# Thin-Shell Gravastar Model in $f(Q,T)$ Gravity

Sneha Pradhan 0000-0002-3223-4085<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Debasmita Mohanty
0009-0006-8118-5327<EMAIL_ADDRESS>Department of Mathematics, Birla
Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Faculty of Mathematics & Computer
Science, Transilvania University of Brasov, Eroilor 29, Brasov, Romania

###### Abstract

In the last few decades, gravastars have been proposed as an alternative to
black holes. The stability of the gravastar has been studied in many modified
theories of gravity along with Einstein’s GR. The $f(Q,T)$ gravity, a
successfully modified theory of gravity for describing the current accelerated
expansion of the Universe, has been used in this article to study gravastar in
different aspects. According to Mazur and Mottola [1, 2], it has three regions
with three different equations of state. Here in this work, we have studied
the interior of the gravastar by considering the $p=-\rho$ EoS to describe the
dark sector for the interior region. The next region is a thin shell of
ultrarelativistic stiff fluid, in which we have investigated several physical
properties, viz., the proper length, energy, entropy, surface energy density,
etc. In addition, we have studied the surface redshift and speed of sound to
check the potential stability of our proposed thin-shell gravastar model.
Apart from that, we have used the entropy maximization technique to verify the
stability of the gravastar model. The gravastar’s outer region is a complete
vacuum described by exterior Schwarzschild geometry. Finally, we have
presented a stable gravastar model which is singularity-free and devoid of any
incompleteness in classical black hole theory.

Keywords: Gravastar; Stability; $f(Q,T)$ gravity.

##  I Introduction

There has been a large scientific interest in understanding the problems in
both cosmology and astrophysics during the past few decades. Compact objects
are a crucial source for this reason because they provide a platform to test
many pertinent ideas in the high-density domain. The Gravitationally Vacuum
Condense Star, or simply gravastar, is an excellent notion for an extremely
compact object that addresses the singularity problems in classical black hole
(CBH) theory. It was first postulated by Mazur and Mottola [1, 2]. They
construct a cold, compact object with an internal de Sitter condensate phase
and an exterior Schwarzschild geometry of any total mass M that is free of all
known limitations on the known CBH. As a result, this hypothesis has gained
popularity among researchers and it could be seen as an alternative for the
CBH.

The gravastar, in particular, has three separate zones with different
equations of states (EoS), according to Mazur and Mottola’s model:

  1. 1.

An internal region that is full of dark energy with an isotropic de Sitter
vacuum situation.

  2. 2.

An intermediate thin shell consists of stiff fluid matter.

  3. 3.

The outer area is completely vacuum, and Schwarzschild geometry represents
this situation appropriately.

Recent studies on the brightness of type Ia distant supernovae [3, 4, 5]
indicate that the universe is expanding more quickly than previously thought,
which suggests that the universe’s pressure $p$ and energy density $\rho$
should contradict the strong energy condition, that is, $\rho+3p<0$. ”Dark
energy” is the substance that causes this requirement to be fulfilled at some
point in the evolution of the universe [6, 7, 8]. There are several substances
for the status of dark energy. The most well-known contender is a non-
vanishing cosmological constant, which is equivalent to the fluid that
satisfies the EoS $p=-\rho$. There are two interfaces (junctions) located at
$R_{1}$ and $R_{2}$ apart from the center, where $R_{1}$ and $R_{2}$ stand for
the thin shell’s interior and outer radii. The presence of stiff matter on the
shell with thickness $R_{2}-R_{1}=\epsilon<<1$ is required to provide the
system’s stability, which is achieved by exerting an inward force to
counteract the repulsion from within.

Astrophysicists proposed a new solution of a compact, spherically symmetric
astrophysical phenomenon, known as a gravastar, to solve the singularity
problem in black hole geometry. There are several arguments for and against
the theory that gravitational waves (GW), detected by LIGO, are the
consequence of merging gravastars or black holes, despite the fact that no
experimental observations or discoveries of gravastars have yet been made. A
method for identifying gravastar was devised by Sakai et al. [9] by looking at
the gravastar shadows. Since black holes don’t exhibit microlensing effects of
maximal brightness, Kubo and Sakai [10] hypothesized that gravitational
lensing may be used to find gravastars. The finding of GW150914 [11, 12] by
interferometric LIGO detectors increased the likelihood that ringdown signals
originated from sources without an event horizon. In a recent examination of
the picture taken by the First M87 Event Horizon Telescope (EHT), a shadow
that resembled a gravastar has been discovered [13].
One could observe that there are numerous publications on the gravastar
available in the literature that focuses on various mathematical and physical
problems in the framework of general relativity postulated by Albert Einstein
[14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Bilic et al. [16] replaced the de
Sitter interior with a Chaplygin gas equation of state and saw the system as a
Born-Infield phantom gravastar to examine the gravastar’s interior, whereas
Lobo [17] replaces the inner vacuum with dark energy. Although it is commonly
known that Einstein’s general relativity is an exceptional tool for revealing
many hidden mysteries of nature, certain observable evidence of the expanding
universe and the existence of dark matter has put a theoretical challenge to
this theory. Hence, a number of modified theories have been put over time,
like $f(R),f(Q),f(T),f(R,T),f(Q,T)$ gravity, etc. The $f(R),f(R,T)$ gravity
which is based upon the Riemannian geometry in which Ricci scalar curvature
plays an important role. Another way to represent the gravitational
interaction between two particles in space-time is by torsion and non-
metricity upon which $f(T)$ and $f(Q)$ gravity theory has been built,
respectively. In the current project, our objective is to investigate the
gravastar using one of the alternative theories of gravity, $f(Q,T)$ gravity,
and to examine many physical characteristics and stability of the object. The
$f(Q,T)$ gravity is the extension of the symmetric teleparallel gravity in
which the gravitational action is determined by any function $f$ of the
nonmetricity $Q$ and the trace of the matter energy-momentum tensor $T$, such
that $L=f(Q,T)$. There are very few articles in which compact object has been
studied under the framework of $f(Q,T)$ gravity [24]. Xu et al. have
investigated the cosmological implication of this theory, and they have
obtained the cosmological evolution equation for isotropy, homogeneous, flat
geometry [25]. In [26], the author has investigated the different FRW models
with three specific forms of $f(Q,T)$ gravity models. One could see the
references to the recent work of gravastar in the framework of modified
gravity [27, 28, 29, 30, 31, 32]. In the article, [33] researchers have
studied the gravastar model in $f(Q)$ gravity. Ghosh et al. [34] has studied
the gravastar in Rastall gravity. In the work [35], the author has studied
traversable wormhole solutions in the presence of the scalar field. Wormhole
solutions in $f(R,T)$ gravity have been studied in [36]. Elizalde et al. [37]
discussed the cosmological dynamics in $R^{2}$ gravity with logarithmic trace
term. Godani and Samanta [38] discussed the gravitational lensing effect in
traversable wormholes. In [39], the researchers have investigated wormhole
solutions with scalar field and electric charge in modified gravity. In [40],
the authors studied the cosmologically stable $f(R)$ model and wormhole
solutions. Salvatore et al. [41] studied the non-local gravity wormholes, and
they obtained stable and traversable wormhole solutions. Shamir et al. [42]
has explored the behavior of anisotropic compact stars in $f(R,\phi)$ gravity.
The Bardeen compact stars in Modified $f(R)$ gravity have been researched in
the work [43].

Our paper is organized as follows: In sec I we have given a brief introduction
to the gravastar model and the recent research work regarding that. After that
in sec II we provide the geometrical aspects of $f(Q,T)$ gravity. In sec III
we have derived the modified field equation and the modified energy
conservation equation in $f(Q,T)$ gravity. Sec IV gives the solution of the
field equation for different regions using different EoS. After that in sec V
we have studied the junction requirement and EoS and we have obtained the
limiting range for the radius of the gravastar. The physical features of the
model has been analyzed in sec VI. The most important thing is to check the
stability of the model which is given in sec VII. Finally, we provide the
conclusion of our analysis in sec VIII.

##  II Construction of $f(Q,T)$Gravity

The $f(Q,T)$ theory of gravity which introduces an arbitrary function of
scalar non-metricity $Q$ and trace $T$ of the matter energy-momentum tensor,
is an intriguing modification to Einstein’s theory of gravity. The action of
$f(Q,T)$ theory coupled with matter Lagrangian $\mathcal{L}_{m}$ is given by
[44]

$S=\int\sqrt{-g}\left[\frac{1}{16\pi}f(Q,T)+\mathcal{L}_{m}\right]d^{4}x,$ (1)

where $g$ represents the determinant of $g_{\mu\nu}$. The non-metricity and
disformation tensor is defined as

$\displaystyle
Q\equiv-g^{\mu\nu}\left(L^{\alpha}_{\,\,\beta\mu}L^{\beta}_{\,\,\nu\alpha}-L^{\alpha}_{\,\,\beta\alpha}L^{\beta}_{\,\,\mu\nu}\right),$
(2) $\displaystyle
L^{\lambda}_{\,\,\,\,\mu\nu}=-\frac{1}{2}g^{\lambda\gamma}\left(\nabla_{\nu}g_{\mu\gamma}+\nabla_{\mu}g_{\gamma\nu}-\nabla_{\gamma}g_{\mu\nu}\right).$
(3)

The non-metricity tensor is defined as the covariant derivative of the metric
tensor, and its explicit form is

$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}.$ (4)

with the trace of a non-metricity tensor as

$Q_{\lambda}=Q_{\lambda\,\,\,\,\,\mu}^{\,\,\,\mu},\quad\quad\tilde{Q}_{\lambda}=Q^{\mu}_{\,\,\,\,\lambda\mu}.$

The Superpotential $P_{\,\,\mu\nu}^{\lambda}$ is defined as

$P_{\,\,\,\,\mu\nu}^{\lambda}=-\frac{1}{2}L^{\lambda}_{\,\,\,\,\mu\nu}+\frac{1}{4}\left(Q^{\lambda}-\tilde{Q^{\lambda}}\right)g_{\mu\nu}-\frac{1}{4}\delta^{\lambda}_{\,\,(\mu\,}Q_{\nu)},$
(5)

giving the relation of scalar non-metricity as

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$ (6)

The field equations of $f(Q,T)$ theory by varying the action (1) with respect
to the metric tensor inverse $g^{\mu\nu}$ is obtained as

$-\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(f_{Q}\sqrt{-g}\,P^{\lambda}_{\,\,\,\,\mu\nu}\right)-\frac{1}{2}f\,g_{\mu\nu}+f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)\\\
-f_{Q}\left(P_{\mu\lambda\alpha}Q_{\nu}^{\,\,\,\lambda\alpha}-2Q^{\lambda\alpha}_{\,\,\,\,\,\,\,\,\mu}\,P_{\lambda\alpha\nu}\right)=8\pi
T_{\mu\nu}.$ (7)

The terms used in the above are defined as

$\displaystyle\Theta_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{\mu\nu}},\quad
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}},$ (8) $\displaystyle f_{T}$ $\displaystyle=$
$\displaystyle\frac{\partial f(Q,T)}{\partial T},\quad f_{Q}=\frac{\partial
f(Q,T)}{\partial Q}.$ (9)

Where $T_{\mu\nu}$ is known as the energy-momentum tensor.

##  III Modified Field Equation in $f(Q,T)$

To derive the modified field equation let’s take the static, spherically
symmetric line element given by,

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (10)

To describe the fluid distribution we are going to take the energy-momentum
tensor in the form :

$T_{\mu\nu}=(\rho+p_{t})u_{\mu}u_{\nu}-p_{t}\delta_{\mu\nu}+(p_{r}-p_{t})v_{\mu}v_{\nu},$
(11)

where $\rho$ is the density of the fluid, $p_{r}$ and $p_{t}$ are the
pressures of the fluid in the direction of $u_{\mu}$(radial pressure) and
orthogonal to $u_{\nu}$ (tangential pressure) respectively. $u_{\mu}$ is the
time like four-velocity vector. $v_{\mu}$ is the unit space-like vector in the
direction of the radial coordinate. Therefore the stress energy momentum
tensor $T_{\mu\nu}$ and the components of $\Theta_{\mu\nu}$ can be expressed
as,

$\begin{gathered}T_{\mu\nu}=diag(e^{\nu}\rho,e^{\lambda}p_{r},r^{2}p_{t},r^{2}p_{t}sin^{2}\theta)\\\
\Theta_{11}=-e^{\nu}(P+2\rho),\,\Theta_{22}=e^{\lambda}(P-2p_{r}),\\\
\Theta_{33}=r^{2}(P-2p_{t}),\,\,\Theta_{44}=r^{2}sin^{2}\theta(P-2p_{t}).\end{gathered}$
(12)

Where we have taken the Lagrangian matter density
$\mathcal{L}_{\mathrm{m}}=-P=-\frac{p_{r}+2p_{t}}{3}$. By utilizing the
aforementioned constraints the derived modified field equation for spherically
symmetric metric in $f(Q,T)$ gravity is,

$8\pi\rho=\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime})\\\
+(1+e^{\lambda})r\lambda^{\prime}]+fr^{2}e^{\lambda}]-f_{T}[P+\rho],$ (13)
$8\pi
p_{r}=-\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)\\\
(2+r\nu^{\prime}+r\lambda^{\prime})-2r\nu^{\prime}]+fr^{2}e^{\lambda}]+f_{T}[P-p_{r}],$
(14) $8\pi
p_{t}=-\frac{1}{4re^{\lambda}}[-2rf_{QQ}Q^{\prime}\nu^{\prime}+f_{Q}[2\nu^{\prime}(e^{\lambda}-2)-r\nu^{\prime
2}+\lambda^{\prime}(2e^{\lambda}+r\nu^{\prime})-2r\nu^{\prime\prime}]+2fre^{\lambda}]+f_{T}[P-p_{t}].$
(15)

Now here we are going to take a particular functional form of $f(Q,T)$ gravity
as $f(Q,T)=\alpha\,Q+\beta\,T$. One can see there are many references [44, 24,
25] in which this cosmological models has been studied widely. Then we can
rewrite the field equation in a as like :

$e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=\rho^{eff},$
(16)
$e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=p_{r}^{eff},$
(17)
$e^{-\lambda}\left(\frac{\nu"}{2}-\frac{\lambda^{\prime}\nu^{\prime}}{4}+\frac{\nu^{\prime
2}}{4}+\frac{\nu^{\prime}-\lambda^{\prime}}{2r}\right)=p_{t}^{eff},$ (18)

Where,

$\rho^{eff}=\frac{8\pi\rho}{\alpha}+\frac{\beta}{3\alpha}(3\rho+p_{r}+2p_{t})-\frac{\beta}{2\alpha}(\rho-
p_{r}-2p_{t}),$ (19) $p_{r}^{eff}=\frac{8\pi
p_{r}}{\alpha}-\frac{2\beta}{3\alpha}(p_{t}-p_{r})+\frac{\beta}{2\alpha}(\rho-
p_{r}-2p_{t}),$ (20) $p_{t}^{eff}=\frac{8\pi
p_{t}}{\alpha}-\frac{\beta}{3\alpha}(p_{r}-p_{t})+\frac{\beta}{2\alpha}(\rho-
p_{r}-2p_{t}).$ (21)

One can verify that for $\alpha=1,\beta=0$ i.e. for $f=Q$ the above field
equation reduces to Einstein’s GR. However in this article, we limit ourselves
to the isotropic scenario in order to establish the simplest possibility where
$p_{r}=p_{t}$. Now the energy conservation equation is given by,

$\frac{dp^{eff}}{dr}+\frac{\nu^{\prime}}{2}(p^{eff}+\rho^{eff})=0.$ (22)

By using the equation (19) and (20) we get the modified energy conservation
equation in $f(Q,T)$ gravity as:

$\frac{dp}{dr}+\frac{\nu^{\prime}}{2}\left[(1+\frac{\beta}{8\pi})(p+\rho)\right]+\frac{\beta}{16\pi}(\rho^{\prime}-3p^{\prime})=0.$
(23)

The above equation is different from that obtained in GR and can be retrieved
in the limit $\beta\to 0$.

##  IV Geometry of Gravastar

We are specifically interested in the geometrical interpretation and their
related analytical solution in the three different zones for the gravastar
under study. It is simple to imagine the idea that the inside of the star is
considered to be encircled by a thin shell made of ultrarelativistic stiff
fluid, but the outside area is completely vacuum. Schwarzschild’s measure is
therefore assumed to be appropriate for this outer area. The shell’s structure
is believed to be extremely thin, with a limited width ranging $R_{1}=R\leq
r\leq R+\epsilon=R_{2}$, where $r$ is the radial coordinate and $R_{1},R_{2}$
denotes the inner and outer radius of the shell.

###  IV.1 Interior Region

In the primary model proposed by Mazur and Mottola[1, 2] the three different
zones obey the standard cosmological EoS $p=\omega\rho$, where $\omega$ is the
EoS parameter takes different value for different region. Here, we suppose
that an enigmatic gravitational source is present in the interior area. Dark
matter and dark energy are typically assumed to be separate entities, although
there is the possibility that they are both just various representations of
the same thing. For describing the dark sector in the interior region we are
interested to consider the EoS is given by,

$p=-\rho.$ (24)

For the aforementioned EoS obtaining constant critical density $\rho_{c}$ from
the energy conservation Eq.(23), we get the pressure for the interior region
as,

$p=-\rho_{c}.$ (25)

Using Eq.(25) in field equations (16) and (19) we obtained the final
expression for metric potential $e^{-\lambda(r)}$ as,

$e^{-\lambda(r)}=\frac{2(\beta-4\pi)\rho_{c}r^{3}-3c_{1}}{3\alpha r}+1.$ (26)

To make our solution regular at center we set the integrating constant
$c_{1}=0$. Thus we have,

$e^{-\lambda(r)}=\frac{2(\beta-4\pi)r^{2}\rho_{c}}{3\alpha}+1.$ (27)

Again using (27) we get another metric potential from (17,20) as,

$e^{\nu(r)}=C_{1}\left[2(4\pi-\beta)\rho_{c}r^{2}-3\alpha\right].$ (28)

It is clear from the aforementioned results that there is no singularity in
the inner solutions, which overcomes the issue of a classical black hole’s
central singularity. For more clearance, we have plotted the variation of the
metric potential $e^{\lambda}$ with respect to the radial parameter $r$ in
Fig.(1).

Figure 1:  Variation of the metric potential ($e^{\lambda}$) with regard to
the radial parameter $r$ for $\alpha=-4.5$, $\beta=3.4$, $\rho_{c}=0.001$ .

One could physically extrapolate from the figure that there is no central
singularity, along with that the metric potential is regular at $r=0$, and it
is finite and positive across the whole interior area. Additionally, The
following equation could be used to determine the active gravitational mass of
the internal region:

$\mathcal{M(R)}=\int_{0}^{\mathcal{R}}4\pi r^{2}\rho
dr=\frac{4}{3}\pi\mathcal{R}^{3}\rho_{c}.$ (29)

Where $\mathcal{R}$ represents the radius for the interior area and $\rho_{c}$
is the critical density.

###  IV.2 Shell

The shell is made of ultra-relativistic stiff matter and abides by the EoS
$p=\rho$. Zel’dovich [45, 46] was the pioneer of the concept of this extremely
relativistic fluid known as the stiff fluid in correspondence to the cold
baryonic universe. We can claim that in the current situation, this could
result from thermal excitation with a very low chemical potential or from the
preserved number density of the gravitational quanta at absolute zero. This
kind of fluid has been widely explored by several researchers to investigate
different cosmological [47, 48, 49] and astrophysical [50, 51, 52] aspects.
One may note that it is extremely challenging to solve the field equations in
the non-vacuum area or the shell. However, an analytical solution could be
found within the specifications of the thin shell limit, i.e.
$0<e^{-\lambda(r)}<<1$. We can argue that the interior area between the two
space-times must be a thin shell, as suggested by Israel [55]. Moreover, in
general, any parameter that is a function of $r$ could be considered $<<1$ as
$r\to 0$. By considering this type of approximation our field Eq.(16)-Eq.(18)
along with the Eq.(19)-Eq.(21) reduces to:

$\alpha\left(\frac{e^{-\lambda}\lambda^{\prime}(r)}{r}+\frac{1}{r^{2}}\right)=8\pi\rho+\frac{\beta}{2}(5p+\rho),$
(30) $\alpha\left(\frac{-1}{r^{2}}\right)=8\pi p+\frac{\beta}{2}(\rho-3p),$
(31)
$\alpha\left(\frac{-\lambda^{\prime}\nu^{\prime}e^{-\lambda(r)}}{4}-\frac{e^{-\lambda}\lambda^{\prime}}{2r}\right)=8\pi
p+\frac{\beta}{2}(\rho-3p).$ (32)

Utilizing the Eqs.(30)-(32) we achieve the two metric potential as

$e^{-\lambda(r)}=\frac{2(\beta+8\pi)\log(r)}{8\pi-\beta}-C_{2},$ (33)

$e^{\nu(r)}=C_{3}\left(r(\beta+8\pi)\right)^{-\frac{32\pi}{\beta+8\pi}}.$ (34)

Where $C_{2}$ and $C_{3}$ are integrating constants. Furthermore, by plugging
the EoS $p=\rho$ and using Eq.(34) into the energy conservation equation (23)
we have obtained the pressure/matter density for the shell region as,

$p(r)=\rho(r)=\rho_{0}\left(8\pi r-\beta r\right)^{\frac{32\pi}{8\pi-\beta}}.$
(35)

Where $\rho_{0}$ is the constant of integration. Fig.(2) shows the variation
of pressure or matter density. One can see that the matter density of the
shell is monotonically growing up toward the outer boundary of the shell. The
shell is made of ultra-relativistic stiff fluid, since the pressure or matter
density is monotonically increasing towards the outer surface we can
physically interpret that the amount of stiff matter is rising towards the
outer border rather than the internal region of the shell.

Figure 2:  Variation of the pressure or matter density $p=\rho$
($\text{km}^{-2}$) with regard to the thickness $\epsilon(km)$ of the shell
for $\beta=3.4$, $\rho_{0}=1$.

###  IV.3 Exterior Region

The EoS $p=\rho=0$ is believed to be obeyed by the outside of the gravastar,
indicating that the external portion of the shell is entirely vacuum. Thus,
utilizing Eq.(16)-Eq.(17) along with the Eq.(19)-Eq.(20), we obtain

$\lambda^{\prime}+\nu^{\prime}=0.$ (36)

The line element for the outside region may be seen as the well-known
Schwarzschild metric, which is provided by the solution to Eq.(35) given by,

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}d\Omega^{2},$
(37)

where $d\Omega^{2}=(d\theta^{2}+sin^{2}\theta d\phi^{2})$ and $M$ denotes the
total mass of the object.

###  IV.4 Boundary Condition

There are two junctions/interfaces in a gravastar configuration. Let us denote
the interface between interior space-time and intermediate thin shell (at
$r=R_{1}$) by junction-$I$ and the interface between the intermediate thin
shell and exterior space-time (at $r=R_{2}$) by junction-$II$. It is necessary
that the metric functions at these interfaces must be continuous for any
stable arrangement. We matched the metric functions at these borders in order
to find the unknown constants of our current study like $C_{1}$, $C_{2}$, and
$C_{3}$, and we ultimately discovered the values of these constants.

  * •

Junction-I :

$\frac{2(\beta+8\pi)\log
R_{1}}{8\pi-\beta}-C_{2}=\frac{2\rho_{c}(\beta-4\pi)R_{1}^{2}}{3\alpha}+1,$
(38)

$C_{3}\left(R_{1}(\beta+8\pi)\right)^{-\frac{32\pi}{\beta+8\pi}}=C_{1}\left[2\rho_{c}(4\pi-\beta)R_{1}^{2}-3\alpha\right].$
(39)

  * •

Junction-II :

$\frac{2(\beta+8\pi)\log R_{2}}{8\pi-\beta}-C_{2}=1-\frac{2M}{R_{2}},$ (40)
$C_{3}\left(R_{2}(\beta+8\pi)\right)^{-\frac{32\pi}{\beta+8\pi}}=1-\frac{2M}{R_{2}}.$
(41)

  * •

Obtained Constants :

$C_{3}=-\frac{(2M-R_{2})((\beta+8\pi)R_{2})^{\frac{32\pi}{\beta+8\pi}}}{R_{2}},$
(42)
$C_{2}=\frac{2(\beta+8\pi)\log(R_{1})}{8\pi-\beta}-\frac{2\rho_{c}(\beta-4\pi)R_{1}^{2}}{3\alpha}-1,$
(43)

$C_{1}=\frac{(2M-R_{2})((\beta+8\pi)R_{1})^{-\frac{32\pi}{\beta+8\pi}}((\beta+8\pi)R_{2})^{\frac{32\pi}{\beta+8\pi}}}{R_{2}\left(3\alpha+2\beta\rho_{c}r^{2}-8\pi\rho_{c}R_{1}^{2}\right)}.$
(44)

Now to find the numerical values of constants $C_{1}$,$C_{2}$ and $C_{3}$ we
have considered the astrophysical object PSR J1416- 2230 [53] with
$M=1.97M_{\odot}$, internal radius $R_{1}=10$ and the exterior radius
$R_{2}=10.01$. Apart from that by varying a number of values of model
parameter $\alpha$ and $\beta$ we have determined a bunch of numerical values
of $C_{1}$, $C_{2}$ and $C_{3}$ which is shown in table-1.
With relation to the example of those numerical solutions of constants for
some particular parameter choices given above, let’s talk about the parameter
space of our solution. One could inquire about the following associated
problems:

  1. 1.

For particular choices of $M,R_{1},R_{2}$, will we get a singular free
solution always ?

  2. 2.

If one varies the model parameter then the results will be unique or not ?

We provide some arguments in answer to these concerns: In the current work, we
have selected values for a number of factors to examine the physical behavior
of gravastar. It will provide a unique solution for a given value of $M$,
$R_{1}$, and $R_{2}$, but we have chosen these values in order to satisfy the
ratios $\frac{2M}{R_{1}}<1,\frac{2M}{R_{2}}<1$ for a stable gravastar model.
Besides there are some other criteria like the surface redshift
$\mathcal{Z_{\mathrm{s}}}<2$ and the square of the speed of sound($v_{s}^{2}$)
must satisfy the inequality $0<v_{s}^{2}<1$. Apart from that for avoiding
central singularity, we should maintain
$\frac{2(\beta-4\pi)\rho_{c}r^{2}}{3\alpha}+1\neq 0$. Moreover, we have taken
$\rho_{0}=1$ and $\rho_{c}=0.001$ in order to maintain $\rho_{0}>>\rho_{c}$ .
We are free to choose any $M$, $R_{1}$, and $R_{2}$ combination that would
provide the same findings as those presented in this research as long as the
aforementioned requirements are valid.

Table 1: Different numerical values of constants for PSR J1416-223 assuming $R_{1}=10\,$ km and $R_{2}=10.01$ km. $\alpha$ | $\beta$ | $C_{1}$ | $C_{2}$ | $C_{3}$
---|---|---|---|---
$-4.5$ | $3.4$ | $0.0396871$ | $4.91029$ | $2.72477\times 10^{8}$
$-4.6$ | $3.3$ | $0.0388762$ | $4.86301$ | $2.88649\times 10^{8}$
$-4.7$ | $3.2$ | $0.0380979$ | $4.81611$ | $3.05892\times 10^{8}$
$-4.8$ | $3.1$ | $0.0373501$ | $4.76958$ | $3.24284\times 10^{8}$
$-4.9$ | $3.0$ | $0.0366311$ | $4.72344$ | $3.4391\times 10^{8}$

##  V Junction Condition and Equation of States

It is established that the gravastar is divided into three regions viz. the
interior (I), the intermediate thin shell (II), and the exterior (III). This
shell keeps the internal area and outer region connected. So, This region is
crucially significant in the construction of the gravastar. According to the
fundamental junction requirement, regions I and III must match smoothly at the
junction. The derivatives of these metric coefficients may not be continuous
at the junction surface, despite the fact that the metric coefficients are
continuous there. In order to calculate the surface stresses at the junction,
we will now employ the Darmois-Israel [54, 55, 56] condition. The Lanczos
equation [57, 58, 59, 60] provides the intrinsic surface stress-energy tensor
$S_{ij}$ in the following manner:

$\displaystyle S_{ij}=-\frac{1}{8\pi}(k_{ij}-\delta_{ij}k_{\gamma\gamma}).$
(45)

In the above expression, $k_{ij}=K^{+}_{ij}-K^{-}_{ij}$ denotes the
discontinuity in some second fundamental expression. Where the second
fundamental expression is given by,

$K^{\pm}_{ij}=-n^{\pm}_{\sigma}\left(\frac{\partial
x_{\sigma}}{\partial\phi^{i}\partial\phi^{j}}+\Gamma^{l}_{km}\frac{\partial
x^{l}}{\partial\phi^{i}}\frac{\partial x^{m}}{\partial\phi^{j}}\right),$ (46)

where $\phi^{i}$ denotes the intrinsic co-ordinate in the shell area, along
with $n^{\pm}$ represents the two-sided unit normal to the surface, which can
be written as,

$n^{\pm}=\pm\left|g^{lm}\frac{\partial f}{\partial x^{l}}\frac{\partial
f}{\partial x^{m}}\right|^{-1/2}\frac{\partial f}{\partial x^{\sigma}},$ (47)

with $n^{\gamma}n_{\gamma}=1$. Utilizing the Lanczos method [57] the surface
energy tensor can be written as $S_{ij}=diag(-\sum,P)$, where the surface
energy density and surface pressure are denoted by $\sum$ and $P$ respectively
and are defined by,

$\sum=-\frac{1}{4\pi\mathcal{R}}\left[\sqrt{e^{-\lambda}}\right]^{+}_{-},$
(48)
$P=-\frac{\sum}{2}+\frac{1}{16\pi}\left[\frac{(e^{-\lambda})^{\prime}}{\sqrt{e^{-\lambda}}}\right]^{+}_{-},$
(49) $\text{Also the EoS}\,(\omega)=\frac{P}{\sum}.$ (50)

Here $-$ and $+$ represent the interior space-time and Schwarzschild space-
time respectively. Calculating the Eq. (48)-(50) we get the expression of the
above quantities as ,

$\sum=\left(-\frac{1}{4\pi
R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\text{{\hbox{\rho}}c}R^{2}}{3\alpha}+1}\right)$
(51)

$P=\frac{1}{16\pi}\left(\frac{2M}{R^{2}\sqrt{1-\frac{2M}{R}}}-\frac{4(\beta-4\pi)\rho_{c}R}{3\alpha\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}}\right)-\frac{1}{2}\left(-\frac{1}{4\pi
R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right),$
(52)

$\omega=\frac{\frac{1}{16\pi}\left(\frac{2M}{R^{2}\sqrt{1-\frac{2M}{R}}}+\frac{4(4\pi-\beta)\rho_{c}R}{\alpha\sqrt{\frac{6(\beta-4\pi)\rho_{c}R^{2}}{\alpha}+9}}\right)}{\left(-\frac{1}{4\pi
R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right)}-\frac{1}{2}.$
(53)

There is some set of conditions known as energy conditions that must be
applied in order for a geometric structure to be physically viable. The well-
recognized energy criterion are :

  1. 1.

NEC: $\sum+P>0$,

  2. 2.

WEC: $\sum>0$, $\sum+P>0$ ,

  3. 3.

SEC: $\sum+P>0$,$\sum+3P>0$,

  4. 4.

DEC:$\sum>0,\sum\pm P>0$ .

The presented model is physically feasible if these energy criteria are
established. Here, we’re investigating to see if the null energy requirement,
which guarantees the presence of ordinary or exotic matter in the thin shell,
is satisfied or not. In this context, it is noteworthy to say that violation
of null energy conditions (NEC) leads to violation of other energy conditions.
It is illustrated in Fig.(3) that the NEC is satisfied over a range of model
parameter values throughout the entire region.

Figure 3: Evolution of the NEC ($\sum+P$) by varying model parameter $\alpha$
. Figure 4:  variation of the Surface energy density ($\sum$) with respect to
thickness $\epsilon(\text{in km})$ for $\alpha=-4.5,\beta=3.4$.

Besides that, we have plotted the variation of surface energy density with
respect to the thickness parameter($\epsilon$) which shows that the surface
energy density is monotonically decreasing towards the boundary of the shell.
The mass of the thin shell now is easily determined using the equation for the
surface energy density given by,

$m_{s}=4\pi
R^{2}\sum=-R\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right).$
(54)

Now for determining the real value of shell mass, we have the inequality,
$m_{s}>0$ from which we get the upper bound of the radius as
$R<\left(\frac{3M\alpha}{\rho_{c}(4\pi-\beta)}\right)^{\frac{1}{3}}.$ Thus we
get the limiting value on the radius as

$2M<R<\left(\frac{3M\alpha}{\rho_{c}(4\pi-\beta)}\right)^{\frac{1}{3}}.$ (55)

##  VI Physical Features of the Model

###  VI.1 Proper thickness

According to Mazur and Mottola’s hypotheses [1, 2], the stiff fluid of the
shell is positioned between the meeting of two space-times. The length of the
shell ranges from $R_{1}=R$ (which is the phase barrier between the interior
area and intermediate thin shell) up to $R_{2}=R+\epsilon$ (which is the phase
border between the exterior space time and intermediate thin shell). So, using
the following formula, one can find the required length or proper thickness of
the shell as well as the proper thickness between these two interfaces:

$\begin{gathered}\textit{l}=\int^{R+\epsilon}_{R}\sqrt{e^{\lambda}}dr,\\\
=\int^{R+\epsilon}_{R}\sqrt{\frac{\beta-8\pi}{(8\pi-\beta)C_{2}-2(\beta+8\pi)\log(r)}}\,dr,\\\
=\left[-e^{-\frac{(\beta-8\pi)C_{2}}{2(\beta+8\pi)}}\sqrt{\frac{\pi(\beta-8\pi)}{2(\beta+8\pi)}}\text{Erf}\sqrt{\frac{(8\pi-\beta)C_{2}}{2(\beta+8\pi)}-\log(r)}\right]_{R}^{R+\epsilon}\end{gathered}$
(56)

The variation of the proper length with respect to the thickness parameter
$\epsilon$ is given in Fig.(5). The figure demonstrates that the proper length
rises monotonically as shell thickness increases.

Figure 5:  variation of the proper length ($l$) with respect to thickness
$\epsilon(\text{in km})$ for $\alpha=-4.5$ and $\beta=3.4$.

###  VI.2 Energy

The energy of the shell can be calculated by the formula,

$\begin{gathered}E=\int^{R+\epsilon}_{R}4\pi r^{2}\rho\,dr,\\\
E=\int^{R+\epsilon}_{R}4\pi r^{2}\rho_{0}\left(8\pi r-\beta
r\right)^{\frac{32\pi}{8\pi-\beta}}\,\,dr,\\\
E=\frac{4\pi\rho_{0}r^{2}((8\pi-\beta)r)^{\frac{32\pi}{8\pi-\beta}+1}}{56\pi-3\beta}.\end{gathered}$
(57)

Figure 6: variation of energy $(E)$ with respect to
thickness$\epsilon(\text{in km})$ for $\alpha=-4.5$ and $\beta=3.4$.

.

The variation of the shell energy is illustrated in Fig.(6). In this graph, it
can be observed that the energy rises as the shell’s thickness increases. The
fluctuation of energy is comparable to the fluctuation in matter density. It
meets the requirement that the energy of the shell must be increased as the
radial distance increases.

###  VI.3 Entropy

The stable configuration for a single condensate area is zero entropy density,
which is present in the gravastar’s innermost region. Entropy on the
intermediate thin shell can be calculated using the formula according to Mazur
and Mottola’s work [1, 2],

$\mathcal{S}=\int^{R+\epsilon}_{R}4\pi r^{2}s(r)\sqrt{e^{\lambda}}dr,$ (58)

here the entropy density at local temperature $T(R)$ is given by the
expression
$s(r)=\frac{\gamma^{2}K_{B}^{2}T(R)}{4\pi\hbar^{2}}=\gamma\sqrt{p/{2\pi}}$
where $\gamma$ is the dimensionless parameter. In this work, we have
considered geometrical units i.e. $G=c=1$ as well as Planckian units
$K_{B}=1,\hbar=1$. Our estimates of the entropy of the thin shell are limited
to the second-order term of the thickness parameter, i.e. the order
$\epsilon^{2}$, using Taylor series approximation. Ultimately, we have
calculated the intermediate thin shell’s entropy as follows:

$\begin{gathered}\mathcal{S}=2\sqrt{2\pi}\gamma
r^{2}\epsilon\sqrt{\frac{(\beta-8\pi)\rho_{0}(8\pi r-\beta
r)^{\frac{32\pi}{8\pi-\beta}}}{(8\pi-\beta)C_{2}-2(\beta+8\pi)\log(r)}}\,\,+\\\
\frac{\epsilon^{2}\left(2\sqrt{2\pi}\gamma r((8\pi-\beta)(\beta-2\beta
C_{2}+8\pi(4C_{2}+1))-4(16\pi-\beta)(\beta+8\pi)\log(r))\sqrt{\frac{(\beta-8\pi)\rho_{0}((8\pi-\beta)r)^{-\frac{32\pi}{\beta-8\pi}}}{(8\pi-\beta)C_{2}-2(\beta+8\pi)\log(r)}}\right)}{2\left((\beta-8\pi)^{2}C_{2}+2\left(\beta^{2}-64\pi^{2}\right)\log(r)\right)}.\end{gathered}$
(59)

Figure 7:  variation of the entropy ($\mathcal{S}$) with respect to thickness
$\epsilon(\text{in km})$ for $\beta=3.4$, $\alpha=-4.5$.

Fig.(7) depicts the evolution of the shell entropy, which shows the growing
behavior of the shell entropy with regard to thickness ($\epsilon$). Another
acceptable condition is that entropy should reach its greatest value on the
surface for a stable gravastar configuration which is demonstrated in our
analysis.

##  VII Stability of the stellar model

In this section, we are interested to investigate the stability of the thin
shell gravastar model by analyzing some physical parameters.

###  VII.1 Study of Herrera’s cracking concept

Recent observational data appear to indicate that the cosmos is expanding more
quickly than before[3, 4, 5]. If general relativity is taken to be the right
theory of gravity characterizing the behavior of the universe on a large
scale, then the energy density and pressure of the cosmos should violate the
strong energy condition. The stable or unstable configuration of gravastars
could be analyzed through the nature of $\eta$, where $\eta$ is an effective
parameter that can be interpreted as the square of the speed of sound i.e.
$\eta=v_{s}^{2}$ [27, 61]. For a stable system $\eta$ should satisfy
$0<\eta\leq 1$. The speed of sound shouldn’t be higher than the speed of
light, as is clear. However, this restriction might not be met on the surface
layer to test the gravastar’s stability. The square of the speed of sound is
defined by,

$\eta=v_{s}^{2}=\frac{P^{\prime}}{\sum^{\prime}}.$ (60)

Where ’ represents the derivative w.r.t the radial coordinate. As a result, by
using (51,52) we examine the parameter’s sign to determine the stability of
gravastar configurations. We utilize the graphical behavior since the
mathematical expression of $\eta$ is complicated.

Figure 8:  variation of $\eta$ with respect to thickness $\epsilon(\text{in
km})$ for different values of $\alpha$

From Fig.(8) it could be noticed that the effective parameter $\eta$ satisfies
the inequality $0<\eta\leq 1$ throughout the entire shell region. Here we have
varied the model parameter $\alpha$ and we observed for each value of $\alpha$
our model behaves physically stable. One more important observation to mention
is that whenever the value of $\alpha$ increases the parameter $\eta\to 1$. So
as the model parameter value rises our proposed gravastar model approaches the
unstable situation.

###  VII.2 Surface Redshift

The study of a gravastar’s surface redshift is one of the most basic ways to
understand the stability and detection of the object. The formula
$Z_{s}=\frac{\Delta\lambda}{\lambda_{e}}=\frac{\lambda_{0}}{\lambda_{e}}$
could be used for determining the gravitational surface redshift of the
gravastar, where $\lambda_{0}$ and $\lambda_{e}$ represents the wavelength
detected by the observer and emitted from the source. Buchdahl [62, 63]
proposed that the value of surface redshift should not be more than 2 for an
isotropic, stable, perfect fluid distribution. However, Ivanov [64] claimed
that for anisotropic fluid dispersion, it might go as high as $3.84$. Other
than that Barraco and Hamity [65] showed that for an isotropic fluid
distribution, $Z_{s}\leq 2$ holds when the cosmological constant is absent.
Bohmer and Harko [66] though, showed that in the presence of the anisotropic
star’s cosmological constant, $Z_{s}\leq 5$. Now in our case, we have obtained
the surface redshift by the following formula,

$Z_{s}=-1+\frac{1}{\sqrt{g_{tt}}}=\frac{1}{\sqrt{C_{3}((\beta+8\pi)r)^{-\frac{32\pi}{\beta+8\pi}}}}-1.$
(61)

Figure 9:  variation of $Z_{s}$ with respect to thickness $\epsilon(\text{in
km})$ for different values of model parameter $\alpha\,\text{and}\,\beta$. Red
Dotdashed ($\alpha=-4.1,\beta=3.9$), Blue Dotted ($\alpha=-4.4,\beta=3.6$),
Green Tan ($\alpha=-4.7,\beta=3.3$), Black Dashed ($\alpha=-5.0,\beta=3.0$),
Magenta Thickness ($\alpha=-5.3,\beta=2.7$)

The graphical analysis of $Z_{s}$ is given in Fig.(9). We have varied the
model parameter $\alpha$ and $\beta$ for analyzing the maximum possibility
case of $Z_{s}$ and in each case, it is noticed that $Z_{s}<1$. Consequently,
we can thus assert that the current gravastar model is both physically stable
and appropriate in the $f(Q,T)$ framework.

###  VII.3 Entropy Maximization:

Each quasi-black hole (QBH) candidate must be stable to constitute a
physically feasible endpoint of gravitational collapse [67]. Now, in order to
verify the stability of the current investigation of gravastar in $f(Q,T)$
gravity, we have used the entropy maximization method recommended by Mazur and
Mottola [1, 2]. Since the shell region is only the non vacuum region with
stiff fluid and contains the positive heat capacity so the solution should
thermodynamically stable here. To check the stability we will use the entropy
maximization technique in the shell region. For maximizing the entropy
function at first the first variation of the entropy function should vanish at
the boundaries of the shell i.e. $\delta\mathcal{S}=0$ at $r=R_{1}$ and
$r=R_{2}$. After that we have checked the nature of the second derivative i.e.
of $\delta^{2}\mathcal{S}$ by it’s sign for all the variation of $M(r)$. The
entropy function is given by,

$\mathcal{S}=\frac{\gamma k_{B}}{\hbar
G}\int_{R_{1}}^{R_{2}}rdr\left(2\frac{dM}{dr}\right)^{\frac{1}{2}}\frac{1}{\sqrt{1-\frac{2M(r)}{r}}}.$
(62)

A necessary and sufficient condition for the dynamic stability of a static,
spherically symmetric solution of the field problem is thermodynamic stability
in the context of a hydrodynamic treatment. The second derivative of the
entropy function is given by,

$\begin{gathered}\delta^{2}\mathcal{S}=\frac{\gamma k_{B}}{\hbar
G}\int_{R_{1}}^{R_{2}}rdr\left(2\frac{dM}{dr}\right)^{-\frac{3}{2}}\left(1-\frac{2M}{r}\right)^{-\frac{1}{2}}\\\
\left\\{-\left[\frac{d(\delta M)}{dr}\right]^{2}+2\frac{(\delta
M)^{2}}{r^{2}\left(1-\frac{2M}{r}\right)^{2}}\frac{dM}{dr}\left(1+2\frac{dM}{dr}\right)\right\\}.\end{gathered}$
(63)

With the help of equation 33, 37 and 43 we have determined the functional
value of $M(r)$ as,

$\begin{gathered}M(r)=\frac{r\left(3\alpha(\beta+8\pi)\ln(\frac{R_{1}}{r})+\left(\beta^{2}-12\pi\beta+32\pi^{2}\right)\rho_{c}R_{1}^{2}\right)}{3\alpha(8\pi-\beta)}.\end{gathered}$
(64)

Now if we consider the linear combination of $M(r)$ as $\delta M=\chi_{0}\psi$
where $\psi$ becomes vanish at the boundaries $R_{1}$ and $R_{2}$, then
integrating Eq.(63) partially by using the diminishing of the variation
$\delta M$ we get,

$\delta^{2}\mathcal{S}=-\frac{\gamma k_{B}}{\hbar
G}\int_{R_{1}}^{R_{2}}\frac{rdr}{\sqrt{\left(1-\frac{2M}{r}\right)}}\left(2\frac{dm}{dr}\right)^{-\frac{3}{2}}\chi_{0}^{2}\left(\frac{d\psi}{dr}\right)^{2}<0.$
(65)

It is evident from the above expression that for any radial variations that
vanish at the endpoints of the shell’s boundaries, the entropy function in
$f(Q,T)$ gravity reaches its maximum value. We may thus draw the conclusion
that a perturbation in the gravastar’s intermediate shell area’s fluid leads
to a decrease in entropy in region II, which indicates the idea that our
solutions are stable against minor perturbations with the specified endpoints.
In essence, the stability of the gravastar is unaffected by the effect of
$f(Q,T)$ gravity.

##  VIII Discussion and Conclusion

Following the model put forward by Mazur-Mottola [1, 2] within the context of
general relativity, we have developed a unique stellar model of a gravastar
under the theory of $f(Q,T)$ gravity in this research. There are three
different regions namely interior region, intermediate thin shell, and
exterior space-time with three different EoS. The interior region fully
consists of dark energy as hypothesized by [1, 2]. The following are some of
the gravastars’ crucial characteristics:

  * •

Interior Region : Using the EoS (25) we have derived two non-singular metric
potentials (27,28) from the described field equation in $f(Q,T)$ gravity. The
metric potentials are finite and remain positive throughout the entire
interior region. This confirms our proposed gravastar model in $f(Q,T)$
gravity is able to devoid of the concept of central singularity in CBH.

  * •

Intermediate thin shell: We have estimated the metric potentials in the region
of the shell by using the thin shell approximation. Eq. (33) and Eq.(34)
indicate that two metric potentials remain finite as well as positive
throughout the entire shell.

    * –

Pressure or Matter density: Apart from that using the energy conservation
equation (23), we have derived the pressure or matter density (35) in the
shell. Fig.(2) represents the variation of the pressure or matter density with
respect to the thickness parameter ($\epsilon$). One can see that the matter
density of the shell is monotonically growing up toward the outer boundary of
the shell. The shell is made of ultra-relativistic stiff fluid, since the
pressure or matter density is monotonically increasing towards the outer
surface we can physically interpret that the amount of stiff matter is rising
towards the outer border rather than the internal region of the shell. That is
why the shell’s outer boundary becomes denser than the interior border.

  * •

Junction Condition and EoS : The junction requirement for the formation of a
thin shell is taken into account between the interior and external space-
times. We analyze the variation in surface energy density with respect to the
thickness parameter ($\epsilon$) using the Darmois-Israel junction condition,
as shown in Fig.(4). The surface energy density increases towards the outer
boundary of the shell. Besides that, in Fig.(3), we have verified that the NEC
is satisfied over a range of model parameter values throughout the entire
shell. It confirms the presence of ordinary or exotic matter in the shell.
Apart from that, we get the limiting value of radius (55) using the concept of
determining the real value of shell mass.

  * •

Physical Features of the Model: Using the geometrical quantity of the
intermediate thin shell we have analyzed some physical properties of the thin
shell.

    * –

Proper length: The variation of the proper length with respect to the
thickness parameter $\epsilon$ is given in Fig.(5) and in Eq.(56). The figure
demonstrates that the appropriate length rises monotonically as shell
thickness increases. This monotonically increasing behavior of proper length
of gravastar is similar to the work which has been done in modified gravity
[21, 20].

    * –

Energy : The variation of the shell energy is illustrated in Fig.(6). In this
graph, it can be observed that the energy rises as the shell’s thickness
increases. The fluctuation of energy is comparable to the fluctuation in
matter density. It meets the requirement that the energy of the shell
increases as the radial distance increases.

    * –

Entropy : Fig.(7) depicts the evolution of the shell entropy, which shows the
growing behavior of the shell entropy with regard to thickness ($\epsilon$).
Another acceptable condition is that entropy should reach its greatest value
on the surface for a stable gravastar configuration which is demonstrated in
our analysis. For comparison of the energy and entropy of the gravastar model
with the previous work [34] one can see that the energy and entropy should
reach their maximum value at the boundary of the shell which is established in
our study.

  * •

Stability of stellar model: Finally we have verified the stability of our
proposed stellar model through the study of the Herreras cracking concept and
by the study of the surface redshift analysis method. After that we have used
the entropy maximization technique to determine the stability of the
gravastar.

    * –

Herrera’s cracking concept: We have analyzed the stability of gravastar by the
nature of the effective parameter $\eta$. In Fig.(8) it is clear that for each
value of $\alpha$ the square of the speed of the sound remains positive and
not exceeding 1. Moreover, we can see that for rising the value of the
$\alpha$ parameter the model approaches instability.

    * –

Surface Redshift: Lastly, we used surface redshift analysis to check the
stability of our recently suggested model. The surface redshift ($Z_{s}$) for
any physically stable star arrangement should always be smaller than 2. By
varying the model parameter $\beta$ we have plotted the surface redshift with
respect to the thickness parameter ($\epsilon$) which is given in Fig.(9) and
in each case $Z_{s}<1$. It demonstrates that our suggested model is stable
under $f(Q,T)$ gravity.

    * –

Entropy Maximization : Here we have applied the entropy maximization technique
for checking the stability of the gravastar system. For maximizing the entropy
function at first the first variation of the entropy function is set to be
zero at the boundaries of the shell i.e. $\delta\mathcal{S}=0$ at $r=R_{1}$
and $r=R_{2}$. After that, we checked the nature of the second derivative i.e.
of $\delta^{2}\mathcal{S}$ by its sign for all the variations of $M(r)$. Eq.65
takes a negative value which represents that the entropy attains its maximum
value for all variations of the radial parameter. This further indicates the
stability of our gravastar model in $f(Q,T)$ gravity. One can check the
stability of the gravastar model for the entropy maximization technique in [1,
34].

We can draw the conclusion that the gravastar might exist within the
constraints of $f(Q,T)$ gravity. In comparison to past work on gravastars, we
have extended the thin shell approximation up to the second order, which
provides a more accurate analytical solution for determining the physical
parameters of the shell. As well as we have applied a new technique of
Herrera’s cracking concept to check the stability of our proposed model in
$f(Q,T)$ gravity. We may conclude that the $f(Q,T)$ theory of gravity was
effectively used in the current study on the gravastar by making this finding.
The problem of the black hole’s event horizon and the central singularity is
promptly solved by a group of physically plausible, non-singular gravastar
solutions.

Data availability There are no new data associated with this article.

###### Acknowledgements.

SP & PKS acknowledges the National Board for Higher Mathematics (NBHM) under
the Department of Atomic Energy (DAE), Govt. of India for financial support to
carry out the Research project No.: 02011/3/2022 NBHM(R.P.)/R & D II/2152
Dt.14.02.2022. PKS thanks Transilvania University of Brasov for Transilvania
Fellowship for Visiting Professors. We are very grateful to the honorable
referees and the editor for the illuminating suggestions that have
significantly improved our research quality and presentation.

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