Deformation of contour and Hawking temperature

0.17 MB·English

by  Chikun Ding

Deformation of contour and Hawking temperature
Chikun Ding and Jiliang Jing
∗ †
Institute of Physics and Department of Physics,
0 Hunan Normal University, Changsha,
1
0 Hunan 410081, P. R. China
2
and
n
a Key Laboratory of Low Dimensional Quantum Structures
J

9 and Quantum Control of Ministry of Education,
1
Hunan Normal University , Changsha,

]
c Hunan 410081, P.R. China
q
-
r Abstract
g
[

It was found that, in an isotropic coordinate system, the tunneling approach brings a factor of 1
2 2
v
6 for the Hawking temperature of a Schwarzschild black hole. In this paper, we address this kind of
4
9 problem by studying the relation between the Hawking temperature and the deformation of integral
2
. contour for the scalar and Dirac particles tunneling. We find that correct Hawking temperature can
1
0
0 be obtained exactly as long as the integral contour deformed corresponding to the radial coordinate
1
: transform if the transformation is a non-regular or zero function at the event horizon.
v
i
X
Keywords: black hole, Hawking temperature, integral contour.
r
a
PACS numbers: 04.70.Dy, 04.62.+v
∗ Email: [email protected]
†Corresponding author, Electronic address: [email protected]
1
I. INTRODUCTION
A semi-classical Hamilton-Jacobi method [1]-[19] for controlling Hawking radiation as a
tunneling effect has been developed recently. In this method a semiclassical propagator
K(~x ,t ;~x ,t ) in a spacetime is described by N exp i(I(~x ,t ;~x ,t ) + C in which the ac-
2 2 1 1 ~ 2 2 1 1
tion I(~x ,t ;~x ,t ) acquires a singularity at the event horizon. This singularity can be reg-
2 2 1 1 (cid:2) (cid:3)
ularized by specifying a suitable complex contour [1]. After integrating around the pole, we
find that the action I(~x ,t ;~x ,t ) is complex. Thus, we know that the probabilities are
2 2 1 1
Γ[emission] e 2Im[I++C] and Γ[absorption] e 2Im[I−+C] = 1, and the ratio is
− −
∝ ∝
Γ[emission] = e 2[ImI+ ImI−]Γ[absorption], (1.1)
− −
where I are the square roots of the relativistic Hamilton-Jacobi equation corresponding to
±
outgoing and ingoing particles. In a system with a temperature T , the absorption and the
H
emission probabilities are related by Γ[emission] = e E/THΓ[absorption]. Then, from the rela-
−
tion
e E/TH = e 2[ImI+ ImI−], (1.2)
− − −
we can obtain the Hawking temperature.
It is well known that the Hawking temperature is an attribution of the black hole and is
independent of coordinates. This can be seen from its definition: T = κ [20], where κ is
H 2π
the surface gravity of the black hole. However, to calculation the Hawking temperature by
tunneling approach, we need to regularize the singularity by specifying a suitable complex
contour to bypass the pole. For the Schwarzschild black hole in the standard coordinate
representation, we should take the contour to be an infinitesimal semicircle below the pole
r = r for outgoing particles from inside of the horizon to outside; similarly, the contour
H
is above the pole for the ingoing particles from outside to inside. But, if we use another
coordinate representations, we find that the calculation of the Hawking temperature is related
to the choice of the integral contour and improper contour would give incorrect result. For
example, if a semi-circular contour is still employed in the isotropic coordinate system, the
temperature calculated by the Hamilton-Jacobi method is one-half of the standard result (the
so-called “factor of 1 problem”, see Appendix A); and if we use a semi-circular contour in a
2
general coordinate (2.3), we can prove that the temperature would be (α + 1) times of the
standard result (see Appendix B).
The “ factor of 1 problem” of the Schwarzschild black hole in the isotropic coordinates is
2
studied by Aknmedov et al [21, 22] by deforming the contour, i.e. using a quarter-circular
contour instead of the semi-circular contour. How to extend it to a general case? In this
2
manuscript we will study the problem in a general coordinate [5, 6] for a Kerr-Newman black
hole via the scalar and Dirac particles tunneling.
This paper is organized as follows. In Sec.II, the different coordinate representations for the
Kerr-Newman black hole are presented. In Sec. III, the Hawking temperature of the Kerr-
Newman black hole from scalar particles tunneling in a general coordinate is studied. In Sec.
IV, the Hawking temperature of the Kerr-Newman black hole from Dirac particles tunneling
is studied. The last section is devoted to a summary.
II. COORDINATE REPRESENTATIONS FOR A KERR-NEWMAN BLACK HOLE
The no-hair theorem postulates that all black hole solutions of the Einstein-Maxwell equa-
tions of gravitation and electromagnetism in general relativity can be completely characterized
by only three externally observable classical parameters: the mass, the electric charge, and
the angular momentum. The final state of a collapsing star is described by the Kerr-Newman
black hole. In the Boyer-Lindquist coordinates, its line element reads
2Mr Q2 2(2Mr Q2)asin2θ ρ2
ds2 = 1 − dt2 − dt dϕ + dr2
− − ρ2 s − ρ2 s s
(cid:18) (cid:19) △
(2Mr Q2)a2sin2θ
+ρ2dθ2 + r2 +a2 + − sin2θdϕ2, (2.1)
ρ2 s
(cid:18) (cid:19)
with
ρ2 = r2 +a2cos2θ, = r2 2Mr +a2 +Q2 = (r r )(r r )
+
△ − − − −
r = M + M2 a2 Q2, r = M M2 a2 Q2,
+
− − − − − −
p p
where M, Q and a are the mass, electric charge and angular momentum of the black hole,
and r and r are the inner and outer horizons. The spacetime has a timelike Killing vector
+
−
ξ˜µ = (1,0,0,0), and a spacelike Killing vector ξ˜µ = (0,0,0,1).
(t) (ϕ)
We note that the Painlev´e-type [23], advanced Eddington-Finkelstein [24] and Boyer-
Lindquist coordinate representations can be casted into an united form which is given by a
general coordinate transform
u = drF(r), v = ηt +η (r2 +a2)G(r)dr, ϕ = δϕ +δa G(r)dr, (2.2)
s s
Z Z Z
where (t , r, θ, ϕ ) are the Boyer-Lindquist coordinates; v, u and ϕ represent the time,
s s
radial and angular coordinates respectively, θ remains the same; η and δ are arbitrary nonzero
constants which re-scale the time and angle; and G and F are arbitrary functions of r only.
3
The line element (2.1) in the new coordinate system becomes
1 2Mr Q2 ηG(r2 +a2) 2
ds2 = 1 − dv du
−η2 − ρ2 − F
(cid:18) (cid:19)(cid:20) (cid:21)
2(2Mr Q2)asin2θ ηG(r2 +a2) δaG
− dv du dϕ du
− ηδρ2 − F − F
(cid:20) (cid:21)(cid:20) (cid:21)
ρ2 (2Mr Q2)a2sin2θ sin2θ δaG 2
+ du2 +ρ2dθ2 + r2 +a2 + − dϕ du .(2.3)
F2 ρ2 δ2 − F
△ (cid:18) (cid:19) (cid:20) (cid:21)
The timelike and spacelike Killing vectors of the spacetime are
∂xµ ∂xµ
ξµ = ξ˜ν = (η,0,0,0), ξµ = ξ˜ν = (0,0,0,δ). (2.4)
(t) ∂x˜ν (t) (ϕ) ∂x˜ν (ϕ)
Painlev´e-type coordinate representation: In the transformation (2.2), if we take η = δ = 1,
G(r) = 1 2Mr Q2 and F(r) = 1, the line element (2.3) becomes the Painlev´e-type coordinate
r2+−a2
△
representaqtion [23], which has no coordinate singularity at (r) = 0.
△
Advanced Eddington-Finkelstein coordinate representation: In the transformation (2.2), if
we let η = 1, δ = 1, G(r) = 1 and F(r) = 1, the line element (2.3) becomes the ad-
−
△
vanced Eddington-Finkelstein representation, which has no coordinate singularity just as in
the Painlev´e-type coordinates [24].
Boyer-Lindquist coordinate representation: In the transformation (2.2), if we let η = δ =
F(r) = 1, G(r) = 0, the line element (2.3) becomes the Boyer-Lindquist coordinate represen-
tation (2.1).
III. TEMPERATURE OF KERR-NEWMAN BLACK HOLE FROM SCALAR TUN-
NELING IN THE GENERAL COORDINATE SYSTEM
Now we study the scalar tunneling in the general coordinates (2.3). Applying the WKB
approximation
i
φ(v,u,θ,ϕ) = exp I(v,u,θ,ϕ)+I (v,u,θ,ϕ)+ (~) (3.1)
~ 1
O
h i
to the charged Klein-Gordon equation
1 iq iq µ2
(∂ A ) √ ggµ¯ν¯(∂ A )φ φ = 0, (3.2)
√ g µ¯ − ~ µ¯ − ν¯ − ~ ν¯ − ~2
−
(cid:2) (cid:3)
then, to leading order in ~, we obtain the relativistic Hamilton-Jacobi equation
gµ¯ν¯(∂ I∂ I +q2A A 2qA ∂ I)+µ2 = 0, (3.3)
µ¯ ν¯ µ¯ ν¯ µ¯ ν¯
−
4
where µ is the mass of tunneling particles. From the symmetries of the metric (2.3), we know
that there exists a solution of the form (see Appendix C)
1 1
I = Ev+W(u)+ mϕ+J(θ)+C. (3.4)
−η δ
Substituting the metric (2.3) and Eq. (3.4) into the Hamilton-Jacobi equation (3.3), we obtain
2
qQr ma
2 FW (u) (r2 +a2)G E
′
△ − − r2 +a2 − r2 +a2
(cid:20) (cid:21)
(cid:16) (cid:17)
2
2 qQr ma
r2 +a2 E + λ = 0, (3.5)
− − r2 +a2 − r2 +a2 △
(cid:20) (cid:21)
(cid:16) (cid:17)
with
2
m
λ = µ2ρ2 +J 2(θ)+ aEsinθ , (3.6)
′
− sinθ
(cid:18) (cid:19)
where W (u) = dW(u), and J (θ) = dJ(θ). Then, W (u) can be expressed as
′ du ′ dθ ′
G qQr ma
W (u) = (r2 +a2) E
′
± F − r2 +a2 − r2 +a2
(cid:16) (cid:17)
1 qQr ma 2
2
r2 +a2 E λ. (3.7)
±F − r2 +a2 − r2 +a2 −△
△r (cid:16) (cid:17)
(cid:0) (cid:1)
One solution of Eq. (3.7) corresponds to the scalar particles moving away from the black
hole (i.e. “+” outgoing), and the other solution corresponds to particles moving toward the
black hole (i.e. “-” incoming). Without loss of generality, the function G can be expressed as
G(r(u)) = A(r(u)) +B(r(u)), where A(r(u)) and B(r(u)) are regular functions. Thus, we have
(r(u))
△
B A qQr ma
ImW (u) = Im du + (r2 +a2) E
± F F − r2 +a2 − r2 +a2
Z (cid:20)(cid:18) △(cid:19) (cid:16) (cid:17)
1 qQr ma 2
2
r2 +a2 E λ . (3.8)
± F − r2 +a2 − r2 +a2 −△
△r (cid:16) (cid:17) (cid:21)
(cid:0) (cid:1)
Imaginary part of the action can only come from the pole at the horizon. We will work out
the integral in two cases: A) F is a regular and non-zero function at the horizon, and B) F is
a singular or zero function at the horizon.
A. F is a regular and non-zero function at the horizon
If F is a regular and non-zero function at the horizon, using the law of residue we obtain
r2 +a2
ImW (u) = A(r ) 1 + (E mΩ qV )π, (3.9)
+ + +
± ± 2(r+ M) − −
h i −
5
where V = Qr+ is the electromagnetic potential, and Ω = a is the angular velocity.
+ r2+a2 + r2+a2
+ +
Then, Eqs. (1.1) and (1.2) show us that the total probability is
r2 +a2
Γ = exp 2π + (E mΩ qV ) , (3.10)
+ +
− (r M) − −
(cid:20) + − (cid:21)
and the Hawking temperature is
r M
+
T = − , (3.11)
H 2π(r2 +a2)
+
which is the same as previous work [1, 2, 23, 25].
B. F is a singular or zero function at the horizon
If F is a non-regular or zero function at the horizon, without loss of generality, we set
F = αX(r), where α is a non-zero constant and X(r) is a regular and non-zero function.
△
Thus, Eq. (3.8) becomes
B A qQr ma
ImW (u) = Im du + (r2 +a2) E
± αX α+1X − r2 +a2 − r2 +a2
Z (cid:20)(cid:18)△ △ (cid:19) (cid:16) (cid:17)
1 qQr ma 2
2
r2 +a2 E λ . (3.12)
± α+1X − r2 +a2 − r2 +a2 −△
△ r (cid:16) (cid:17) (cid:21)
(cid:0) (cid:1)
From which we know
Im[W (u) W (u)]
+
− −
1 qQr ma 2
2
= 2Im du r2 +a2 E λ . (3.13)
α+1X − r2 +a2 − r2 +a2 −△
Z △ r (cid:16) (cid:17)
(cid:0) (cid:1)
We now study two cases: 1) α = 1 and 2) α = 1.
6 − −
1. α= 1
6 −
The Laurent expansion for the factor 1 is
α+1(r(u))
△
1 X(r(u )) 1 ∞
= + + a (u u )n. (3.14)
α+1(r(u)) 2(α+1)(r M)u u n − +
+ +
△ − − n=0
X
Then Eq. (3.13) can be written as
1 1 1 ∞
Im[W (u) W (u)] = 2Im du + a (u u )n
+ n +
− − Z (cid:20)α+1 · 2(r+ −M)(u−u+) X n=0 − (cid:21)
X
qQr ma 2
2
r2 +a2 E λ . (3.15)
· − r2 +a2 − r2 +a2 −△
r
(cid:16) (cid:17)
(cid:0) (cid:1)
6
Now, we need to choose a contour to bypass the pole u = u . We note that, in the Boyer-
+
Lindquist coordinate, the contour can be constructed by taking r = r +ǫeiθ, (ǫ is a positive
+
small real quantity, θ [0,π] for outgoing particle, θ [π,2π] for ingoing particle). Thus, in
∈ ∈
the general coordinate (2.3), by substituting r = r + ǫeiθ into u = αX(r)dr = [(r
+
△ −
r )(r r )]αX(r)dr, we have
+ R R
− −
u = [ǫeiθ(r r +ǫeiθ)]αX(r +ǫeiθ)dǫeiθ
+ +
− −
Z
= u +f(u )ǫα+1ei(α+1)θ, (3.16)
+ +
where f(u+) = (r+−rα−+)α1X(r+). Eq. (3.16) indicates that the contour is different from semi-circle
now. The integral contours for outgoing particles corresponding to r and u complex plane are
showed in figure (1). Using Eqs. (3.15), (3.16) and residue theorem, we have
C(cid:13)
1(cid:13)
C(cid:13)
1(cid:13)
C(cid:13)
2(cid:13)
((cid:13)
a(cid:13)
C(cid:13) +(cid:13)
C(cid:13) 3(cid:13) C(cid:13) C3(cid:13) (cid:13) 1)(cid:13)(cid:13)p(cid:13) R(cid:13)
2(cid:13) 4(cid:13) R(cid:13)
C(cid:13)
4(cid:13)
(a(cid:13))(cid:13)(cid:13)
(b(cid:13))(cid:13)(cid:13)
FIG.1: Thefigure(a) isthesemicircle integrate contour foroutgoing particles in ther complex plane;
and the figure (b) is the deformation contour in the u complex plane when α = 1. For α = 1,
6 − −
its contour is still a semicircle. The tick mark on the real axis denotes the position of the black hole
event horizon.
Im[W (u) W (u)]
+
− −
0 1 f(u )(ǫeiθ)α+1(α+1) ∞
= 2Imlim idθ + + a fn(u )(ǫeiθ)(α+1)n
n +
− ǫ→0Zπ (cid:20)2(r+ −M) X n=0 (cid:21)
X
qQ(r +ǫeiθ)+ma 2
(r +ǫeiθ)2 +a2 2 E + λ
·s + − (r+ +ǫeiθ)2 +a2 −△
h i
(cid:2) (cid:3)
r2 +a2
= π + (E mΩ qV ), (3.17)
+ +
(r M) − −
+
−
which gives the Hawking temperature (3.11).
7
2. α = 1
−
It is the tortoise-like coordinate transformation if α = 1
−
u = X(r) 1dr. (3.18)
−
△
Z
By using r = r +ǫeiθ, we know
+
u = u +iθg(u ), (3.19)
+ +
where g(u ) = X(r+). Substituting it into Eq. (3.13), we obtain
+ r+ r−
−
0 idθg(u )
+
Im[W (u) W (u)] = 2Imlim
+ − − − ǫ→0Zπ (r+ +ǫeiθ)2 +a2
qQ(r +ǫeiθ)+ma 2
(r +ǫeiθ)2 +a2 2 E + λ
·s + − (r+ +ǫeiθ)2 +a2 −△
h i
(cid:2) (cid:3)
r2 +a2
= π + (E mΩ qV ), (3.20)
+ +
(r M) − −
+
−
which also presents the Hawking temperature (3.11).
Above discussions show us that: i) the integral contour needs to be deformed corresponding
to the radial coordinate transformation if this transformations are non-regular or zero at the
event horizon; ii) theHawking temperature isinvariant inthegeneral coordinaterepresentation
(2.3) for the scalar particle tunneling.
IV. TEMPERATURE OF KERR-NEWMAN BLACK HOLE FROM DIRAC PARTI-
CLE TUNNELING
In this section, we study the Dirac particle tunneling of the Kerr-Newman black hole in the
coordinates (2.3). The Dirac equation is [26]
iq µ
γαeµ¯(∂ +Γ A )+ ψ = 0, (4.1)
α µ¯ µ¯ − ~ µ¯ ~
(cid:20) (cid:21)
with
1
Γ = [γa,γb]eν¯e ,
µ¯ 8 a bν¯;µ¯
where γa is the Dirac matrix, and eµ¯ is the inverse tetrad defined by eµ¯γa, eν¯γb = 2gµ¯ν¯ 1.
a { a b } ×
For the Kerr-Newman metric in the general coordinate system (2.3), the tetrad eµ¯ can be taken
a
8
as
eva = √χ−△2ηρ2√(r2+a2)2G2, 0, 0, 0 ,
△
(cid:16) (cid:17)
eu = 1 △2FGη(r2+a2) , 1 △F√χ , 0, 0 ,
a −ρ√ √χ 2η2(r2+a2)2G2 ρ√ √χ 2η2(r2+a2)2G2
(cid:18) △ −△ △ −△ (cid:19)
eθ = 0, 0, 1, 0 ,
a ρ
(cid:16) (cid:17)
eϕ = aηδ (2Mr−Q2)−△2(r2+a2)G2, aδG√△ χ−(2Mr−Q2)η2(r2+a2) , 0, ηδρ ,
a ρ√ √χ 2η2(r2+a2)2G2 ρ √χ(χ 2η2(r2+a2)2G2) sinθ√χ
(cid:18) △ −△ −△ (cid:19)
χ = η2 r2 +a2 2 a2sin2θ . (4.2)
−△
h i
(cid:0) (cid:1)
Without loss of generality, we can choose the following ansatz for spin up and spin down Dirac
particles according to [27],
A(v,u,θ,ϕ)
A(v,u,θ,ϕ)ξ i  0  i
ψ = ↑ exp I (v,u,θ,ϕ) = exp I (v,u,θ,ϕ) ,
~ ~
↑ B(v,u,θ,ϕ)ξ ↑ B(v,u,θ,ϕ) ↑
(cid:18) (cid:19)
 
↑ (cid:0) (cid:1) (cid:0) (cid:1)
 
 0 
 
 
0
C(v,u,θ,ϕ)ξ i  C(v,u,θ,ϕ)  i
ψ = ↓ exp I (v,u,θ,ϕ) = exp I (v,u,θ,ϕ) ,
~ ~
↓ D(v,u,θ,ϕ))ξ ↓  0  ↓
(cid:18) (cid:19)
 
↓ (cid:0) (cid:1) (cid:0) (cid:1)
 
D(v,u,θ,ϕ)
 
 
(4.3)
where “ ”and“ ”represent thespinupandspindowncases, andξ andξ aretheeigenvectors
↑ ↓ ↑ ↓
of σ3. Inserting Eqs. (4.2) and (4.3) into Eq. (4.1), and employing the ansatz
1 1
I = Ev +W(u)+ mϕ+J(θ)+C, (4.4)
↑ −η δ
9
to the lowest order in ~, we obtain
1 qQr 1 qQr
ev E +euW (u)+eϕ m asin2θ +µ A
− 0η − ρ2 0 ′ 0 δ − ρ2
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) 1 qQr (cid:1)
+B euW (u)+eϕ m asin2θ = 0 , (4.5)
1 ′ 1 δ − ρ2
(cid:20) (cid:21)
1(cid:0) qQr (cid:1)
B eθJ (θ)+ieϕ m asin2θ = 0 , (4.6)
2 ′ 3 δ − ρ2
(cid:20) (cid:21)
1 qQr 1 (cid:0)qQr (cid:1)
ev E +euW (u)+eϕ m asin2θ µ B
− − 0η − ρ2 0 ′ 0 δ − ρ2 −
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) 1 qQr (cid:1)
A euW (u)+eϕ m asin2θ = 0 , (4.7)
− 1 ′ 1 δ − ρ2
(cid:20) (cid:21)
1(cid:0) qQr (cid:1)
A eθJ (θ)+ieϕ m asin2θ = 0 . (4.8)
− 2 ′ 3 δ − ρ2
(cid:20) (cid:21)
(cid:0) (cid:1)
Eqs. (4.6) and (4.8) both yield eθJ (θ) + ieϕ1(m qQrasin2θ) = 0, regardless of A or B.
2 ′ 3 δ − ρ2
Then substituting tetrad elements (4.2) into (4.5)–(4.8), after tedious calculating, we obtain
(cid:2) (cid:3)
2
qQr ma
2 FW (u) G(r2 +a2) E
′
△ − − r2 +a2 − r2 +a2
(cid:20) (cid:21)
(cid:16) (cid:17)
2
2 qQr ma
r2 +a2 E
− − r2 +a2 − r2 +a2
(cid:20) (cid:21)
(cid:16) (cid:17)
2
m
+ µ2ρ2 +J 2(θ)+ aEsinθ = 0, (4.9)
′
△ − sinθ
" #
(cid:18) (cid:19)
which is the same as Eq. (3.5). Therefore, it is easy to find the Hawking temperature (3.11).
The spin-down calculation is similar to the spin-up case discussed above, and the result is
the same.
V. SUMMARY
Wefirstlycastthreewell-known coordinaterepresentations, i.e. thePainlev´e-type, advanced
Eddington-Finkelstein and Boyer-Lindquist coordinate representations for the Kerr-Newman
black hole, into an united and general coordinate representation (2.3). Then, based on this
coordinate representation, we study the relation between the Hawking temperature and the
deformation of integral contour for the scalar and Dirac particle tunneling. We find that
correct Hawking temperature can be obtained exactly as long as the integral contour deformed
corresponding to the radial coordinate transform if the transformation is a non-regular or zero
function at the event horizon.
10