk , + Qµ (x) g = Q (x) µ=1 µ=1 µ k=0 (b) D = 2n + 1 g
(2n+1)
=
n X dx2µ
µ=1
Qµ (x)
+
n X
Qµ (x)
µ=1
n−1 X
A(k) µ dψk
k=0
!2
c + (n) A
n X
(k)
A
dψk
k=0
!2
.
(2)
The functions Qµ (µ = 1, 2, · · · , n) are given by Xµ , Uµ
Qµ (x) =
Uµ =
n Y
ν=1 (ν6=µ)
(x2µ − x2ν ),
(3)
(k)
where Xµ is a function depending only on the coordinate xµ , and A(k) and Aµ are the elementary symmetric functions of {x2ν } and {x2ν }ν6=µ respectively: n Y
(t − x2ν ) = A(0) tn − A(1) tn−1 + · · · + (−1)n A(n) ,
(4)
ν=1 n Y
ν=1 (ν6=µ)
n−1 n−2 (t − x2ν ) = A(0) − A(1) + · · · + (−1)n−1 Aµ(n−1) . µ t µ t
(5)
The metrics are Einstein if Xµ takes the form [1, 11] (a) D = 2n n X
Xµ =
c2k x2k µ + bµ xµ ,
(6)
k=0
(b) D = 2n + 1 Xµ =
n X
c2k x2k µ + bµ +
k=0
where c, c2k and bµ are free parameters.
(−1)n c , x2µ
(7)
1. D=2n For the metric (1) we introduce the following orthonormal basis {ea } = {eµ , en+µ } (µ = 1, 2, · · · , n): dxµ , eµ = p Qµ
en+µ =
q
Qµ
n−1 X
A(k) µ dψk .
(8)
k=0
The dual vector fields are given by eµ =
q
Qµ
∂ , ∂xµ
en+µ =
2
n−1 X
2(n−1−k)
(−1)k xµ p Qµ Uµ k=0
∂ . ∂ψk
(9)
The spin connection is calculated as [11] p √ xν Qν µ xµ Qµ ν ωµν = − 2 e − 2 e , xµ − x2ν xµ − x2ν q
n+µ
ωµ,n+µ = −(∂µ Qµ ) e
−
X xµ
x2 ρ6=µ ρ
(µ 6= ν)
p
Qρ n+ρ e , − x2µ
(no sum over µ),
(10)
p √ xµ Qν n+µ xµ Qµ n+ν = e − 2 e , (µ 6= ν) x2µ − x2ν xµ − x2ν p √ xµ Qν µ xν Qµ ν = − 2 e − 2 e , (µ 6= ν). xµ − x2ν xµ − x2ν
ωµ,n+ν ωn+µ,n+ν
Then, the Dirac equation is written in the form (γ a Da + m)Ψ = 0,
(11)
1 Da = ea + ωbc (ea )γ b γ c . 4
(12)
where Da is a covariant differentiation,
From (9),(10) and (12), we obtain the explicit expression for the Dirac equation n X
γ
µ=1
+
n X
µ=1
γ
µ
q
Qµ
n+µ
!
′ n xµ 1 Xµ 1 X ∂ + + Ψ 2 ∂xµ 2 Xµ 2 xµ − x2ν
ν=1 (ν6=µ)
2(n−1−k)
n−1 X
(−1)k xµ Xµ k=0
q
Qµ
(13) !
n ∂ 1 X xν + (γ ν γ n+ν ) Ψ + mΨ = 0. 2 ∂ψk 2 xµ − x2ν ν=1 (ν6=µ)
Let us use the following representation of γ-matrices: {γ a , γ b } = 2δab , γ µ = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗σ1 ⊗ I ⊗ · · · ⊗ I, γ
n+µ
|
{z
}
µ−1
(14)
= σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗σ2 ⊗ I ⊗ · · · ⊗ I, |
{z
}
µ−1
where I is the 2 × 2 identity matrix and σi are the Pauli matrices. In this representation, we write the 2n components of the spinor field as Ψǫ1 ǫ2 ···ǫn (ǫµ = ±1), and it follows that (γ µ Ψ)ǫ1 ǫ2 ···ǫn (γ n+µ Ψ)ǫ1 ǫ2 ···ǫn
=
µ−1 Y ν=1
ǫν Ψǫ1···ǫµ−1 (−ǫµ )ǫµ+1 ···ǫn ,
= −iǫµ
µ−1 Y ν=1
By the isometry the spinor field takes the form
ǫν Ψǫ1 ···ǫµ−1 (−ǫµ )ǫµ+1 ···ǫn .
ˆ ǫ ǫ ···ǫn (x) exp i Ψǫ1 ǫ2 ···ǫn (x, ψ) = Ψ 1 2
n−1 X k=0
3
(15)
Nk ψk
!
(16)
with arbitrary constants Nk . Substituting (15) into (13), we obtain n X
µ=1
µ−1 Y
p
Qµ
+
ǫρ
ρ=1
!
!
′
n ∂ 1 1 Xµ 1 ǫµ Yµ 1 X ˆ ǫ ···ǫ (−ǫ )ǫ ···ǫ + + + Ψ µ µ+1 n 1 µ−1 ∂xµ 2 Xµ 2 Xµ 2 xµ − ǫ µ ǫ ν xν ν=1 (ν6=µ)
ˆ ǫ ǫ ···ǫn = 0, mΨ 1 2
(17)
where we have introduced the function Yµ =
n−1 X
(−1)k xµ2(n−1−k) Nk ,
(18)
k=0
which depends only on xµ . Consider now the region xµ − xν > 0 for µ < ν and xµ + xν > 0. Let us define Φǫ1ǫ2 ···ǫn (x) =
Y
1≤µ
