# Thermodynamic stability and $P$–$V$ criticality of nonsingular-AdS black
holes endowed with clouds of strings

Ashima Sood<EMAIL_ADDRESS>Centre for Theoretical Physics, Jamia
Millia Islamia, New Delhi 110025, India  Department of Mathematics, Netaji
Subhas University of Technology, New Delhi-110078, India  Arun Kumar
<EMAIL_ADDRESS>Centre for Theoretical Physics, Jamia Millia Islamia,
New Delhi 110025, India  J. K. Singh<EMAIL_ADDRESS>Department of
Mathematics, Netaji Subhas University of Technology, New Delhi-110078, India
Sushant G. Ghosh<EMAIL_ADDRESS>Centre for Theoretical Physics, Jamia
Millia Islamia, New Delhi 110025, India  Department of Mathematics, Netaji
Subhas University of Technology, New Delhi-110078, India  Astrophysics and
Cosmology Research Unit, School of Mathematics, Statistics and Computer
Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South
Africa

###### Abstract

We investigate the extended phase space thermodynamics of nonsingular-AdS
black holes minimally coupled to clouds of strings in which we consider the
cosmological constant ($\Lambda$) as the pressure ($P$) of the black holes and
its conjugate variable thermodynamical volume ($V$) of the black holes. Owing
to the background clouds of strings parameter ($a$), we analyse the Hawking
temperature, entropy and specific heat on horizon radius for fixed-parameter
$k$. We find that the strings clouds background does not alter small/large
black hole (SBH/LBH) phase transition but occurs at a larger horizon radius,
and two second-order phase transitions occur at a smaller horizon radius.
Indeed, the $G$–$T$ plots exhibit a swallowtail below the critical pressure,
implying that the first-order phase transition is analogous to the liquid-gas
phase transition at a lower temperature and lower critical pressure. To
further examine the analogy between nonsingular-AdS black holes and a liquid-
gas system, we derive the exact critical points and probe the effects of a
cloud of strings on $P-V$ criticality to find that the isotherms undergo
liquid-gas like phase transition for $\tilde{T}\,<\,\tilde{T}_{c}$ at lower
$\tilde{T}_{c}$. We have also calculated the critical exponents identical with
Van der Walls fluid, i.e., same as those obtained before for arbitrary other
AdS black holes, which implies that the background clouds of strings do not
change the critical exponents.

##  I Introduction

The black hole’s thermodynamics investigation has communicated a deep and
fundamental relationship among gravitation, thermodynamics, and quantum
theory. Interestingly, classical and semiclassical analysis has given rise to
most of our present physical understandings of the nature of quantum phenomena
[1, 2]. An interesting phenomenon in the AdS black hole thermodynamics is the
existence of the transition between AdS–Schwarzschild black hole and thermal
AdS, unlocked in the pioneering work of Hawking and Page [3], which has
attracted numerous astrophysicists towards studying black holes thermodynamics
in AdS spacetimes. Indeed, the Hawking-Page phase transition between large
black holes (LBH) and thermal gas in the AdS space is a topic of intense
research [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], e.g., an
analogy between phase structures of various AdS black holes and statistical
models associated with Van der Waals like phase transitions has been suggested
[19, 20, 21]. The analysis of AdS black hole phase transition has been
generalized to the extended phase space where the cosmological constant has
been treated as the pressure of the black hole revealing several exciting
results[24, 25, 22, 23].

In an interesting paper, the authors [26] explicitly bring out how the effect
of a background of clouds of strings can modify black hole solutions and their
properties aspects in non-extended thermodynamics. A cascade of subsequent
interesting work analysed black hole solutions in clouds of strings model [29,
28, 30, 27, 31] and these black holes have been shown to also exhibit Hawking-
Page type phase transitions [29]. The theoretical developments signal toward a
scenario in which the fundamental building blocks of the Universe are extended
objects instead of point objects can be exciting and have been taken quite
seriously [32] with the most natural candidate being a one-dimensional strings
object. In addition, the intense level of activity in string theory has led to
the idea that the static Schwarzschild black hole (point mass) may have
atmospheres composed of fluid or field of strings. The clouds of strings
background for the Schwarzschild black hole was proposed by Letelier [33] to
demonstrate that event horizon has modified radius $r_{H}={2M}/{(1-a)}$ with
$0<a<1$ being the string cloud parameter [33], thereby enlarging the
Schwarzschild radius of the black hole by the factor $(1-a)^{-1}$. Many
authors generalized the pioneering work of Letelier [33], for instance, in GR
[30], for EGB models [34, 29, 31] and in Lovelock gravity [28, 27]. Thus, the
study of black holes surrounded by clouds of strings, in general relativity or
modified theories, may be critical because relativistic strings at a classical
level can be used to construct applicable models [32].

The primary purpose of this paper is to look for an exact spherically
symmetric nonsingular black hole solution endowed with clouds of strings in
AdS spacetime, and explicitly mention how background clouds of strings can
alter black hole solutions and their extended phase thermodynamic qualitative
features we know from our experience in general relativity. We shall, in turn,
probe the effects of background on $P-V$ criticality, the critical behaviour
of the thermodynamic quantities and demonstrate that there exists a phase
transition and critical phenomena similar to the ones in a Van der Waals
liquid-gas system. Thus, it is our goal to connect nonsingular-AdS black holes
with the concept of black hole chemistry [40, 35, 39]. The black hole
chemistry – a new perspective on black hole thermodynamics, with the
interpretation of black hole mass as enthalpy, the cosmological constant
$\Lambda$ and it’s conjugate variable, respectively, as a pressure term and
the thermodynamical volume of the black holes has led to a new understanding
of Van der Waals fluids phase transitions from a gravitational viewpoint [40].
In particular, we investigate the analogy between nonsingular-AdS black holes
endowed with clouds of strings and a liquid-gas system and find exact
solutions of the critical points.

This paper is organized as follows. In Section II, we consider nonsingular–AdS
black holes surrounded by clouds of strings, discuss the effect of clouds of
strings parameter $a$, nonlinear electrodynamics (NED) parameter $k$, on
horizon structure. The thermodynamic properties of the solution derived and
discussed in Sec. III. In Sec. IV, the thermodynamical stability analysis and
$P-V$ criticality of the black holes has been discussed, and the critical
exponents have been calculated in Sec. V. Finally, we summarize our main
findings in Section VI.

We use geometrized units $8\pi G=c=1$, unless units are specifically defined.

##  II Nonsingular black holes with clouds of strings

The AdS black holes are natural tools to investigate AdS/CFT correspondence
and black hole thermodynamics in extended phase space. Here, we are interested
in a nonsingular-AdS black hole solution endowed with clouds of strings, and
the energy-momentum tensor of clouds of strings resembles the global monopole
[31]. The nonsingular-AdS black holes can be derived from general relativity
minimally coupled to NED whose action reads

$\mathcal{S}=\frac{1}{16\pi}\int\mathrm{d}^{4}x\sqrt{-g}[\mathcal{R}+6l^{-2}-\mathcal{L(F)}]+{S_{M}},$
(1)

where $\mathcal{R}$ is the scalar curvature, $g$ is the determinant of the
metric tensor and $l$ is positive AdS radius related to cosmological constant
$\Lambda$ through the relation $\Lambda=-3/l^{2}$. The Lagrangian density
$\mathcal{L(F)}$ is a function of
$\mathcal{F}=\frac{1}{4}\mathcal{F}^{\textit{ab}}\mathcal{F}_{\textit{ab}}$
with
$\mathcal{F}_{\textit{ab}}=\partial_{\textit{a}}\mathcal{A}_{\textit{b}}-\partial_{\textit{b}}\mathcal{A}_{\textit{a}}$
and the solution we are interested, has the form

$\mathcal{L}(F)=F\exp[{{-k/e}~{}(2e^{2}F)^{\frac{1}{4}}}],\,\,\,\,\text{with}~{}~{}~{}~{}F=\frac{e^{2}}{2r^{4}}.$
(2)

where $e$ is NED charge and $k>0$ is constant. To obtain our black hole
solution, we assume the static spherically symmetric metric of the form [35]

$ds^{2}=-{f}(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega,$ (3)

where the function $\textit{f}(r)$ is the metric function to be determined and
$\textit{d}\Omega=\textit{d}\theta^{2}+\sin^{2}\theta\textit{d}\phi^{2}$. The
Nambu-Goto action of strings evolving in spacetime is given by [33, 31]

${S_{M}}=\int_{\Sigma}\mathcal{M}\,(\gamma)^{-\frac{1}{2}}\,d\lambda^{0}d\lambda^{1}=\int_{\Sigma}\mathcal{M}\left[-\frac{1}{2}\Sigma_{\mu\nu}\Sigma^{\mu\nu}\right]^{\frac{1}{2}}d\lambda^{0}d\lambda^{1},$
(4)

where the world sheet is parameterized by spacelike and timelike parameters
[32] represented by $\lambda^{0}$ and $\lambda^{1}$ and $\mathcal{M}$ is a
dimensionless positive constant which characterises each string. The quantity
$\gamma$ is determinant of $\gamma_{ab}$ given by

$\gamma_{ab}=g_{\mu\nu}\frac{\partial\textit{x}^{\mu}}{\partial\lambda^{a}}\frac{\partial\textit{x}^{\nu}}{\partial\lambda^{b}}.$
(5)

The movement of a string in time sweeps out area in two dimension which is
termed as it’s world sheet $\Sigma$ [41] and has associated with it a bivector
given by [33]

$\Sigma^{\mu\nu}=\epsilon^{ab}\frac{\partial\textit{x}^{\mu}}{\partial\lambda^{a}}\frac{\partial\textit{x}^{\nu}}{\partial\lambda^{a}},$
(6)

where $\epsilon^{ab}$ is the Levi-Civita tensor in two dimensions, which is
anti-symmetric in the indices $a$ and $b$ given by
$\epsilon^{01}=-\,\epsilon^{10}=1$. By varying the action (1) with respect to
$g_{ab}$, we obtain the following equations of motion

$\textit{G}_{ab}-\frac{3}{\textit{l}^{2}}g_{ab}=\mathcal{T}_{\textit{ab}}=2\left(\frac{\partial\mathcal{L(F)}}{\partial\mathcal{F}}\mathcal{F}_{\textit{ac}}\mathcal{F}^{\textit{c}}_{\textit{b}}-g_{ab}\mathcal{L(F)}\right)+T_{ab}^{\mathrm{cs}},$
(7)

and

$\Delta_{\textit{a}}\left(\frac{\partial\mathcal{L(F)}}{\partial\mathcal{F}}\mathcal{F}_{\textit{ac}}\right)=0,$
(8)

where $\textit{G}_{ab}$ is the Einstein tensor and

$T_{ab}^{\mathrm{cs}}=2\partial L/\partial
g_{ab}=\rho\,(\gamma)^{-1/2}\Sigma^{h}_{a}\Sigma_{hb},$ (9)

is energy-momentum tensor of clouds of strings [34]. The associated quantity
$\rho$ is the proper density of the string cloud [34] and
$\rho\,(\gamma^{1/2})$ is the gauge invariant density. The conservation of
energy and momentum yields the equation

$\nabla_{a}\,T_{ab}\,=\,0,$ (10)

which leads to

$\partial_{a}\,(\sqrt{-g}\,\rho\,\Sigma^{ab})\,=\,0.$ (11)

---
Figure 1: The parameter space ($a$, $\tilde{k}$) for the existence of black
holes. The black solid line corresponds to the extremal black holes.

For a spherically symmetric solution the density $\rho$ and bivector
$\Sigma_{ab}$ are functions of radial and temporal components, hence the
energy momentum tensor for a string cloud is [33]

$T^{t}_{t}\,=\,T^{r}_{r}\,=\,\frac{a}{r^{2}}.$ (12)

On using Eqs. (3), (7) and (12), the $(r,r)$ component of Einstein equations
can be integrated to

$f(r)=1-\frac{2Me^{-k/r}}{r}+\frac{r^{2}}{l^{2}}-a,$ (13)

where $M$ is the integration constant related to the mass of the black hole
and $k$ is related to charge $e$ via $e^{2}=2Mk$. Clearly solution (13) in the
absence of clouds of strings background $(a\,=\,0)$, exactly goes over to
solution in [35] and encompasses the Schwarzschild black holes [34, 43, 42]
when ($1/l^{2}\,=\,k\,=\,a\,=\,0$).

We know that the metric (3) with (13), in the absence of clouds of strings, is
a regular or nonsingular black hole [35]. However, with the background strings
of clouds, we find the invariant Ricci scalar $R=R_{ab}\,R^{ab}$, ($R_{ab}$
Ricci tensor) and the $K=R_{abcd}\,R^{abcd}$($R_{abcd}$ Reimann tensor) takes
the form

$R~{}=\frac{2a}{r^{2}}+~{}\frac{2~{}M~{}k^{2}~{}e^{-k/r}}{r^{5}},$ (14)
$K=\frac{4a^{2}}{r^{4}}+\frac{16e^{-k/r}Ma}{r^{5}}+$
$\frac{4M^{2}e^{-k/r}(k^{4}-8k^{3}r+24k^{2}r^{2}-24kr^{3}+12r^{4})}{r^{10}}$
(15)

which obviously diverges, when $a,M\neq 0$ and $r\to 0$. Thus, the spacetime
with the background strings of clouds is no more regular, but it becomes
singular.

### Energy Conditions

Next, we check the status of the various energy conditions using the
prescription of Hawking Ellis [36, 37, 38]. The Einstein equations governing
the stress energy tensor $T_{\mu\nu}$ is given by the Eq. (7), which leads to

$\displaystyle\rho=\frac{2\,M\,k\,e^{-k/r}}{r^{4}}+\frac{a}{r^{2}}=-P_{r},$
(16) $\displaystyle
P_{\theta}=P_{\phi}=-\frac{M~{}k~{}e^{-k/r}(k-2r)}{r^{5}}.$

The weak energy condition (WEC) demands that $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$
everywhere, for any timelike vector $t^{\mu}$, which is equivalent to

$\rho\geq 0,\;\;\rho+P_{i}\geq 0\,(i=r,\;\theta,\;\phi),\\\ $ (17)

and hence

$\rho+P_{\theta}=\frac{a}{r^{2}}-\frac{Mk(k-4r)e^{-k/r}}{r^{5}}.$ (18)

Hence, the weak energy condition is satisfied when $k\leq 4r$. Next, the null
energy condition (NEC) requires that $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$ in the
entire spacetime, for any null vector $t^{\mu}$. The null energy condition
demands $\rho+P_{r}\geq 0$ and $\rho+P_{\theta}\geq 0$. The former becomes
identically zero and $\rho+P_{\theta}$ becomes positive for $k\leq 4r$. Hence
the null energy condition is also satisfied for $k\leq 4r$.

Finally, the strong energy condition (SEC) states that
$T_{\mu\nu}t^{\mu}t^{\nu}\geq 1/2T^{\mu}_{\mu}t^{\nu}t_{\nu}$ globally, for
any timelike vector $t^{\mu}$, which requires

$\rho+P_{1}+2~{}P_{2}\geq 0.$ (19)

We find that

$\displaystyle\rho+P_{1}+2~{}P_{2}=\frac{a}{r^{2}}-\frac{3Mke^{-k/r}\left(k-\frac{8r}{3}\right)}{r^{5}}.$
(20)

Thus, the strong energy condition is satisfied for $k\leq 3/8~{}r$. We find
that the energy conditions are satisfied when $r>k/4$ and hence they are
violated in the deep core.

### Extremal black holes

In order to discuss the properties of the solution (13) we rewrite it as

$f(r)=1-\frac{2me^{-\tilde{k}/x}}{x}+x^{2}-a,$ (21)

such that $x=r/l$, $m=M/l$ and $\tilde{k}=k/l$. Thus the black hole (13) is
characterised by three parameters $M$, $k$ and $a$. The metric (3) is singular
at the points where $f(x)\,=\,0$ and it corresponds to coordinate singularity,
signifying the presence of event horizon. The numerical solution of $f(x)=0$
reveals existence of maximum two roots as shown in Fig. 2. It is possible to
find critical value $m_{0}$, for a given $a$ and $\tilde{k}$, such that for
$m>m_{0}$ there exists two roots of $f(x)\,=\,0$ viz. ${\pm}\,x\,$
corresponding to Cauchy horizon ($x_{-}$) and event horizon ($x_{+}$) (cf.
Fig. 2). When $m=m_{o}$ the two horizon degenerate on ($x_{+}=x_{-}=x_{0}$)
and we have extremal black holes. Similarly, for given $\tilde{k}$ and $m$, we
have found critical value of $a$ such that $a=a_{0}$ corresponds to extremal
black hole (cf. Fig. 3) and $a>a_{0}$ corresponds to regular black holes with
two horizons ($x_{\pm}$). In Fig. 1, we have depicted the parametric space
($a$, $\tilde{k}$) for $m=0.3$, in which the solid black curve is
corresponding to the extremal black holes. The grey region is showing black
holes with two horizons whereas the white region corresponds to no black hole
solutions. In absence of clouds of strings parameter ($a=0$), we only obtain
critical values $m_{0}$ and $\tilde{k}_{0}$ which correspond to extremal black
hole at $m=m_{0}$ ($\tilde{k}=\tilde{k}_{0}$) and regular black hole with two
horizons for $m>m_{0}$ ($\tilde{k}<\tilde{k}_{0}$) [35]. Interestingly the
radius of an extremal black hole is slightly smaller when we introduce the
clouds of strings parameter.

---
Figure 2:  The metric function $f(x)$ vs $x$. When $m=m_{0}=0.5039$ (left) and
$m=m_{0}=0.16139$ (right) corresponding to extremal black hole with degenerate
horizon. The black hole with Cauchy and event horizons exist when $m>m_{0}$.
---
Figure 3: The metric function $f(x)$ vs $x$ showing extremal black hole with
degenerate horizon for $a=a_{0}=0.5602$ (left) and
$\tilde{k}=\tilde{k}_{0}=0.42206$ (right) and the black hole with Cauchy and
event horizons exist for $a>a_{0}$ and $\tilde{k}<\tilde{k}_{0}$.

The existence of extremal black hole confirms that the black hole undergoing
evaporation will result in stable black hole remnant characterised by mass
($m_{0}$) and radius size ($x_{0}$), can be determined by solving systems [35,
39]

$f(x)\,=\,0\,=\,\frac{\partial f(x)}{\partial x}|_{x=x_{o}}.$ (22)

Solving Eq. (22), we obtain

$x_{o}=\frac{1}{9\upsilon}\left[\upsilon^{2}+\tilde{k}\upsilon+\tilde{k}^{2}-9(1-a)\right],$
(23)

and

$m_{o}=e^{9\tilde{k}\upsilon/\mu}\left[\frac{81(1-a)\upsilon+\left(\upsilon^{3}+\tilde{k}\upsilon+\tilde{k}^{2}-9(1-a)\right)\mu}{81\upsilon}\right],$
(24)

where
$\upsilon=\left[\tilde{k}^{3}-108(1-a)\tilde{k}+9\sqrt{3}\sqrt{\left(1-a\right)\beta}\right]^{\frac{1}{3}}$,
$\mu=\upsilon^{2}+\tilde{k}\upsilon+\tilde{k}^{2}-9(1-a)$ and
$\beta=\tilde{k}^{4}-3\left(1-a\right)^{2}+47\left(1-a\right)\tilde{k}^{2}$.
The above results, in the limit $a\to 0$ reduce to those obtained in Ref.
[35].

##  III Black hole thermodynamics

Black hole thermodynamics conceived as a discipline, was formulated with the
advent of quantum field theory in curved spacetime which suggested analogies
of thermodynamic quantities like internal energy, temperature and entropy with
black hole mass, surface gravity and event horizon area, respectively [43, 46,
44, 45]. The nomenclature in black hole chemistry, has been given in a new
scientific era and is simply regarded as the thermodynamics of black holes in
extended AdS spacetime, [40, 47] by demonstrating cosmological constant
$\Lambda$ as dynamical thermodynamical pressure [48, 49] via $P=3/8\pi l^{2}$.
This inclusion of $\Lambda$ as thermodynamical pressure ($P$) leads to
interpretation of black mass as enthalpy $(H\,=\,m)$ instead of internal
energy [50, 51]. Then the generalised first law of black hole thermodynamics
reads as [52, 47, 40, 53, 21]

$dm=TdS+VdP+\phi dk,$ (25)

where $m$ $(S,k,P)$ is black hole mass function with dynamical variables
namely the entropy $S$, the pressure $P$ and the magnetic charge parameter
$k$. While the temperature $T$, the thermodynamical volume $V$ and the
chemical potential $\phi$ are the conjugate variables of $S$, $P$ and $k$,
respectively, which can be defined as [52, 47, 40, 53, 21]

$T=\left(\frac{\partial m}{\partial
S}\right)_{k,P},~{}~{}~{}~{}\phi=\left(\frac{\partial m}{\partial
k}\right)_{S,P},~{}~{}~{}~{}V=\left(\frac{\partial m}{\partial
P}\right)_{k,S}.$ (26)

The cosmological term leads the Smarr relation [54] to the following
generalized form [55, 40]

$m=2TS-2PV+{\phi}k.$ (27)

The analysis of all thermodynamic quantities of the black hole is associated
with the horizon $x_{+}$. We want to mention that $k$ and $e$ are not two
independent parameters as in the case of Born-Infeld black holes [62]. Indeed,
our solution is characterized by only one parameter $k$ apart from mass $M$,
and in principle, we can work with only parameter $k$. The parameter $e$ is
introduced if one wishes to compare our solution with the Reissner-Nordstrom
metric via $e^{2}=2Mk$ for $r\gg k$. Hence, there should only be the
thermodynamic conjugate $\phi$ of parameter $k$ in the first law of black hole
thermodynamics. The mass of the black hole is obtained by solving $f(x_{+})=0$
as

$m=H=\frac{x_{+}}{2}\,[e\,^{k/x_{+}}(1-a+x_{+}^{2})].$ (28)

The Hawking temperature of the black hole can be derived from the definition
of surface gravity ($\kappa$) [47] as follows

$\tilde{T}=\frac{\kappa}{2\pi}=\frac{1}{4\pi
x_{+}}[\,3x{{}_{+}}^{2}-\frac{\tilde{k}}{x_{+}}(1-a+x{{}_{+}}^{2})+(1-a)].$
(29)

${T}={1}/{4\pi x_{+}}$ is the temperature of the Schwarzschild black holes.
Obviously, the Hawking temperature of nonsingular-AdS black holes surrounded
by clouds of strings is modified and it evident that the temperature is
sensitive to the clouds of strings (cf. Fig. 4). It is evident from the Fig.
4, for a given $\tilde{k}$ ($a$), there is a phase transition with $0<a<a_{c}$
(($0<\tilde{k}<\tilde{k}_{c}$)). Fig. 4 shows that the Hawking temperature has
local maxima and minima, respectively, at critical radii $x_{c1}=0.25578$ and
$x_{c2}=0.31547$ when $a<a_{c}=0.189999$ (for $\tilde{k}=0.1$) (cf. Fig. 4).
The critical value $a_{c}$ ($\tilde{k}_{c}$) for some given value of
$\tilde{k}$ ($a$), can be obtained by solving $\partial\tilde{T}/\partial
x|_{x=x_{c}}=0$. The Hawking temperature plot is showing positive slope for
small black hole (SBH) and LBH whereas for the intermediate black hole (IBH)
it has negative slope, mimicking the Van der Walls-like first-order phase
transition between SBH–LBH [21, 56]. When $a=a_{c}$
($\tilde{k}=\tilde{k}_{c}$), the Hawking temperature’s local maxima and minima
merge and they increases monotonically when $a>a_{c}$
($\tilde{k}>\tilde{k}_{c}$) (cf. Fig. 4). Also, due to background clouds of
string, the $\tilde{T}_{max}$ increases.

---
Figure 4: The Hawking temperature $\tilde{T}$ vs horizon $x_{+}$ for $a=0.6$
varying $\tilde{k}$ with $\tilde{T}_{max}=0.215369$ ,
$\tilde{T}_{min}=0.160236$ for $\tilde{k}=0.04$ (left), and for
$\tilde{k}=0.1$ varying $a$ with $\tilde{T}_{max}=0.223639$,
$\tilde{T}_{min}=0.221622$ for $a=0.1$ (right). For $a=0.6$, $\tilde{T}_{max}$
= $\tilde{T}_{min}$ at $x=0.210$ for $\tilde{k}=0.0702728$ whereas for
$\tilde{k}=0.1$, $\tilde{T}_{max}$ = $\tilde{T}_{min}$ at $x=0.3$ for
$a_{c}=0.1899999$.

The Bekenstein–Hawking entropy of the black hole is obtained from the first
law of black hole thermodynamics (25), which comes out to be

$\tilde{S}=\frac{A}{4}\left[{\frac{e~{}^{\tilde{k}/{x_{+}}}}{x_{+}}}\left(\tilde{k}+x_{+}\right)-\frac{\tilde{k}}{x{{}_{+}}^{2}}\mathrm{Ei}\left[\frac{\tilde{k}}{x_{+}}\right]\right],$
(30)

where $A=4\pi x_{+}^{2}$ and $\mathrm{Ei}$ is the exponential integral
function. Eq. (30) reduces to the usual area law ($S=A/4$) up to $\mathrm{Ei}$
corrections in the classical limit. It is interesting to note that the entropy
(30) does not explicitly depend on the clouds of strings parameter $a$, but is
endowed in the horizon $x_{+}$. The cosmological term permits the definition
of the system’s pressure and its conjugate quantity, the black hole volume,
and we obtain pressure, volume and $\phi$ associated with the black holes as

$P=\frac{3}{8\pi
l^{2}},~{}~{}~{}\tilde{V}=\frac{4}{3}e~{}^{\tilde{k}/{x_{+}}}\pi
x_{+}^{3},~{}~{}~{}\bar{\phi}=\frac{e~{}^{\tilde{k}/{x_{+}}}}{2}\left[1-a+x_{+}^{2}\right].$
(31)

In the classical limit ($a,k\to 0$), all above variables coincide with the
conventional variables of a Schwarzschild AdS black hole [21], and in the case
when only $a\,\to\,0$, they go over to that of the nonsingular-AdS black holes
[35].

##  IV Stability and $P-V$ critically

Here, we investigate the thermal properties and the phase structure of black
holes in canonical ensemble (fixed NED charge). Hawking and Page [3]
investigated the thermodynamic properties of black holes in asymptotically AdS
spacetimes. The thermodynamic stability of the black holes requires
investigation of the Gibbs free energy behaviour[35, 39]. We are interested in
the regions where the free energy is negative, and identify where black holes
are thermally favoured over the reference background. The free energy of the
black hole is calculated from [35, 39],

$G=m-\tilde{T}\tilde{S}$

which comes out to be complicated to present and hence depicted in Fig. 5

---
Figure 5: The plot of Gibbs free energy $\tilde{G}$ vs horizon $x_{+}$ for
nonsingular-Ads black hole surrounded by clouds of strings for different
values of clouds of strings parameter $a$.
---
Figure 6: The plot of Gibbs free energy $\tilde{G}$ vs temperature $\tilde{T}$
exhibiting a liquid-gas phase transition for $\tilde{P}<\tilde{P}_{c}$ and no
transition for $\tilde{P}>\tilde{P}_{c}$ obtained for a particular value of
$a$ ($\tilde{k}$) and corresponding critical pressure $\tilde{P}_{c}$. The
swallowtail occurs at lower temperature for higher values of $a$.
$\tilde{P}_{c}=0.00368414$, $0.00184207$, respectively, for $a=0.6$, $0.8$.

Since the global stability of the system is measured by the Gibbs free energy,
its global minimum is estimated to be the preferred state of the black hole
[47]. Gibbs free energy plays a crucial role by identifying the first-order
phase transition via the swallowtail behaviour in the $\tilde{G}$– $\tilde{T}$
plots (Fig. 6), and $G$-$X_{+}$ plots (cf. Fig. 5) throw light on the SBH/LBH
transition. As it is shown in Fig. 5, the free energy $G_{+}$ for various $a$,
have local minimum and local maximum, respectively, at horizon radii $x_{c1}$
and $x_{c2}$ with $x_{c2}>x_{c1}$, which can be identified as the extremal
points of the Hawking temperature shown in Fig. 4 and where the specific heat
capacity $C_{+}$ diverges (cf. Fig. 7). The first-order phase transition
occurs at $x_{+}=x_{c}$, where the free energy turns negative viz., $G_{+}<0$
for $x_{c}>x_{c2}$ . Thus the LBH, with horizon radii $x_{+}>x_{c}$, are
thermodynamically globally stable. However, at minimal horizon radii, the
Hawking temperature is negative and hence not physical for global stability.
The minimum value of the $\tilde{G}$ decreases (increases), occur at the lower
(higher) radius with increasing $a$ ($\tilde{k}$), and there will be no black
hole for radii smaller than this. (cf. Fig. 5).

Fig. 6 shows the plot of the Gibbs free energy vs temperature. For the values
of pressure below the critical pressure $\tilde{P}_{c}$, the
$\tilde{G}-\tilde{T}$ diagram exhibits a swallowtail structure [58] where the
Gibbs free energy of the black hole intersects with itself which is indicative
of first-order phase transition between SBH and LBH (cf. Fig. 6). By analyzing
the Gibbs free energy, one can observe that at a pressure lower than the
critical value ($\tilde{P}<\tilde{P}_{c}$), the Gibbs free energy exhibits a
characteristic swallowtail behaviour which implies that SBH/LBH first-order
phase transition occurs. Indeed, for $\tilde{P}<\tilde{P}_{c}$, the $G$ – $T$
graph exhibit three black holes states: SBH stable, intermediate BH (IBH)
unstable and LBH stable (cf. Fig. 8 ). Further, the SBH / IBH black hole
transition occurs at the inflection point $T_{2}$ , the IBH/ LBH meet at
$T_{1}$. Thus, LBH is the preferred state when
$\tilde{T}\in[\tilde{T}_{0},\tilde{T}_{2}]$, and for
$\tilde{T}<\tilde{T}_{0}$, the preferred state is stable SBH. Thus, the SBH
undergoes first-order phase transition to LBH at $\tilde{T}=\tilde{T}_{0}$
(cf. Fig. 8). The second order phase transition is shown through the
discontinuity of specific heat (c.f. Fig. 8) and through the cusp in $G-T$
plot when we take $P=P_{C}$.

After discussing the global thermodynamical stability of nonsingular-AdS black
holes endowed with clouds of string, we next turn to the local stability or
thermal stability. The reason to consider local stability is that even when a
black hole configuration is globally stable, it can be locally unstable [26].
One can investigate the thermodynamic stability from the specific heats
$\mathcal{C_{V}}$ and $\mathcal{C_{P}}$ of the system [21], where
$\mathcal{C_{V}}=T\left({\partial S}/{\partial T}\right)_{V.k}$ is the
specific heat at constant volume and $\mathcal{C_{P}}=T\left({\partial
S}/{\partial T}\right)_{P,k}$ is the specific heat at constant pressure [47].
The black hole with positive specific heat is at least locally thermodynamical
stable, whereas the negativity of specific heat signifies the thermodynamical
instability of the black hole to the thermal fluctuations [59, 60]. A phase
transition is known to exist in the case when the specific heat diverges [39].
The specific heat at constant volume, $\mathcal{C_{V}}$, is zero for our case.
The specific heat at constant pressure, $\mathcal{\tilde{C_{P}}}$ is
calculated to be

$\mathcal{\tilde{C_{P}}}=2\pi
e^{\tilde{k}/{x_{+}}}x_{+}^{2}\left[\frac{\left(1-a+3x_{+}^{2}\right)-\frac{\tilde{k}}{x_{+}}\left(1-a+x_{+}^{2}\right)}{\left(1-a\right)\frac{2\tilde{k}}{x_{+}}-\left(1-a-3x_{+}^{2}\right)}\right].$
(32)

---
Figure 7: The plot of specific heat $\mathcal{\tilde{C_{P}}}$ vs horizon
$x_{+}$ for different values of parameter $a$ and NED parameter $\tilde{k}$.

The specific heat of the Schwarzchild AdS black holes can be obtained in the
classical limits $a=\;\tilde{k}\,=\,0$ [21] whereas when only $a\,=\,0$ it
resembles the specific heat of nonsingular-AdS black holes [35]

$\mathcal{\tilde{C_{P}}}=-2\pi
x_{+}^{2}\left[\frac{1+3x_{+}^{2}}{1-3x_{+}^{2}}\right]$ (33)

---
Figure 8: The generic plot of Gibbs free energy $\tilde{G}$ vs horizon
$\tilde{T}$ for $\tilde{P}<\tilde{P}_{c}$.

To analyse the behaviour, we depict the heat capacity $\tilde{C_{P}}$ against
$x_{+}$ in Fig. 7 for different values of $\tilde{k}$ and $a$. It turns out
that there exists $a_{c}(\tilde{k}_{c})$, for given $\tilde{k}(a)$, such that
the heat capacity is discontinuous at the critical radii $x_{c1}$ and
$x_{c2}$, corresponding to the local maxima and minima of Hawking temperature,
with $x_{c1}<x_{c2}$, when $a<a_{c}$ ($\tilde{k}<\tilde{k}_{c}$) . This
signals a second-order phase transition [35, 3]. As can be seen from the Fig.
7, we have three different cases:

  * •

When $a<a_{c}$ ($\tilde{k}<\tilde{k}_{c}$), we have the second-order phase
transition between SBH – IBH and IBH–LBH such that the black holes with
$x<x_{c1}$ (SBH) and $x>x_{c2}$ (LBH) are locally stable with positive
specific heat whereas the black holes with $x_{c1}<x<x_{c2}$ having negative
specific heat are locally unstable .

  * •

For $a>a_{c}$ ($\tilde{k}>\tilde{k}_{c}$), no phase transition occurs and the
specific heat is always positive signifying the local stability of the black
holes.

  * •

The $a=a_{c}$ case for $\tilde{k}=0.1$ corresponds to existence of two stable
black holes (SBH and LBH) that coexist at the inflection point $x\approx
0.2988$.

The equation of state to investigate the $P-V$ criticality of the black hole,
by using the equations of pressure, volume and temperature [40], is found to
be

$\tilde{\textit{P}}=\frac{3\tilde{\textit{T}}}{2\left[3x{{}_{+}}-\tilde{k}\right]}+\frac{3\left(1-a\right)\left[x_{+}-\tilde{k}\right]}{8\pi
x{{}_{+}}^{2}\left[3x_{+}-\tilde{k}\right]},$ (34)

where $\tilde{\textit{P}}=\textit{P}l^{2}$. The $\tilde{P}-\tilde{V}$ diagram
depicted in Fig. 9 plots the various isotherms of the equation of state Eq.
(34). For values of temperature $\tilde{T}$ higher than the critical
temperature $\tilde{T}_{c}$, the plot shows the isotherms follow ideal gas
behaviour. The isotherms undergo liquid-gas like phase transition [40] for
$\tilde{T}\,<\,\tilde{T}_{c}$ governed by the Maxwell’s equal area law [61]
and have a point of inflection at $\tilde{T_{c}}$, which can be obtained from
the equation

$\left.\frac{\partial\tilde{P}}{\partial
x_{+}}\right|_{\tilde{T}}\,=\,0\,=\,\left.\frac{\partial^{2}\tilde{P}}{\partial
x_{+}^{2}}\right|_{\tilde{T}}.$ (35)

The LBH are depicted by the branch with low pressure and SBH are shown by the
branch with high pressure. The oscillating branch between SBH and LBH show the
black hole is going through Van der Walls-like first order phase transition
[61]. The Eq. (35) yields the expression for the critical temperature
$\tilde{T}_{c}$

$\tilde{T_{c}}\,=\dfrac{(1-a)(3x_{c}^{2}-5\tilde{k}x_{c}+\tilde{k^{2}})}{6\pi
x_{c}^{3}},$ (36)

and further we can get the equation for critical pressure $\tilde{P}_{c}$ in
terms of critical radius $x_{c}$ as

$\tilde{P}_{c}\,=\,\frac{(1-a)(81\tilde{k}x_{c}-81\tilde{k}^{2}-26x_{c}^{2})}{216\tilde{k}\pi(\tilde{k}-3x_{c})x_{c}^{2}}.$
(37)

Inserting the value of critical radius $x_{c}=3\tilde{k}$ in the above
expressions for critical temperature, critical pressure and critical volume we
obtain the following expressions

$\tilde{P}_{c}=\frac{1-a}{216\pi\tilde{k}^{2}},~{}~{}~{}~{}~{}\tilde{V}_{c}=36\,e^{1/3}\pi\tilde{k}^{3},~{}~{}~{}~{}~{}\tilde{T}_{c}=\frac{13(1-a)}{162\pi\tilde{k}}.$
(38)

The universal constant $\epsilon$ for the nonsingular AdS black hole endowed
with clouds of string calculated as

$\epsilon=\frac{\tilde{P}_{c}\,{\tilde{V}_{c}}^{1/3}}{{\tilde{T}_{c}}}=0.311,$
(39)

which is slightly less than $3/8$ – the value for Van der Walls fluid [35]. It
turns out that the temperature of isotherms decreases from top to bottom (cf.
Fig. 9), and ${\tilde{T}_{c}}$ decreases with increasing $a$. Clearly, the
quantities $\tilde{P}_{c}$ $\tilde{V}_{c}$ ${\tilde{T}_{c}}$ are corrected
because background clouds of strings.

|
---|---
Figure 9: The isotherms of black holes in $\tilde{P}$ \- $\tilde{V}$ plane for
two different values of parameter $a=0.6$ where $\tilde{T}_{c}=0.0340578$
(left) and $a=0.8$ where $\tilde{T}_{c}=0.0170289$ (right) with fixed
$\tilde{k}=0.3$. The temperature of isotherms decreases from top to bottom

##  V Critical Exponents

We analyse the thermodynamic quantities near the critical point with the help
of critical exponents $\alpha$, $\beta$, $\gamma$, and $\delta$ [62]. These
exponents do not depend on the physical system instead can depend on the
dimension of the system or the range of the interaction. The exponents
$\alpha$ and $\beta$ determine the behaviour of specific heat at constant
volume [21] and behaviour of the order parameter $\eta$ which is the
difference between the volume of the gas phase and the volume of the liquid
phase respectively. $\gamma$ explains the behaviour of isothermal
compressibility $\kappa_{\tilde{T}}$ and $\delta$ governs the behaviour of the
critical isotherm $\tilde{P}\,-\,\tilde{P}_{c}$ [58]. The critical exponents
can be calculated by using following relations [35, 39]

$\displaystyle\tilde{C}_{\tilde{V}}$ $\displaystyle=$
$\displaystyle\tilde{T}\frac{\partial\tilde{S}}{\partial\tilde{T}}\Big{|}_{\tilde{V}}\propto|t|^{-\alpha}\,,\;\;\;\;\;\eta=\tilde{V}_{l}-\tilde{V}_{s}\propto|t|^{\beta}\,,$
(40) $\displaystyle\kappa_{\tilde{T}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\tilde{V}}\frac{\partial\tilde{V}}{\partial\tilde{P}}\Big{|}_{\tilde{T}}\propto|t|^{-\gamma}\,,\;|\tilde{P}-\tilde{P}_{c}|_{\tilde{T}=\tilde{T}_{c}}\propto|\tilde{V}-\tilde{V}_{c}|^{\delta},$
(41)

where $\tilde{C}_{\tilde{V}}$ is the specific heat at constant volume, $\eta$
is the order parameter, $\kappa_{\tilde{T}}$ is the isothermal
compressibility. $\tilde{V}_{l}$ and $\tilde{V}_{s}$ are the volumes of LBH
and SBH. The specific heat at constant volume is independent of t hence the
critical exponent $\alpha\,=\,0$ [61]. In order to study the critical
exponents $\delta$, $\beta$ and $\gamma$ we define the following set of
dimensionless quantities [35, 58, 61]

$p=\frac{\tilde{P}}{\tilde{P_{C}}},~{}~{}~{}1+\epsilon=\frac{x_{+}}{x_{c}},~{}~{}~{}1+\omega=\frac{\tilde{V}}{\tilde{V}_{c}},~{}~{}~{}1+t=\frac{\tilde{T}}{\tilde{T}_{c}},$
(42)

where $\mid\epsilon\mid$, $\mid\omega\mid$, $\mid t\mid\,\ll 1$. Our analysis
requires a relation between $\epsilon$ and $\omega$ which is obtained by using
Eqs. (31) and (42) as

$\omega=4\left(1-\frac{\tilde{k}}{2x_{c}}\right)\epsilon.$ (43)

The equation of state given by Eq. (34) in terms of dimensionless quantities
defined by Eq. (42) becomes

$p=1+\mathcal{U}t-\mathcal{V}t\epsilon-\mathcal{W}\epsilon^{3}+\mathcal{O}(t\epsilon^{2},\epsilon^{4}),$
(44)

where
$~{}\mathcal{U}=\dfrac{3}{2\left(3x_{c}-\tilde{k}\right)}\frac{\tilde{T}_{c}}{\tilde{P}_{c}}$,
$\mathcal{V}=\dfrac{9x_{c}}{2\,\left(3\,\mathrm{x_{c}}-\tilde{k}\right)^{2}}\frac{\tilde{T}_{c}}{\tilde{P}_{c}}$,
and

$\mathcal{W}=\dfrac{162\pi x_{c}^{5}\tilde{T}_{c}\tilde{P}_{c}}{4\pi
x_{c}^{2}\left(3x_{c}-\tilde{k}\right)^{4}}\left[3(-1+a)\left[2\tilde{k}^{4}+x_{c}\left(-23\tilde{k}^{3}+6x_{c}(16\tilde{k}^{2}-27x_{c}\tilde{k}+9x_{c})\right)\right]\right].$
(45)

The pressure of the system remains constant during transition of phase from
SBH to LBH when $t\,<\,0$. Hence, from Eq. (34), we can write

$1+\mathcal{U}t+\mathcal{V}\epsilon_{s}-\mathcal{W}\epsilon_{s}^{3}=1+\mathcal{U}t+\mathcal{V}\epsilon_{l}-\mathcal{W}\epsilon_{l}^{3},$
(46)

where $\epsilon_{s}$ and $\epsilon_{l}\,$, respectively, are related to the
radii of SBH and LBH. To calculate the exponent $\beta$, the well known
Maxwell’s area law is used [58, 61].

$\int_{\omega_{s}}^{\omega_{l}}\omega d\tilde{P}\,=\,0.$ (47)

By using Eq. (44) and Eq. (43) in Eq. (47), we get

$\frac{1}{2}\mathcal{V}\epsilon_{l}^{2}+\frac{3}{4}\mathcal{W}\epsilon_{l}^{4}=\frac{1}{2}\mathcal{V}t\epsilon_{s}^{2}+\frac{3}{4}\mathcal{W}\epsilon_{s}^{4}.$
(48)

By solving Eqs. (46) and (34), one gets

$-\epsilon_{s}\,=\,\epsilon_{l}\,=\,\sqrt{\dfrac{-\mathcal{V}t}{\mathcal{W}}}.$
(49)

Hence, the order parameter $\eta$ can be calculated as

$\eta\,=\,\tilde{V_{l}}-\tilde{V_{s}}\,=\,\tilde{V}_{c}(\omega_{l}-\omega_{s})\,\propto\,\sqrt{-t},$
(50)

which means $\beta=1/2$. To calculate $\gamma$ we use the equation

$\kappa_{\tilde{T}}\,=\,-\dfrac{1}{\tilde{V}}\,\dfrac{\partial\tilde{V}}{\partial\tilde{P}}\,|_{\,\tilde{T}}\,=\,\dfrac{4}{\tilde{P}\left(1+\omega\right)}\left(1-\dfrac{\tilde{k}}{2\,x_{c}^{2}}\right)\left(\dfrac{\partial\epsilon}{\partial
p}\,|_{\,t}\right)\,\propto\,\dfrac{1}{\mathcal{V}\,t},$ (51)

which leads to $\gamma\,=\,1$. To compute $\delta$ we find the value of
$|\,\tilde{P}\,-\,\tilde{P}_{c}|$ at $\tilde{T}\,=\,\tilde{T}_{c}$ or
$t\,=\,0$, with the help of the equation

$|\tilde{P}-\tilde{P}_{c}|_{\tilde{T}_{c}}\varpropto\epsilon^{3}\propto|\tilde{V}-\tilde{V}_{c}|^{3},$
(52)

which implies $\delta=3.$ The values of the critical exponents of the
nonsingular-AdS black holes with clouds of strings background are similar to
the Van der Walls fluid.

##  VI Conclusion

The idea that the Schwarzschild black hole (point mass) may have atmospheres
composed of clouds of strings because of the intense level of activity in
string theory. The clouds of strings are the one-dimensional analogue of a
dust cloud, have the same energy-momentum tensor as that of the global
monopole, which could be effective on gravitational fields such as black
holes. Further, it could describe a globular cluster with components of dark
matter. Further, it is firmly believed that the one-dimensional objects known
as strings can describe the Universe; hence the study of black hole solution
in clouds of strings background has a tremendous physical significance.
Motivated by this and AdS/CFT correspondence, we have derived an exact
nonsingular-AdS black hole solution endowed with clouds of strings background.
In turn, we have investigated the extended phase space of thermodynamics and
studied the critical phenomena of the nonsingular-AdS black holes surrounded
by clouds of strings background by treating the cosmological constant as a
thermodynamical pressure.

Interestingly, the thermodynamical quantities viz. temperature, pressure,
specific heat and the Gibbs free energy are corrected except the entropy,
which is not directly affected by the background clouds of strings, and then
we probe the thermodynamic stability of the black hole. We find an analogy
between the nonsingular-AdS black holes endowed with clouds of strings and Van
der Waals fluid. Investigation of the Hawking temperature revealed critical
values of NED parameter $\tilde{k}$ and clouds of strings parameter ($a$) and
the critical value of horizon radius $x_{c}$ where the temperature has local
extrema and also divergence of specific heat thereby confirming the existing
of two second order phase transition. Interestingly, the heat capacity
diverges at two critical radii $x_{c1}$ and $x_{c2}$, for $a<a_{c}$,
respectively, at which the temperature has the local maximum and the local
minimum values indicate that two second-order phase transitions exist in the
canonical ensemble. The black hole unstable due to thermal fluctuation when
$x_{c1}<x<x_{c2}$ with negative specific heat and stable otherwise. The
behaviour of Gibbs free energy revealed that the LBH with less Gibbs free
energy are more stable than the SBH. We also analysed isobars on
$\tilde{G}-\tilde{T}$ plane and found that when pressure $\tilde{P}$ is less
than critical pressure $\tilde{P}_{c}$, the SBH underwent first-order phase
transition (exhibiting swallow tail) to LBH at temperature $\tilde{T}_{0}$,
and it was also found that an unstable IBH possible. Thus, our model still
exhibits SBH/LBH phase transition but at a larger horizon radius, and also two
second-order phase transitions occur at a smaller horizon radius. We found
that the background clouds of strings impact the $P-V$ critically and
calculated the values of critical exponents to see that these values match
that of Van der Walls fluid.

Thus, we demonstrated that background clouds of strings profoundly influence
the $P$–$V$ criticality and thermodynamic properties of black holes, which may
have several astrophysical consequences, for example, on wormholes and
accretion onto black holes. Some of the results presented here generalise
previous discussions on nonsingular AdS black holes [35] and Schwarzschild AdS
black holes [20] to a more general setting. The possibility of further
extensions of these results to higher curvature gravity is an interesting
problem for the future.

##  VII Acknowledgments

S.G.G. would like to thank SERB-DST for the ASEAN project
IMRC/AISTDF/CRD/2018/000042.

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