# On the Thermodynamical Black Hole Stability in the Space-time of a Global
Monopole in $f(R)$-Gravity

F. B. Lustosa1<EMAIL_ADDRESS>M.E.X.Guimarães2<EMAIL_ADDRESS>Cristine N. Ferreira3<EMAIL_ADDRESS>J. L. Neto4<EMAIL_ADDRESS>J. A.
Helayel-Neto5<EMAIL_ADDRESS>1\. Instituto de Física, Departamento de Física
Teórica, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier,
524, 20550-013, Maracanã, Rio de Janeiro, RJ, Brazil  2\. Instituto de Física,
Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza, s/n -
Campus da Praia Vermelha 24210-346 Niterói, RJ, Brazil  3\. Núcleo de Estudos
em Física, Instituto Federal de Educação Ciência e Tecnologia Fluminense, Rua
Dr Siqueira, RJ, Brazil  4\. Instituto de Física, Universidade Federal do Rio
de Janeiro, Caixa Postal 68528 Rio de Janeiro 21941-972, RJ, Brazil  5\.
Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio
de Janeiro, Brazil

###### Abstract

In this work, we re-assess a class of black hole solutions in a global
monopole spacetime in the framework of an $f(R)$-gravity model. Our main line
of investigation consists in considering a region close enough to the black
hole, but such that the weak field approximation is still valid. The stability
of the black hole is studied in terms of its thermodynamical properties, with
the radial coordinate written as a power law function with the status of the
main factor underneath the stability of the model. We obtain the explicit
expressions for the thermodynamical quantities of the black hole as functions
of the event horizon, by considering both the Hawking and the local
temperatures. The phase transitions that may occur in this system, including
the Hawking-Page phase transition, are inspected with particular attention. We
work out and contemplate a solution of special interest in which one of the
parameters is related to the cosmological constant. Our main result sets out
to establish a comparison between both the Hawking and the local formalisms
for the black hole in the framework of the $f(R)$-gravity in the particular
space-time adopted here.

###### pacs:

?

##  I Introduction

There are still some important issues for which current cosmological standard
theories in Physics have not yet found complete answers. General Relativity
(GR) has not been able to clarify questions such as the non-renormalization of
the gravity theory, the singularity problems in black hole physics and the
physics of the early Universe, leading us to the necessity of finding
alternative approaches. In this context, one of the most intriguing facts is
the accelerating Universe. Despite of the existence of some alternative
approaches to explain this behavior, one of those without adding dark matter
or dark energy Buchdahl:1983zz ; Morais:2015ooa ; Takahashi:2015ifa , and the
other with $f(R)$ gravities Nojiri:2010wj ; Sotiriou:2008rp that has received
much attention as one of the strongest candidates to explain the current
accelerating universe Perlmutter:1998np . The $f(R)$ gravity is constructed by
replacing the Ricci scalar in the Einstein-Hilbert action by an arbitrary
function of the Ricci scalar $f(R)$. Such theory is well known to lead to an
extra scalar degree of freedom and it has been shown, for example in Hu:2007nk
; Appleby:2007vb ; Starobinsky:2007hu , that observationally acceptable models
may be built up. In some new contributions, f(R)-gravity can exhibit
dependence on higher powers of the curvature scalar Ohta:2015efa . Indeed, it
must be stressed that the recent starting of the gravitational wave astronomy
through the LIGO detections could be, in principle, an important tool to test
the validity of the f(R) theories of gravity, as stated in the ref. corda .

In 1972, Bekenstein published his first article demonstrating the relation
between thermodynamic quantities and gravitational properties of a black hole.
His works Bekenstein:1972tm -Bekenstein:1974ax were followed by the model of
particle creation around BH proposed by Hawking Hawking:1974sw . Since then,
the thermodynamical behavior of quasi-classical systems has been widely
explored in other gravitational frameworks. Most recent contributions were
made in the study of the thermodynamics of the black hole in modified
Schwarszchild Cai:2009ua , Born-Infeld-anti-de Sitter Myung:2008eb , Horava-
Lifshitz Biswas:2010zzb and $f(R)$ theory delaCruzDombriz:2009et .

It has been shown that topological defects such as cosmic strings, monopoles
and domain walls could be formed as a result of spontaneously broken symmetry
in a vacuum phase transition of the early universe Kibble:1976sj
-Vilenkin:1984ib . In the context of the GR, Barriola and Vilenkin
Barriola:1989hx studied the gravitational effects of a global monopole as a
spherically symmetric topological defect. In the context of $f(R)$-theories,
global monopoles heve been recently studied by Carames:2011uu -Carames:2011xi
, and references therein. In this solution, there appears a term which
corresponds to a black hole solution. It is natural, then, to imagine that
there is a region with a global monopole that could have been swallowed by a
black-hole. Another interesting situation where the monopole appears as an
important ingredient involves boson stars. When these compact objects are
present along with a non-minimally coupled global monopole, a black-hole-type
solution can be mimicked. This approach is important to unveil the non-linear
gravitational effects and the gravitational back reaction Marunovic:2013eka .

On the other hand, the analysis of the thermodynamical properties of that
case, in the context of GR, was carried out in Yu:1994fy and, more recently,
in the context of an $f(R)$-theory in Man:2013sf , where the authors adopt the
weak field approximation solution presented in Carames:2011uu with a specific
ansatz for $f^{\prime}(R)=F(R)=\frac{df(R)}{dR}$ as a linear function of the
radial coordinate.

In the present paper, we consider thermodynamical aspects of black holes in
the space-time of a global monopole in the framework of $f(R$)-gravity. We
study a general case in which $f(R)$ is a power law function of the radial
coordinates. We anticipate that we obtain the explicit expressions for the
local thermodynamical quantities of the black hole as a function of the event
horizon, the parameter describing the monopole and the measurable corrections
on the usual GR gravity due to the $f(R)$-extension. The paper is organized as
follows. In Section 2, motivated by the analysis of thermodynamic aspects, we
revise the solution for a black hole in the region containing a global
monopole with the use of a $f(R)$ theory in the weak field approximation
Carames:2011xi . We include as the original result for this section the case
where $\psi_{n}<0$. In Section 3, new features were introduced. Firstly, the
stability of the black hole was analyzed considering thermodynamic aspects by
the analysis of the regions of temperature positivity and heat capacity of the
system, after this analysis, a comparison was made between Hawking and local
formalisms. We also include in this version the stability analysis by
comparing the heat capacity and the temperature of the black hole. We also
inspect, in this Section, the case with $\psi>0$, where use is made of the
local prescription, also valid for the case in which $\psi<0$. In Section 4,
we consider the thermodynamical properties of the black hole in an $f(R)$
global monopole space-time. Finally, in the Section 5, we cast our Closing
Remarks, discuss the stability of our solution and analyze the particular case
of $n=2$ in more details; in this special case, the parameter $\psi_{2}$ can
be related with a positive cosmological constant, that, in the presence of the
monopole presence exhibits a deficit solid angle.

##  II Field Equations Solution for the $f(R)$ Gravity in the Metric
Formalism: A Review

In this Section, for the sake of understanding, we make a small review of the
solution of the field equations for an $f$(R)-theory with a spherically
symmetric space-time obtained in Carames:2011xi . In this reference it is
shown that it is possible to find a black hole solution in a global monopole
region by using an $f$(R) modification of the GR gravity within the weak field
approximation. For an $f$(R) theory, the action associated to the matter field
coupled with gravity is given by:

$S=\frac{1}{2\kappa}\int{d^{4}x\sqrt{-g}f(R)}+S_{m}(g_{\mu\nu},\psi),$ (1)

where $\kappa=8\pi G$, $G$ is the Newton constant, $g$ is the determinant of
the metric $g_{\mu\nu}$ with $\mu,\nu=0,1,2,3$, $R$ is the curvature scalar,
$S_{m}$ is the action associated with the matter fields and $f(R)$ is an
analytic function of the Ricci scalar. In this model, the Ricci scalar in
Einstein -Hilbert action is replaced by $f(R)$. The monopole which introduces
an angular deficit in the space-time metric, gives us some interesting effects
that we shall discuss in the sequel. Here, we assume that the Christoffel
symbol is a function of the metric, its derivatives and its inverse. The
general form of the time-independent metric with spherical symmetry in (1+3)
dimensions is given by

$ds^{2}=B(r)dt^{2}-A(r)dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$ (2)

The $g_{\mu\nu}$ field equations read as below::

$R_{\mu\nu}F(R)-{1\over
2}f(R)g_{\mu\nu}-(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)F(R)=\kappa
T_{\mu\nu}$ (3)

with $F(R)={df(R)\over dR}$, and $\Box$ is the usual notation for covariant D’
Alembert operator $\Box\equiv\nabla_{\mu}\nabla^{\mu}$. The only term
associated to the matter action is related to the global monopole described by
the Lagrangian density

${\cal
L}=\frac{1}{2}(\partial_{\mu}\phi^{a})(\partial^{\mu}\phi^{a})-\frac{1}{4}\lambda(\phi^{a}\phi^{a}-\eta^{2})^{2},$
(4)

where $\lambda$ and $\eta$ are the monopole field parameters and the triplet
field that will result in a monopole configuration that can be described by
$\phi^{a}=\eta\frac{h(r)}{r^{2}}x^{a}$, with $a=1,2,3$ and $x^{a}x^{a}=r^{2}$.
The function $h$(r) is dimentionless and is constrained by the conditions
$h(0)=0$ and $h(r>\eta)\approx 1$ Barriola:1989hx . The energy-momentum tensor
associated with that field configuration is

$\displaystyle T_{t}^{t}$ $\displaystyle=$ $\displaystyle
T^{r}_{r}\approx\frac{\eta^{2}}{r^{2}}$ $\displaystyle T^{\theta}_{\theta}$
$\displaystyle=$ $\displaystyle T^{\phi}_{\phi}=0$ (5)

We can now focus on the gravitational effects of the global monopole in the
$f(R)$ approach to gravity, by taking the trace of the field equations (3) in
the presence of the monopoleCarames:2011uu ; Multamaki:2006zb

$f(R)={1\over 2}RF(R)+{3\over 2}\Box F(R)-{\kappa\over 2}T$ (6)

where $F(R)={df(R)\over dR}$. By replacing (6) in (3), we have

$F(R)R_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F(R)-T_{\mu\nu}=g_{\mu\nu}\,C$ (7)

where the quantity $C={1\over 4}(F(R)\,R-\Box F(R)-\kappa T)$ is a scalar
quantity.

In a spherically symmetric space-time, with the metric given by (2) and with
the energy-momentum tensor expressed by (5), we find, by virtue of (7), that
the equations may be written as

$\displaystyle 2rF^{\prime\prime}-\beta(rF^{\prime}+2F)=0,$ (8) $\displaystyle
4B(A\\!-\\!1)\\!\\!+\\!r\Big{(}\\!2\\!{F^{\prime}\over
F}\\!-\\!\beta\\!\Big{)}\\!(rB^{\prime}\\!-\\!2B)\\!+\\!2r^{2}B^{\prime\prime}\\!-\\!{4\alpha
AB\over F}=0$ (9)

where $\alpha$ and $\beta$ are:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
2\kappa\eta^{2},\,\,\mbox{with}\,\,\kappa=8\pi G$ (10) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle{(AB)^{\prime}\over AB}\,\,.$ (11)

We make the assumption of time-independent solutions, i.e. $B=B(r)$ and
$A=A(r)$, which yields the metric as given in (2). With these assumptions, we
consider $F(R)=F(r)$ and $F^{\prime}$ and $F^{\prime\prime}$ are the first and
the second derivatives with respect to $r$, respectively. The solution to
these equations is the exact description of the global monopole in $f(R)$
theories. This prescription has however an analytical solution only in the
weak field approximation. In this approximation, the field equations read as
follows:

$\displaystyle F(R)=F(r)=1+\psi(r)$ $\displaystyle
B(r)=1+b(r),\,\,\,A(r)=1+a(r)$ (12)

with $|b(r)|$, $|a(r)|$ and $|\psi(r)|$ much smaller than one. These
redefinitions are used in equations (8) and (9), and with the help of

$\displaystyle{F^{\prime}\over F}\sim\psi^{\prime},$
$\displaystyle{F^{\prime\prime}\over F}\sim\psi^{\prime\prime}$ (13)
$\displaystyle{A^{\prime}\over A}\sim a^{\prime},$
$\displaystyle{B^{\prime}\over B}\sim b^{\prime}$ (14)

the field equations (8) and (9) can then be written in a linear form as:

$\beta\sim r\psi^{\prime\prime}(r),$ (15)
$2a(r)-2r\psi^{\prime}(r)+r\beta+r^{2}b^{\prime\prime}(r)-2\kappa\eta^{2}=0,$
(16)

where $\beta$ has been defined in (11). This is the solution to our equations
in the weak field approximation for a metric with spherical symmetry in an
$f$(R)-theory with $\frac{df(R)}{dR}=1+\psi(r)$ where R is the function of the
radial coordinate, r. Following in Carames:2011xi , we assume that the
function $\psi(r)$ is a power law-like function of the radial coordinate,
namely $\psi(r)=\psi_{(n)}r^{n}$, where $\psi_{n}$ is a constant parameter in
$r$. Notice, however, that in our work it shows a dependence on $n$. With this
ansatz, we can find solutions for the equations above. Noting that equation
(11) with (14) can be rewritten as:

$a^{\prime}(r)+b^{\prime}(r)=n(n-1)\psi_{n}r^{(n-1)}$ (17)

we can solve (16) for $b(r)$, obtaining:

$b(r)=\frac{c_{1}}{r}-\psi_{n}r^{n}-\kappa\eta^{2}.$ (18)

We define the integration constants $c_{1}=-2GM$ and, for convenience, we take
$c_{2}=0$. This term corresponds to the cosmological constant term. Equations
(11) and (15) yield the relation:

$A(r)B(r)=a_{0}e^{(n-1)\psi_{n}r^{n}}.$ (19)

Defining the integration constant $a_{0}=1$, we then have the full form of the
metric as

$ds^{2}=B(r)dt^{2}-e^{(n-1)\psi_{n}r^{n}}B(r)^{-1}dr^{2}-r^{2}d\Omega^{2},$
(20)

$B(r)=1-\kappa\eta^{2}-\frac{2GM}{r}-\psi_{n}r^{n}$ (21)

In Figure 1, we plot the function $B(r,n)$ for $n$ varying from 1 to 4. It can
be seen, in the case of the figure on the left, with $\psi_{n}>0$, that the
function $B(r,n)$ grows to a value and then decreases for all orders of n. In
the case $\psi_{n}<0$, the function transits to ever increasing values.

Figure 1: Graphs for B(r, n) in function of position for each n- orders going
from 1 to 4, where $GM=1$ and $\kappa\eta^{2}=0.3\cdot 10^{-5}$. In the figure
a) $\psi_{n}=0.2\cdot 10^{-2n}$ and b) $\psi_{n}=-0.2\cdot 10^{-2n}$

Considering the mass term as cosmologically relevant is equivalent to adopting
the scenario with a black hole in the space-time of a global monopole
Barriola:1989hx . To confirm that our solution is actually a black-hole-type
solution for all degrees $n$ of the $\psi(r)$-function, we have to find if all
of them have an event horizon. To do this, we have to keep in mind the
approximations that we have used to solve the field equations, namely
$h\approx 1$, which we have employed to define the energy-momentum tensor
outside the core of the global monopole, and $|\psi(r)|<<1$. These
approximations require that we restrict our analysis to a region defined by:

$\delta<r<\frac{1}{|\psi_{n}r^{n}|}$ (22)

where $\delta\approx(\lambda\eta^{1/2})$ is the order of magnitude of the
monopole’s core.

In the following Sections, we shall concentrate on the cases where
$\psi_{n}<0$ and $\psi_{n}>0$ in the limit given by (22). These solutions do
not contemplate the asymptotic case when $n>0$; however, its analysis is
important if we wish to consider regions closer to the black hole.

##  III The Thermodynamics of the Black Hole in an $f(R)$ Global Monopole for
Generic $n$.

In this Section, the thermodynamic behavior is reported for two cases. In one
of the cases, with $\psi_{<}0$, the stability of the BH is ensured for all
powers; the other case, where $\psi>0$, the technique of local thermodynamics
has been adopted to analyze the stability of the BH .

###  III.1 The Hawking Thermodynamics and the Power Law Series for a Generic n

In this Sub-section, we shall be considering the thermodynamical behavior of a
black hole in a region with a topological defect, by following the Hawking
procedure. We can analytically with prove that the metric has an event horizon
for the solution of an $(n+1)$-degree equation:

$1-\kappa\eta^{2}-\frac{2GM}{r_{H}}-\psi_{n}r^{n}_{H}=0.$ (23)

We have written the parameter $\psi_{n}$ as a power law-like function of
radial coordinate for a generic n. This is essential to respect the condition
(22) without narrowing the region that we are going to analyze. Considering
only the region where $|\psi(r)|<<1$ implies that the other roots of equation
(23) are out of the region where our solution is valid. In addition to the
limit for $\psi_{n}$ and $\kappa\eta^{2}$, used in the previous graphs, the
black hole (BH) stability was analyzed in detail considering the functions of
temperature, heat capacity and its phase.

From (23), we get the dependence of the energy (GM) on the horizon $r_{H}$

$GM={1\over 2}\Big{[}(1-\kappa\eta^{2})r_{H}-\psi_{n}r^{n+1}_{H}\Big{]}$ (24)

The Hawking temperature can be readily obtained from the GM expression, which
corresponds to the energy of the black hole. The Hawking temperature is,
therefore, the derivative of the energy with respect to the entropy of the
black hole Faraoni:2010yi

$S=F(R)A/4\equiv{1\over 4}(1+\psi_{n}r_{H}^{n})\pi r_{H}^{2}\,,$ (25)

where $A$ is the area of the event horizon.

Figure 2: Graph for $S(r_{H})$ for n going from 2 to 4 and Schwarzschild black
hole with global monopole , $\kappa\eta^{2}=0.3\cdot 10^{-5}$ where in left
panel $\psi_{n}=0.4\cdot 10^{-2n}$ and in the right panel $\psi_{n}=-0.4\cdot
10^{-2n}$

In Figure 2, which shows graphically the behavior of the entropy with the
radius of the horizon, it indicates that there are critical points for
different values of n given by the expression to $r_{H}$:

$r_{H_{max}}=2^{1/n}\left(-\psi_{n}(n+2)\right)^{-1/n}$ (26)

In this expression it can be easily verified that $r_{H_{max}}$ applies only
when $\psi<0$ , as can be seen by the graphs of Figure 2. This fact has given
us an important result which was related to the critical range between allowed
and forbidden region, but throughout this work will be analyzed if this
transition is within the limit of validity of the solution found.

Figure 3: Graphs for $GM_{HW}(S,n)$ for n going from 2 to 4 and the
Schwarzschild black hole case, $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and
$\psi_{n}=\pm 0.4\cdot 10^{-2n}$.

In the Figure 3, we plot the energy in terms of the entropy of the black hole.
It is interesting to investigate how the analogous relations for gravity and
thermodynamics become; with the development of a quasi-classical theory
Bardeem73 ; Bekenstein:1973ur ; Hawking74 , in a widely accepted relation in
which the area of the black hole corresponds in fact to its entropy and
consequently, its superficial gravity is responsible for measuring the
temperature that the radiation emitted by the black hole will have. Although
we are far from testing these results experimentally, the apparent lack of
inconsistencies in the theoretical model and its application to several cases
of interest [4] make it widely accepted.

The prescription for the Hawking’s temperature is given by the second low
$T={dGM\over dS}$ given by

$T_{HW}={dGM\over dr_{H}}\Big{(}{dS\over dr_{H}}\Big{)}^{-1}$ (27)

where

$\displaystyle{dS\over dr_{H}}$ $\displaystyle=$ $\displaystyle{\pi r_{H}\over
2}\Big{[}1+(1+{n\over 2})\psi_{n}r_{H}^{n}\Big{]}$ (28)
$\displaystyle{dGM_{HW}\over dr_{H}}$ $\displaystyle=$ $\displaystyle{1\over
2}\Big{(}1-\kappa\eta^{2}-(n+1)\psi_{n}r_{H}^{n}\Big{)}$ (29)

$\displaystyle
T_{HW}(\eta,\psi_{n})=\frac{1}{\pi\,r_{H}}{\Big{[}1-\kappa\eta^{2}-(n+1)\psi_{n}r_{H}^{n}\Big{]}\over\Big{[}1+(1+{n\over
2})\psi_{n}r_{H}^{n}\Big{]}},$ (30)

which, for $n=1$, recovers the result obtained in Man:2013sf , and, for
$\eta=0$, $\psi(r)=0$, the Schwarzschild result is reproduced. In this work,
we concentrate in the case $n>1$ with a Schwarzschild-like solution in
presence of a global monopole.

Figure 4: Graph for the Hawkings temperature, $T$ in function of the horizn
radio $r_{H}$, with $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and n going from 2 to 4,
with Schwarzschild black hole with global monopole $\psi_{n}=-0.4\cdot
10^{-2n}$.

Figure 4 shows that, for all values of $n$, there are two types of the black
hole temperature: one with the temperature decreasing with the horizon,
$r_{H}$, and the other with the temperature growing with the horizon. The
minimum point between these two regimes is calculated in this model by the
derivative of the Hawking temperature as a function of the horizon radius
${dT_{HW}\over r_{H}}=0$. The expression resulting from this calculation is a
simple polynomial of degree n, for $n\neq 1$. In this graph it can also be
seen that the temperature behavior in this region has the same shape and
transition. This fact has not been applied to the whole region of space.

Despite the positivity of the temperature, to analyze the stability of the
black hole, it is necessary to verify the prescription for the heat capacity.
Using the laws of thermodynamics for black holes, it has been found that it is
possible to obtain the heat capacity of the black hole by considering the
entropy of the system, i.e., $dE=TdS$. In the case where the volume is
constant, the energy, E, can be related to the mass of the black hole.

We shall now look for the heat capacity, which can be calculated from the
energy. The expression is obtained as follows:

$C_{HW}=\Big{(}\frac{dGM}{dT_{H}}\Big{)}_{r}=\Big{[}\frac{dGM_{H}}{dr_{H}}\cdot\Big{(}\frac{dT_{H}}{dr_{H}}\Big{)}^{-1}\Big{]}_{r}$
(31)

$\displaystyle u$ $\displaystyle={dGM_{H}\over dr_{H}}$ $\displaystyle={1\over
2}\Big{(}1-\kappa\eta^{2}-(n+1)\psi r_{H}^{n}\Big{)}$ $\displaystyle v$
$\displaystyle={dT_{H}\over dr_{H}}$ $\displaystyle=-{1\over 4\pi[1+(1+{n\over
2})\psi_{n}r_{H}^{n}]^{2}r_{H}^{2}}\Big{\\{}1-\kappa\eta^{2}+(n^{2}-1)\Big{[}1+{(1+{n\over
2})\over(n-1)}(1-\kappa\eta^{2}-\psi_{n}r_{H}^{n})\Big{]}\psi_{n}r_{H}^{n}\Big{\\}}$

$C_{H}={u\over v}$ (32)

By considering equations (30) and (24), we obtain the expression:

$C_{HW}=-2\pi
r_{H}^{2}{\Big{[}1-\kappa\eta^{2}-(n+1)\psi_{n}r_{H}^{n}]\Big{[}1+(1+{n\over
2})\psi_{n}r_{H}^{2}\Big{]}^{2}\over
1-\kappa\eta^{2}+(n^{2}-1)\Big{[}1+{(1+{n\over
2})\over(n-1)}(1-\kappa\eta^{2}-\psi_{n}r_{H}^{n})\Big{]}\psi_{n}r_{H}^{n}}$
(33)

In Figure 5, we see the heat capacity plotted for the classical case and the
$n=2$ to $4$ and compared with the Schwarzschild black hole.

Figure 5: Graph for heat capacity, $C$, with $\kappa\eta^{2}=0.3\cdot 10^{-5}$
with $\psi_{n}=-0.4\cdot 10^{-2n}$ and panel right.

By inspecting Figures 4 and 5, one can discuss the the validity regions of our
solution of the BH in terms of the temperature and heat capacity. It can be
seen that, for the region $\delta\leq r_{H}<r_{H}^{*}$, the temperature is
positive but the specific heat is negative. This region is therefore
prohibited because it has a negative heat capacity because it is not in
accordance with the laws of thermodynamics. There is a discontinuity in the
heat capacity equation at the point $r_{H}=r^{*}_{H}$, which can be seen in
the graph of Figure 5. This discontinuity is the point where the specific heat
changes behavior, becoming positive. In this region both temperature and heat
capacity are positive, with the possibility of obtaining an allowable solution
for the black hole.

Due to the maximum point given by the equation (26) and which can be seen
graphically in Figure 2, we have another phase transition in this system. For
$r_{H}>r_{max}$ the black hole becomes unstable again, although the heat
capacity is positive. The heat capacity at this point goes from decreasing to
the radius of the horizon, to increasing with the radius of the horizon, and
this destabilizes the BH presenting negative temperatures, in this way this
region becomes forbidden as we can saw in Figure 6. In the next section we
will discuss the local case, where we will compare the local temperature with
the Hawking temperature, this comparison gives us new results.

Figure 6: Graph for temperature and heat specific, $C$ in function of the
horizon ratio for n=2, with $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and
$\psi_{n}=-0.4\cdot 10^{-2n}$ .

In the graph of Figure 6 it is easy to see that the transition is located
exactly at the maximum point of the entropy curve given by the expression
(26).

It may be verified that, whenever $\psi_{n}>0$, both regions are prohibited,
because, for a positive temperature, the heat capacity is negative, and when
the heat capacity is positive, the temperature is negative. Then, in the case
of $\psi_{n}>0$ it is necessary to work with another prescription that, in
this paper, we take as the local prescription.

###  III.2 The Local Thermodynamics of the Black Hole for a Generic $n$.

In the previous Section, we have treated the thermodynamical properties by
considering the Hawking formalism, where it is possible to investigate the
properties in the case $\psi_{n}<0$. In the present Section, we go through a
formulation where it is possible to discuss the stability conditions
independently of the signal $\psi_{n}$. We have nevertheless to recall the
limitations imposed in the metric analysis (20). The solution that led us to
this configuration was found by using weak field approximations, which
consists in disregarding any cross-terms. In addition, the metric (20) is only
valid locally, for a region $\delta<r<1/|\psi_{n}|$, also we analyzed in
Hawking’s case, which forces us to work with local thermodynamical quantities.
The Schwarzschild metric does not present such limitations, but for the
purposes of comparison, in this Section the local thermodynamic quantities are
going to be derived. We use the prescription given in Tolman:1930zza to obtain
the local temperature of the black body radiation:

$T_{loc}={T\over\sqrt{g_{00}}}$ (34)

with this relation the temperature was obtained as a function of the position
as

$\displaystyle
T_{loc}=\frac{1}{\pi}{\Big{[}\frac{1-\kappa\eta^{2}}{r_{H}}-(n+1)\psi_{n}r_{H}^{n-1}\Big{]}\over\Big{[}1+(1+{n\over
2})\psi_{n}r_{H}^{n}\Big{]}}\sqrt{\\!\frac{r}{\psi_{n}r_{H}^{n+1}\\!\\!\\!-\\!r_{H}(1\\!-\\!8\pi
G\eta^{2})\\!\\!+\\!r(1\\!-\\!8\pi G\eta^{2})\\!-\\!\psi_{n}r^{n+1}}}\,.$ (35)

The local temperature for $n=2$ to $4$ and the Schwarzschild black hole is
plotted in Figure 7 as a function of the event horizon where we fixed the
position at $r=10$. The graph is zoomed to show the slight variation
difference between the minimum values of the temperature for different values
of the degree $n$.

Figure 7: Graph for $T_{loc}(r_{H})$ for $n=2,3,4$ and Schwarzschild, for
$r=10$.

An important characteristic to observe at this temperature is the existence of
a minimum in its variation with respect to the event horizon. This minimum can
be called the critical temperature [29], which would be the minimum
temperature in the region around the black hole. In Figure 7 the local
temperature curves for n = 2, 3, 4 and the Schwarzschild were calculated
numerically. Displays that, for all values of $n$, there are two possible
black holes: one for which the temperature diminishes with the horizon, to
$r_{H}>r^{*}_{H}$, and the other with the temperature growing with the
horizon, to $r_{H}>r_{min}$. These phase transitions were analyzed in more
detail in the local case in connection with the heat capacity formalism.

It is important to notice, from this and the following graphs, that we fix a
position for the measurement of the temperature and we are analyzing how the
temperature in that position varies as the event horizon increases. That
means, we are finding what size the black hole has at the moment in which the
temperature in the position r = 10 is minimal. In this sense, we see that as
the horizon approaches the chosen position the temperature decreases rapidly
to the minimum and then increases asymptotically when it approaches r = 10.

From the first law of thermodynamics $dGM_{loc}=T_{loc}dS$, the
thermodynamical local energy, $GM_{loc}$, can be derived. Following the
prescription discussed in last section, we can assume that
$S=F(R)A/4\equiv{1\over 4}(1+\psi_{n}r_{H}^{n})\pi r_{H}^{2}$, where A is the
area of the event horizon.

$GM_{loc}=E_{0}+\int_{S_{0}}^{S}T_{loc}dS=E_{0}+{\pi\over
2}\int_{r_{H}(M=0)}^{r_{H}(M)}T_{loc}(\xi)\Big{[}1+(1+{n\over
2})\psi_{n}\,\xi^{n}\Big{]}\,\xi\,d\xi$ (36)

where we choose $E_{0}=0$ for simplicity. Using the expression for the local
temperature in the integral using

$r_{H}(M=0)=[(1-\kappa\eta^{2})/\psi_{n})]^{-1/n}$ (37)

With the integration constants conveniently, we get

$\displaystyle GM_{loc}=r\sqrt{(1-8\pi
G\eta^{2}-\psi_{n}r^{n})}-\sqrt{r}\sqrt{\\!r(1\\!-\\!8\pi
G\eta^{2}\\!-\\!\psi_{n}r^{n})\\!-\\!r_{H}(1\\!-\\!8\pi
G\eta^{2}\\!-\\!\psi_{n}r_{H}^{n})}\ ,$ (38)

In Figure 8, we plot the Energy as a function of the entropy. It can be seen
that, in the local framework, the energy and the temperature $T$ are both
positive.

Figure 8: Graph for $GM(S,n)_{loc}$ for n going from 2 to 4 and and
Schwarzschild r= 15, $\kappa\eta^{2}=10^{-5}$ and $\psi_{n}=0.4\cdot
10^{-2n}$.

We are now going to look for the local heat capacity, which can be calculated
from the energy. The expression is obtained as follows:

$C_{loc}=\Big{(}\frac{dGM_{loc}}{dT_{loc}}\Big{)}_{r}=\Big{[}\frac{dGM_{loc}}{dr_{H}}\cdot\Big{(}\frac{dT_{loc}}{dr_{H}}\Big{)}^{-1}\Big{]}_{r}$
(39)

The subscript is there to show that we are calculating this quantity in a
fixed position. We work with Eq. (39) and give us

$\displaystyle{dGM_{loc}\over dr_{H}}$ $\displaystyle=$
$\displaystyle{\sqrt{r}\over 2}\Big{(}1-8\pi
G\eta^{2}-(n+1)\psi_{n}r_{H}^{n}\Big{)}$ (40)
$\displaystyle\times\Big{[}r(1-8\pi G\eta^{2}-\psi_{n}r^{n})-r_{H}(1-8\pi
G\eta^{2}-\psi_{n}r_{H}^{n})\Big{]}^{-1/2},$ $\displaystyle{dT_{loc}\over
dr_{H}}$ $\displaystyle=$ $\displaystyle v\sqrt{r}\times\Big{[}r(1-8\pi
G\eta^{2}-\psi_{n}r^{n})-r_{H}(1-8\pi
G\eta^{2}-\psi_{n}r_{H}^{n})\Big{]}^{-1/2}$ (41) $\displaystyle+{T_{H}\over
2}\sqrt{r}\Big{(}1-8\pi G\eta^{2}-(n+1)\psi_{n}r_{H}^{n+2}\Big{)}$
$\displaystyle\times\Big{[}r(1-8\pi G\eta^{2}-\psi_{n}r^{n})-r_{H}(1-8\pi
G\eta^{2}-\psi_{n}r_{H}^{n})\Big{]}^{-3/2}$

Figure 9: Graph for $C_{loc}(r_{H},n)$ for the $n=2$, $n=3$ and $n=4$ and the
Schwarzschild for black hole cases, for fixed $r=10$ ,
$\kappa\eta^{2}=10^{-5}$ and $\psi_{n}=\mp 0.4\cdot 10^{-2n}$.

In Figure 9, we see the local heat capacity plotted for the classical case and
the $n=2$ to $4$ and the local form for the Schwarzschild black hole. It can
be observed that for the local case, it is possible to obtain an analysis for
the thermodynamics of black holes for both $\psi<0$ and $\psi>0$ to r=10. It
can be seen that the curves for heat capacity are similar for the case $n>2$.
In fact in both cases the r limit of truncates the function.

The transition from negative to positive values happens as the horizon is
closer to the chosen position, $r$ also vanishes present only the positive
part. We can notice this feature more clearly by analyzing the heat capacity
as a function of the position, with a fixed horizon. The expected negative
heat capacity will be observed as $r$ grows and the transition to positive
values happens even for bigger black hole. This transition is yet to be
explained and it occurs even for the classical Schwarzschild case in the local
analysis, but not in the most usual global calculation where it remains
negative for every value of $r_{H}$. It can be observed that for the local
case, it is possible to obtain case for $\psi>0$, that to $n>2$ has been
similar to case $\psi<0$, in the case of small r.

Figure 10: Local heat capacity for n= 2 as a function of the position,
$\kappa\eta^{2}=10^{-5}$ and $\psi_{n}=-0.4\cdot 10^{-2n}$.

With Figure 10 it can be seen more easily that the function of the specific
heat is stopped precisely at the point $r_{limit}$. Now we can analyze the
case $n=2$, where another phase transition appears where the specific heat
changes direction and direction in case $\psi>0$. It is found that when r is
large this would happen for all n. However, when r is very large, the specific
heat becomes totally negative and BH will not even form.

Turning to the transition of the entropy phase, due to the correction
introduced by the $f(R)$ theory discussed in Faraoni:2010yi , we can have two
cases the first for $\psi<0$, which is analyzed in a shown in Figure 11 on the
left to n=2. The Hawking temperature introduces a maximum limit for the
horizon radius $r_{H}$, which divides a permitted region with, $T_{HW}>0$, of
a forbidden region $T_{HW}<0$. This maximum limit in the radius of the horizon
$r_{H}$ can be calculated using the Eq. (26) and corresponds to the solid line
of the graph.

Figure 11: Local temperature in the vicinity of $r_{H_{max}}$ for n= 2 in
panel left with $\psi_{n}=-0.4\cdot 10^{-2n}$ compared with Hawking
temperature where $r_{H_{max}}=111.8033989$ and in panel right the heat
capacity.

An interesting analysis is that the limit of $r_{Max}$ that appears in the
temperature of Hawking is also present in the local case. In the local case,
this boundary also divides a permitted region of a forbidden region, and this
happens when the position $r=r_{H_{max}}$. When $r>r_{H_{Max}}$, the minimum
temperature decreases, but the allowed region always has the same size as
$r_{H_{Max}}$.

This behavior can be seen in the graph of Figure 11 on the right, showing the
value of $r_{H_{Max}}$ for n = 2, considering different values of r. It can be
shown that for $r<r_{H_{Max}}$ there is no forbidden region.

The existence of a $r_{H}$ limit for black holes that has been studied in the
case $\psi<0$ repeats to $\psi>0$ existing a value of $r_{H}$ that can not be
passed $r_{H_{Max}}$ . The Hawking temperature for $\psi>0$ has no minimum
point becoming negative, the $r_{H_{max}}$ can be found when that temperature
is zero by doing the zero expression (30) that gives us:

$r_{H_{Max}}=\left({1-\kappa\eta^{2}\over\psi_{n}(n+1)}\right)^{1/2}$ (42)

In the local case we have a behavior similar to $\psi<0$ the maximum value for
r is $r_{H_{Max}}$, however in this case there will always be a transition
from the allowed region to the forbidden region, even when $r<r_{H_{Max}}$ in
the vicinity of $r_{max}$. If we analyze the heat capacity we can observe that
the validity limit to $r$ is very small and far from $r_{r_{H_{M}ax}}$ because
the heat capacity became negative and the black hole is not stable.

###  III.3 The Thermodynamical Phase Transitions for the Black Hole Systems

We shall take care in this Section of the possible phase transitions of the
system we are considering. The first one is the Hawking-Page phase transition.
It occurs whenever the function that gives the behavior of the Helmholtz free
energy with the radius of the horizon has a root which is a minimum of this
function.

$\displaystyle F_{HP}^{HW}|_{r_{H}=r_{HP}}$ $\displaystyle=$ $\displaystyle
GM-T_{HP}S={1\over
2}\Big{[}(1-\kappa\eta^{2})-\psi_{n}r^{n}_{HP}\Big{]}r_{HP}-{1\over
4}(1+\psi_{n}r_{HP}^{n})\pi T_{HP}r_{HP}^{2}=0$ $\displaystyle\left({\partial
F_{HP}^{HW}\over\partial r_{H}}\right)_{r_{H}=r_{HP}}$ $\displaystyle=$
$\displaystyle{1\over
2}\Big{[}(1-\kappa\eta^{2}-(n+1)\psi_{n}r_{HP}^{n}\Big{]}-{\pi\over
2}T_{HP}\Big{[}1+({n\over 2}+1)\psi_{n}r_{HP}^{n}\Big{]}r_{HP}=0$ (43)

where $T_{HP}$ is the Hawking-Page temperature. We find the radius and the
Hawking-Page temperature as given below:

$\displaystyle T_{HP}^{HW}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\,r_{HP}}{\Big{[}1-\kappa\eta^{2}-(n+1)\psi_{n}r_{HP}^{n}\Big{]}\over\Big{[}1+(1+{n\over
2})\psi_{n}r_{HP}^{n}\Big{]}}$ (44) $\displaystyle r_{HP}$ $\displaystyle=$
$\displaystyle\left[{n-{1\over 2}\alpha(n+1)\pm{1\over
2}\Delta^{1/2}\over\psi_{n}}\right]^{1/n}$ (45) $\displaystyle\Delta$
$\displaystyle=$
$\displaystyle\alpha^{2}(n+1)^{2}-4\alpha(n^{2}+n+1)+4(n^{2}+1)$ (46)

In Figure 12, the free energy question can be analyzed by considering the
Hawking-Page phase transition showing its phase change for each value of n
using Hawking’s formalism. In the local case we can have this transition for
both $\psi<0$ and $\psi>0$. Repeating what happens in the case of the Hawking-
Page radius, these values are very close, so we chose $n=2$ to do the analysis
as shown in Figure 13.

Figure 12: Graph of $F_{HW}$ for the n-order going from 2 to 5, where $GM=1$,
$\kappa\eta^{2}=0.3\cdot 10^{-5}$ and $\psi_{n}=-0.4\cdot 10^{-2n}$ .

In Figure 13, we present the possible regimes for the free energy in the case
$n=2$. The same analysis applies for the others orders of n. The Hawking case
that correspond to $\psi_{2}<0$ was plotted in the panel of the left. In this
case, we have a negative cosmological constant. In the right plot, we have
$\psi_{2}>0$ that have similar results to $\psi_{2}<0$ in the local case,
which can represents when $\psi>0$ the case with a positive cosmological
constant.

Figure 13: Graph for F for each n=2, $\kappa\eta^{2}=0.3\cdot 10^{-5}$. The
solid line is the Hawking-Page free energy, the dashed line is the minimal
energy the dashed dotted line is the energy when $T>T_{HP}$ and the dotted
line is the energy when $T<T_{min}$ and in a) the Hawking framework
$\psi_{n}=-0.4\cdot 10^{-2n}$ b) The local framework $\psi_{n}=0.4\cdot
10^{-2n}$ .

The region of temperatures higher than the Hawking-page temperature of the
Black Hole will always be stable. The minimum temperature puts into evidence a
black hole’s unstable region. In both cases, for $T<T_{min}$, where $T_{min}$
is represented by the dashed curve, we have a region of pure radiation. The
solid curve represents the Hawking-Page temperature that gives us the Hawking-
Page Phase transition. The expression, for the minimum of the temperature in
Hawking case, can be readily worked out and gives us

$\displaystyle T_{min}$ $\displaystyle=$ $\displaystyle{1\over
4\pi}\Big{(}{1-\kappa\eta^{2}\over
r_{min}}-(n+1)\psi_{2}r_{min}^{(n-1)}\Big{)}$ (47) $\displaystyle r_{min}$
$\displaystyle=$
$\displaystyle\Big{(}{-(1-\kappa\eta^{2})\over(n^{2}-1)\psi_{2}}\Big{)}^{1/2}$
(48)

As previously discussed, this work considers the boundary of the weak field
approximation in a region in the vicinity of the BH with the defect. For this
reason, we do not contemplate the asymptotic limits. However, whenever $n=2$
and $\psi$ is negative, the behavior of BH in the f(R)-model considering the
Hawking framework resembles the one which occurs in the case of the negative
cosmological factor corresponding to the Anti-de-Sitter case. This case is
extremely important for the study of the quantum behavior of gravitational
systems.

Figure 14: Graph for T for each n=2, $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and
$\psi_{n}=-0.4\cdot 10^{-2n}$ with r=105.57.

So far, the case analyzed considers $\psi_{n}<0$, which can be treated by the
Hawking’s formalism. In the case of the local formalism, it is possible to
verify that independently of $\psi_{n}$, it was possible to study both
behaviors.

In the graph of Figure 14, the study of the compatibility between the two
formalisms can be found. It can be verified that both present the same
critical behavior if we choose the value in the local quantities of the fixed
point where local quantities are calculated.

In Figure 14, it can be seen that both the case of $\psi_{n}<0$ to local and
Hawking cases, can present the same set of minima and Hawking-Page
temperature, showing compatibility between the two theories. It is always
possible to find the same set of values if we regulate the values of $r$ in
local formalism. In this Figure, this happens when, for a given Hawking-Page
temperature, in local formalism, we set r so that the Hawking-Page temperature
value is consistent with Hawking’s formalism.

Figure 15: Graph for the comparation between $C_{loc}$ and $C_{H}$ for each
n=2, $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and $\psi_{n}=-0.4\cdot 10^{-2n}$ with
$T_{HP}=1.3408\times 10^{-2}$ for both cases.

In Figure 15, a comparison was made between the heat capacity calculated in
Hawking’s formalism ($C_{HW}$) and Local ($C_{loc}$) in the case of psi ¡0. In
the figure to the left it can be seen that in the case of Hawking formalism,
as indicated in the graph, there are three distinct regions, the first
comprises a negative $C_{HW}$ for $r_{H}<r^{*}$. The other region corresponds
to a positive $C_{HW}$ for $r^{*}<r_{H}<r^{c}$ and the latter comprises the
region given by $r_{H}>r^{c}$. In the figure to the right, upper part, this
behavior was further detailed. Analyzing the graph to the right, it can be
clearly seen that $r^{*}$ coincides with the forbidden / allowed transition
region. The region $rH<r^{*}$ is a forbidden layer because it has specific
heat negative. The region $r^{*}<r_{H}<r^{c}$ is permissive where we find a
stable black hole. In this region the temperature starts from the minimum and
grows with the radius of the horizon while the heat capacity decreases. In
this region the black hole is stable. When $r_{H}>r^{c}$ we have again a
forbidden region.

The other curve of the specific heat of Figure 15 left corresponds to the
local case. For the comparison the parameters of the local model were adjusted
to $r=105.57$. With these conditions, the Hawking-Page temperature can be
obtained. In figure 15 the right panel down can analyze the detail of the
local case. It can be seen that in this case the solution to the black hole
presents two possibilities corresponding to both heat capacity and temperature
greater than zero. It can be seen that there is a region between
$r*<r_{H}<r_{min}$, where a black hole solution is possible, but this is
unstable because the temperature decreases with the radius of the horizon
going to a more stable location that is $r_{min}$. this region is called a
small black hole. The other region presents, as in the case of the Hawking
formalism, the temperature increasing with the increase of $r_{h}$, showing
that for the increase of the black hole this is giving more energy, for that
reason warmer. This region is called the Great Black Hole.

In Figure 16, for n = 2, the comparison of the black hole in the local
formalism with $\psi<0$ and $\psi>0$ for the same Hawking-Page temperature was
analyzed. As in the case of figure 15, where the $\psi<0$ case was analyzed,
it can be seen that the limits for large $r_{H}$ depend on the value of r.
Comparing these two cases, it can be analyzed that when $\psi>0$, has only one
allowed region for $r*<r_{H}<r^{c}$, where we have both temperature and
calorific capacity positive. In this region there may be several
possibilities, the temperature decreases as the heat capacity decreases, for
increasing $r_{H}$, small black hole. A region where temperature increases and
specific heat decreases and a region where both specific heat and temperature
increase with increasing horizon radius $r_{H}$ this region is the big black
hole.

Figure 16: Graph for the comparation between $C_{loc}$ for $\psi_{n}<0$ and
$C_{loc}$ for $\psi_{n}>0$ each n=2, $\kappa\eta^{2}=0.3\cdot 10^{-5}$ and
$\psi_{n}=\mp 0.4\cdot 10^{-2n}$ r=10 and $T_{HP}=0.1078$.

##  IV Closing Remarks

In this work, the solution of a black hole in a region containing a global
monopole in a class of f(R) gravity models has been investigated using a power
law approach. To pursue our investigation, we have considered the weak field
approximation for small radial coordinate values, assuming the validity of the
condition of the eq. (22). In this his region, it is possible to analyze the
behavior of BH solutions for $n>0$ and $\psi_{n}$ positive and negative.

The very goal of this work is the analysis of the thermodynamical behavior of
the black hole, considering two possible formalisms: Hawking’s, on the one
hand, and local formulation on the other hand.

In the case of Hawking’s approach, it was possible to study the case where
$\psi_{n}<0$ as being stable, by studying the positivity of the temperature
with the positivity of the heat capacity. This black hole was called a large
black hole, located for the radius of the horizon, $r_{H}>r_{min}$. For
$r_{H}<r_{min}$, we have an unstable black hole despite having a region
between $r_{H}^{*}<r_{H}<r_{min}$, where we the heat capacity is positive and
the temperature is positive.

The case with $\psi_{n}>0$ has been tackled in the local formalism; this
framework is actually valid for both $\psi_{n}$ positive and negative. The
local case was studied in more details for $\psi_{n}>0$ and a BH with similar
characteristics as in the Hawking’s treatment was obtained, if we work in the
region of validity. In this work, we also compared these two regimes, by
analyzing not only the relationship between heat capacity and temperature, but
also the phase transitions that the system may undergo. It has been found
that, for small values of $r_{H}$, the two formalisms are compatible. When
$r_{H}$ grows, it reaches a value where the BH undergoes another transition,
now related to a change of behavior in the heat capacity. In the case of the
thermodynamics with the Hawking formalism, the heat capacity passes from a
region where it decreases with increasing temperature to a region where it
increases with temperature. In the local case the heat capacity for
$r_{H}>r_{min}$ always decreases with temperature going to negative values.

Based on the assumption that heat capacity and temperature are not sufficient
to guarantee stability, the Hawking-Page phase transition, which applies to
both Hawking’s formalism and local formalism, was also considered. It has been
checked that, in both formalisms, a Hawking-Page phase transition exists as
another attribute in favor of the stability of BH.

The case with $n=2$ has also been contemplated. The relevance of this case
lies in the fact that the power law parameter, $\psi_{2}$, can act as a
cosmological constant, that can be positive or negative in this region.

The idea now is to inspect the asymptotic region that occurs for the case
$n<0$. With this analysis, it will be possible to study the limits of-Sitter
and Anti-de Sitter and the behavior on the borders. Besides the asymptotic
behaviors, it is also interesting to study the potential generators of the
power law presage that can be clarified when discussing the problem all over
the space. In future works, we intend to investigate this case, that might be
important for the AdS/CFT context for the $n=2$ situation. These aspects are
under study and shall be the subject of a forthcoming work. It is possible to
solve the stability problem for a negative $\psi_{2}$ and to use the same
thermodynamical procedure that we have used here. The importance, in the AdS
context, is the fact that this black hole presents an angular deficit, and it
is important to study the holographic principle of this object. We have shown
in Bayona:2010sd that the structure of the angular deficit in a theory in
$D=(1+3)$ is preserved on the $(1+2)$ boundary and, in our case, the defect is
preserved in $D=(1+2)$. The AdS4/CFT3 is very important in studying planar
materials like graphene and topological insulators. The introduction of the
defect in this type of systems gives us current properties in these materials
and it may become a relevant aspect of our work, opening up new application
possibilities for our model. Another issue that we can pick up to study is the
behavior of the objects near these systems BallonBayona:2013gx with both
regimes for $\psi_{2}$. We can intensively study quark/anti-quark interactions
and discuss the confining and non-confining transitions as well as the chiral
symmetry breaking critical temperature BallonBayona:2008uc .

ACKNOWLEDGEMENTS:  F. B. Lustosa would like to thank CNPq-Brasil for his MSc
Fellowship at the Instituto de Física of Universidade Federal Fluminense. The
authors express their gratitude to T.R.P. Caramês for clarifying discussions.

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