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

0.11 MB·English

by  Takeshi Oota

November, 2007
OCU-PHYS 280
Separability of Dirac equation
in higher dimensional Kerr-NUT-de Sitter spacetime
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0 Takeshi Ootaa1 and Yukinori Yasuib2
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2

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a aOsaka City University Advanced Mathematical Institute (OCAMI)
J
3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN

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bDepartment of Mathematics and Physics, Graduate School of Science,

] Osaka City University
h
t 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN
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p
e
h
[

2
v
Abstract
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It is shown that the Dirac equations in general higher dimensional Kerr-NUT-de Sitter space-
0
times are separated into ordinary differential equations.
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a
1 [email protected]
2 [email protected]
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Recently, the separability of Klein-Gordon equations in higher dimensional Kerr-NUT-de Sitter
spacetimes [1] was shown by Frolov, Krtouˇs and Kubiznˇa´k [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 [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
n dx2 n n−1 2
g(2n) = µ + Q (x) A(k)dψ , (1)
µ=1Qµ(x) µ=1 µ k=0 µ k!
X X X
(b) D = 2n+1
n dx2 n n−1 2 c n 2
g(2n+1) = µ + Q (x) A(k)dψ + A(k)dψ . (2)
µ=1 Qµ(x) µ=1 µ k=0 µ k! A(n) k=0 k!
X X X X
The functions Q (µ = 1,2, ,n) are given by
µ
···
n
X
Q (x) = µ, U = (x2 x2), (3)
µ U µ µ− ν
µ
νY=1
(ν6=µ)
where X is a function depending only on the coordinate x , and A(k) and A(k) are the elementary
µ µ µ
symmetric functions of x2 and x2 respectively:
ν ν ν6=µ
{ } { }
n
(t x2)= A(0)tn A(1)tn−1+ +( 1)nA(n), (4)
ν
− − ··· −
ν=1
Y
n
(t x2) = A(0)tn−1 A(1)tn−2+ +( 1)n−1A(n−1). (5)
ν µ µ µ
− − ··· −
νY=1
(ν6=µ)
The metrics are Einstein if X takes the form [1, 11]
µ
(a) D = 2n
n
X = c x2k +b x , (6)
µ 2k µ µ µ
k=0
X
(b) D = 2n+1
n ( 1)nc
X = c x2k +b + − , (7)
µ 2k µ µ x2
k=0 µ
X
where c,c and b are free parameters.
2k µ
1. D=2n
For the metric (1) we introduce the following orthonormal basis ea = eµ, en+µ (µ = 1,2, ,n):
{ } { } ···
n−1
dx
eµ = µ , en+µ = Q A(k)dψ . (8)
Q µ µ k
µ
q kX=0
The dual vector fields are given by p
∂ n−1( 1)kx2(n−1−k) ∂
µ
e = Q , e = − . (9)
µ µ n+µ
∂x Q U ∂ψ
µ µ µ k
q kX=0
p
2
The spin connection is calculated as [11]
x √Q x Q
ω = ν ν eµ µ µ eν, (µ = ν)
µν −x2 x2 − x2 x2 6
µ ν µp ν
− −
x Q
ω = (∂ Q ) en+µ µ ρ en+ρ, (no sum over µ), (10)
µ,n+µ − µ µ − x2 x2
q ρX6=µ ρp− µ
x √Q x Q
ω = µ ν en+µ µ µ en+ν, (µ = ν)
µ,n+ν x2 x2 − x2 x2 6
µ ν µp ν
− −
x √Q x Q
ω = µ ν eµ ν µ eν, (µ = ν).
n+µ,n+ν −x2 x2 − x2 x2 6
µ ν µp ν
− −
Then, the Dirac equation is written in the form
(γaD +m)Ψ = 0, (11)
a
where D is a covariant differentiation,
a
1
D = e + ω (e )γbγc. (12)
a a bc a
4
From (9),(10) and (12), we obtain the explicit expression for the Dirac equation
n ∂ 1X′ 1 n x
γµ Q + µ + µ Ψ (13)
µX=1 q µ ∂xµ 2Xµ 2 νX=1 x2µ−x2ν!
(ν6=µ)
n n−1 ( 1)kx2(n−1−k) ∂ 1 n x
+ γn+µ Q − µ + ν (γνγn+ν) Ψ+mΨ = 0.
µX=1 q µ kX=0 Xµ ∂ψk 2 νX=1 x2µ−x2ν !
(ν6=µ)
Let us use the following representation of γ-matrices: γa,γb = 2δab,
{ }
γµ = σ σ σ σ I I, (14)
3 3 3 1
⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
µ−1
γn+µ = |σ3 σ3 {z σ3} σ2 I I,
⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
µ−1
| {z }
where I is the 2 2 identity matrix and σ are the Pauli matrices. In this representation, we write
i
×
the 2n components of the spinor field as Ψ (ǫ = 1), and it follows that
ǫ1ǫ2···ǫn µ ±
µ−1
(γµΨ) = ǫ Ψ , (15)
ǫ1ǫ2···ǫn  ν ǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn
ν=1
Y
 
µ−1
(γn+µΨ) = iǫ ǫ Ψ .
ǫ1ǫ2···ǫn − µ ν ǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn
ν=1
Y
 
By the isometry the spinor field takes the form
n−1
Ψ (x,ψ) = Ψˆ (x)exp i N ψ (16)
ǫ1ǫ2···ǫn ǫ1ǫ2···ǫn k k
!
k=0
X
3
with arbitrary constants N . Substituting (15) into (13), we obtain
k
n µ−1 ∂ 1X′ 1ǫ Y 1 n 1
Q ǫ + µ + µ µ + Ψˆ
µX=1 p µ ρY=1 ρ! ∂xµ 2Xµ 2 Xµ 2 νX=1 xµ−ǫµǫνxν! ǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn
(ν6=µ)
+ mΨˆ = 0, (17)
ǫ1ǫ2···ǫn
where we have introduced the function
n−1
Y = ( 1)kx2(n−1−k)N , (18)
µ µ k
−
k=0
X
which depends only on x .
µ
Consider now the region x x > 0 for µ < ν and x +x > 0. Let us define
µ ν µ ν
−
1
Φ (x)= . (19)
ǫ1ǫ2···ǫn √xµ+ǫµǫνxν
1≤µ<ν≤n
Y
Then, one can obtain an equality
Φǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn(x) =( ǫ )µ−1 µ−1ǫ (−1)µ−1Uµ . (20)
Φǫ1ǫ2···ǫn(x) − µ ρY=1 ρ! nq(xµ ǫµǫνxν)
−
νY=1
(ν6=µ)
Now we show that the Dirac equation allows a separation of variables by setting
n
Ψˆ (x) = Φ (x) χ(µ)(x ). (21)
ǫ1ǫ2···ǫn ǫ1ǫ2···ǫn ǫµ µ
µ=1
Y
It should be noticed that
n
∂ d 1 1
logΨˆ = logχ(µ) . (22)
∂x ǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn dx −ǫµ − 2 x ǫ ǫ x
µ µ µ µ ν ν
νX=1 −
(ν6=µ)
By using (20) and (22), the substitution of (21) into (17) leads to
n (µ)
P (x )
ǫµ µ +m = 0, (23)
n
µX=1 (ǫµxµ ǫνxν)
−
νY=1
(ν6=µ)
(µ)
where P is a function of the coordinate x only,
ǫµ µ
′
1 d 1X ǫ Y
P(µ) = ( 1)µ−1(ǫ )n−µ ( 1)µ−1X + µ + µ µ χ(µ) . (24)
ǫµ − µ q − µχǫ(µµ) dxµ 2Xµ Xµ ! −ǫµ
Putting
n−2
Q(y) = myn−1+ q yj (25)
j
−
j=0
X
4
with arbitrary constants q , we find
j
P(µ)(x ) =Q(ǫ x ). (26)
ǫµ µ µ µ
(µ)
Thus, the functions χ satisfy the ordinary differential equations
ǫµ
′
d 1X ǫ Y ( 1)µ−1(ǫ )n−µQ(ǫ x )
+ µ + µ µ χ(µ) − µ µ µ χ(µ) = 0. (27)
dxµ 2Xµ Xµ ! −ǫµ − ( 1)µ−1X ǫµ
µ
−
2. D=2n+1 q
For the metric (2) we introduce the orthonormal basis eˆa = eˆµ, eˆn+µ, eˆ2n+1 (µ = 1,2, ,n):
{ } { } ···
n
eˆµ = eµ, eˆn+µ = en+µ, eˆ2n+1 = √S A(k)dψ (28)
k
k=0
X
with S = c/A(n). The 1-forms eµ and en+µ are defined by (8). The dual vector fields are given by
( 1)n ∂ 1 ∂
eˆ = e , eˆ = e + − , eˆ = (29)
µ µ n+µ n+µ x2µ QµUµ∂ψn 2n+1 √SA(n)∂ψn
with (9). The spin connection is calculated asp[11]
√S
ωˆ = ω , ωˆ = ω +δ eˆ2n+1, ωˆ = ω ,
µν µν µ,n+ν µ,n+ν µν n+µ,n+ν n+µ,n+ν
x
µ
√S Q √S
ωˆ = eˆn+µ µeˆ2n+1, ωˆ = eˆµ. (30)
µ,2n+1 n+µ,2n+1
x − x − x
µ pµ µ
A similar calculation to the even dimensional case yields the following Dirac equation,
n ∂ 1X′ 1 1 n x
γµ Q + µ + + µ Ψ
µ
µX=1 q ∂xµ 2Xµ 2xµ 2 νX=1 x2µ−x2ν!
(ν6=µ)
n n−1( 1)kx2(n−1−k) ∂ ( 1)n ∂ 1 n x
+ γn+µ Q − µ + − + ν (γνγn+ν) Ψ
µX=1 q µ kX=0 Xµ ∂ψk x2µXµ ∂ψn 2 νX=1 x2µ−x2ν !
(ν6=µ)
n
1 1 ∂
+ γ2n+1√S (γµγn+µ)+ Ψ+mΨ= 0. (31)
− 2x c∂ψ 
µ=1 µ n
X
 
We use the representation of γ-matrices given by (14) together with
γ2n+1 = σ σ σ . (32)
3 3 3
⊗ ⊗···⊗
Thus, the spinor field Ψˆ defined by
ǫ1ǫ2···ǫn
n
Ψ (x,ψ) = Ψˆ (x)exp i N ψ (33)
ǫ1ǫ2···ǫn ǫ1ǫ2···ǫn k k
!
k=0
X
satisfies the equation
n µ−1 ∂ 1X′ 1ǫ Yˆ
µ µ µ
Q ǫ + +
µ ρ
µ=1 ρ=1 ! ∂xµ 2Xµ 2 Xµ
X p Y
n
1 1 1
+ + Ψˆ
2xµ 2 xµ ǫµǫνxν! ǫ1···ǫµ−1(−ǫµ)ǫµ+1···ǫn
νX=1 −
(ν6=µ)
n n
ǫ N
+ i√S ǫ µ + n +m Ψˆ = 0, (34)
 ρ=1 ρ!−µ=1 2xµ c   ǫ1ǫ2···ǫn
Y X
   
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where
n
Yˆ = ( 1)kx2(n−1−k)N . (35)
µ µ k
−
k=0
X
We find that the Dirac equation above allows a separation of variables
n (µ)
χ (x )
Ψˆ (x) = Φ (x) ǫµ µ (36)
ǫ1ǫ2···ǫn ǫ1ǫ2···ǫn µ=1 √xµ !
Y
with Φ defined by (19). Indeed, (34) becomes
ǫ1ǫ2···ǫn
n (µ) n
P (x ) i√c ǫ N
ǫµ µ + µ + n +m = 0 (37)
n n − 2x c 
µX=1 (ǫµxµ ǫνxν) (ǫρxρ) µX=1 µ
−  
νY=1 ρY=1
(ν6=µ)
with the help of (24). Let us introduce the function
n−1
Qˆ(y) = q yj (38)
j
j=−2
X
where
i i
q = m, q = ( 1)n−1√c, q = ( 1)nN . (39)
n−1 −1 −2 n
− 2 − √c −
Using the identities
n 1 ( 1)n−1 n 1
= − (40)
µX=1yµ2 (yµ−yν) n yµ νX=1 yν
ν
(νY6=µ) µY=1
n 1 ( 1)n−1
= − (41)
n
y (y y )
µX=1 µ µ− ν yµ
ν
(νY6=µ) µY=1
(µ)
we can confirm that the functions χ satisfy the ordinary differential equations (27) by the replace-
ǫµ
ments Y Yˆ and Q(ǫ x ) Qˆ(ǫ x ).
µ µ µ µ µ µ
→ →
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 andKlein-Gordonequations weknowthattheexistence ofseparablecoor-
dinates 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 [12, 13, 14]. 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. In-
deed, as shown in [1, 15, 16, 17], 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.
Acknowledgements
We thank G.W. Gibbons for attracting our attention to the references [12, 13, 14]. This work
is supported by the 21 COE program “Construction of wide-angle mathematical basis focused on
6
knots”. The work of Y.Y. is supported by the Grant-in Aid for Scientific Research (No. 19540304
andNo. 19540098) fromJapanMinistryofEducation. Theworkof T.O.issupportedbytheGrant-in
Aid for Scientific Research (No. 18540285 and No. 19540304) from Japan Ministry of Education.
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