Charged vector particle tunneling from a pair of accelerating and rotating and 5D gauged supergravity black holes
Eur. Phys. J. C
Charged vector particle tunneling from a pair of accelerating and rotating and 5D gauged supergravity black holes
0 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur , Pakistan
1 Division of Science and Technology, University of Education , Township Campus, Lahore 54590 , Pakistan
The aim of this paper is to study the quantum tunneling process for charged vector particles through the horizons of more generalized black holes by using the Proca equation. For this purpose, we consider a pair of charged accelerating and rotating black holes with NewmanUntiTamburino parameter and a black hole in 5D gauged supergravity theory, respectively. Further, we study the tunneling probability and corresponding Hawking temperature for both black holes by using the WKB approximation. We find that our analysis is independent of the particles species whether or not the background black hole geometries are more generalized.

A black hole (BH) is considered as an object which absorbs
all the matter/energy from the environing area into it due to
its intense gravitational field. General relativity (GR) depicts
that a BH swallows all particles that collides the horizon of
the BH. In 1974, Hawking predicted that a BH behaves like
a black body having a specific temperature, known as the
Hawking temperature, which allows a BH to emit radiation
(called Hawking radiation) from its horizon by assuming
quantum field hypotheses in the background of the curved
spacetime.
A particle’s action of a quantum mechanical nature is used
in order to calculate the Hawking radiation spectrum from
different BHs [1,2]. The analysis of Hawking radiation as
a quantum tunneling phenomenon and accretion onto some
particular BHs has attracted the attention of many researchers
[3–8]. Various efforts have been made to examine this
radiation spectrum from BHs by considering the quantum
mechana emails: ;
b emails: ;
c email:
ics of scalar, Dirac, fermion and photon particles etc. Many
researchers [9–12] have studied vector particle tunneling to
obtain more information as regards the Hawking temperature
and the radiation spectrum from different BHs. The charged
vector particle tunneling from Kerr–Newman BH [13] and
charged black string [14] are important contributions toward
BH physics.
The charged fermions tunneling from
Reissner–Nordström–de Sitter BH with a global monopole [15] is studied by
using the WKB approximation and Dirac equation to
evaluate the tunneling process for charged particles as well as the
Hawking temperature. In this paper the authors have
evaluated the tunneling probability and the Hawking temperature
for charged fermion tunneling from event horizon. The
tunneling process for Plebanski–Demianski BHs is determined
by the graphical behavior of the Hawking temperature of an
ingoing and outgoing charged fermion from the event horizon
[16]. The Hawking temperature for charged NUT (Newman–
Unti–Tamburino) BH solutions to the field equations is
considered with rotation and acceleration. A BH can be studied
by a small measurement through quantum field theory on
a curved background [17]. The tunneling probability for an
outgoing particle is ruled by the imaginary part of the
particle’s action. A large number of attempts [18–26] have been
made to calculate the tunneling of charged and uncharged
scalar and Dirac particles with different BH configurations.
The tunneling of spin 21 particles by the event horizon of the
Rindler spacetime was explained and the Unruh temperature
has been calculated [27]. Kraus and Wilczek [28,29] took
a semiclassical process to analyze Hawking radiation as a
tunneling event. This process contains the calculation for the
phenomenon of swave emission across the event horizon. In
[30], it has been shown that the Hawking radiation from a
rotating wormhole may emit all types of particles.
This paper deals with the study of the Hawking radiation
process of charged vector particles from the horizons of a
pair of accelerating and rotating BHs and a BH in 5D gauged
supergravity.
Vector particles (spin1 bosons) such as Z (uncharged) and
W ± (charged) bosons are of great importance in the Standard
Model. In the background of BH geometries, the behavior
of the bosons can be determined by using the Proca
equation. First, we formulate the field equations of charged W
±bosons by using the Lagrangian of the Glashow–Weinberg–
Salam model [31]. Then we shall investigate particle
emission process by using the Hamilton–Jacobi definition and
WKB approximation to the derived equation for the charged
case in the considered BH geometries. By putting the
determinant (of the coefficient matrix) equal to zero, we can solve
for the radial function. Consequently, we compute the
tunneling rate of the charged vector particles from the horizons of
BHs and find the corresponding Hawking temperature values
in both cases.
The paper is planned as follows: we discuss in Sect. 2 the
tunneling rate and Hawking temperature for charged
accelerating and rotating BH solutions with NUT parameter.
Section 3 is devoted to an investigation of charged vector
particle tunneling and the Hawking temperature for BH in 5D
gauged supergravity spacetime, by investigating the W ±
bosons observation. Section 4 provides a summary of the
results for both cases.
2 Accelerating and rotating black holes with NUT
parameter
In general, the NUT parameter is affiliated with the
gravitomagnetic monopole, related to the bending properties of the
environing spacetime due to the fundamental mass, its
accurate physical significance could not be determined. The
generalization for multi dimensional Kerr–NUT–de Sitter
spacetime [32,33] and its physical implication [34] are also
investigated. As a BH, the dominance on the NUT parameter,
the revolution parameter sets the spacetime free of
bending singularities and the result can be thought of as a
NUTlike result. If the revolution parameter commands the NUT
parameter, the result is Kerrlike and a closed chain of
bending singularity forms. The behavior of this form of the
singularity structure is independent of the existence on the
cosmology constant.
There are lots of BHs which are associated with the NUT
parameter and lots of investigations have been made to
examine their physical effects in the space of colliding waves. The
significance of the NUT parameter makes itself accurately
felt when a motionless Schwarzschild mass is absorbed in
a stationary source and allows for electromagnetism [35].
The NUT parameter refers to the bend of the
electromagnetism leaving out the fundamental Schwarzschild mass. In
the absence of an electromagnetic field, it reduces to the
bending of the vacuum spacetime [36]. The bend of the
surrounding space pair with the mass of reference yields the NUT
parameter.
The line element for accelerating and rotating BHs with
NUT parameter is defined as [37]
ds2 = −
Q =
= 1 − ωα (l + a cos θ )r, ρ2 = r 2 + (l + a cos θ )2,
(ω2k˜ + e˜2 + g˜2)(1 + 2αl ωr ) − 2Mr + aω22−k˜rl22
a4 = −
a3 = 2M
Here, M denotes the mass of pairs of BHs, e and g indicate
the electric and magnetic charges, respectively, while l is the
NUT parameter of BH, α and ω indicate acceleration and
rotation of the sources, respectively. Also, a is the Kerrlike
rotation parameter and k˜ is given by
Here, α, ω, M, e˜, g˜ and k˜ are arbitrary real parameters. We
would like to mention that ω depends on the NUT
parameter l and the Kerrlike rotation parameter a. The α twisting
property of BHs is proportional to the rotation ω. Also, ω
depends on rotation parameters l and a. The parameters α,
ω, M, e˜, g˜ and k˜ vary independently. If α is equal to zero,
then the metric in Eq. (2.1) leads to the Kerr–Newman–NUT
solution. If l = 0, then the metric in Eq. (2.1) gives the
couple of charged and rotating BHs. In this case, if e˜ and g˜ are
equal to zero, we have a Schwarzschild BH and if l and a are
equal to zero it leads to a Cmetric.
The metric (2.1) can be rewritten as
− Q a sin2 θ + 4l sin2 θ .
2
The electromagnetic potential for these BHs is given by
1
A = a(r 2 + (l + a cos θ )2)
− (l2 + a2 cos2 θ + 2al cos θ ))
− g˜ l + a cos θ adt − dφ r 2 + (l + a)2) .
The event horizons are obtained for g(r, θ ) = Qρ2 2 = 0,
which implies that = 0, so Q = 0, which yields the
following real roots of r :
− (ω2k˜ + e˜2 + g˜2) αωl − M
r± =
where rα1 and rα2 are acceleration horizons and r± represent
the outer and inner horizons, respectively, such that
(ω2k˜ + e˜2 + g˜2) αωl − M
−(ω2k˜ + e˜2 + g˜2) a2ω−2k˜l2 > 0.
where f (r, θ ), g(r, θ ), (r, θ ), K (r, θ ) and H (r, θ ) are given
by the following equations:
Proca equation by using the Lagrangian of the Wbosons of
Glashow–Weinberg–Salam model [10]
ˇ = r +2 + (a + l)2
In order to investigate the tunneling spectrum for charged
vector particles through the BH horizon, we will consider
Proca equation with electromagnetic effects. In a curved
spacetime with electromagnetic field, the motion of
massive spin1 charged vector fields is depicted by the given
where g is the determinant of the coefficient matrix, m is
particles mass and ψ μν is an antisymmetric tensor, i.e.,
i i
ψνμ = ∂ν ψμ − ∂μψν + h e Aν ψμ − h e Aμψν andF μν
Here, Aμ is considered as the electromagnetic potential of
the BH, e denotes the charge of the Wbosons and ∇μ is
the geometrically covariant derivative. Since the equation of
motion for the W + and W − bosons is similar, the tunneling
processes should be similar too. For simplification, here we
will consider the W + boson case; the results of this case can
be extended to W − bosons due to the digitalization of the
line element. For W + field, the values of the components of
ψ μ and ψ νμ are obtained as follows:
ψ 12 = g −1ψ12, ψ 13 =
The electromagnetic vector potential for this BH is given by
[38]
1
A = a[r 2 + (l + a cos θ )2] [−e˜r [adt
−dφ (l + a)2 − (l2 + a2 cos2 θ + 2la cos θ )]
−g˜(l + a cos θ )[adt − dφr 2 + (l + a)2]].
Applying the WKB approximation [39], i.e.,
to the Proca equation (2.6) and neglecting the terms for n =
1, 2, 3, 4, . . ., we obtain the following set of equations:
kg[c1(∂1 S0)((∂1 S0) + e A0)
− c0(∂1 S0)2] − H g c3(∂1 S0)2
− c1(∂1 S0)((∂3 S0) + e A3) +
c2(∂2 S0)((∂0 S0)
+ e A3) − c0(∂2 S0)2
+ [c3(∂3 S0)((∂0 S0) + e A0)
c3(∂2 S0)2 − c2(∂2 S0)((∂3 S0) + e A3)
[c3(∂2 S0)(∂3 S0) − c2(∂3 S0)2 − e A3c2(∂3S0)]
− c1((∂3 S0) + e A3)] = 0,
kg[c1(∂0 S0)((∂0 S0) + e A0 − c0(∂1 S0)(∂0 S0)]
− H g[c3(∂1 S0)(∂0 S0)
− c1(∂0 S0)((∂3 S0) + e A3)] +
× c2(∂1 S0)(∂2 S0) − c1(∂2 S0)2
+ g f [c3(∂1 S0)(∂3 S0) − c1(∂3 S0)((∂3 S0) + e A3)]
+ g H [c1(∂3 S0)((∂0 S0)
− e A0) − c0(∂1 S0)(∂3 S0)] + e A0kg[c1((∂0 S0)
−c1((∂3 S0) − e A3)]
+ e A3g f [c3(∂1 S0) − c1((∂3 S0)
+ e A3)] + e A3 H [c1((∂0 S0) + e A0)
− c0(∂1 S0)] = 0,
c2(∂2 S0)2 − c0(∂2 S0)(∂0 S0) + e A0(∂0 S0)c2
[c3(∂1 S0)(∂2 S0)
− c2(∂1 S0)(∂3 S0)] −
[c2(∂1 S0)2
[(∂3 S0)(∂0 S0)c2
− c0(∂2 S0)(∂3 S0) + c2e A0(∂2 S0)(∂3 S0)]
− m2c2 −1( f k − H )
k
+ e A0 [c2(∂0 S0) − c0(∂2 S0)
H
+ e A0c2] − e A0 [c3(∂2 S0) − c2(∂3 S0)
f
− c2e A3] + e A3 [c3(∂2 S0) − c2(∂3 S0)
− c0(∂2 S0) + e A0c2] = 0,
[c3(∂0 S0)2 − c0(∂3 S0)(∂0 S0) + e A0c3(∂0 S0)
− e A3c0(∂0 S0)]
− f g[c3(∂2 S0)2
− c0(∂2 S0)2
[c3(∂2 S0)2
− c1(∂1 S0)(∂3 S0) − e A3c1(∂0 S0)] −
[c2(∂0 S0)(∂2 S0)
[c2(∂0 S0)(∂2 S0) − c0(∂2 S0)2 + e A0c2(∂2 S0)]
− c2(∂2 S0)(∂3 S0) − e A3c2(∂2 S0)]
+ e A0[c3(∂0 S0) − c0(∂3 S0)
+ e A0c3 − e A3c0 + m2[H c0 + c3 f ] = 0.
Using the technique of separation of variables, we can choose
S0 = −(E − j ˇ )t + W (r ) + N φ +
where E and j represent the particle’s energy and angular
momentum, respectively. From Eqs. (2.9)–(2.12), we can
obtain a matrix equation,
G(c0, c1, c2, c3)T = 0,
which implies a 4×4 matrix labeled “G”, whose components
are given as follows:
G11 = −W˙ 2kg −
− ˙ 2 − ˙ e A3 − m2k − e A3kgW˙ ,
G12 = −W˙ kg(E − j ˇ ) + kgW˙ e A0 + H gW˙ ˙
+ H gW˙ e A3 − e A3kg(E − j ˇ ) + kge2 A3 A0
+ e A3 H g ˙ + H ge2 A3,
k k H H
G13 = −
(E − j ˇ )N +
G14 = −W˙ 2 H g −
N 2 − ˙ 2(E − j ˇ ) + ˙ e A0
− m2 H − e A3g H W˙ ,
N 2( f k − H ) − g f ˙ ( ˙ + e A3))
+ g H ˙ (−(E − j ˇ ) − e A0) + e A0kg(−(E − j ˇ )
+e A0g H ( ˙ − e A3) − e A3g f ( ˙ + e A3),
G23 =
W˙ N ( f k − H 2),
Expanding the functions f (r ) and g(r ) in Taylor’s series
near the horizon, we get
Using the above expressions in Eq. (2.14), one can see that
the resulting equation has poles at r = r+. For the
calculation of the Hawking temperature by using the tunneling
method, it is required that we regularize the singularity by a
specific complex contour to bypass the pole. For our standard
coordinates of the BH metric, the tunneling of outgoing
particles can be obtained by taking an infinitesimal half circle
below the pole r = r+, while for the ingoing particle such a
contour is taken above the pole. Further, in order to calculate
the semiclassical tunneling probability, it is required that the
resulting wave equation must be multiplied by its complex
conjugate. In this way, the part of the trajectory that starts
from outside of the BH and continues to the observer will
not contribute to the calculation of the final tunneling
probability and can be ignored because it will be completely real.
Therefore, the only part of the trajectory that contributes to
the tunneling probability is the contour around the BH
horizon.
Hence using Eqs. (2.14) and (2.15), and integrating the
resulting equation around the pole, we get
and the surface gravity is [36]
αωl (ω2k˜ + e˜2 + g˜2) − M + aω2−2k˜l2 r+
G31 =
G32 =
(E − j ˇ )N −
N ˙ − e A3 H N ,
W˙ N (E − j ˇ ),
G33 = −
[(E − j ˇ )2 − e A0(E − j ˇ )] +
[W˙ 2( f k + H 2)] −
[ ˙ ((E − j ˇ ) + e A0 N )]
k
− m2 −1( f k + H 2) − e A0 [(E − j ˇ )
− e A0] + e A0
G34 = −
N ˙ − e A0 N
+ m2 H + (E − j ˇ )e A0
G43 =
(E − j ˇ )N −
N − m2 f − e A0[(E − j ˇ ) − e A0],
where W˙ = ∂r S0, ˙ = ∂θ S0 and N = ∂φ S0. For the
nontrivial solution, the absolute value G equals zero, and we
solve the resultant equation for the radial part so that we get
the following integral:
I mW ± = ±
(E − e A0 − j ˇ )2 + X dr
f (r )g(r )
where + and − represent the radial functions of outgoing
and incoming particles, respectively, while the function X
Prob[emission] exp[−2(I mW + + I m )] = exp[−4I mW +]
= Prob[absorption] = exp[−2(I mW − − I m )]
⎡
⎢
= exp ⎢⎢ −2π
⎢
⎣
E − e A0 − j ˇ
× 1 + α(aω−l) r+ × 1 −
can be defined as X = − −1 f N − m2 f − H g(E − j ˇ ) −
g f ˙ + e A0g H − e A3g f ; ˇ is the angular velocity on the
event horizon.
Now, finally we can calculate the Hawking temperature by
comparing the above result with the Boltzmann formula
B = e−(E−e A0− j ˇ )/TH , to get
TH = ⎣
The Hawking temperature depends on A0, the vector
potential, E , the energy, ˇ , the angular momentum; M is the mass
of the pair of BHs, e and g are electric and magnetic charges,
respectively, a is the rotation of a BH, l is the NUT
parameter, α represents the acceleration of the sources and ω the
rotation of the sources.
We would like to mention that the Hawking temperature
of charged vector particles given in Eq. (2.17) is the same as
the Hawking temperature of fermion particles in Eq. (4.20)
of [36]. Thus the Hawking temperature is independent of the
particle species.
3 Black holes in 5D gauged supergravity
The gauged theory is stated as a supergravity theory in
which the gravitino, the superpartner of the graviton is
charged under some internal gauge group. However, the
gauged supergravity is more significant as compared to the
ungauged case, because this theory has a negative
cosmological constant, so it is defined on an antide Sitter space.
Here, for the discussion of charged vector particles tunneling
spectrum from a BH in 5D gauged supergravity, we evaluate
the tunneling probability of particles and the corresponding
Hawking temperature at the BH horizon. Such BH solutions
occur in D = 5 N = 8 gauged supergravity (symmetry)
[40]. Firstly, this solution was formulated in [41] as a
particular case (STU model) of solutions of the D = 5N = 2
gauged supergravity equations of motion. The metric for the
BH in 5D gauged supergravity is [40]
and d 23,k is the metric on S3 with unit radius if k = 1, or the
metric on R3 if k = 0; here μ is the nonextremality
parameter [41], which is related to the ADM mass, g = 1/L is the
inverse radius of Ad S5 related to the cosmological constant
= −6g2 = −6/L2, and qi are charges entering the
metric. The three gauge field potentials Aiμ from the solution of
equation of motion are of the form
(for i = 1, 2, 3),
where the q˜i are the physical charges, which are conserved
and the Gauss law is applicable to such charges.
The line element can be rewritten as
ds2 = − A˜(r )dt 2 + B˜ −1(r )dr 2 + C˜ (r )dθ 2
The horizons of the metric (3.2) can be determined when
f (r ) = 0. For this purpose we follow [40] and assume that
g2 = 1 (by the choice of units as in [40]). Hence, in this case
the outer horizon is located at
r+ =
(1 + qi )2 + 4μ > (1 + qi ) and i = 1, 2, 3.
In the Proca equation (2.6) the components of ψ ν and ψ μν
are given by
ψ 0 = − A˜−1ψ0, ψ 1 = B˜ ψ1, ψ 2 = C˜ −1ψ2,
ψ 3 = D˜ −1ψ3, ψ 4 = E˜ −1ψ4,
ψ o1 = −B˜ A˜−1ψ01, ψ 02 = −( A˜C˜ )−1ψ02,
ψ 04 = −( A˜ E˜ )−1ψ04, ψ 12 = B˜ C˜ −1ψ12,
ψ 13 = B˜ D˜ −1ψ13, ψ 14 = B˜ E˜ −1ψ14,
ψ 23 = (C˜ D˜ )−1ψ23, ψ 24 = (C˜ E˜ )−1ψ24,
By using Eq. (2.6), we obtain the following set of equations
(for simplicity, we assume A0 ≡ Ai0 for all i ):
B˜ [c0(∂1 S0)2 − c1(∂0 S0)(∂1 S0) − e A0c1(∂1 S0)]
+ D˜ −1[C0(∂3 S0)2 − c3(∂3 S0)(∂0 S0)
− e A0c4(∂4 S0)] + m2c0 = 0,
A˜−1[c0(∂1 S0)(∂0 S0) − c1(∂0 S0)2 − e A0c1(∂0 S0)]
− D˜ −1(∂3 )2 − E˜ −1 N 2
+ e A0 A˜−1[e A0 − (E − j ˇ 1)] − m2,
23 = D˜ −1(∂2 )(∂3 ),
24 = E˜ −1(∂2 )N ,
30 = A˜−1(E − j ˇ 1)(∂3 ) − e A0 A˜−1(∂3 ),
31 = B˜ W˙ (∂3 ),
32 = C˜ −1(∂2 )(∂3 ),
33 = A˜−1(E − j ˇ 1)2 − e A0 A˜−1(E − j ˇ 1) − B˜ W˙ 2
−E˜ −1 N 2 − C˜ −1(∂2 )2 − m2
−e A0 A˜−1[(E − j ˇ 1) − e A0],
34 = E˜ −1 j (∂3 ),
−e A0 A˜−1 j,
42 = C˜ −1(∂2 )N ,
40 = A˜−1((E − j ˇ 1)N
41 = B˜ W˙ N ,
43 = D˜ −1(∂3 )N ,
44 = A˜−1(E − j ˇ 1)2 − e A0 A˜−1(E − j ˇ 1)
−B˜ W˙ 2 − C˜ −1(∂2 )2
− D˜ −1(∂3 )2 − m2 − e A0 A˜−1[(E − j ˇ 1) − e A0].
For the nontrivial solution, the determinant is equal to zero
and using the same technique as discussed in the previous
section, we get
I mW ± = ±
X˜ = − A˜C˜ −1(∂2 )2 − A˜ D˜ −1(∂3 )2 − A˜m2
−E˜ −1(∂2 )N .
− c3(∂3 S0)(∂1 S0)] + E˜ −1[c1(∂4 S0)2
− c4(∂1 S0)c0(∂4 S0)] + e A0 A˜−1[c0(∂1 S0)
− c1(∂0 S0)] + m2c1 = 0,
A˜−1[c2(∂0 S0)2 − c0(∂0 S0)(∂2 S0) + e A0c2(∂0 S0)]
−B˜ [c2(∂1 S0)2 − c1(∂1 S0)(∂2 S0)]
+D˜ −1[c3(∂2 S0)(∂3 S0)
−c2(∂3 S0)2] + E˜ −1[c4(∂2 S0)(∂4 S0)
− c2(∂4 S0)2] + e A0 A˜−1[c2(∂0 S0) − c0(∂2 S0)
+ e A0c2] − m2c2 = 0,
A˜−1[c3(∂0 S0)2 − c0(∂0 S0)(∂3 S0) + e A0c3(∂0 S0)]
− c2(∂2 S0)(∂3 S0)] + E˜ −1[c4(∂4 S0)(∂3 S0)
−c3(∂4 S0)2] + e A0 A˜−1[c3(∂0 S0) − c0(∂3 S0)
+e A0c3] − m2c3 = 0,
A˜−1[c4(∂0 S0)2 − c0(∂0 S0)(∂4 S0) + e A0c4(∂0 S0)]
−c2(∂2 S0)(∂4 S0)]
− D˜ −1[c4(∂3 S0)2 − c3(∂3 S0)(∂4 S0)]
+e A0 A˜−1[c4(∂0 S0) − c0(∂4 S0) + e A0c4] − m2c4 = 0.
(3.7)
We carry out the separation of variables:
S0 = −(E − j ˇ 1)t + W (r ) +
where ˇ 1 is the angular velocity for BH given by Eq. (3.2).
For the above S0 the preceding set of equations (3.3)–(3.7)
can be written in terms of a matrix equation, (c0, c1, c2, c3,
c4)T = 0, and the elements of the required matrix have the
following form:
01 = B˜ [(E − j ˇ 1)W˙ − e A0W˙ ],
02 = C˜ −1(E − j ˇ 1)(∂2 )
03 = D˜ −1(E − j ˇ 1)(∂3 ) − D˜ −1e A0(∂3 ),
04 = E˜ −1(E − j ˇ 1)N − E˜ −1 j e A0,
10 = − A˜−1(E − j ˇ 1)W˙ + e A0 A˜−1W˙ ,
11 = − A˜−1(E − j ˇ 1)2 + e A0(E − j ˇ 1) A˜−1
+C˜ −1(∂2 )2 + D˜ −1(∂3 )2 + E˜ −1 N 2
+e A0 A˜−1(E − j ˇ 1) + m2,
12 = −C˜ −1W˙ (∂2 ),
13 = −D˜ −1W˙ (∂3 ),
14 = −E˜ −1W˙ N ,
21 = B˜ W˙ (∂2 ),
Since the BH given by Eq. (3.2) is nonrotating, ˇ 1 = 0. The
surface gravity for this BH is given by [40]
i3=1(r +2 + qi )
The required tunneling probability as discussed in the
previous section is
˜ =
= e−4I mW + = e
The Hawking temperature in this case is given by
T˜H =
The Hawking temperature is related to the energy E, the
potential A0, the angular momentum j , the radial coordinate
at the outer horizon r+ and the charge qi . We would like
to mention that the Hawking temperature of charged vector
particles given by Eq. (3.12) is the same as the Hawking
temperature of 5D gauged supergravity BH in Eq. (9) in
Ref. [40].
4 Outlook
During the tunneling process when a particle with
electropositive energy crosses the horizon, it appears as Hawking
radiation. Likewise, a particle with electronegative energy burrows
in weave, it is assimilated by the BH, so its mass falls and
finally disappears. Thus, the movement of the particles may
be in the configuration of outgoing and incoming,
performing the particle’s action turns out to be complex and real,
respectively. The emission rate of a tunneling particles from
the BH is associated with the imaginary component of the
particles’ action, which in fact is related to the Boltzmann
factor based on the Hawking temperature.
In this paper, we have extended the work of vector
particle tunneling for more generalized BHs in 4D and 5D spaces
and recovered their corresponding Hawking temperatures at
which particles tunnel through horizons. For this purpose, we
have used the Proca equation with the background of
electromagnetism to investigate the tunneling of charged vector
particles from accelerating and rotating BHs in 4D and 5D BHs
having electric and magnetic charges with a NUT
parameter. We have implemented the WKB approximation to the
Proca equation, which leads to the set of field equations; then
we use separation of variables to solve these equations. We
solve for the radial part by using the determinant of the
coefficient matrix being equal to zero. Using the surface gravity,
we have formulated the tunneling probability and the
Hawking temperature for both BHs at the outer horizon. All these
quantities depend on the defining parameters of the BHs. It is
worthwhile to mention here that the backreaction effects of
the emitted particle on the BH geometry and selfgravitating
effects have been neglected, the derived Hawking
temperature is only a leading term. Thus one does not need to
calculate the appropriate solution of the semiclassical Einstein
field equations for the geometry of the background BH in
equilibrium with its Hawking radiation [42].
From our analysis we have concluded that the Hawking
temperature at which particles tunnel through the horizon
is independent of the species of particles. In particular the
nature of the background BH geometries, for the particles
having different spins (either spin up or down) or zero spin,
the tunneling probabilities will be seen to be the same by
considering semiclassical effects. Thus, their
corresponding Hawking temperatures must be the same for all kinds of
particles. For both cases, we have carried out the calculations
for more general BHs, i.e., a pair of charged accelerating and
rotating BHs with NUT parameter (which is a more general
case of BHs as compared to the BH taken in [43]) and a BH
in 5D gauged supergravity. Our findings are similar to the
statement that the temperature of tunneling particles is
independent of the species of the particles, and this result is also
valid for different coordinate frames by using specific
coordinate transformations. The authors of Ref. [43] have proved
it for the Kerr BH (which only is rotating), while we have
proved it for more generalized BHs. Hence, the conclusion
still holds if background BH geometries are more general.
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