Greybody factor and Hawking radiation with non-minimal derivative coupling in the slowly rotating black hole spacetime

Abstract

We study the greybody factor and Hawking radiation with a non-minimal derivative coupling between the scalar field and the curvature in the background of the slowly rotating Kerr-Newman black hole. Our results show that both the absorption probability and luminosity of Hawking radiation of the scalar field increase with the coupling. Moreover, we also find that for the weak coupling $\eta<\eta_c$, the absorption probability and luminosity of Hawking radiation decrease when the black hole's Hawking temperature decreases; while for stronger coupling $\eta>\eta_c$, the absorption probability and luminosity of Hawking radiation increase on the contrary when the black hole's Hawking temperature decreases. This feature is similar to the Hawking radiation in a $d$-dimensional static spherically-symmetric black hole surrounded by quintessence \cite{chensong}. Comment: 15 pages, 6 figures, 1 table, Appendix added, JHEP style

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arXiv:1007.4240v2 [gr-qc] 13 Aug 2010
Greybody factor and Hawking radiation with non-minimal derivative coupling
in the slowly rotating black-hole spacetime
Chikun Ding∗
Department of Physics and Information Engineering,
Hunan Institute of Humanities Science and Technology, Loudi, Hunan 417000, P. R. China
Jiliang Jing†
Institute of Physics and Department of Physics,
Hunan Normal University, Changsha, Hunan 410081, P. R. China
Key Laboratory of Low Dimensional Quantum Structures
and Quantum Control of Ministry of Education, Hunan Normal University,
Changsha, Hunan 410081, People’s Republic of China
Abstract
We study the greybody factor and Hawking radiation with a non-minimal derivative coupling
between the scalar field and the curvature in the background of the slowly rotating Kerr-Newman
black hole. Our results show that both the absorption probability and luminosity of Hawking
radiation of the scalar field increase with the coupling. Moreover, we also find that for the weak
coupling η < ηc, the absorption probability and luminosity of Hawking radiation decrease when
the black hole’s Hawking temperature decreases; while for stronger coupling η > ηc, the absorption
probability and luminosity of Hawking radiation increase on the contrary when the black hole’s
Hawking temperature decreases. This feature is similar to the Hawking radiation in a d-dimensional
static spherically-symmetric black hole surrounded by quintessence [1]. All these features can help
us understand more about these “scalar-tensor” theories.
PACS numbers:04.70.Dy, 95.30.Sf, 97.60.Lf
∗Electronic address: dingchikun@163.com
†Electronic address: jljing@hunnu.edu.cn

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I.INTRODUCTION
Scalar fields in General Relativity has been a topic of great interest in the latest years. One of the main
reasons is that the models with scalar fields are relatively simple, which allows us to probe the detailed features
of the more complicated physical system. In cosmology, scalar fields can be considered as candidate (inflaton,
quintessence, phantom fields, etc.) to explain the inflation of the early Universe [2] and the accelerated
expansion of the current Universe [3–5]. In the Standard Model of particle physics, the scalar field presents as
the Higgs boson [6], which would help to explain the origin of mass in the Universe. Moreover, it has been found
that scalar field plays the important roles in other fundamental physical theories, such as, Jordan-Brans-Dicke
theory [7], Kaluza-Klein compactification theory [8] and superstring theory [9], and so on.
In another side, including nonlinear terms of the various curvature tensors (Riemann, Ricci, Weyl) and
nonminimally coupled terms in the effective action of gravity has become a very common trend from quantum
field theory side and cosmology. These theories cover the f(R) modified gravity, the Gausss-Bonnet gravity,
the tachyon, dilaton, and so on. The nonminimal coupling between scalar field and higher order terms in
the curvature (the so-called “scalar-tensor” theory) naturally give rise to inflationary solutions improve the
early inflationary models and could contribute to solve the dark matter problem. The new coupling between
the derivative of scalar field and the spacetime curvature may appear firstly in some Kaluza-Klein theories
[10–12]. Amendola [13] considered the most general theory of gravity with the Lagrangian linear in the Ricci
scalar, quadratic in ψ, in which the coupling terms have the forms as follows
R∂µψ∂µψ, Rµν∂µψ∂νψ, Rψ∇2ψ, Rµνψ∂µψ∂νψ, ∂µR∂νψ, ∇2Rψ.(1)
And then he studied the dynamical evolution of the scalar field in the cosmology by considering only the
derivative coupling term Rµν∂µψ∂νψ and obtained some analytical inflationary solutions [13] . Capozziello
et al. [14] investigated a more general model of containing coupling terms R∂µψ∂νψ and Rµν∂µψ∂νψ, and
found that the de Sitter spacetime is an attractor solution in the model. Recently, Daniel and Caldwell [15]
obtained the constraints on the theory with the derivative coupling term of Rµν∂µψ∂νψ by Solar system tests.
In general, a theory with derivative couplings could lead to that both the Einstein equations and the equation
of motion for the scalar are the fourth-order differential equations. However, Sushkov [16] studied recently
the model in which the kinetic term of the scalar field only coupled with the Einstein’s tensor and found that
the equation of motion for the scalar field can be reduced to second-order differential equation. This means
that the theory is a “good” dynamical theory from the point of view of physics. Gao [17] investigated the

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cosmic evolution of a scalar field with the kinetic term coupling to more than one Einstein’s tensors and found
the scalar field presents some very interesting characters. He found that the scalar field behaves exactly as
the pressureless matter if the kinetic term is coupled to one Einstein’s tensor and acts nearly as a dynamic
cosmological constant if it couples with more than one Einstein’s tensors. The similar investigations have been
considered in Refs.[18, 19]. These results will excite more efforts to be focused on the study of the scalar field
coupled with tensors in the more general cases.
Since black hole is another fascinating object in modern physics, it is of interest to extend the study the
properties of the scalar field when it is kinetically coupled to the Einstein’s tensors in the background of a black
hole. This extension to Reissner-Nordstr¨ om black hole spacetime is studied by S. Chen et al [20]. In this paper,
we will investigate the greybody factor and Hawking radiation of the scalar field coupling to the Einstein’s
tensor Gµνin the slowly rotating Kerr-Newman black hole spacetime. We find that the presence of the coupling
terms enhances both the absorption probability and luminosity of Hawking radiation of the scalar field in the
black hole spacetime. Moreover, we also find that for the weak coupling η < ηc, the absorption probability
and luminosity of Hawking radiation decrease when the black hole’s Hawking temperature decreases; while
for stronger coupling η > ηc, the absorption probability and luminosity of Hawking radiation increase on
the contrary when the black hole’s Hawking temperature decreases. This feature is similar to the Hawking
radiation in a d-dimensional static spherically-symmetric black hole surrounded by quintessence [1], i.e. when
0 < −ωq< (d − 3)/(d − 1), Hawking temperature deceases and the luminosity of Hawking radiation both in
the bulk and on the brane decrease naturally; when (d − 3)/(d − 1) < −ωq< 1, Hawking temperature still
deceases, but the luminosity of Hawking radiation both in the bulk and on the brane increase conversely.
The paper is organized as follows: in the following section we will introduce the action of a scalar field
coupling to Einstein’s tensor and derive its master equation in the slowly rotating Kerr-Newman black hole
spacetime. In Sec. III, we obtain the expression of the absorption probability in the low-energy limit by
using the matching technique. In section IV, we will calculate the absorption probability and the luminosity
of Hawking radiation for the coupled scalar field. Finally in the last section we will include and discuss our
conclusions.

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II.MASTER EQUATION WITH NON-MINIMAL DERIVATIVE COUPLING IN THE SLOWLY
ROTATING BLACK HOLE SPACETIME
Let us consider the action of the scalar field coupling to the Einstein’s tensor Gµνin the curved spacetime
[16],
S =
?
d4x√−g
?
R
16πG+1
2∂µψ∂µψ +η
2Gµν∂µψ∂νψ
?
.(2)
The coupling between Einstein’s tensor Gµνand the scalar field ψ is represented by
η
2Gµν∂µψ∂νψ, where η
is the coupling constant with dimensions of length-squared. In general, the presence of such a coupling term
brings some effects to the original metric of the background. However, we can treat the scalar filed as a
perturbation so that the backreaction effects on the background can be ignored, and then we can study the
effects of the coupling constant η on the greybody factor and Hawking radiation of the scalar filed in a black
hole spacetime.
Varying the action with respect to ψ, one can obtain the modified Klein-Gordon equation
1
√−g∂µ
?√−g
?
gµν+ ηGµν
?
∂νψ
?
= 0, (3)
which is a second order differential equation. Obviously, all the components of the tensor Gµνvanish in the
Kerr black hole spacetime because it is the vacuum solution of the Einstein’s field equation. Thus, we cannot
probe the effect of the coupling term on the greybody factor and Hawking radiation in the Kerr black-hole
background. The simplest rotating black hole with the non-zero components of the tensor Gµνis Kerr-Newman
one. In this paper, we consider a slowly rotating Kerr-Newman black hole, whose element reads [21]
ds2= −∆
r2dt2−2(2Mr − Q2)asin2θ
r2
dtdϕ +r2
∆dr2+ r2dΩ2+ O(a2) (4)
with
∆ = r2− 2Mr + Q2, (5)
where M, a, Q are the mass, angular momentum and charge of the black hole. The Einstein’s tensor Gµνfor
the metric (4) has a form
Gµν=Q2
r4




−r2
0
0
∆
0
∆
r2
0 −1
0
0
−
a
r2∆(r2+ ∆)
0
0
−
r2
0−
a
r2∆(r2+ ∆) 0
1
r2sin2θ



. (6)
Adopting to the spherical harmonics
φ(t,r,θ,ϕ) = e−iωteimϕRωℓm(r)Tm
ℓ(θ,aω),(7)

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we can obtain the radial part of the equation (3)
d
dr
?
∆?1 +ηQ2
r4
?dRωℓm
dr
?
+
?r2ω(r2ω − 2am)
∆
?1 +ηQ2
r4
?− [l(l + 1) − 2amω]?1 −ηQ2
r4
??
Rωℓm= 0. (8)
Clearly, the radial equation (8) contains the coupling constant η, which means that the presence of the coupling
term will change the evolution of the scalar field in the Kerr-Newman black hole spacetime.
The solution of the radial function Rωℓm(r) will help us to obtain the absorption probability |Aℓm|2and
the luminosity of Hawking radiation for a scalar field coupling with Einstein’s tensor in the slowly rotating
Kerr-Newman black hole spacetime.
III.GREYBODY FACTOR IN THE LOW-ENERGY REGIME
In order to study the effects of the coupling constant η on the absorption probability |Aℓm|2and the
luminosity of Hawking radiation of a scalar field in the background spacetime, we must first get an analytic
solution of the radial equation (8). In general, it is very difficult because that the equation (8) is nonlinear.
However, as in ref.[22–31], we can provide an approximated solution of the radial equation (8) by employing
the matching technique. Firstly, we must derive the analytic solutions in the near horizon (r ≃ r+) and
far-field (r ≫ r+) regimes in the low-energy limit. Finally, we smoothly match these two solutions in an
intermediate region. In this way, we can construct a smooth analytical solution of the radial equation valid
throughout the entire spacetime.
Now, we focus on the near-horizon regime and perform the following transformation of the radial variable
as in Refs. [29–31]
r → f(r) =∆(r)
r2
=⇒df
dr= (1 − f)A(r)
r
, (9)
with
A = 1 −
Q2
2Mr − Q2.(10)
The equation (8) near the horizon (r ∼ r+) can be rewritten as
f(1 − f)d2R(f)
df2
+ (1 − D∗f)dR(f)
df
+
?
K2
∗
A2
∗(1 − f)f−
Λm
ℓ
∗(1 − f)A2
?1 − η∗Q2
∗
1 + η∗Q2
∗
??
R(f) = 0,(11)
where
a∗= a/r+, Q∗= Q/r+, η∗= η/r2
+, K∗= ωr+− a∗m, A∗= 1 − Q2
2Q2
∗
(1 − Q2
∗,
Λm
ℓ= l(l + 1) − 2maω, D∗= 1 −
∗)2+
4η∗Q2
∗)[1 + η∗Q2
∗
(1 − Q2
∗].(12)

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Making the field redefinition R(f) = fα(1−f)βF(f), one can find that the equation (11) can be rewritten as
a form of the hypergeometric equation
f(1 − f)d2F(f)
df2
+ [c − (1 + ˜ a + b)f]dF(f)
df
− ˜ abF(f) = 0, (13)
with
˜ a = α + β + D∗− 1,b = α + β,c = 1 + 2α.(14)
Considering the constraint coming from coefficient of F(f), one can easy to obtain that the power coefficients
α and β satisfy
α2+K2
∗
A2
∗
= 0,(15)
and
β2+ β(D∗− 2) +
1
A2
∗
?
K2
∗− Λm
ℓ
?1 − η∗Q2
∗
1 + η∗Q2
∗
??
= 0, (16)
respectively. These two equations admit that the parameters α and β have the forms
α±= ±iK∗
A∗
,(17)
β±=1
2
?
(2 − D∗) ±
?
(D∗− 2)2−
4
A2
∗
?
K2
∗− Λm
ℓ
?1 − η∗Q2
∗
1 + η∗Q2
∗
? ?
,(18)
Following the operation in Refs. [29–31] and using the boundary condition that no outgoing mode exists near
the horizon, we can obtain that the parameters α = α− and β = β−. Thus the asymptotic solution near
horizon has the form
RNH(f) = A−fα(1 − f)βF(˜ a,b,c;f),(19)
where A−is an arbitrary constant.
Let us now to stretch smoothly the near horizon solution to the intermediate zone. We can make use of the
property of the hypergeometric function [33] and change its argument in the near horizon solution from f to
1 − f
RNH(f) = A−fα(1 − f)β
+ (1 − f)c−˜ a−bΓ(c)Γ(˜ a + b − c)
?Γ(c)Γ(c − ˜ a − b)
Γ(c − ˜ a)Γ(c − b)F(˜ a,b,˜ a + b − c + 1;1 − f)
F(c − ˜ a,c − b,c − ˜ a − b + 1;1 − f)
Γ(˜ a)Γ(b)
?
.(20)
As r ≫ r+, the function (1 − f) can be approximated as
1 − f =2Mr − Q2
r2
≃2M
r
,(21)

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and then the near horizon solution (20) can be simplified further to
RNH(r) ≃ C1r−β+ C2rβ+D∗−2, (22)
with
C1= A−(2M)βΓ(c)Γ(c − ˜ a − b)
Γ(c − ˜ a)Γ(c − b),(23)
C2= A−(2M)−(β+D∗−2)Γ(c)Γ(˜ a + b − c)
Γ(˜ a)Γ(b)
.(24)
Nextly in order to obtain a solution in the far field region, we expand the wave equation (8) as a power
series in 1/r and keep only the leading terms
d2RFF(r)
dr2
+2
r
dRFF(r)
dr
+
?
ω2−l(l + 1)
r2
?
RFF(r) = 0.(25)
This is usual Bessel equation. Thus the solution of radial master equation (8) in the far-field limit can be
expressed as
RFF(r) =
1
√r
?
B1Jν(ω r) + B2Yν(ω r)
?
, (26)
where Jν(ω r) and Yν(ω r) are the first and second kind Bessel functions, ν = l + 1/2. B1 and B2 are
integration constants. In order to stretch the far-field solution (26) towards small radial coordinate, we take
the limit r → 0 and obtain
RFF(r) ≃
B1(ω r
√r Γ(ν + 1)−
2)ν
B2Γ(ν)
π√r (ω r
2)ν.(27)
In the low-energy and low-angular momentum limit (ωr+)2≪ 1 and (a/r+)2≪ 1, the two power coefficients
in Eq.(22) can be approximated as
− β ≃ l + O(ω2,a2,aω),(28)
(β + D∗− 2) ≃ −(l + 1) + O(ω2,a2,aω).(29)
By using the above results, one can easily show that both Eqs. (22) and (27) reduce to power-law expressions
with the same power coefficients, rland r−(l+1). By matching the corresponding coefficients between Eqs.
(22) and (27), we can obtain two relations between C1, C2 and B1, B2. Removing A−, we can obtain the
ratio between the coefficients B1, B2
B ≡B1
B2
= −1
π
?
1
ωM
Γ(c − ˜ a − b)Γ(˜ a)Γ(b)
Γ(˜ a + b − c)Γ(c − ˜ a)Γ(c − b).
?2l+1
νΓ2(ν)
× (30)

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In the asymptotic region r → ∞, the solution in the far-field can be expressed as
RFF(r) ≃
B1+ iB2
√2π ω re−iω r+B1− iB2
e−iω r
r
√2π ω reiω r
eiω r
r
(31)
= A(∞)
in
+ A(∞)
out
. (32)
The absorption probability can be calculated by
|Aℓm|2= 1 −
????
A(∞)
out
A(∞)
in
????
2
= 1 −
????
B − i
B + i
????
2
=
2i(B∗− B)
BB∗+ i(B∗− B) + 1. (33)
Inserting the expression of B (30) into Eq.(33), we can probe the properties of absorption probability for the
scalar field coupled with Einstein’s tensor in the slowly rotating black hole spacetime in the low-energy limit.
IV.THE ABSORPTION PROBABILITY AND HAWKING RADIATION WITH NON-MINIMAL
DERIVATIVE COUPLING
We are now in a position to calculate the absorption probability and discuss Hawking radiation of a scalar
field coupling to Einstein’s tensor in the background of a slowly rotating Kerr-Newman black hole.
In Fig. 1, we fix the coupling constant η, and angular momentum a, and plot the change of the absorption
probability of a scalar particle with the charge Q for the first partial waves (ℓ = 0) in the slowly rotating
Kerr-Newman black hole. One can easily see that for the smaller η the absorption probability Aℓ=0decreases
with the charge Q of the black hole, which is similar to that for the usual scalar field without coupling to
Einstein’s tensor. However, for the larger η, the absorption probability Aℓ=0increases as the charge Q increase.
These properties mean that the stronger coupling between the scalar field and Einstein’s tensor changes the
properties of the absorption probability of scalar field in the black hole spacetime. In Fig.2, we also find that
the absorption probability increases with the increase of the coupling constant η for fixed values of charge
q = 0.3, and a = 0.1. In Fig. 3, we show the Hawking temperature of the slowly rotating Kerr-Newman
black hole with various charge Q. Therefore, for weak coupling, when Hawking temperature decreases, the
greybody factor decrease naturally, but for stronger coupling the greybody factor increases on the contrary.
The above results about the absorption probability also hold true for other values of ℓ,m. In this case,
there has superradiation region when m = 1,2,··· ,ℓ, which is similar to [29]. It is shown in Fig. 4, in which
we plotted the dependence of the absorption probability on the angular index ℓ and m with different η, a and
Q. From the above two figures in Fig. 4, we can obtain that the influence of charge Q on the usual radiation
(m = −1,0) is similar to that on the first partial wave. While for the super-radiation, the charge enhance it
both for weak and strong coupling. And the angular momentum a enhance the usual radiation (m = −1) and

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9
00.02 0.04
ΩM
0.060.08
0
0.005
0.01
0.015
0.02
0.025
?A?2
Η?0.1 a?0.1 l?m?0
q?0.3
q?0.2
q?0.1
q?0
0 0.020.04
ΩM
0.060.08
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
?A?2
Η?1.1 a?0.1 l?m?0
q?0.3
q?0.2
q?0.1
q?0
FIG. 1: Variety of the absorption probability |Aℓm|2of a scalar field with the charge Q and angular momentum in the
slowly rotating Kerr-Newman black hole for fixed ℓ = m = 0. The coupling constant η is set by η = 0.1 in the left and
by η = 1.1 in the right. We set 2M = 1.
00.020.04
ΩM
0.060.08
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
?A?2
q?0.3 a?0.1 l?m?0
Η?1.1
Η?0.8
Η?0.4
Η?0
FIG. 2: The dependence of the absorption probability |Aℓm|2of a scalar field on the coupling constant η in the slowly
rotating Kerr-Newman black hole for fixed ℓ = m = 0 and Q = 0.3. We set 2M = 1.
the super-radiation (m = 1) both for weak and strong coupling. Moreover, we see the suppression of |Aℓm|2
as the values of the angular index increase. This means that the first partial wave dominates over all others
in the absorption probability. It is similar to that of the scalar field without coupling to Einstein’s tensor as
shown in refs.[22–31].
Now let us turn to study the luminosity of the Hawking radiation for the mode ℓ = m = 0 which plays a
dominant role in the greybody factor. Performing an analysis similar to that in [29–31], we can obtain that
the greybody factor (33) in the low-energy limit has a form
|Aℓ=0|2≃
4ω2r2
(r2
+(r2
++ Q2)
+− Q2)(2 − D∗).(34)
Combining it with Hawking temperature TH[32] of the Kerr-Newman black hole,
TH=
r+− M
2π(r2
++ a2)=
r2
4πr+(r2
+− a2− Q2
++ a2)=r2
+− Q2
4πr3
+
+ O(a2),(35)

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00.10.2
Q
0.30.4
0.0791
0.0792
0.0793
0.0794
0.0795
TH
FIG. 3: The Hawking temperature of the slowly rotating Kerr-Newman black hole with various charge Q. We set
2M = 1.
0.02 0.04 0.06 0.080.10.12 0.14
ΩM
2?10?6
4?10?6
6?10?6
?A?2
Η?0.1 a?0.1 l?1
m?1m??1m?0
q?0.1
0.2
0.3
0.02 0.04 0.06 0.080.10.12 0.14
ΩM
?4?10?6
?2?10?6
2?10?6
4?10?6
6?10?6
8?10?6
?A?2
Η?1.1 a?0.1 l?1
m?1m??1m?0
q?0.1
0.2
0.3
0.020.040.060.080.10.12
ΩM
?1?10?6
1?10?6
2?10?6
3?10?6
?A?2
Η?0.1 q?0.2 l?1
m?1m??1m?0
a?0.050.08
0.1
0.020.040.06 0.080.10.12
ΩM
?1?10?6
1?10?6
2?10?6
3?10?6
4?10?6
?A?2
Η?1.1 q?0.2 l?1
m?1m??1m?0
a?0.050.08
0.1
FIG. 4: Variety of the absorption probability |Aℓm|2of a scalar field with the charge Q and angular momentum a in
the slowly rotating Kerr-Newman black hole spacetime for fixed ℓ = 1,m = 1,0,−1. The coupling constant η is set by
η = 0.1 on the left and η = 1.1 on the right. In the above two figures, the solid lines represent Q = 0.1, the dashed
lines represent Q = 0.2, the dashed-dotted lines represent Q = 0.3; in the lower two figures, the solid lines represent
a = 0.1, the dashed lines represent a = 0.08, the dashed-dotted lines represent a = 0.05. We set 2M = 1.
the luminosity of the Hawking radiation for the scalar field with coupling to Einstein’s tensor is given by
L =
?∞
0
dω
2π|Aℓ=0|2
ω
eω/TH− 1. (36)
The integral expressions above are just for the sake of completeness by writing the integral range from 0 to
infinity. However, as our analysis has focused only in the low-energy regime of the spectrum, an upper cutoff

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will be imposed on the energy parameter so that the low-energy conditions ω ≪ THand ωr+≪ 1 are satisfied.
In the low-energy limit, the luminosity of the Hawking radiation for the mode ℓ = 0 can be approximated as
L ≈2π3
15GT4
H,(37)
with
G =
r2
+(r2
+− Q2)(2 − D∗).
++ Q2)
(r2
(38)
In Fig. 5 and 6, we show the dependence of the luminosity of Hawking radiation on the charge Q and the
coupling constant η, respectively. From Fig. 5, one can easily obtain that with increase of Q the luminos-
ity of Hawking radiation L decreases for the smaller η and increases for the larger η. In other words, for
the weak coupling, the luminosity of Hawking radiation decreases naturally when the black hole’s Hawking
temperature decreases; while for stronger coupling, the luminosity of Hawking radiation increases on the con-
trary when the black hole’s Hawking temperature decreases. This is similar to the behavior of the absorption
probability discussed previously. This feature is similar to the Hawking radiation in a d-dimensional static
spherically-symmetric black hole surrounded by quintessence [1], i.e. when 0 < −ωq< (d−3)/(d−1), Hawk-
ing temperature deceases, the luminosity of Hawking radiation both in the bulk and on the brane decrease
naturally; when (d − 3)/(d − 1) < −ωq< 1, Hawking temperature still deceases, the luminosity of Hawking
radiation both in the bulk and on the brane increase conversely. In Fig. 6, we show that the luminosity of
Hawking radiation L increases monotonously with the coupling constant η for the all Q.
00.050.1 0.15
Q ?a?0.1?
0.2 0.250.3
0.00015
0.000155
0.00016
0.000165
0.00017
0.000175
0.00018
L
Η?0.9
Η?0.8
Η?0.4
Η?0.1
Η?0
FIG. 5: Variety of the luminosity of Hawking radiation L of scalar particles with the charge Q and angular momentum
a in the slowly rotating Kerr-Newman black hole for fixed ℓ = 0 and different values of η. We set 2M = 1
From the Fig. 5, there should exist a critical coupling constant ηc, so in the table I, we list the critical
coupling constant ηcfor different Q by using equation ∂L/∂Q = 0. The results show that different charge Q
of the slowly rotating Kerr-Newman black hole correspond to different critical coupling constant ηc.

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0 0.20.40.6
Η ?a?0.1?
0.811.2 1.4
0.00014
0.00016
0.00018
0.0002
0.00022
L
q?0.3
q?0.2
q?0.1
q?0
FIG. 6: The dependence of the luminosity of Hawking radiation L of a scalar field on the coupling constant η in the
slowly rotating Kerr-Newman black hole for fixed ℓ = 0 and different values of Q and angular momentum a. We set
2M = 1.
Q
ηc
0.10.20.3
0.6000.5100.541
TABLE I: The critical value of coupling constant ηc for different Q. We set 2M = 1 and ℓ = m = 0.
V.SUMMARY AND DISCUSSION
In this paper, we have studied the greybody factor and Hawking radiation with a non-minimal derivative
coupling between a scalar field and the curvature in the background of the slowly rotating Kerr-Newman black
hole spacetime in the low-energy approximations. We have found that the presence of the coupling enhances
both the absorption probability and the luminosity of Hawking radiation of the scalar field in the black hole
spacetime. Moreover, we also find that for the weak coupling η < ηc, the absorption probability and the
luminosity of Hawking radiation decrease when the black hole’s Hawking temperature decreases; while for
stronger coupling η > ηc, the absorption probability and the luminosity of Hawking radiation increase on
the contrary when the black hole’s Hawking temperature decreases. This feature is similar to the Hawking
radiation in a d-dimensional static spherically-symmetric black hole surrounded by quintessence [1], i.e. when
0 < −ωq < (d − 3)/(d − 1), Hawking temperature deceases, the luminosity of Hawking radiation both in
the bulk and on the brane decrease naturally; when (d − 3)/(d − 1) < −ωq< 1, Hawking temperature still
deceases, the luminosity of Hawking radiation both in the bulk and on the brane increase conversely.
Discussion: This very amazing and interesting similarity show us that the scalar field which is coupling to
the Einstein’s tensor of a black hole spacetime can be taken as a dark energy in the universe. In the Ref.
[17], the authors also point out that this scalar field which is coupling to the Einstein’s tensor behaves exactly
as pressureless matter (without a scalar potential), plays the role of both cold dark matter and dark energy

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(with a scalar potential). If there exist dark energy or dark matter in our cosmology, then they would be
a scalar field which has a non-minimal derivative coupling to the curvature of a spacetime. And our results
could provide a way to detect whether there exist a coupling between the scalar field and Einstein’s tensor or
not. So that these conjectures will be conformed in the near future.
Acknowledgments
This work was partially supported by the Scientific Research Foundation for the introduced talents of Hunan
Institute of Humanities Science and Technology. Jing’s work was partially supported by the National Natural
Science Foundation of China under Grant No.10675045, No.10875040 and No.10935013; 973 Program Grant
No. 2010CB833004and the Hunan Provincial Natural Science Foundation of China under Grant No.08JJ3010.
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