- Email: [email protected]
Fermions tunneling of higher-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter
Physics Letters B 674 (2009) 127–130
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Fermions tunneling of higher-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter Kai Lin ∗ , ShuZheng Yang Institute of Theoretical Physics, China West Normal University, NanChong, SiChuan 637002, China
a r t i c l e
i n f o
Article history: Received 11 December 2008 Received in revised form 24 February 2009 Accepted 27 February 2009 Available online 6 March 2009 Editor: M. Cvetiˇc
a b s t r a c t The 1/2 spin fermions tunneling at the horizon of n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter is researched via semi-classical approximation method, and the Hawking temperature and fermions tunneling rate are obtained in this Letter. Using a new method, the semiclassical Hamilton–Jacobi equation is gotten from the Dirac equation in this Letter, and the work makes several quantum tunneling theories more harmonious. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
PACS: 04.70.Dy 04.62.+v 03.65.Sq Keywords: n-Dimensional Kerr–Anti-de Sitter black hole with one rotational parameter Hawking radiation Fermions tunneling
1. Introduction Recently, based on the work of Parikh and Wilczek et al. [1–6], Kerner and Mann put forward a method to research on the fermions tunneling of black holes via semi-classical approximation theory. In the method, the Hawking radiation is regarded as the quantum tunneling process, and then Kerner and Mann classify the spinor function as spin-up case and spin-down case. Next, the Dirac equation is simplified in semi-classical approximation theory, and the equations could be decompounded as radial equations and non-radial equations respectively. In the precondition that the nonradial equations must be true, researcher just studies the radial equations because the tunneling is radial property. Applying the WKB approximation theory, the relation between tunneling rate Γ and the classical action S is Γ ∼ exp(−2 Im S ), both fermions tunneling rate and Hawking temperature could be gotten via semiclassical approximation method. Using this method, Kerner and Mann have researched on the fermions tunneling of static black holes and Kerr–Newman black hole [7,8], and Chen et al. have studied the charged dilatonic black hole case and de Sitter hori-
*
Corresponding author. E-mail addresses: [email protected] (K. Lin), [email protected] (S.Z. Yang).
0370-2693/$ – see front matter Crown Copyright doi:10.1016/j.physletb.2009.02.055
© 2009 Published
zon case [9,10]; fermions tunneling from the horizon of BTZ black hole and Kerr black hole have been researched by Li et al. [11, 12]; Lin and Yang generalized the work to the Finsler space–time case and (3 + 1)-dimensional non-stationary black hole case [13, 14]; Jiang’s work proved the method also can be applied in 5dimensional space–time [15,16]. However, up to now, the method of fermions tunneling fails to deal with in very high- (even any-) dimensional stationary space– time case, because there are too much equations in very high dimensional space–time, and resolving the equations is very troublesome. In this Letter, we put forward a method to resolve the difficulty. We introduce the reasonable gamma matrices firstly, and then get the Hamilton–Jacobi equation in the high-dimensional Kerr–AdS space–time via semi-classical approximation method. Decompounding the variable of the Hamilton–Jacobi equation, the radial term equation, angular term equation and extra dimension term equation can be gotten respectively. Because quantum tunneling at the horizon of black hole is radial, we can resolve the radial equation and get the fermions tunneling rate and Hawking temperature. This Letter includes the following four sections: An introduction; next the Dirac equation in high-dimensional Kerr– AdS space–time is treated in Section 2, and the fermions tunneling of the black hole is researched in Section 3. In Section 4, some conclusions and discussions are included.
by Elsevier B.V. All rights reserved.
128
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
2. The Dirac equation in n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter As the conclusion of Einstein equation, the researches about black holes are very significant in theory and astronomical observations, because they have some singular property [17–21]. On the other hand, several modern physics theories, such as string theory and TeV-scale gravity theory, need to introduce the concept of extra dimension [21–24], so some high-dimensional black holes are brought forward. The metric of n-dimensional Kerr–Anti-de Sitter black hole in [25,26] is given by ds2 = −
Δr
dt −
ρ2
a
Ξ
2 sin2 θ dφ
+
ρ2
dr 2 +
ρ2
dθ 2
Δr Δθ 2 2 2 2 r +a Δθ sin θ + a dt − dφ + r 2 cos2 θ dΩn2−4 , Ξ ρ2
In high-dimensional Kerr–AdS space–time, we choose the gamma matrices as t m×m
γ
=
γmr ×m = γmθ ×m =
0 g tt ξm ×m
2
2
ρ = r + a cos θ,
Δr = r 2 + a2 r 2l−2 + 1 − 2Mr 5−n , 2 −2
Δθ = 1 − a l
2
cos θ,
Ξ = 1 − a2l−2 .
= =
M, a are mass, angular momentum respectively, and l is an inverse cosmological constant; dΩn2−4 denotes the standard metric on the (n − 4)-sphere; the inverse metric elements are g tt =
a2 sin2 θ
g φφ =
ρ
2Δ
θ
−
(7)
1 γmn− ×m =
aΞ Ξ a(r 2 + a2 ) − 2 , ρ 2 Δr ρ Δθ Δr g rr = 2 ,
ρ
ρ2
(8) (9) (10)
g τ τ = r −2 cos−2 θ hτ τ ,
(11)
g tt
h¯
D μ = ∂μ +
2
Γ α μ β Πα β ,
i Πα β = γ α , γ β , 4
(19)
0
2
g φφ g tt − ( g t φ )2
ξm2 ×m
−I m × m 2
2
− ( g t φ )2
ξ 2m × m
0
2
ξ 2m × m 2
2
0
(20)
,
2
ξ km × m 2 2 ,
0
ξ km × m
4 k n − 2,
0
(21)
2
−i I m × m
0
2
iI m ×m
0
2
(22)
,
2
where m = 2n/2 (m = 2(n−1)/2 ) is the order of matrix in even(odd-) dimensional space–time; 0 is the zero matrix with m × m2 2 m ν orders; I m × m is a unit matrix with m × orders; γ and m 2 2 ×m 2
2
ξ νm × m are the ν th gamma matrices with
m 2
2
×
m 2
2
2
orders in curved
and flat space–time respectively (the gamma matrices in flat space–time should satisfy {ξ ν , ξ μ } = 2δνμ ). Especially, the (2 × 2)order matrices are
0 1 , 1 0
(23)
0 −i i 0
(24)
, 1 0 . 0 −1
(13)
Using the Dirac equation, we will research on the fermions behavior near the horizon in this space–time via semi-classical approximation method.
(14) (15)
(25)
3. Fermions tunneling of the high-dimensional Kerr–AdS space–time Now, let us rewritte the spinor function as
Ψ=
A m ×1 (t , r , θ, φ, . . . , τ , . . . , xn−1 ) 2
n−1
B m ×1 (t , r , θ, φ, . . . , τ , . . . , x 2
2
(16)
)
n −1
i
e h¯ S (t ,r ,θ,φ,...,τ ,...,x
)
, (26)
where A m ×1 (t , r , θ, φ, . . . , τ , . . . , xn−1 ) and B m ×1 (t , r , θ, φ, . . . , τ ,
. . . , xn−1 ) are the matrices with
and the gamma matrices must satisfy
μ ν γ , γ = 2g μν I .
(18)
ξ23×2 = σ 3 =
where m is a mass of particle; t = x0 and r = x3 are time coordinate and radial coordinate respectively; θ = x1 and φ = x2 are angular coordinates, and . . . , τ , . . . , xn−1 are extra-dimensional coordinates. In Eq. (13), i
,
(12)
In the high-dimensional space–time, the Dirac equation is
μ = t , r , θ, φ, . . . , τ , . . . , xn−1 ,
(17)
ξ 1m × m 2 2 ,
ξ 1m × m
0
g (n−1)(n−1)
ξ22×2 = σ 2 =
= 0,
0
2
0
g tt
dination. Obviously, the horizon r0 in the space–time satisfies the equation
m
m × m2 2
g tt
I m×m 2 2 0
g φφ g tt
gτ τ
ξ21×2 = σ 1 =
γ μ DμΨ +
k g τ τ ξm ×m
where, the g τ τ is extra-dimensional element of inverse metric, and hτ τ is the part that only depends on the extra-dimensional coor-
Δr |r =r0 = r02 + a2 r02l−2 − 1 − 2Mr05−n = 0.
0 2
,
−I m × m 2 2 ξ3
ξ 3m × m
g θθ
2
2
,
2
Ξ 2 a2 , ρ 2 Δr
ρ 2 Δθ sin2 θ
Δθ
= .. .
gtφ =
g θθ =
γmτ ×m =
(6)
−
0
.. .
(r 2 + a2 )2 , ρ 2 Δr
Ξ2
ξm0 ×m +
g tt
+
(2)
(5)
I m×m 2 2 0
g rr
gtφ gtφ
(1)
(4)
g tt
2
γmφ×m
(3)
1 g θθ ξm ×m =
2
=
3 g rr ξm ×m =
where 2
m 2
2
× 1 elements. Substituting
Eq. (26) into Eq. (13), near the horizon, the resulting equations to leading order in h¯ are
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
Ψ=
C E
D F
A m ×1 2 B m ×1
,
(27)
2
where C=
g tt
∂S gtφ ∂ S I m × m − imI m × m , I m×m + 2 2 2 2 ∂t g tt ∂φ 2 2
(28)
∂S 1 g φφ g tt − ( g t φ )2 ∂ S 2 ξm×m + ξm m D = g θθ ∂θ 2 2 g tt ∂φ 2 × 2 ∂S ∂S k + g rr ξ 3m × m + · · · + g τ τ ξm m − ··· ∂r 2 2 ∂τ 2 × 2 ∂S − i g (n−1)(n−1) n−1 I m × m , 2 2 ∂x E=
g θθ
∂S 1 ξm m + ∂θ 2 × 2
g φφ g tt − ( g t φ )2 ∂ S g tt
∂φ
(29)
−1
F − EC
∂S gtφ ∂ S I m × m − imI m × m . I m×m − 2 2 2 2 ∂t g tt ∂φ 2 2
F C − E D = 0.
(34)
Notice the relation of flat gamma matrices: {ξ ν , ξ μ } = 2δνμ , and we can get the equation
2
2 ∂S ∂S ∂S + 2g t φ ∂φ ∂ t ∂φ 2 2 ∂ ∂ S S + g θθ + · · · + gτ τ + · · · + m2 = 0. ∂θ ∂τ ∂S ∂r
+ g φφ
(35)
Distinctly, the equation is no other than the Hamilton–Jacobi equation in this space–time. Next, we decompound the action S defined in Eq. (26) as
S = −ωt + R (r ) + j φ + Y (θ) + Θ . . . , τ , . . . , xn−1 ,
(36)
where ω and j are energy and angular momentum of emitted particles respectively. From the inverse metric and the Hamilton– Jacobi equation, we can get
Δr2
∂R ∂r
2
a2 sin2 θ
Δθ
2 λa2 − r 2 + a2 ω + Ξ aj + Δr r 2m + 2 + η = 0, (37) r
ω2 −
Ξ2 2
Δθ sin θ λ + m2 a2 cos2 θ + 2
j2 −
cos θ
−λ + Στ hτ τ
∂Θ ∂τ
2aΞ
Δθ
= η,
+ η)
.
ω j + Δθ
∂Y ∂θ
2
(41)
,
Δr r 2 + a2
(42)
.
ω − Ω0 j f (r0 )
,
(43)
(44)
ω − Ω0 j Γ = exp(−2 Im S ) = exp −4π . f (r0 )
(45)
Now, we have obtained the fermions tunneling rate and Hawking temperature in the n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter, and the result is reasonable and right. 4. Conclusions In this Letter, we generalize the semi-classical approximation fermions tunneling method into the stationary high-dimensional Kerr–AdS space–time. From the study process, it is very clear that we have never divided the spinor function into spin-up case and spin-down case respectively, so the method makes the fermions tunneling theory more elegant, and makes the research about fermions tunneling in any dimensional space–time more simple. On the other hand, we can get the Hamilton–Jacobi equation via the method, and the fact implies that Hamilton–Jacobi equation is a basal semi-classical quantum equation in curved space–time. In fact, the method can also be used in the research of fermions tunneling of (3 + 1)-dimensional space–times and lower-dimensional black holes. Clearly, fermions tunneling rate and Hawking temperature in this Letter could become the results in (3 + 1)-dimensional Kerr–AdS space–time when n = 4. We have got the fermions tunneling rate and Hawking temperature in high-dimensional Kerr– AdS space–time, but the work bases on the semi-classical approximation theory, so there is some correction in further successful quantum gravity theory. Work on these areas is in progress. Acknowledgements
(38)
This work is supported by National Natural Science Foundation of China (No. 10773008). References
2 = 0,
(40)
Im S = Im R = Im R + − Im R − .
If we want to get non-trivial solutions of A m × m and B m × m , the 2 2 2 2 coefficient matrices of Eq. (32) and Eq. (33) must vanish. Due to the fact that C D = DC , the condition becomes
+ a2 )
(31)
(33)
+ g rr
/(r 2
Im stands for the imaginary part. Therefore, the tunneling rate is
2
Δr
2 Δr (r 2m + λra2 (r 2 +a2 )2
where R + is the radial part of out-coming solution, while R − is the radial part of in-going solution, so imaginary part of the total action is
E − F D −1 C A m × m = 0. 2 2
∂S ∂t
−
(30)
(32)
f (r ) =
R ± = ±i π
D B m × m = 0, 2 2
g tt
Ξ aj 2 ] r 2 +a2
The answer of Eq. (40) is
From the equations, we obtain
−Ξ a r02 + a2
2
∂S ∂S k + g rr ξ 3m × m + · · · + g τ τ ξm m + ··· ∂r 2 2 ∂τ 2 × 2 ∂S + i g (n−1)(n−1) n−1 I m × m , 2 2 ∂x
[ω + ∂R =± ∂r
Ω0 =
F = − g tt
where Eq. (37) is a radial equation, and Eqs. (38) and (39) are angular equation and extra-dimensional equation respectively. λ and η are constants. However, we only study the radial equation (37), because the tunneling is radial. From the radial equation, we can get
Near the horizon, we let
ξ 2m × m 2
129
(39)
[1] P. Kraus, F. Wilczek, Nucl. Phys. B 433 (1995) 403, arXiv:gr-qc/9408003.
130
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
[2] S.P. Robinson, F. Wilczek, Phys. Rev. Lett. 95 (2005) 011303, arXiv:gr-qc/ 0502074. [3] Q.Q. Jiang, S.Q. Wu, X. Cai, Phys. Rev. D 75 (2007) 064029, arXiv:hep-th/ 0512351. [4] J.Y. Zhang, Z. Zhao, Phys. Lett. B 638 (2006) 110, arXiv:gr-qc/0512153. [5] R. Kerner, R.B. Mann, Phys. Rev. D 73 (2006) 104010. [6] S.Z. Yang, H.L. Li, Q.Q. Jiang, M.Q. Liu, Sci. Chin. G 37 (2007) 66 (in Chinese). [7] R. Kerner, R.B. Mann, Class. Quantum Grav. 25 (2008) 095014, arXiv:0710.0612 [hep-th]. [8] R. Kerner, R.B. Mann, Phys. Lett. B 665 (2008) 277, arXiv:0803.2246 [hep-th]. [9] D.Y. Chen, Q.Q. Jiang, X.T. Zhu, Class. Quantum Grav. 25 (2008) 205022, arXiv: 0803.3248 [hep-th]. [10] D.Y. Chen, Q.Q. Jiang, X.T. Zhu, Phys. Lett. B 665 (2008) 106, arXiv:0804.0131 [hep-th]. [11] R. Li, J.R. Ren, S.W. Wei, Class. Quantum Grav. 25 (2008) 125016, arXiv: 0803.1410 [gr-qc].
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
R. Li, J.R. Ren, Phys. Lett. B 661 (2008) 370, arXiv:0802.3954 [gr-qc]. K. Lin, S.Z. Yang, Chin. Phys. Lett. 26 (2009) 010401. K. Lin, S.Z. Yang, Acta Phys. Sin. 58 (2009) 744 (in Chinese). Q.Q. Jiang, Phys. Rev. D 78 (2008) 044009. Q.Q. Jiang, Phys. Lett. B 666 (2008) 517. J.G. Russo, Phys. Lett. B 339 (1994) 35. S.K. Rama, Phys. Lett. B 321 (1994) 338. W.L. Smith, R.B. Mann, Phys. Rev. D 56 (1997) 4942. R.B. Mann, Class. Quantum Grav. 14L (1997) 109. R.G. Cai, K.S. Soh, Phys. Rev. D 59 (1997) 044013. A. Strominger, C. Vafa, Phys. Lett. B 379 (1996) 99, arXiv:hep-th/9601029. M. Cavaglia, Int. J. Mod. Phys. A 18 (2003) 1843, arXiv:hep-ph/0210296. M.H. Dehghani, Phys. Rev. D 65 (2002) 124002, arXiv:hep-th/0203049. D. Klemm, JHEP 9811 (1998) 019, arXiv:hep-th/9811126. S.W. Hawking, C.J. Hunter, M.M. Taylor-Robinson, Phys. Rev. D 59 (1999) 064005, arXiv:hep-th/9811056v2.
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Fermions tunneling of higher-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter Kai Lin ∗ , ShuZheng Yang Institute of Theoretical Physics, China West Normal University, NanChong, SiChuan 637002, China
a r t i c l e
i n f o
Article history: Received 11 December 2008 Received in revised form 24 February 2009 Accepted 27 February 2009 Available online 6 March 2009 Editor: M. Cvetiˇc
a b s t r a c t The 1/2 spin fermions tunneling at the horizon of n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter is researched via semi-classical approximation method, and the Hawking temperature and fermions tunneling rate are obtained in this Letter. Using a new method, the semiclassical Hamilton–Jacobi equation is gotten from the Dirac equation in this Letter, and the work makes several quantum tunneling theories more harmonious. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
PACS: 04.70.Dy 04.62.+v 03.65.Sq Keywords: n-Dimensional Kerr–Anti-de Sitter black hole with one rotational parameter Hawking radiation Fermions tunneling
1. Introduction Recently, based on the work of Parikh and Wilczek et al. [1–6], Kerner and Mann put forward a method to research on the fermions tunneling of black holes via semi-classical approximation theory. In the method, the Hawking radiation is regarded as the quantum tunneling process, and then Kerner and Mann classify the spinor function as spin-up case and spin-down case. Next, the Dirac equation is simplified in semi-classical approximation theory, and the equations could be decompounded as radial equations and non-radial equations respectively. In the precondition that the nonradial equations must be true, researcher just studies the radial equations because the tunneling is radial property. Applying the WKB approximation theory, the relation between tunneling rate Γ and the classical action S is Γ ∼ exp(−2 Im S ), both fermions tunneling rate and Hawking temperature could be gotten via semiclassical approximation method. Using this method, Kerner and Mann have researched on the fermions tunneling of static black holes and Kerr–Newman black hole [7,8], and Chen et al. have studied the charged dilatonic black hole case and de Sitter hori-
*
Corresponding author. E-mail addresses: [email protected] (K. Lin), [email protected] (S.Z. Yang).
0370-2693/$ – see front matter Crown Copyright doi:10.1016/j.physletb.2009.02.055
© 2009 Published
zon case [9,10]; fermions tunneling from the horizon of BTZ black hole and Kerr black hole have been researched by Li et al. [11, 12]; Lin and Yang generalized the work to the Finsler space–time case and (3 + 1)-dimensional non-stationary black hole case [13, 14]; Jiang’s work proved the method also can be applied in 5dimensional space–time [15,16]. However, up to now, the method of fermions tunneling fails to deal with in very high- (even any-) dimensional stationary space– time case, because there are too much equations in very high dimensional space–time, and resolving the equations is very troublesome. In this Letter, we put forward a method to resolve the difficulty. We introduce the reasonable gamma matrices firstly, and then get the Hamilton–Jacobi equation in the high-dimensional Kerr–AdS space–time via semi-classical approximation method. Decompounding the variable of the Hamilton–Jacobi equation, the radial term equation, angular term equation and extra dimension term equation can be gotten respectively. Because quantum tunneling at the horizon of black hole is radial, we can resolve the radial equation and get the fermions tunneling rate and Hawking temperature. This Letter includes the following four sections: An introduction; next the Dirac equation in high-dimensional Kerr– AdS space–time is treated in Section 2, and the fermions tunneling of the black hole is researched in Section 3. In Section 4, some conclusions and discussions are included.
by Elsevier B.V. All rights reserved.
128
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
2. The Dirac equation in n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter As the conclusion of Einstein equation, the researches about black holes are very significant in theory and astronomical observations, because they have some singular property [17–21]. On the other hand, several modern physics theories, such as string theory and TeV-scale gravity theory, need to introduce the concept of extra dimension [21–24], so some high-dimensional black holes are brought forward. The metric of n-dimensional Kerr–Anti-de Sitter black hole in [25,26] is given by ds2 = −
Δr
dt −
ρ2
a
Ξ
2 sin2 θ dφ
+
ρ2
dr 2 +
ρ2
dθ 2
Δr Δθ 2 2 2 2 r +a Δθ sin θ + a dt − dφ + r 2 cos2 θ dΩn2−4 , Ξ ρ2
In high-dimensional Kerr–AdS space–time, we choose the gamma matrices as t m×m
γ
=
γmr ×m = γmθ ×m =
0 g tt ξm ×m
2
2
ρ = r + a cos θ,
Δr = r 2 + a2 r 2l−2 + 1 − 2Mr 5−n , 2 −2
Δθ = 1 − a l
2
cos θ,
Ξ = 1 − a2l−2 .
= =
M, a are mass, angular momentum respectively, and l is an inverse cosmological constant; dΩn2−4 denotes the standard metric on the (n − 4)-sphere; the inverse metric elements are g tt =
a2 sin2 θ
g φφ =
ρ
2Δ
θ
−
(7)
1 γmn− ×m =
aΞ Ξ a(r 2 + a2 ) − 2 , ρ 2 Δr ρ Δθ Δr g rr = 2 ,
ρ
ρ2
(8) (9) (10)
g τ τ = r −2 cos−2 θ hτ τ ,
(11)
g tt
h¯
D μ = ∂μ +
2
Γ α μ β Πα β ,
i Πα β = γ α , γ β , 4
(19)
0
2
g φφ g tt − ( g t φ )2
ξm2 ×m
−I m × m 2
2
− ( g t φ )2
ξ 2m × m
0
2
ξ 2m × m 2
2
0
(20)
,
2
ξ km × m 2 2 ,
0
ξ km × m
4 k n − 2,
0
(21)
2
−i I m × m
0
2
iI m ×m
0
2
(22)
,
2
where m = 2n/2 (m = 2(n−1)/2 ) is the order of matrix in even(odd-) dimensional space–time; 0 is the zero matrix with m × m2 2 m ν orders; I m × m is a unit matrix with m × orders; γ and m 2 2 ×m 2
2
ξ νm × m are the ν th gamma matrices with
m 2
2
×
m 2
2
2
orders in curved
and flat space–time respectively (the gamma matrices in flat space–time should satisfy {ξ ν , ξ μ } = 2δνμ ). Especially, the (2 × 2)order matrices are
0 1 , 1 0
(23)
0 −i i 0
(24)
, 1 0 . 0 −1
(13)
Using the Dirac equation, we will research on the fermions behavior near the horizon in this space–time via semi-classical approximation method.
(14) (15)
(25)
3. Fermions tunneling of the high-dimensional Kerr–AdS space–time Now, let us rewritte the spinor function as
Ψ=
A m ×1 (t , r , θ, φ, . . . , τ , . . . , xn−1 ) 2
n−1
B m ×1 (t , r , θ, φ, . . . , τ , . . . , x 2
2
(16)
)
n −1
i
e h¯ S (t ,r ,θ,φ,...,τ ,...,x
)
, (26)
where A m ×1 (t , r , θ, φ, . . . , τ , . . . , xn−1 ) and B m ×1 (t , r , θ, φ, . . . , τ ,
. . . , xn−1 ) are the matrices with
and the gamma matrices must satisfy
μ ν γ , γ = 2g μν I .
(18)
ξ23×2 = σ 3 =
where m is a mass of particle; t = x0 and r = x3 are time coordinate and radial coordinate respectively; θ = x1 and φ = x2 are angular coordinates, and . . . , τ , . . . , xn−1 are extra-dimensional coordinates. In Eq. (13), i
,
(12)
In the high-dimensional space–time, the Dirac equation is
μ = t , r , θ, φ, . . . , τ , . . . , xn−1 ,
(17)
ξ 1m × m 2 2 ,
ξ 1m × m
0
g (n−1)(n−1)
ξ22×2 = σ 2 =
= 0,
0
2
0
g tt
dination. Obviously, the horizon r0 in the space–time satisfies the equation
m
m × m2 2
g tt
I m×m 2 2 0
g φφ g tt
gτ τ
ξ21×2 = σ 1 =
γ μ DμΨ +
k g τ τ ξm ×m
where, the g τ τ is extra-dimensional element of inverse metric, and hτ τ is the part that only depends on the extra-dimensional coor-
Δr |r =r0 = r02 + a2 r02l−2 − 1 − 2Mr05−n = 0.
0 2
,
−I m × m 2 2 ξ3
ξ 3m × m
g θθ
2
2
,
2
Ξ 2 a2 , ρ 2 Δr
ρ 2 Δθ sin2 θ
Δθ
= .. .
gtφ =
g θθ =
γmτ ×m =
(6)
−
0
.. .
(r 2 + a2 )2 , ρ 2 Δr
Ξ2
ξm0 ×m +
g tt
+
(2)
(5)
I m×m 2 2 0
g rr
gtφ gtφ
(1)
(4)
g tt
2
γmφ×m
(3)
1 g θθ ξm ×m =
2
=
3 g rr ξm ×m =
where 2
m 2
2
× 1 elements. Substituting
Eq. (26) into Eq. (13), near the horizon, the resulting equations to leading order in h¯ are
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
Ψ=
C E
D F
A m ×1 2 B m ×1
,
(27)
2
where C=
g tt
∂S gtφ ∂ S I m × m − imI m × m , I m×m + 2 2 2 2 ∂t g tt ∂φ 2 2
(28)
∂S 1 g φφ g tt − ( g t φ )2 ∂ S 2 ξm×m + ξm m D = g θθ ∂θ 2 2 g tt ∂φ 2 × 2 ∂S ∂S k + g rr ξ 3m × m + · · · + g τ τ ξm m − ··· ∂r 2 2 ∂τ 2 × 2 ∂S − i g (n−1)(n−1) n−1 I m × m , 2 2 ∂x E=
g θθ
∂S 1 ξm m + ∂θ 2 × 2
g φφ g tt − ( g t φ )2 ∂ S g tt
∂φ
(29)
−1
F − EC
∂S gtφ ∂ S I m × m − imI m × m . I m×m − 2 2 2 2 ∂t g tt ∂φ 2 2
F C − E D = 0.
(34)
Notice the relation of flat gamma matrices: {ξ ν , ξ μ } = 2δνμ , and we can get the equation
2
2 ∂S ∂S ∂S + 2g t φ ∂φ ∂ t ∂φ 2 2 ∂ ∂ S S + g θθ + · · · + gτ τ + · · · + m2 = 0. ∂θ ∂τ ∂S ∂r
+ g φφ
(35)
Distinctly, the equation is no other than the Hamilton–Jacobi equation in this space–time. Next, we decompound the action S defined in Eq. (26) as
S = −ωt + R (r ) + j φ + Y (θ) + Θ . . . , τ , . . . , xn−1 ,
(36)
where ω and j are energy and angular momentum of emitted particles respectively. From the inverse metric and the Hamilton– Jacobi equation, we can get
Δr2
∂R ∂r
2
a2 sin2 θ
Δθ
2 λa2 − r 2 + a2 ω + Ξ aj + Δr r 2m + 2 + η = 0, (37) r
ω2 −
Ξ2 2
Δθ sin θ λ + m2 a2 cos2 θ + 2
j2 −
cos θ
−λ + Στ hτ τ
∂Θ ∂τ
2aΞ
Δθ
= η,
+ η)
.
ω j + Δθ
∂Y ∂θ
2
(41)
,
Δr r 2 + a2
(42)
.
ω − Ω0 j f (r0 )
,
(43)
(44)
ω − Ω0 j Γ = exp(−2 Im S ) = exp −4π . f (r0 )
(45)
Now, we have obtained the fermions tunneling rate and Hawking temperature in the n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter, and the result is reasonable and right. 4. Conclusions In this Letter, we generalize the semi-classical approximation fermions tunneling method into the stationary high-dimensional Kerr–AdS space–time. From the study process, it is very clear that we have never divided the spinor function into spin-up case and spin-down case respectively, so the method makes the fermions tunneling theory more elegant, and makes the research about fermions tunneling in any dimensional space–time more simple. On the other hand, we can get the Hamilton–Jacobi equation via the method, and the fact implies that Hamilton–Jacobi equation is a basal semi-classical quantum equation in curved space–time. In fact, the method can also be used in the research of fermions tunneling of (3 + 1)-dimensional space–times and lower-dimensional black holes. Clearly, fermions tunneling rate and Hawking temperature in this Letter could become the results in (3 + 1)-dimensional Kerr–AdS space–time when n = 4. We have got the fermions tunneling rate and Hawking temperature in high-dimensional Kerr– AdS space–time, but the work bases on the semi-classical approximation theory, so there is some correction in further successful quantum gravity theory. Work on these areas is in progress. Acknowledgements
(38)
This work is supported by National Natural Science Foundation of China (No. 10773008). References
2 = 0,
(40)
Im S = Im R = Im R + − Im R − .
If we want to get non-trivial solutions of A m × m and B m × m , the 2 2 2 2 coefficient matrices of Eq. (32) and Eq. (33) must vanish. Due to the fact that C D = DC , the condition becomes
+ a2 )
(31)
(33)
+ g rr
/(r 2
Im stands for the imaginary part. Therefore, the tunneling rate is
2
Δr
2 Δr (r 2m + λra2 (r 2 +a2 )2
where R + is the radial part of out-coming solution, while R − is the radial part of in-going solution, so imaginary part of the total action is
E − F D −1 C A m × m = 0. 2 2
∂S ∂t
−
(30)
(32)
f (r ) =
R ± = ±i π
D B m × m = 0, 2 2
g tt
Ξ aj 2 ] r 2 +a2
The answer of Eq. (40) is
From the equations, we obtain
−Ξ a r02 + a2
2
∂S ∂S k + g rr ξ 3m × m + · · · + g τ τ ξm m + ··· ∂r 2 2 ∂τ 2 × 2 ∂S + i g (n−1)(n−1) n−1 I m × m , 2 2 ∂x
[ω + ∂R =± ∂r
Ω0 =
F = − g tt
where Eq. (37) is a radial equation, and Eqs. (38) and (39) are angular equation and extra-dimensional equation respectively. λ and η are constants. However, we only study the radial equation (37), because the tunneling is radial. From the radial equation, we can get
Near the horizon, we let
ξ 2m × m 2
129
(39)
[1] P. Kraus, F. Wilczek, Nucl. Phys. B 433 (1995) 403, arXiv:gr-qc/9408003.
130
K. Lin, S.Z. Yang / Physics Letters B 674 (2009) 127–130
[2] S.P. Robinson, F. Wilczek, Phys. Rev. Lett. 95 (2005) 011303, arXiv:gr-qc/ 0502074. [3] Q.Q. Jiang, S.Q. Wu, X. Cai, Phys. Rev. D 75 (2007) 064029, arXiv:hep-th/ 0512351. [4] J.Y. Zhang, Z. Zhao, Phys. Lett. B 638 (2006) 110, arXiv:gr-qc/0512153. [5] R. Kerner, R.B. Mann, Phys. Rev. D 73 (2006) 104010. [6] S.Z. Yang, H.L. Li, Q.Q. Jiang, M.Q. Liu, Sci. Chin. G 37 (2007) 66 (in Chinese). [7] R. Kerner, R.B. Mann, Class. Quantum Grav. 25 (2008) 095014, arXiv:0710.0612 [hep-th]. [8] R. Kerner, R.B. Mann, Phys. Lett. B 665 (2008) 277, arXiv:0803.2246 [hep-th]. [9] D.Y. Chen, Q.Q. Jiang, X.T. Zhu, Class. Quantum Grav. 25 (2008) 205022, arXiv: 0803.3248 [hep-th]. [10] D.Y. Chen, Q.Q. Jiang, X.T. Zhu, Phys. Lett. B 665 (2008) 106, arXiv:0804.0131 [hep-th]. [11] R. Li, J.R. Ren, S.W. Wei, Class. Quantum Grav. 25 (2008) 125016, arXiv: 0803.1410 [gr-qc].
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
R. Li, J.R. Ren, Phys. Lett. B 661 (2008) 370, arXiv:0802.3954 [gr-qc]. K. Lin, S.Z. Yang, Chin. Phys. Lett. 26 (2009) 010401. K. Lin, S.Z. Yang, Acta Phys. Sin. 58 (2009) 744 (in Chinese). Q.Q. Jiang, Phys. Rev. D 78 (2008) 044009. Q.Q. Jiang, Phys. Lett. B 666 (2008) 517. J.G. Russo, Phys. Lett. B 339 (1994) 35. S.K. Rama, Phys. Lett. B 321 (1994) 338. W.L. Smith, R.B. Mann, Phys. Rev. D 56 (1997) 4942. R.B. Mann, Class. Quantum Grav. 14L (1997) 109. R.G. Cai, K.S. Soh, Phys. Rev. D 59 (1997) 044013. A. Strominger, C. Vafa, Phys. Lett. B 379 (1996) 99, arXiv:hep-th/9601029. M. Cavaglia, Int. J. Mod. Phys. A 18 (2003) 1843, arXiv:hep-ph/0210296. M.H. Dehghani, Phys. Rev. D 65 (2002) 124002, arXiv:hep-th/0203049. D. Klemm, JHEP 9811 (1998) 019, arXiv:hep-th/9811126. S.W. Hawking, C.J. Hunter, M.M. Taylor-Robinson, Phys. Rev. D 59 (1999) 064005, arXiv:hep-th/9811056v2.