# Hawking black body spectrum from tunneling mechanism

Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We obtain, using a reformulation of the tunneling mechanism, the Hawking black
body spectrum with the appropriate temperature for a black hole. This is a new
result in the tunneling formalism of discussing Hawking effect. Our results
are given for a spherically symmetric geometry that is asymptotically flat.

Introduction: After Hawking’s observation [1] that black holes radiate, there
were several approaches [2, 3, 4, 5, 6, 7] to study this effect. A
particularly intuitive and widely used approach is the tunneling mechanism [4,
5]. The essential idea is that a particle-antiparticle pair forms close to the
event horizon which is similar to pair formation in an external electric
field. The ingoing mode is trapped inside the horizon while the outgoing mode
can quantum mechanically tunnel through the event horizon. It is observed at
infinity as a Hawking flux. So this effect is totally a quantum phenomenon and
the presence of an event horizon is essential. However, in the literature [4,
5, 8, 9, 10, 11, 12], the analysis is confined to obtention of the Hawking
temperature only by comparing the tunneling probability of an outgoing
particle with the Boltzmann factor. There is no discussion of the spectrum.
Hence it is not clear whether this temperature really corresponds to the
temperature of a black body spectrum associated with black holes. One has to
take recourse to other results to really justify the fact that the temperature
found in the tunneling approach is indeed the Hawking black body temperature.
In this sense the tunneling method, presented so far, is incomplete.

In this paper we rectify this shortcoming. Using density matrix techniques we
will directly find the spectrum from a reformulation of the tunneling
mechanism. For both bosons and fermions we obtain a black body spectrum with a
temperature that corresponds to the familiar semiclassical Hawking expression.
Our results are valid for black holes with spherically symmetric geometry.
Finally, we show the connection of our formulation with usual tunneling
formulations [4, 5] by exploiting the principle of detailed balance.

General formulation: Consider a black hole characterised by a spherically
symmetric, static space-time and asymptotically flat metric of the form,

$\displaystyle ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}d\Omega^{2}$ (1)

whose event horizon $r=r_{H}$ is defined by $F(r_{H})=0$. For discussing
Hawking effect by tunneling, the radial trajectory is relevant [4, 5]. We
therefore consider only the $(r-t)$ sector of the metric (1).

Now consider the massless Klein-Gordon equation
$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ which, in the ($r-t$) sector,
reduces to,

$\displaystyle-\frac{1}{F(r)}\partial^{2}_{t}\phi+F^{{}^{\prime}}(r)\partial_{r}\phi+F(r)\partial^{2}_{r}\phi=0$
(2)

in the black hole space-time (1). Taking the standard WKB ansatz

$\displaystyle\phi(r,t)=e^{-\frac{i}{\hbar}S(r,t)}$ (3)

and substituting the expansion for $S(r,t)$

$\displaystyle S(r,t)=S_{0}(r,t)+\sum_{i=1}^{\infty}\hbar^{i}S_{i}(r,t)$ (4)

in (2) we obtain, in the semiclassical limit (i.e. $\hbar\rightarrow 0$),

$\displaystyle\partial_{t}S_{0}(r,t)=\pm F(r)\partial_{r}S_{0}(r,t)$ (5)

This is the usual semiclassical Hamilton-Jacobi equation [4, 9] which can also
be obtained in a similar way from Dirac [10] or Maxwell equations [11]. Also,
this equation is a natural consequence if the chirality (holomorphic)
condition on the scalar field with the WKB ansatz (3) is imposed with the
$+(-)$ solutions standing for the left (right) movers [14].

Now since the metric (1) is stationary, it has a timelike Killing vector.
Therefore we choose an ansatz for $S_{0}(r,t)$ as

$\displaystyle S_{0}(r,t)=\omega t+{\tilde{S}}_{0}(r)$ (6)

where $\omega$ is the conserved quantity corresponding to the timelike Killing
vector. This is identified as the effective energy experienced by the particle
at asymptotic infinity. Substituting this in (5) a solution for
${\tilde{S}}_{0}(r)$ is obtained. Inserting this back in (6) yields,

$\displaystyle S_{0}(r,t)=\omega(t\pm
r_{*});\,\,\,\,r_{*}=\int\frac{dr}{F(r)}$ (7)

For further discussions it is convenient to introduce the sets of null
tortoise coordinates which are defined as,

$\displaystyle u=t-r_{*},\,\,\,v=t+r_{*}.$ (8)

It is important to note that expressing (7) in these coordinates, defined
inside and outside the event horizon, and then substituting in (3) one can
obtain the right and left modes for both sectors:

$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega
u_{\textrm{in}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega
v_{\textrm{in}}}$
$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega
u_{\textrm{out}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega
v_{\textrm{out}}}$ (9)

Now in the tunneling formalism a virtual pair of particles is produced in the
black hole. One member of this pair can quantum mechanically tunnel through
the horizon. This particle is observed at infinity while the other goes
towards the center of the black hole. While crossing the horizon the nature of
the coordinates changes. This can be accounted by working with Kruskal
coordinates which are viable on both sides of the horizon. The Kruskal time
($T$) and space ($X$) coordinates inside and outside the horizon are defined
as [13],

$\displaystyle
T_{\textrm{in}}=e^{K(r_{*})_{\textrm{in}}}~{}{\textrm{cosh}}(Kt_{\textrm{in}});\,\,\,X_{\textrm{in}}=e^{K(r_{*})_{\textrm{in}}}~{}{\textrm{sinh}}(Kt_{\textrm{in}})$
$\displaystyle
T_{\textrm{out}}=e^{K(r_{*})_{\textrm{out}}}~{}{\textrm{sinh}}(Kt_{\textrm{out}});\,\,\,X_{\textrm{out}}=e^{K(r_{*})_{\textrm{out}}}~{}{\textrm{cosh}}(Kt_{\textrm{out}})$
(10)

where, as usual, $K=\frac{F^{\prime}(r_{H})}{2}$ is the surface gravity of the
black hole. These two sets of coordinates are connected by the relations,

$\displaystyle t_{\textrm{in}}\rightarrow
t_{\textrm{out}}-i\frac{\pi}{2K};\,\,\,\,(r_{*})_{\textrm{in}}\rightarrow(r_{*})_{\textrm{out}}+i\frac{\pi}{2K}$
(11)

so that, with this mapping, $T_{\textrm{in}}\rightarrow T_{\textrm{out}}$ and
$X_{\textrm{in}}\rightarrow X_{\textrm{out}}$. In particular, for the
Schwarzschild metric, $K=\frac{1}{4M}$ so that the extra term connecting
$t_{\textrm{in}}$ and $t_{\textrm{out}}$ is given by ($-2\pi iM$). Such a
result (for the Schwarzschild case) was earlier discussed in [12]. Now,
following the definition (8), we obtain the relations connecting the null
coordinates defined inside and outside the horizon,

$\displaystyle
u_{\textrm{in}}=t_{\textrm{in}}-(r_{*})_{\textrm{in}}\rightarrow
u_{\textrm{out}}-i\frac{\pi}{K}$ $\displaystyle
v_{\textrm{in}}=t_{\textrm{in}}+(r_{*})_{\textrm{in}}\rightarrow
v_{\textrm{out}}$ (12)

Under these transformations the inside and outside modes are connected by,

$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}\rightarrow
e^{-\frac{\pi\omega}{\hbar K}}\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}$
$\displaystyle\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}\rightarrow\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}$
(13)

Using the above transformations the density matrix operator for an observer
outside the event horizon will be constructed in the next section which will
lead to the black body spectrum and thermal flux corresponding to the
semiclassical Hawking temperature.

Black body spectrum and Hawking flux: Now to find the black body spectrum and
Hawking flux, we first consider $n$ number of non-interacting virtual pairs
that are created inside the black hole. Each of these pairs is represented by
the modes defined in the first set of (9). Then the physical state of the
system, observed from outside, is given by,

$\displaystyle|\Psi>=N\sum_{n}|n^{(L)}_{\textrm{in}}>\otimes|n^{(R)}_{\textrm{in}}>\rightarrow
N\sum_{n}e^{-\frac{\pi n\omega}{\hbar
K}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}>$ (14)

where use has been made of the transformations (13). Here
$|n^{(L)}_{\textrm{out}}>$ corresponds to $n$ number of left going modes and
so on while $N$ is a normalization constant which can be determined by using
the normalization condition $<\Psi|\Psi>=1$. This immediately yields,

$\displaystyle N=\frac{1}{\Big{(}\displaystyle\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar K}}\Big{)}^{\frac{1}{2}}}$ (15)

The above sum will be calculated for both bosons and fermions. For bosons
$n=0,1,2,3,....$ whereas for fermions $n=0,1$. With these values of $n$ we
obtain the normalization constant (15) as

$\displaystyle N_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}^{\frac{1}{2}}$ (16) $\displaystyle
N_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}^{-\frac{1}{2}}$ (17)

Therefore the normalized physical states of the system for bosons and fermions
are respectively,

$\displaystyle|\Psi>_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}^{\frac{1}{2}}\sum_{n}e^{-\frac{\pi n\omega}{\hbar
K}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}>$ (18)
$\displaystyle|\Psi>_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}^{-\frac{1}{2}}\sum_{n}e^{-\frac{\pi n\omega}{\hbar
K}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}>$ (19)

From here on our analysis will be only for bosons since for fermions the
analysis is identical. For bosons the density matrix operator of the system is
given by,

$\displaystyle{\hat{\rho}}_{(\textrm{boson})}$ $\displaystyle=$
$\displaystyle|\Psi>_{(\textrm{boson})}<\Psi|_{(\textrm{boson})}$ (20)
$\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}\sum_{n,m}e^{-\frac{\pi n\omega}{\hbar K}}e^{-\frac{\pi
m\omega}{\hbar
K}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}><m^{(R)}_{\textrm{out}}|\otimes<m^{(L)}_{\textrm{out}}|$

Now tracing out the ingoing (left) modes we obtain the density matrix for the
outgoing modes,

$\displaystyle{\hat{\rho}}^{(R)}_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}\sum_{n}e^{-\frac{2\pi n\omega}{\hbar
K}}|n^{(R)}_{\textrm{out}}><n^{(R)}_{\textrm{out}}|$ (21)

Therefore the average number of particles detected at asymptotic infinity is
given by,

$\displaystyle<n>_{(\textrm{boson})}={\textrm{trace}}({\hat{n}}{\hat{\rho}}^{(R)}_{(\textrm{boson})})$
$\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar
K}}\Big{)}\sum_{n}ne^{-\frac{2\pi n\omega}{\hbar K}}$ (22) $\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}(-\frac{\hbar
K}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar K}}\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}(-\frac{\hbar
K}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\frac{1}{1-e^{-\frac{2\pi\omega}{\hbar
K}}}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\frac{2\pi\omega}{\hbar K}}-1}$

where the trace is taken over all $|n^{(R)}_{\textrm{out}}>$ eigenstates. This
is the Bose distribution. Similar analysis for fermions leads to the Fermi
distribution:

$\displaystyle<n>_{(\textrm{fermion})}=\frac{1}{e^{\frac{2\pi\omega}{\hbar
K}}+1}$ (23)

Note that both these distributions correspond to a black body spectrum with a
temperature given by the Hawking expression,

$\displaystyle T_{H}=\frac{\hbar K}{2\pi}$ (24)

Correspondingly, the Hawking flux can be obtained by integrating the above
distribution functions over all $\omega$’s. For fermions it is given by,

$\displaystyle{\textrm{Flux}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\omega~{}d\omega}{e^{\frac{2\pi\omega}{\hbar
K}}+1}=\frac{\hbar^{2}K^{2}}{48\pi}$ (25)

Similarly, the Hawking flux for bosons can be calculated, leading to the same
answer.

Connection with usual approaches: For completeness and for revealing the
connection with usual approaches [4, 5, 8] to the tunneling formalism we will
show below how one can find only the Hawking temperature using the principle
of detailed balance.

Since the left moving mode travels towards the center of the black hole, its
probability to go inside, as measured by an external observer, is expected to
be unity. This is easily seen by computing,

$\displaystyle
P^{(L)}=|\phi^{(L)}_{\textrm{in}}|^{2}\rightarrow|\phi^{(L)}_{\textrm{out}}|^{2}=1$
(26)

where we have used (13) to recast $\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}$ in
terms of $\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}$ since measurements are done
by an outside observer. This shows that the left moving (ingoing) mode is
trapped inside the black hole, as expected.

On the other hand the right moving mode ($\phi^{(R)}_{\textrm{in}}$) tunnels
through the event horizon. So to calculate the tunneling probability as seen
by an external observer one has to use the transformation (13) to recast
$\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}$ in terms of
$\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}$. Then we find,

$\displaystyle
P^{(R)}=|\phi^{(R)}_{\textrm{in}}|^{2}\rightarrow|e^{-\frac{\pi\omega}{\hbar
K}}\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}|^{2}=e^{-\frac{2\pi\omega}{\hbar
K}}$ (27)

Finally, using the principle of “detailed balance” [4, 9],
$P^{(R)}=e^{-\frac{\omega}{T_{H}}}P^{(L)}=e^{-\frac{\omega}{T_{H}}}$ and
comparison with (27) immediately reproduces the Hawking temperature (24).

Conclusions: To conclude, we have provided a novel formulation of the
tunneling formalism to highlight the role of coordinate systems. A particular
feature of this reformulation is that explicit treatment of the singularity in
(7) is not required since we do not carry out the complex path integration. Of
course, the singularity at the event horizon is manifested in the
transformations (11). In this way our formalism, contrary to the traditional
approaches [4, 5, 8], avoids explicit complex path analysis. It is implicit
only in the definition (7). Computations were done in terms of the basic
modes. From the density matrix constructed from these modes we were able to
directly reproduce the black body spectrum, for either bosons or fermions,
from a black hole with a temperature corresponding to the standard Hawking
expression. We feel that the lack of such an analysis was a gap in the
existing tunneling formulations [4, 5, 8, 9, 10, 11, 12, 14] which yield only
the temperature rather that the actual black body spectrum. Finally, the
connection of our approach with these existing formulations was revealed
through the use of the detailed balance principle.

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# Thermodynamic phase transition of a Schwarzschild black hole
with global monopole under GUP

Yang Chen  College of Physics Science and Technology, Shenyang Normal
University, Shenyang 110034, China  Hui-Ling Li<EMAIL_ADDRESS>College of
Physics Science and Technology, Shenyang Normal University, Shenyang 110034,
China

###### Abstract

Abstract:

In this paper, considering the influence of the principle of generalized
uncertainty (GUP), the phase transition of a Schwarzschild black hole with
global monopole is discussed. First, we use the generalized Dirac equation to
obtain corrected Hawking temperature, local temperature, black hole residue,
black hole entropy, thermal capacity, and other thermodynamic quantities.
Then, we use images to analyze the effects of generalized uncertainty
parameters on phase transitions and the effects of magnetic monopole
parameters on phase transitions. Finally, we study the thermodynamic stability
and phase change structure under the influence of the principle of generalized
uncertainty. The results show that, in the black hole with global monopole,
there are first-order and second-order phase transitions. In addition, the
general uncertainty parameters and monopole parameters will affect the black
hole residues.

Keywords: global monopole, thermodynamic phase transition, GUP

PACS: 04.70.-s, 04.70.Dy, 97.60.Lf

##  I Introduction

In general relativity, black holes are the most surprising objects. In
classical (non-quantum) physics, any matter (including light) entering a black
hole cannot escape. With the continuous exploration of black holes, scientists
have become extremely interested in the thermodynamics of black holes. In
1972, Beckenstein proposed the “hairless theorem” for black holes. He believed
that after the star collapsed into a black hole, only the three basic
conserved quantities of mass, angular momentum and electric charge continued
to play a role. Some other factors (“hair”) disappeared after entering the
black hole. This theorem was later rigorously proved by Hawking and others 1 ;
2 . Bekenstein also conceived that the second law of thermodynamics should be
universally valid 3 ; 4 , and he believed that black holes should have the
entropy proportional to their surface area.

Black holes have rich critical phenomena and phase structures. In 1983,
Hawking et al. 5  found through calculation that in the Schwarzschild–anti de
Sitter space, there was a phase transition between the stable large black hole
phase and the hot gas phase, which is called Hawking–Page phase transition.
Through this discovery, physicists have a thorough understanding of the
relationship between black hole physics and traditional thermodynamic systems,
and opened the door to the study of black hole phase transitions. The location
of the Hawking–Page transition between the charged black hole and the pure
anti-de Sitter space–time was clearly discuss in 1999 6 . It is studied
whether the Hawking–Page transition can occur in three dimensions in 2005 7 .
The thermodynamic phase transitions of high–dimensional single–spin asymptotic
AdS black holes in a canonical (fixed J) set of extended phase space are
studied in 2013 8 . In addition, the criticality index of charged AdS black
hole is also calculated, and the results show that they are consistent with
the criticality index of the van der Waals system 9 .

In the vacuum phase transition of the early universe, the breakdown of local
or global scale symmetry produced topological defects such as domain walls,
cosmic strings, and monopoles 10 ; 11 . It was proved in 12  that during the
phase transition of the universe, the global monopole charge spontaneously
breaks into U (1) with the global O (3) symmetry. In 1994, H. W. Yu et al.
studied the thermodynamics of a static spherically symmetric black hole–global
unipolar system through two methods, surface gravity and Euclidean path
integration 13 . In 14 , Dadhich et al. studied the effect of global monopolar
charge on particle orbits and Hawking radiation, and corrected the global
monopole black hole Hawking radiation in 15 . In addition, Kimet and Gordana
proposed the quantum tunneling effect of the Dirac magnetic monopole in a
black hole with an integral magnetic monopole 16 . In 2016, researchers
studied the mass fraction of the unipolar parameter and found that it can
suppress the maximum accretion rate 17 .

On the other side, the generalized uncertainty principle has been applied to
different studies. The form of GUP is

$\displaystyle\Delta x\Delta p\geqslant\frac{\hbar}{2}[1+\beta{(\Delta
p)^{2}}],$ (1)

where $\Delta x$ and $\Delta p$ represent the uncertainty of position and
momentum, respectively. GUP parameter
$\beta={\beta_{0}}l_{{}_{p}}^{2}/{\hbar^{2}}={\beta_{0}}/{M_{p}}{c^{2}}$,
$l_{p}$ is the Planck length ($\approx{10^{-35}}$), $M_{p}$ is Planck mass and
$\beta_{0}$ is a dimensionless parameter. GUP is a low–energy approximation
theory. There are three general types of GUP. One of them is GUP with minimum
position 18  and the other two are GUP with maximum momentum and minimum
position 19 ; 20 , and GUP with minimum position, minimum momentum, and
maximum momentum 21 . Adler et al. 22  found that after considering the
corrected of GUP, the black holes will not evaporate completely, leaving
residual mass. In black hole physics, many meaningful conclusions are drawn by
applying different forms of GUP 23 ; 24 ; 25 . Based on the GUP, we will
explore the phase transition of a Schwarzschild black hole with a global
monopole.

The structure of this paper is as follows: In Section 2, we review the
modified Dirac equation and use the Dirac equation to derive the relationship
between local temperature and mass in a Schwarzschild black hole with the
global monopole. The effects of GUP on phase transitions are discussed in
Section 3. We finally concluded.

##  II Quantum tunneling and local temperature based on GUP

In this section, firstly, we consider the effects of quantum gravity and
discuss the tunneling behavior of fermions through the Schwarzschild black
hole with the global monopole event horizon. In curved space-time, the
Generalized Dirac equation can be expressed as 26

$\displaystyle[i{\gamma^{0}}{\partial_{0}}+i{\gamma^{i}}{\partial_{i}}(1-\beta{m^{2}})+i{\gamma^{i}}\beta{\hbar^{2}}({\partial_{j}}{\partial^{j}}){\partial_{i}}+\frac{m}{\hbar}(1+\beta{\hbar^{2}}{\partial_{j}}{\partial^{j}}-\beta{m^{2}})$
$\displaystyle+i{\gamma^{\mu}}{\Omega_{\mu}}(1+\beta{\hbar^{2}}{\partial_{j}}{\partial^{j}}-\beta{m^{2}})]\psi=0.$
(2)

The curved space-time metric of Schwarzschild black hole with the global
monopole is

$\displaystyle
d{S^{2}}=-{\rm{f}}\left(r\right)d{t^{2}}+\frac{1}{{g\left(r\right)}}d{r^{2}}+a{r^{2}}\left({d{\theta^{2}}+{{\sin}^{2}}\theta
d{\phi^{2}}}\right),$ (3)

where $a=1-8\pi{\eta^{2}}$ and

$\displaystyle f\left(r\right)=g\left(r\right)=1-\frac{{2GM}}{r}.$ (4)

The mass of black hole is expressed by $M$. We have set $G=c=1$ . This metric
is not asymptotic flat space time due to the existence of global magnetic
monopoles. When $f\left({{r_{+}}}\right){\rm{=0}}$, event horizon
${r_{+}}=2GM$ can be obtained. The fermion s movement follows the generalized
Dirac equation (2). Therefore, in order to describe the motion, we assume that
the wave function state is

$\displaystyle\psi=\left(\begin{array}[]{l}A\\\ 0\\\ B\\\ 0\\\
\end{array}\right)\exp\left({\frac{i}{\hbar}I\left({t,r,\theta,\phi}\right)}\right),$
(9)

where $I$ is the action of the emitted fermions, $A$ and $B$ are functions of
coordinates, that is
$A=A\left({t,r,\theta,\phi}\right),B=B\left({t,r,\theta,\phi}\right)$. Then
the ${\gamma^{\mu}}$ matrices are given by

$\displaystyle{\gamma^{t}}=\frac{1}{{\sqrt{f\left(r\right)}}}\left({\begin{array}[]{*{20}{c}}i&0\\\
0&{-i}\\\
\end{array}}\right),{\gamma^{\theta}}=\sqrt{{g^{\theta\theta}}}\left({\begin{array}[]{*{20}{c}}0&{{\sigma^{1}}}\\\
{{\sigma^{1}}}&0\\\ \end{array}}\right),$ (14)
$\displaystyle{\gamma^{r}}=\sqrt{g\left(r\right)}\left({\begin{array}[]{*{20}{c}}0&{{\sigma^{3}}}\\\
{{\sigma^{3}}}&0\\\
\end{array}}\right),{\gamma^{\phi}}=\sqrt{{g^{\phi\phi}}}\left({\begin{array}[]{*{20}{c}}0&{{\sigma^{2}}}\\\
{{\sigma^{2}}}&0\\\ \end{array}}\right),$ (19)

where
$\sqrt{{g^{\theta\theta}}}=\frac{1}{{\sqrt{a}r}},\sqrt{{g^{\phi\phi}}}=\frac{1}{{\sqrt{a}r\sin\theta}}$.
${\sigma^{\rm{i}}}$ s are the Pauli matrices with $i=1,2,3$. Substituting the
wave function and gamma matrices into the generalized Dirac equation (2),
ignoring the contributions from $\partial A$, $\partial B$ and ignoring high-
order terms of $\hbar$ , we finally receive the following equations of motion

$\displaystyle-
iA\frac{1}{{\sqrt{f}}}{\partial_{t}}I-B\left({1-\beta{m^{2}}}\right)\sqrt{g}{\partial_{r}}I-Am\beta[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]$
$\displaystyle+B\beta\sqrt{g}{\partial_{r}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]+Am\left({1-\beta{m^{2}}}\right)=0,$
(20) $\displaystyle
iB\frac{1}{{\sqrt{f}}}{\partial_{t}}I-A\left({1-\beta{m^{2}}}\right)\sqrt{g}{\partial_{r}}I-Bm\beta[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]$
$\displaystyle+A\beta\sqrt{g}{\partial_{r}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]+Bm\left({1-\beta{m^{2}}}\right)=0,$
(21) $\displaystyle
A\left({1-\beta{m^{2}}}\right)\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}I-A\beta\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]$
$\displaystyle+iA\left({1-\beta{m^{2}}}\right)\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}I-iA\beta\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]=0,$
(22) $\displaystyle
B\left({1-\beta{m^{2}}}\right)\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}I-B\beta\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]$
$\displaystyle+iB\left({1-\beta{m^{2}}}\right)\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}I-iB\beta\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}I[{g^{rr}}{({\partial_{r}}I)^{2}}+{g^{\theta\theta}}{({\partial_{\theta}}I)^{2}}+{g^{\phi\phi}}{({\partial_{\phi}}I)^{2}}]=0.$
(23)

It is difficult to solve the action from the above equations directly.
Following the standard process, we carry out the separation of variables as
follows

$\displaystyle
I=-({\rm{1-8}}\pi{\eta^{\rm{2}}})\omega{\rm{t+W}}\left(r\right)+\Theta\left({\theta,\phi}\right),$
(24)

where $\omega$ is the energy of emitted fermions. We substitute eqn. (11) into
eqn.s (7)-(10) and observe the last two equations. After dividing by $A$ and
$B$ respectively, they are found to be same, so we can find

$\displaystyle\left({\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}\Theta+i\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}\Theta}\right)\left[{\beta{g^{rr}}{{\left({{\partial_{r}}W}\right)}^{2}}+\beta{g^{\theta\theta}}{{\left({{\partial_{\theta}}\Theta}\right)}^{2}}+\beta{g^{\phi\phi}}{{\left({{\partial_{\phi}}\Theta}\right)}^{2}}-\left({1-\beta{m^{2}}}\right)}\right]=0.$
(25)

In the above equation, $\beta$ is a small quantity and represents the effects
from quantum gravity. The value in the square bracket can t vanish. Therefore
eqn. (12) is simplified as follow

$\displaystyle\sqrt{{g^{\theta\theta}}}{\partial_{\theta}}\Theta+i\sqrt{{g^{\phi\phi}}}{\partial_{\phi}}\Theta=0.$
(26)

In the previous study, $\Theta$ has a complex function. However, it has no
effect on the tunneling rate. Then by canceling $A$ and $B$, eqn. (7) and eqn.
(8) are identical and written as

$\displaystyle{B_{6}}{\left({{\partial_{r}}W}\right)^{6}}+{B_{4}}{\left({{\partial_{r}}W}\right)^{4}}+{B_{2}}{\left({{\partial_{r}}W}\right)^{2}}+{B_{0}}=0,$
(27)

where

$\displaystyle{B_{6}}={\beta^{2}}{g^{3}}f;$
$\displaystyle{B_{4}}=\beta{g^{2}}f({m^{2}}\beta+2\beta Q-2);$
$\displaystyle{B_{2}}=gf[{(1-\beta{m^{2}})^{2}}+\beta(2{m^{2}}-2{m^{4}}\beta-2Q+\beta{Q^{2}})];$
(28) $\displaystyle{B_{0}}=-{m^{2}}{(1-\beta{m^{2}}-\beta
Q)^{2}}f-{\omega^{2}}{(1-8\pi{\eta^{2}})^{2}};$ $\displaystyle
Q={g^{\theta\theta}}{\left({{\partial_{\theta}}\Theta}\right)^{2}}+{g^{\phi\phi}}{\left({{\partial_{\phi}}\Theta}\right)^{2}}.$

Ignoring the higher order terms of $\beta$ and solving the above equations at
the event horizon, we get

$\displaystyle{W_{\pm}}$
$\displaystyle=\pm\int{dr\frac{1}{{\sqrt{fg}}}}\sqrt{{\omega^{2}}-{m^{2}}f+2\beta{m^{4}}f}\left[{1+\beta\left({{m^{2}}+\frac{{{\omega^{2}}}}{f}}\right)}\right]$
(29) $\displaystyle=\pm i\pi
2M\omega(1-8\pi{\eta^{2}})(1+\frac{1}{2}\beta(3{m^{2}}+4{(1-8\pi{\eta^{2}})^{2}}{\omega^{2}}))+\text{real
part},$

where ${+\mathord{\left/{\vphantom{+-}}\right.\kern-1.2pt}-}$ signs represent
the outgoing/ingoing solutions. The real part is independent of the tunneling
rate. Therefore, the tunneling rate is written as

$\displaystyle\Gamma$ $\displaystyle=\exp(-4{\mathop{\rm
Im}\nolimits}{W_{+}})$ (30) $\displaystyle=\exp\left\\{{-8\pi
M\omega(1-8\pi{\eta^{2}})\left\\{{1+\frac{1}{2}\beta\left[{3{m^{2}}+4{{(1-8\pi{\eta^{2}})}^{2}}{\omega^{2}}}\right]}\right\\}}\right\\},$

which shows Hawking temperature is written as

$\displaystyle T$
$\displaystyle={\left\\{{8M\pi(1-8\pi{\eta^{2}})\left\\{{1+\frac{1}{2}\beta\left[{3{m^{2}}+4{{(1-8\pi{\eta^{2}})}^{2}}{\omega^{2}}}\right]}\right\\}}\right\\}^{-1}}$
(31)
$\displaystyle\simeq{T_{H}}\left\\{{1-\frac{1}{2}\beta\left[{3{m^{2}}+4{{(1-8\pi{\eta^{2}})}^{2}}{\omega^{2}}}\right]}\right\\}.$

Then, we further discuss on the local temperature based on GUP. According to
27 ; 28 , in order to obtain a local thermodynamic quantity, the Schwarzschild
black hole with global monopole needs to be located at the center of the
spherical cavity of radius $r$. Where ${T_{H}}=\frac{1}{{4\pi{r_{+}}}}$ is
Black hole’s original Hawking temperature, where ${{\rm{r}}_{+}}=2GM$. It is
worth noting that the ADM quality of the magnetic monopole is
$(1-8\pi{\eta^{2}})M$ 29 . So, Hawking temperature can be written as

$\displaystyle{T_{H}}=\frac{1}{{8\pi(1-8\pi{\eta^{2}})GM}}.$ (32)

Local temperature in the Schwarzschild black hole can be written as follows

$\displaystyle{T_{Local}}=\frac{T}{{\sqrt{f(r)}}}=\frac{1}{{\sqrt{1-\frac{{2M}}{r}}}}\frac{1}{{8\pi
G(1-8\pi{\eta^{2}})M}}\left\\{{1-\frac{1}{2}\beta\left[{3{m^{2}}+4{{(1-8\pi{\eta^{2}})}^{2}}{\omega^{2}}}\right]}\right\\}.$
(33)

In Hawking radiation, in order to facilitate the calculation, we can get the
lower limit of the energy of the released particles from the saturated form of
the uncertainty principle,

$\displaystyle\omega^{\prime}\geq\frac{\hbar}{{\Delta x}}.$ (34)

It is noteworthy that $\omega^{\prime}$ in the magnetic monopole black hole is
$(1-8\pi{\eta^{2}})\omega$. In the vicinity of the black hole event horizon,
the uncertainty can be used in the radius of the black hole 30 . So we can be
obtain

$\displaystyle\Delta x\approx{r_{BH}}={r_{+}}.$ (35)

From equations (21) and (22), we have

$\displaystyle\omega^{\prime}=\frac{\hbar}{{{r_{+}}}}.$ (36)

So the local temperature can be written as

$\displaystyle{T_{Local}}=\frac{1}{{\sqrt{1-\frac{{2M}}{r}}}}\frac{1}{{8\pi
G(1-8\pi{\eta^{2}})M}}\left\\{{1-\frac{1}{2}\beta\left[{3{m^{2}}+{{\left({\frac{\hbar}{M}}\right)}^{2}}}\right]}\right\\}.$
(37)

Based on the above equation, we can to find out the critical value of phase
transition. When considering the cavity of the radius r as a constant amount,
and setting $8\pi{\eta^{2}}{\rm{=}}X$, the following three critical values can
be calculated by the following equations:

$\displaystyle{\left({\frac{{\partial{T_{Local}}}}{{\partial
M}}}\right)_{\eta,{\rm{r}}}}={\left({\frac{{{\partial^{2}}{T_{Local}}}}{{\partial{M^{2}}}}}\right)_{\eta,{\rm{r}}}}=0.$
(38)

Setting $r=10,\hbar=1$, and substituting equation (24) into equation (25),
critical mass, critical GUP parameters, and critical local temperature can be
obtained as follows
${\beta_{C{\rm{r}}}}=2.52068,{M_{Cr}}=2.59934,{T_{Cr}}=0.019964$. According to
Eq. (24), the relationship between local temperature versus mass as a function
can be shown as Figures 1-3.

Figure 1: Fixed $G=1$, $r=10$ and $X=0.1$, plot a graph of the relationship
between ${T_{L{\rm{ocal}}}}$ and $M$ for $\beta=1<{\beta_{C}}$ (top),
$\beta{\rm{=}}{\beta_{C}}$(intermediate), and $\beta=5>{\beta_{C}}$(bottom).

Obviously, if $\beta<{\beta_{C}}$, the image curve will fluctuate, which
indicates that there may be a phase change. As $\beta$ increases, the image
curve monotonically increases and the inflection point disappears, which
indicates that no phase change may occur.

Figure 2: Fixed $G=1$, $r=10$ and $\beta{\rm{=}}1$, plot a graph of the
relationship between ${T_{L{\rm{ocal}}}}$ and $M$. The curve is the image from
top to bottom for $X={\rm{}}0.{\rm{3}},0.{\rm{1}},0.05$.

The trend of the image is similar to that in Figure 1, so a phase change
occurs.

Figure 3: Fixed $G=1$, $r=10$ and $X=0.1$, plot a graph of the relationship
between ${T_{L{\rm{ocal}}}}$ and $M$. The curve is the image from top to
bottom for $\beta{\rm{=0}}$ and $\beta{\rm{=1}}$.

Form Figure 3, the relationship between the original local temperature (for
$\beta{\rm{=}}0$) and mass and the relationship between local temperature (for
$\beta{\rm{=}}1$) and mass can be obtained respectively. When
$\beta{\rm{=}}1$, the evaporation process of the Schwarzschild black hole with
global monopole is generally divided into three stages. In the first stage
(${{\rm{M}}_{\rm{2}}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{3}}}$), the local
temperature is reduced by the evaporation process. In the second stage
(${{\rm{M}}_{\rm{1}}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{2}}}$), the local
temperature decreases with the mass increases. In the final stage
(${{\rm{M}}_{0}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{1}}}$), because the negative
temperature violates the laws of thermodynamics, the mass of the black hole
cannot be less than ${{\rm{M}}_{0}}$. When the mass is close to
${{\rm{M}}_{0}}$, the local temperature becomes 0, which results in a residual
black hole, that is ${M_{0}}={M_{res}}=\sqrt{\frac{{{\hbar^{2}}\beta}}{2}}$.
Therefore, a large black hole (LBH) of
${{\rm{M}}_{\rm{2}}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{3}}}$, an intermediate
black hole (IBH) of ${{\rm{M}}_{\rm{1}}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{2}}}$,
and a small black hole (SBH) of
${{\rm{M}}_{0}}\leq{\rm{M}}\leq{{\rm{M}}_{\rm{1}}}$ can be defined.

##  III The effect of GUP on phase transition

According to the first law of thermodynamics, the local energy of the
Schwarzschild black hole with global monopole can be given by

$\displaystyle{\left({\frac{{\partial{T_{Local}}}}{{\partial
M}}}\right)_{\eta,{\rm{r}}}}={\left({\frac{{{\partial^{2}}{T_{Local}}}}{{\partial{M^{2}}}}}\right)_{\eta,{\rm{r}}}}=0.$
(39)

That is

$\displaystyle{E_{Local}}$
$\displaystyle=\mathop{\smallint}\limits_{(1-8\pi{\eta^{2}}){M_{0}}}^{(1-8\pi{\eta^{2}})M}\frac{{{T_{Local}}}}{{{T_{H}}}}d(1-8\pi{\eta^{2}})M$
(40)
$\displaystyle=\frac{{(1-8\pi{\eta^{2}})}}{{2(-1+8\pi{\eta^{2}})}}\sqrt{\frac{{2(-1+8\pi{\eta^{2}})M+r}}{r}}[r(2-3{m^{2}}\beta)+\frac{{\beta{\hbar^{2}}}}{{(-1+d)M}}]$
$\displaystyle-\frac{{(1-8\pi{\eta^{2}})}}{{2(-1+8\pi{\eta^{2}})}}\left[{r(2-3{m^{2}}\beta)+\frac{{\sqrt{2}\sqrt{\beta{\hbar^{2}}}}}{{-1+8\pi{\eta^{2}}}}}\right]\sqrt{\frac{{r+\sqrt{2}(-1+8\pi{\eta^{2}})\sqrt{\beta{\hbar^{2}}}}}{r}}$
$\displaystyle-\frac{{(1-8\pi{\eta^{2}})}}{{2(-1+8\pi{\eta^{2}})}}\frac{{2\beta{\hbar^{2}}ArcTanh[\sqrt{\frac{{2(-1+8\pi{\eta^{2}})M+r}}{r}}]}}{r}$
$\displaystyle+\frac{{(1-8\pi{\eta^{2}})}}{{2(-1+8\pi{\eta^{2}})}}\frac{{2\beta{\hbar^{2}}ArcTanh[\sqrt{\frac{{r+\sqrt{2}(-1+8\pi{\eta^{2}})\sqrt{\beta{\hbar^{2}}}}}{r}}]}}{r}.$

Meanwhile, we can obtain heat capacity

$\displaystyle C$
$\displaystyle=\frac{{\partial{E_{Local}}}}{{\partial{T_{Local}}}}=\frac{{\frac{{\partial{E_{Local}}}}{{\partial
M}}}}{{\frac{{\partial{T_{Local}}}}{{\partial M}}}}$ (41)
$\displaystyle=\frac{{8G{M^{2}}\pi\sqrt{1-\frac{{2M}}{r}}(-2M+r)({{(-1+8\pi{\eta^{2}})}^{2}}{M^{2}}(-2+3{m^{2}}\beta)+\beta{\hbar^{2}})}}{{\sqrt{\frac{{2(-1+8\pi{\eta^{2}})M+r}}{r}}({M^{2}}r(2-3{m^{2}}\beta)+{M^{3}}(-6+9{m^{2}}\beta)+7M\beta{\hbar^{2}}-3r\beta{\hbar^{2}})}}.$

The heat capacity associated with the quality of the Schwarzschild black hole
with global magnetic monopoles is shown in Figure 4.

Figure 4: Fixed $G=1$, $r=10$ and $X=0.1$, plot the relationship between $C$
and $M$. The figure is the $C-M$ image for $\beta{\rm{=0}}$ and
$\beta{\rm{=1}}$ respectively.

As shown in Fig. 4, the solid blue line is an image of the thermal capacity of
a black hole and the mass change of the black hole when the GUP parameter is
taken into account. It can be seen from the figure that there are two stable
regions: ${M_{0}}\leq M\leq{M_{1}}$ and ${M_{2}}\leq M\leq{M_{3}}$. While when
${M_{1}}\leq M\leq{M_{2}}$, the black hole is in an unstable state. Since the
thermal capacity is negative, the red dotted line represents an image of the
original heat capacity as a function of the mass of the black hole. When the
mass is zero, the thermal capacity also approaches 0. As the mass increases, a
divergence point appears at ${M_{2}}$, the thermal capacity on the left side
of the divergence point is negative, the black hole is unstable, the thermal
capacity on the right side of the divergence point is positive, and the black
hole is relatively stable. It can be clearly seen that the images diverge at
${M_{1}}$ and ${M_{2}}$ , proving the existence of a phase transition.
Connecting the Eq.(20), we get the entropy

$\displaystyle
S=\int{\frac{{d(1-8\pi{\eta^{2}})M}}{{{T_{H}}}}}=4G{M^{2}}\pi{(1-8\pi{\eta^{2}})^{2}}.$
(42)

In order to study the phase transition of the Schwarzschild black hole with
the global monopole, we get its free energy as follows 31 ; 32 ,

$\displaystyle F$ $\displaystyle={E_{Local}}-{T_{Local}}S$ (43)
$\displaystyle=\frac{{(1-8\pi{\eta^{2}})\left[{{M^{2}}(-2+3{m^{2}}\beta)+\beta{\hbar^{2}}}\right]}}{{4M\sqrt{1-\frac{{2M}}{r}}}}$
$\displaystyle-\frac{1}{2}\sqrt{\frac{{2(-1+8\pi{\eta^{2}})M+r}}{r}}\left[{r(2-3{m^{2}}\beta)+\frac{{\beta{\hbar^{2}}}}{{(-1+8\pi{\eta^{2}})M}}}\right]$
$\displaystyle+\frac{1}{2}\left[{r(2-3{m^{2}}\beta)+\frac{{\sqrt{2}\sqrt{\beta{\hbar^{2}}}}}{{-1+8\pi{\eta^{2}}}}}\right]\sqrt{\frac{{r+\sqrt{2}(-1+8\pi{\eta^{2}})\sqrt{\beta{\hbar^{2}}}}}{r}}$
$\displaystyle+\frac{{\beta{\hbar^{2}}ArcTanh[\sqrt{\frac{{2(-1+8\pi{\eta^{2}})M+r}}{r}}]}}{r}$
$\displaystyle-\frac{{\beta{\hbar^{2}}ArcTanh[\sqrt{\frac{{r+\sqrt{2}(-1+8\pi{\eta^{2}})\sqrt{\beta{\hbar^{2}}}}}{r}}]}}{r}.$

Figure 5: Plot the relationship between local temperature ${T_{Local}}$ and
free energy $F$. The curve is the image from top to bottom for
$\beta=4>{\beta_{C}}$, $\beta={\beta_{C}}$, $\beta=0.5<{\beta_{C}}$. The
parameters are taken as $r=10$, $\hbar=1$, $X={10^{-5}}$.

From Figure 5, we show free energy $F$ curves with local temperature
${T_{Local}}$ under different generalized uncertainty parament $\beta$. It can
be clearly seen that a dovetail structure appears, when the GUP parameter
$\beta$ at this time is smaller than the critical value ${\beta_{Cr}}$, which
means two phases coexist. That is the first order phase transition occurs.
When the GUP parameter $\beta$ is equal to the critical value ${\beta_{Cr}}$,
an inflection point occurs. The inflection point is the second-order phase
transition point. When $\beta$ is more than ${\beta_{Cr}}$, the inflection
point disappears. The trend of the image corresponds to the trend shown in the
${\rm{T}}-{\rm{M}}$ image in Figure 1. To further explore the effect of
$\beta$ on phase transitions. A plot of local temperature ${T_{Local}}$ and
free energy F with $\beta=0$ is shown in Figure 6.

Figure 6: Plot the local temperature ${T_{Local}}$ versus the free energy F
for $\beta=0$. The curve corresponds to the phase transition. The parameters
are taken as $r=10$, $\hbar=1$, $X={10^{-5}}$.

Figure 6 shows the relationship between free energy and local temperature in
the original black hole. It can be found that the local temperature has a
minimum at ${{\rm{T}}_{1}}^{\prime}$ and there is no black hole solution below
it. When $\beta=0$ and $M=0$, black hole returns to monopole space-time.
Therefore, ${F^{GMS}}$ represents the free energy of global monopole space
time(GMS). For ${\rm{T}}<{{\rm{T}}_{C}}^{\prime}$, the free energy of unstable
black holes and stable large black holes is higher than ${F^{GMS}}$, which
means that GMS is more popular than unstable black holes and stable large
black holes. However, when ${\rm{T>}}{{\rm{T}}_{C}}^{\prime}$, the free energy
of the unstable black hole is higher than ${F^{GMS}}$, and the free energy of
the stable large black hole is lower than ${F^{GMS}}$. This indicates that the
stable large black hole is more popular than GMS. So at
${\rm{T>}}{{\rm{T}}_{C}}^{\prime}$, we can find that the phase transition
tunneling radiation collapses to a stable large black hole, and the unstable
small black hole eventually evolves into a large black hole.

Figure 7: Plot the relationship between local temperature ${T_{Local}}$ and
free energy $F$ for $\beta=0$. The curve is the image for
$\beta=0.5<{\beta_{C}}$. The parameters are taken as $r=10$, $\hbar=1$,
$X={10^{-5}}$.

As can be seen from Figure 7, F is free energy of curve space (CS). The SBH
transforms to IBH at ${T_{1}}$, and the IBH transforms to LBH at ${T_{2}}$.
The dovetail structure corresponds to a first-order phase transition. When
${T_{0}}<T<{T_{C}}$, ${F^{SBH}}<{F^{LBH}}<{F^{IBH}}$, which means that SBH is
more possible than LBH and IBH, therefore LBH and IBH will decay into SBH;
when ${T_{c}}<T<{T_{1}}$, ${F^{LBH}}<{F^{SBH}}<{F^{IBH}}$, which means that
LBH is more possible than SBH and IBH, hence SBH and IBH will decay into LBH.

##  IV Conclusion

Based on GUP, we studies the phase transition and stability of Schwarzschild
black holes with global monopoles. We calculated corrected thermodynamic
quantities such as tunneling temperature, local temperature, heat capacity,
and the entropy of the black hole. From the relationship between these
thermodynamic quantities, we analyzed the phase transition and thermodynamic
behavior of black holes. We find that GUP prevents black holes from completely
evaporating and produces a stable residual mass, that is
${M_{res}}=\sqrt{\frac{{{\hbar^{2}}\beta}}{2}}$. Secondly, when the GUP
parameter $\beta$ and $M$ are close to zero, the heat capacity of the black
hole disappears. When $\beta<{\beta_{C}}$, the heat capacity of a
Schwarzschild black hole with a monopole has two stable regions, and there is
a first-order phase transition, that is, two phases coexist. Finally,
according to the image of free energy vs. local temperature, there are two
phase transitions. One is a dovetail structure, which means that there is a
first-order phase transition in the Schwarzschild black hole with global
monopoles at $\beta<{\beta_{C}}$, and the other is a second-order phase
transition at $\beta={\beta_{C}}$. In addition, we also found that with
increasing temperature, stable SBH and unstable IBH will eventually decay into
stable LBH. It is worth noting that the effects of GUP parameter and monopole
parameter on the thermodynamic properties of Schwarzschild black holes are
discussed with GUP parameter as variant. In the following work, we will
discuss the thermodynamic behavior of a black hole with an integral magnetic
monopole using the magnetic monopole parameters as variant.

## Acknowledgements

This work is supported in part by the National Natural Science Foundation of
China (Grant No. 11703018), Natural Science Foundation of Liaoning Province,
China (Grant No. 20180550275) and Doctoral Scientific Research Foundation of
Shenyang Normal University (Grant No. BS201843).

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