# Anisotropic stars in modified gravity: An extended gravitational decoupling
approach

S. K. Maurya  B. Mishra  Saibal Ray  Riju Nag

###### Abstract

In the work we present investigation on decoupling gravitational sources under
the framework of $f(R,T)$ gravity. Basically the complete geometric
deformation technique has been employed here which facilitates finding exact
solutions to the anisotropic astrophysical system smoothly without imposing
any particular ansatz for deformation function. Along with this we have also
used 5-dimensional Euclidean spacetime in order to describe the embedding
Class I spacetime for getting a solvable spherical physical system. The
solutions thus obtained show physically interesting as well as viable with new
possibilities to sought for. Especially, from the present investigation it is
worthy to note that the mixture $f(R,T)$ \+ CGD translate the scenario beyond
the pure GR realm and hence clearly helps to enhance the features of the
interior astrophysical aspects of compact stellar objects. Therefore to check
the physical acceptability and stability of the stellar system based on the
obtained solutions, we have performed a few physical tests which satisfy all
the stability criteria, including nonsingular nature of the density as well as
pressure.

##  1 Introduction

The modification in the gravitational sector has been proposed from time to
time because of the correction in the gravitation action. These corrections
have become inevitable, when already existing gravitational theories unable to
address certain key issues of the present universe. There are several such
corrections incorporated in recent past such as, for cosmological applications
[1, 2], string theory [3, 4], teleparallel gravity [5], unimodular gravity
[6], $f(R)$ gravity [7, 8], $f(R,G)$ gravity [10, 11], $f(Q,T)$ gravity [12]
and so on. Several cosmological observations have confirmed the late time
cosmic acceleration phenomena and this development has changed our
understanding on the universe. This recent development has been led to new
concepts and ideas. The late time cosmic acceleration phenomena attributed to
an exotic form of energy called the dark energy and general relativity (GR)
has certain limitations in addressing this issue. There are different exotic
matter fields that simulate negative pressure and positive energy density to
explain this bizarre fact, but a suitable geometrical modification without
adding any exotic matter possibly settle this issue. Harko et al. [9] proposed
the $f(R,T)$ gravity by assuming a weak coupling between matter and geometry.
The geometrical part of Einstein-Hilbert action has been modified by assuming
the arbitrary function $f(R,T)$, where $R$ and $T$ respectively denote the
Ricci scalar and the trace of the energy momentum tensor. The trace
incorporated in this function may associate with the existence of exotic
imperfect fluids. Three form of the functions are suggested as: (i)
$f(R,T)=R+2f(T)$, (ii) $f(R,T)=f_{1}(R)+f_{2}(T)$ and (iii)
$f(R,T)=f_{1}(R)+f_{2}(R)f_{3}(T)$, where $f_{1}(R)$, $f_{2}(R)$, $f(T)$,
$f_{2}(T)$ and $f_{3}(T)$ are arbitrary functions of their respective
arguments. The first case can be reduced to GR under certain condition and
this has been widely used to address issues related to cosmology and
astrophysics.

In recent time, $f(R,T)$ gravity theory has been of great interest and been
used widely in literature to address many issues related to astrophysics and
cosmology. The issue of late time cosmic dynamics and the anisotropy behaviour
of the expansion has been partially addressed in $f(R,T)$ gravity. Some of the
key findings in $f(R,T)$ gravity are given here. Alvarenga et al. [13] have
studied, in metric formalism, the evolution of scalar cosmological
perturbations. Balakin and Bochkarev [14] have investigated the rip cosmology
where as Noureen et al. [15] have shown the implications of shear-free
condition on the instability range of an anisotropic fluid. Baffou et al. [16]
have solved the cosmological evolution of the cosmological parameters
numerically in $f(R,T)$ gravity. Mishra et al. [44] have introduced the hybrid
scale factor to study the dynamical behaviour of the cosmological model of the
universe and also introduced the squared trace model [18] in $f(R,T)$ gravity.
In order to address the singularity issue, bouncing cosmology has also been
studied in $f(R,T)$ gravity. Shabani and Ziaie [19] have introduced the
effective fluid by defining the effective energy density and pressure to
present the bouncing cosmological model. Tripathy et al. [20] have studied the
bouncing cosmology in $f(R,T)$ gravity and performed the stability analysis of
the models under linear homogeneous perturbations solved the bouncing. The
$f(R,T)$ gravity theory has also been significant in the study of wormhole
solutions. Zubair et al. [21] have obtained the wormhole solutions in $f(R,T)$
gravity and shown that the wormhole can be constructed without exotic matter
in few regions of space-time. Elizalde and Khurshudyan [22] have obtained the
wormhole solution and explored the observational possibilities for testing
these models. Yousaf et al. [23] have investigated the irregularity factors of
self-gravitating spherical star that evolves in the presence of an imperfect
fluid. Das et al. [24] have studied the gravastar in $f(R,T)$ gravity and
obtained a set of singularity-free and exact solution of the gravastar. Abbas
et al. [25] have studied the charged perfect fluid spherically symmetric
gravitational collapse and commented that the singularity is formed earlier
than the apparent horizons.

In the context of GR, the charged compact objects have been investigated with
the interpretation of anisotropic system from Einstein-Maxwell field
equations. The anisotropic factor has been mimicked from the electric field
intensity and the model with this assumption helps in obtaining the stability
of static fluid in presence of charge. The solution of Einstein-Maxwell
equations has been instrumental in describing the astrophysical compact
objects [26, 27, 28, 29]. However, since last one decade the researchers have
given attention to construct charged compact star model in $f(R,T)$ gravity to
understand its behaviour in the geometrically extended gravity. Moraes et al.
[30] have studied the stellar equilibrium configurations of compact stars in
$f(R,T)$ gravity. Islam and Basu [31] have constructed model of a compact star
in presence of magnetic fluid and suggested that these solutions will enable
to describe the interior of compact objects. With the Krori-Barua solutions
and in an anisotropic distribution, Sharif and Waseem [32] have analysed the
effects of charge on the nature of relativistic compact star candidates. In
the scope of this extended gravity, Yadav et al. [33] have proposed the
existence of non-exotic compact star, that validates the energy conditions and
stability of the model. Biswas et al. [34] have studied the anisotropic
spherically symmetric strange star and have shown the validity of the
conditions used in the model. Maurya et al. [35] have suggested an embedded
approach to study the existence of compact structures that describes
anisotropic matter distributions in the framework of matter geometry coupling.
Also, Maurya and Tello-Ortiz [36] have extended the isotropic Durgapal-Fuloria
model and investigated the high dense charged anisotropic compact structure in
an isotropic background. Rahaman et al. [37] have predicted the exact redii of
the values of the coupling parameter involve in $f(R,T)$ gravity by
considering the observed mass values of six compact stars. Rej et al. [38]
have obtained singularity free model of charged anisotropic compact star in
$f(R,T)$ gravity.

The standard model is based on the homogeneity of space and large scale
isotropy, however a small scale of anisotropy can be expected in the universe
[39, 40, 41, 42, 43, 44]. The isotropy and homogeneity can be observed through
the space-time under consideration. But, the space-time of compact objects
lead to the anisotropic features with the inhomogeneous matter distribution.
The difference between the radial and tangential component pressure leads to
the anisotropic pressure. Due to this anisotropy feature, the physical
properties like gravitational redshift, energy density, total mass etc. are
affected. There are many causes behind the anisotropy, e.g. pion condensation
[45], phase transitions [46], immense magnetic field of neutron star [47],
strong electric field [48] and refs. therein [49, 50, 51]. In addition to
these, the form of gravitational tidal effects believed to be another reason
of anisotropy in the compact star and this has been responsible for the
deformation [52, 53, 54, 55, 56, 57].

A literature survey shows that now-a-days, several modified/extended theories
of gravity are being used to investigate compact stellar models. Therefore, it
is very much needed to understand the inherent geometry of the spacetime and
the process to embed a 4D space-time. It is known that Karmarkar [59] has
embedded 4-dimensional spacetime into 5-dimensional Euclidean space. This
embedding actually simplifies the process of solving the Einstein field
equations. Further, different types of manifolds are linked by embedding
4-dimensional Einstein field equations into 5-dimensional flat spacetime [60,
61]. It is notable that in general relativistic background Maurya et al. [62]
have obtained the exact generalized model for anisotropic compact stars of
embedding Class I and tested the viability of the model by performing
different physical tests, viz. the energy conditions, stability analysis and
mass-radius relation. It is also to note that Salako et al. [63] have studied
the existence of strange starts in $f(\mathcal{T},T)$ gravity, where
$\mathcal{T}$ be the torsion tensor. Waheed et al. [64] have used the
Karmarkar condition [59] to find the physically acceptable solution for
compact star in $f(R,T)$ gravity. In $f(R,T)$ gravity, Ahmed and Abbas [65]
have studied the gravitational collapse using the Karmarkar condition [59] to
the spherically symmetric non-static radiating star.

Under the above discussion we would like to now explain here our technique to
adopt for solving the Einstein field equations in the anisotropic domain as
efficient as possible. The newly adopted method which is known in the
literature as the gravitational decoupling (GD) approach, that allows one to
decouple the Einstein equations [66, 67, 68, 69] 111A detailed discussion on
the origin as well as feature of GD can be obtained in Ref. [70, 71, 72, 73,
74]. This gravitational decoupling approach is applied in system via minimal
geometric deformation (MGD) and its extension, called extended MGD or complete
geometric deformation (CGD). These efficient technique have been employed by
many investigators to tackle the system [75, 76, 77, 78, 79, 80, 81, 82, 83,
84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94]. Here, the extended MGD or
complete geometric deformation (CGD) which basically has been used by us in
the present paper to investigate the exact solution for compact star in
modified gravity theory without imposing any deformation function. The
systematic approach of this technique has been explained in details later on
in a forgoing section.

The paper is organized as follows: In Section 2, some preliminaries of the
$f(R,T)$ gravity for gravitational decoupling along with the Einstein’s field
equations and the Class I condition are discussed. In Section 3, we have
adopted a systematic procedure to solve the field equations for decoupled
system. In Section 4, the matching conditions for the astrophysical system has
been elaborated. The physical analysis of the problem are done in Section 5
and in Section 6 discussions and conclusions are presented.

##  2 Preliminaries of the mathematical frameworks

###  2.1 Generalized $f(R,T)$ gravity for gravitational decoupling system

The generalized integral action for the $f(R,T)$ formulation, one can write by
adding extra source as [70]

$\displaystyle S=\frac{1}{16\pi}\int
f(R,{T})\sqrt{-g}d^{4}x+\int{L}_{m}\sqrt{-g}d^{4}x+\beta\int{L}_{X}\sqrt{-g}d^{4}x,$
(2.1)

where ${L}_{m}$ denotes the matter Lagrangian, $\beta$ denotes a coupling
constant and ${L}_{X}$ be the Lagrangian density of a new sector. Here
${L}_{X}$ is not required to be essentially defined by GR, however this can
create the alterations in GR as argued in [70].

Let us now vary the action with respect to the metric tensor $g^{\mu\nu}$
which yields the field equations as follows

$\displaystyle\left(R_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\right)f_{R}(R,{T})+\Box
f_{R}(R,{T})g_{\mu\nu}-\frac{1}{2}f(R,{T})g_{\mu\nu}$
$\displaystyle=8\pi\left(T_{\mu\nu}+\beta\theta_{\mu\nu}\right)-f_{T}(R,{T})\,\left(T_{\mu\nu}+\Theta_{\mu\nu}\right),$
(2.2)

where $f_{R}(R,T)=\partial f(R,T)/\partial R$ and $f_{T}(R,T)=\partial
f(R,T)/\partial T$, respectively.

Again, the energy-momentum tensor (EMT) $T_{\mu\nu}$ and extra source
$\theta_{\mu\nu}$, along with $\Theta_{\mu\nu}$ can be provided as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}{L}_{m}-2\partial{L}_{m}/\partial g^{\mu\nu},$ (2.3)
$\displaystyle\theta_{\mu\nu}$ $\displaystyle=$ $\displaystyle-
g_{\mu\nu}{L}_{X}+2\partial{L}_{X}/\partial g^{\mu\nu},$ (2.4)
$\displaystyle{\Theta_{\mu\nu}}$ $\displaystyle=$
$\displaystyle{g^{\gamma\,\epsilon}\delta T_{\gamma\,\epsilon}/\delta
g^{\mu\nu}},$ (2.5)

where the conceptual meaning of this special tensor $\Theta_{\mu\nu}$ will be
physically meaningful later on.

Hence the Einstein tensor $G_{\mu\nu}$, after rearranging of Eq. (2.1), can be
given as

$\displaystyle
G_{\mu\nu}=\frac{1}{f_{R}\left(R,{T}\right)}\big{[}8\pi\left(T_{\mu\nu}-\beta\theta_{\mu\nu}\right)+\frac{1}{2}\left(f\left(R,{T}\right)-Rf_{R}\left(R,{T}\right)\right)g_{\mu\nu}$
$\displaystyle-\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)f_{{T}}\left(R,{T}\right)-\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}\left(R,{T}\right)+8\,\pi\,E_{\mu\nu}\big{]}.$
(2.6)

The respective conservation equation now leads to

$\displaystyle\nabla^{\mu}T_{\mu\nu}=\frac{f_{T}(R,{T})}{8\pi-
f_{T}(R,{T})}\bigg{[}(T_{\mu\nu}+\Theta_{\mu\nu})\nabla^{\mu}\ln f_{T}(R,{T})$
$\displaystyle+\nabla^{\mu}\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}{T}\frac{8\pi}{f_{{T}}\left(R,{T}\right)}\big{(}\beta\,\nabla^{\mu}\theta_{\mu\nu}\big{)}\bigg{]}.$
(2.7)

The energy-momentum tensor $T_{\mu\nu}$ for the anisotropic matter
distribution can be taken as

${T}_{\mu\nu}=\left({\rho}+{p}_{t}\right)u_{\mu}u_{\nu}+{p}_{t}g_{\mu\nu}+(p_{r}-p_{t})\,\zeta_{\mu}\zeta_{\nu},$
(2.8)

where ${u^{\mu}}$ is the four velocity, satisfying $u_{\mu}u^{\mu}=-1$ and
$u_{\mu}\nabla^{\nu}u_{\mu}=0$ while $\rho$, $p_{r}$ and $p_{t}$ are
respectively the matter density, radial pressure and tangential pressure of
the system.

Here, the matter Lagrangian considered as, ${L}_{m}=-\mathcal{P}$, where
$\mathcal{P}=-\frac{1}{3}\big{(}p_{r}+2\,p_{t})$. It is worthy to mention here
that the matter Lagrangian appears in the field equations (2.1) and the
conservation equations (2.1) can be realized according to its choice 222See
[96] and references therein for a modern, recent and detailed discussion about
this subject in the framework of modified gravity theories, included $f(R,T)$
gravity.. However, in the context of GR, this choice would not affect the
observational outcomes. So, when the matter field is described a perfect
fluid, it is quite relevant to choose a particular form of the matter
Lagrangian. The choice we mention here is consistent since it provides well-
established Lagrangian density. Moreover, in the limit $\chi\rightarrow 0$, it
reduces to the perfect fluid distribution in the context of GR [97].

So, from Eq. (2.3) one gets

$\frac{\delta\,T_{\mu\nu}}{\delta\,g^{\gamma\epsilon}}=\left(\frac{\delta\,g_{\mu\nu}}{\delta\,g^{\gamma\epsilon}}\right)\,L_{m}+g_{\mu\nu}\,\left(\frac{\partial\,L_{m}}{\partial\,g^{\gamma\epsilon}}\right)-2\,\frac{\partial^{2}\,L_{m}}{\partial\,g^{\mu\nu}\,\partial\,g^{\gamma\epsilon}}.$
(2.9)

Now, using
$\delta\,g_{\mu\nu}/\delta\,g^{\gamma\epsilon}=-g_{\mu\sigma}g_{\nu\chi}\delta^{\sigma\chi}_{\gamma\epsilon}$,
the above equation becomes

$\displaystyle\frac{\delta\,T_{\mu\nu}}{\delta\,g^{\gamma\epsilon}}=g_{\mu\nu}\,\left(\frac{\partial\,L_{m}}{\partial\,g^{\gamma\epsilon}}\right)-g_{\mu\sigma}\,g_{\nu\chi}\,\delta^{\sigma\chi}_{\gamma\epsilon}\,L_{m}-2\,\frac{\partial^{2}\,L_{m}}{\partial\,g^{\mu\nu}\,\partial\,g^{\gamma\epsilon}}.~{}~{}~{}$
(2.10)

Plugging (2.10) in (2.5), we obtain

$\Theta_{\mu\nu}=-2\,T_{\mu\nu}+g_{\mu\nu}\,L_{m}-2\,g^{\gamma\epsilon}\frac{\partial^{2}\,L_{m}}{\partial\,g^{\gamma\epsilon}\,\partial\,g^{\mu\nu}}.$
(2.11)

Again, employing (2.3) and ${L}_{m}=-\mathcal{P}$, we finally get

$\Theta_{\mu\nu}=-2\,T_{\mu\nu}-\mathcal{P}\,g_{\mu\nu}.$ (2.12)

Now, taking into account the work of Harko et al. [9], we consider the linear
form of $f(R,T)$ as

$\displaystyle f(R,{T})=R+2\chi{T},$ (2.13)

where $\chi$ is dimensionless and known as the coupling constant. The linear
form of $f(R,T)$ is quite successful in the context of astrophysical and
cosmological models. We have elaborated its successes in the Introduction.

By substituting the $f(R,T)$ functional (2.13) in Eq. (2.1) we find

$\displaystyle
G_{\mu\nu}=8\pi\left(T_{\mu\nu}-\beta\,\theta_{\mu\nu}\right)+\chi(2T_{\mu\nu}+2\mathcal{P}g_{\mu\nu}+{T}g_{\mu\nu})$
$\displaystyle\hskip
62.59596pt=8\pi\,\big{(}T_{\mu\nu}-\beta\,\theta_{\mu\nu}+\hat{T}_{\mu\nu}\big{)},$
(2.14)

where we denote $\hat{T}_{\mu\nu}$ as

$\displaystyle\hat{T}_{\mu\nu}=\frac{\chi}{8\,\pi}\,(2T_{\mu\nu}+2\,\mathcal{P}g_{\mu\nu}+{T}g_{\mu\nu}),$
(2.15)

with ${T}_{\mu\nu}$ as given by Eq. (2.8).

Hence, from the conservation of (2.1) one obtains

$\displaystyle\hskip
14.22636pt\nabla^{\mu}\big{(}T_{\mu\nu}-\beta\,\theta_{\mu\nu}+\hat{T}_{\mu\nu}\big{)}=0.$
(2.16)

At this stage we would like to mention the terms $\mathcal{P}$ and $T$ as

$\displaystyle\mathbf{\mathcal{P}}=-\frac{1}{3}(p_{r}+2\,p_{t})~{}~{}~{}\text{and}~{}~{}~{}T=-\rho+p_{r}+2\,p_{t}.$

###  2.2 The Einstein field equations for the decoupled system

We consider the most general line element to describe a spherically symmetric
and static spacetime which is given by

$ds^{2}=-e^{y(r)}\,dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(2.17)

where the metric potential $y$ and $\lambda$ are function of the radial
coordinate $r$ only, i.e, $y=y(r)$ and $\lambda=\lambda(r)$. Let us write the
non-zero components of the field equations under the static spherically
symmetric line element (2.17) can be provided as

$\displaystyle\frac{e^{-\lambda}}{8\pi}\left(-\frac{1}{r^{2}}+\frac{\lambda^{\prime}}{r}+\frac{e^{\lambda}}{r^{2}}\right)={\rho}-\hat{T}^{0}_{\
0}+\beta\,\theta^{0}_{\ 0},$ (2.18)
$\displaystyle\frac{e^{-\lambda}}{8\pi}\left(\frac{1}{r^{2}}+\frac{y^{\prime}}{r}-\frac{e^{\lambda}}{r^{2}}\right)={p_{r}}+\hat{T}^{1}_{\
1}-\beta\,\theta^{1}_{\ 1},$ (2.19)
$\displaystyle\frac{e^{-\lambda}}{32\pi}\left(2y^{\prime\prime}+y^{\prime
2}-\lambda^{\prime}y^{\prime}+2\frac{y^{\prime}-\lambda^{\prime}}{r}\right)={p_{t}}+\hat{T}^{2}_{\
2}-\beta\,\theta^{2}_{\ 2}.~{}~{}~{}$ (2.20)

In the above a ‘prime’ as usual denotes differentiation with respect to the
radial coordinate $r$. Moreover, $\hat{T}^{0}_{\ 0}$ and $\hat{T}^{1}_{\ 1}$
are expressed as

$\displaystyle\hat{T}^{0}_{0}=\frac{\chi}{24\,\pi}(-9\rho+p_{r}+2p_{t}),$
(2.21) $\displaystyle\hat{T}^{1}_{\
1}=\frac{\chi}{24\,\pi}(-3\rho+7p_{r}+2p_{t}),$ (2.22)
$\displaystyle\hat{T}^{2}_{\ 2}=\frac{\chi}{24\,\pi}(-3\rho+p_{r}+8p_{t}).$
(2.23)

###  2.3 The Class I condition for the decoupled system

In general, the 4-dimensional spherically symmetric spacetime given by the
metric (2.17), describes a spacetime of Class II. This shows that it is
required a 6-dimensional pseudo-Euclidean space for embedding. In this
connection, Gupta et al. [58] have provided 6-dimensional Euclidean spacetime
in the form

$\displaystyle
ds^{2}=-d\mathbb{Y}_{1}^{2}-d\mathbb{Y}_{2}^{2}-d\mathbb{Y}_{3}^{2}+d\mathbb{X}_{1}^{2}+d\mathbb{X}_{2}^{2}\pm
d\mathbb{X}_{3}^{2},$ (2.24)

with the particular transformation as follows:

$\displaystyle\mathbb{Y}_{1}=r\,\sin\theta\cos\phi,~{}\mathbb{Y}_{2}=r\,\sin\theta\sin\phi,~{}\mathbb{Y}_{3}=r\,\cos\theta,~{}$
$\displaystyle\mathbb{X}_{1}=K\,e^{y(r)/2}\,\cosh\Big{(}\frac{t}{K}\Big{)},~{}\mathbb{X}_{2}=K\,e^{y(r)/2}\,\sinh\Big{(}\frac{t}{K}\Big{)},~{}\mathbb{X}_{3}=\mathbb{Z}(r).$

Then Eq. (2.24) readily takes the form

$\displaystyle ds^{2}=e^{y(r)/2}\,dt^{2}-\Big{(}1+\frac{K^{2}\,y^{\prime
2}(r)\,e^{y(r)}}{4}\,\pm\mathbb{Z}^{\prime}(r)\Big{)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).$
(2.25)

After comparing from Eqs. (2.3) and (2.25), we have

$\displaystyle e^{\lambda(r)}=\Big{(}1+\frac{K^{2}\,y^{\prime
2}(r)\,e^{y(r)}}{4}\,\pm\mathbb{Z}^{\prime}(r)\Big{)}$ (2.26)

and

$\displaystyle
ds^{2}=-d\mathbb{Y}_{1}^{2}-d\mathbb{Y}_{2}^{2}-d\mathbb{Y}_{3}^{2}+d\mathbb{X}_{1}^{2}+d\mathbb{X}_{2}^{2}+d\mathbb{X}_{3}^{2}.$
(2.27)

On the other hand, the Karmarkar condition [59] proposes that for any
spherically symmetric spacetime (static as well as non-static) to be a Class
I, the spacetime must satisfy the following condition

$\displaystyle R_{1010}R_{2323}=R_{1212}R_{3030}+R_{2102}R_{3103},$ (2.28)

subject to $R_{2323}\neq 0$ [95], where the quantities regarding the Riemann
components for the metric (2.17) are given as

$\displaystyle
R_{2323}=-r^{2}\,(1-e^{-\lambda})\,\sin^{2}\theta,~{}~{}R_{1212}=\frac{\lambda^{\prime}r}{2},~{}~{}R_{3103}=0,$
$\displaystyle
R_{1010}=-e^{y}\left[\frac{y\,^{\prime\prime}}{2}+\frac{y\,^{\prime
2}}{4}-\frac{\lambda^{\prime}y\,^{\prime}}{4}\right],~{}~{}R_{2102}=0,$
$\displaystyle R_{3030}=-\sin^{2}{\theta}\frac{y^{\prime}re^{y-\lambda}}{2}.$
(2.29)

After inserting the Riemann components in condition (2.15), we obtain

$\displaystyle
2\frac{y^{\prime\prime}}{y^{\prime}}+y^{\prime}=\frac{\lambda^{\prime}e^{\lambda}}{e^{\lambda}-1},$
(2.30)

with $e^{\lambda}\neq 1$.

The solution of the above differential equation requires for the space time
(2.17) to be a Class I. Hence the integration of (2.16) provides the relation
between the gravitational potentials

$\displaystyle e^{\lambda}=1+A\,y^{\prime 2}(r)\,e^{y(r)},$ (2.31)

where $A$ is the integration constants.

From (2.26) and (2.31), we note that the transformation given below (2.24)
give 5-dimensional Euclidean spacetime in order to describe the embedding
Class I spacetime if and only if $\mathbb{Z}=0$ and $A=\frac{K^{2}}{4}$.

##  3 A systematic procedure for solving the field equations for decoupled
system via CGD technique

A close observation on the field equations (2.18)–(2.20) clearly shows that a
closed exact solution is not a easy and trivial task. Therefore, we employ the
complete geometric deformation (CGD) technique for solving these system of
equations in a unique way. This CGD technique provides a systematic approach
which are: first, split the decoupled system into two subsystems, and secondly
solve these system individually.

###  3.1 Splitting the decoupled system via CGD approach

In this approach, we basically deform the gravitational potentials
${\lambda(r)}$ and ${y(r)}$ over a linear transformation given as

$\displaystyle\lambda(r)$ $\displaystyle\mapsto$
$\displaystyle-\ln[\xi(r)+\beta h(r)],$ (3.1) $\displaystyle y(r)$
$\displaystyle\mapsto$ $\displaystyle\eta(r)+\beta g(r),$ (3.2)

where $h(r)$ and $g(r)$ denote the decoupling functions corresponding to the
radial and the temporal components of the line element (2.17). Here we take
the deformation along with both the radial component and the temporal
components, i.e. $f(r)\neq 0$ and $g(r)\neq 0$ so that it is known the
complete geometric deformation (CGD). Also, It is always possible to separate
the new piece $\theta_{\mu\nu}$ from the seed matter sector for pure $f(R,T)$
system.

Now, inserting Eqs. (3.1) and (3.2) into the system of equations (2.18)–(2.20)
we can have

$\displaystyle{8\,\pi}\,\big{(}\rho-\hat{T}^{0}_{\
0}\big{)}+8\pi\beta\theta^{0}_{\
0}=\bigg{[}\frac{1}{r^{2}}-\frac{\xi}{r^{2}}-\frac{\xi^{\prime}}{r}\bigg{]}-\beta\bigg{[}\frac{h}{r^{2}}+\frac{h^{\prime}}{r}\bigg{]},~{}~{}~{}~{}$
(3.3)
$\displaystyle{8\,\pi}\,\big{(}p_{r}+\hat{T}^{1}_{1}\big{)}-8\pi\beta\theta^{1}_{\
1}=\bigg{[}\xi\left(\frac{1}{r^{2}}+\frac{\eta^{\prime}}{r}\right)-\frac{1}{r^{2}}\bigg{]}+\beta
h\bigg{[}\frac{1}{r^{2}}+\frac{\eta^{\prime}}{r}\bigg{]}+\beta\frac{\xi\,g^{\prime}}{r},$
(3.4)
$\displaystyle{8\,\pi}\,\big{(}p_{t}+\hat{T}^{2}_{2}\big{)}-8\pi\beta\theta^{2}_{\
2}=\bigg{[}\frac{\xi}{4}\left(2\eta^{\prime\prime}+\eta^{\prime
2}+2\frac{\eta^{\prime}}{r}\right)+\frac{\xi^{\prime}}{4}\left(\eta^{\prime}+\frac{2}{r}\right)\bigg{]}$
$\displaystyle\hskip
85.35826pt+\beta\bigg{[}\frac{h}{4}\left(2\eta^{\prime\prime}+\eta^{\prime
2}+2\frac{\eta^{\prime}}{r}\right)+\frac{h^{\prime}}{4}\left(\eta^{\prime}+\frac{2}{r}\right)+\Psi(r)\bigg{]},$
(3.5)

where

$\displaystyle\Psi(r)=\frac{\mu^{\prime}h^{\prime}}{4}+\frac{\mu}{4}\Big{(}2h^{\prime\prime}~{}+\beta
h^{\prime 2}~{}+\frac{2h^{\prime}}{r}~{}+2\xi^{\prime}h^{\prime}\Big{)}$ (3.6)

where $\Psi(r)$ is given by the Eq. (3.6).

$\displaystyle
m(r)=\underbrace{\frac{\chi}{6\,\pi}\,\int_{0}^{r}\big{\\{}9\,\rho(x)-p_{r}(x)-2p_{t}(x)\big{\\}}\,x^{2}\,dx}_{m_{frt}}$
$\displaystyle+\underbrace{4\pi\,\int_{0}^{r}{\rho(x)\,x^{2}\,dx}}_{m_{GR}}+\underbrace{4\pi\,\beta\,\int_{0}^{r}{\theta^{0}_{0}(x)\,x^{2}\,dx}}_{m_{CGD}}.$
(3.7)

Obviously, the limit $\chi\rightarrow 0$ and $\beta\rightarrow 0$ will provide
the usual mass function expression for an anisotropic compact structure in the
arena of GR. From the Eq. (3.1) we extra contributions due to the $f(R,T)$ and
CGD scenarios as $m_{frt}(r)$ and $m_{CGD}(r)$, respectively. It is to be
noted at this point that the mixture $f(R,T)$ \+ CGD going beyond the pure GR
scope and hence clearly helps to enhance the compactness, at least from the
theoretical point of view.

It can be observed from the right hand side of Eqs. (3.3) – (3.1) that the
first members were separated from the new piece containing $\beta$ and the
decoupler functions $h(r)$ and $g(r)$. Now, the separated field equations and
their corresponding conservation law can be written as

$\displaystyle{8\,\pi}\,\big{(}\rho-\hat{T}^{0}_{\
0}\big{)}=\bigg{[}\frac{1}{r^{2}}-\frac{\xi}{r^{2}}-\frac{\xi^{\prime}}{r}\bigg{]},~{}~{}~{}~{}$
(3.8)
$\displaystyle{8\,\pi}\,\big{(}p_{r}+\hat{T}^{1}_{1}\big{)}=\bigg{[}\xi\left(\frac{1}{r^{2}}+\frac{\eta^{\prime}}{r}\right)-\frac{1}{r^{2}}\bigg{]},$
(3.9)
$\displaystyle{8\,\pi}\,\big{(}p_{t}+\hat{T}^{2}_{2}\big{)}=\bigg{[}\frac{\xi}{4}\left(2\eta^{\prime\prime}+\eta^{\prime
2}+2\frac{\eta^{\prime}}{r}\right)+\frac{\xi^{\prime}}{4}\left(\eta^{\prime}+\frac{2}{r}\right)\bigg{]},~{}~{}~{}~{}$
(3.10)

Now Bianchi identities require the following conservation equation related to
the above system of Eqs. (3.8) – (3.10), i.e.
$\nabla^{\mu}\big{(}T_{\mu\nu}+\hat{T}_{\mu\nu}\big{)}=0$ for when $\beta=0$
and provides

$\displaystyle
p^{\prime}_{r}+\frac{\eta^{\prime}}{2}\,(\rho+p_{r})-\frac{2}{r}\,(p_{t}-p_{r})=\frac{\chi\,(3\rho^{\prime}-p^{\prime}_{r}-2p^{\prime}_{t})}{6\,(4\,\pi+\chi)}.~{}~{}~{}~{}$
(3.11)

The Eq. (3.9) can be termed as the modified Tolman-Oppenheimer-Volkoff (TOV)
[98, 99] equation in the arena of $f(R,T)$ gravity theory. However, it is
noticeable that Eq. (3.11) converts into the hydrostatic equilibrium condition
for standard GR for$\chi=0$. However, the corresponding solution can be given
from the following static spacetime as

$ds^{2}=e^{\eta(r)}\,dt^{2}-\xi^{-1}(r)dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).$
(3.12)

In this scenario, the gravitational mass for anisotropic matter distribution
in $f(R,T)$-gravity can be determined by the formula as

$\displaystyle
m_{0}(r)=\underbrace{\frac{\chi}{6\,\pi}\,\int_{0}^{r}\big{\\{}9\,\rho(x)-p_{r}(x)-2p_{t}(x)\big{\\}}\,x^{2}\,dx}_{m_{frt}}+\underbrace{4\pi\,\int_{0}^{r}{\rho(x)\,x^{2}\,dx}}_{m_{GR}}.$
(3.13)

Here $m_{0}$ is the mass function in pure $f(R,T)$ scenario. Let us now look
at the factor $\beta$, so that the field equations for $\theta_{\mu\nu}$
becomes

$\displaystyle\theta^{0}_{\
0}=-\frac{1}{8\pi}\bigg{[}\frac{h}{r^{2}}+\frac{h^{\prime}}{r}\bigg{]},~{}~{}~{}~{}$
(3.14) $\displaystyle\theta^{1}_{\
1}=-\frac{h}{8\pi}\bigg{[}\frac{1}{r^{2}}+\frac{\eta^{\prime}}{r}\bigg{]}+\frac{\xi\,g^{\prime}}{r},$
(3.15) $\displaystyle\theta^{2}_{\
2}=-\frac{1}{8\pi}\bigg{[}\frac{h}{4}\left(2\eta^{\prime\prime}+\eta^{\prime
2}+2\frac{\eta^{\prime}}{r}\right)+\frac{h^{\prime}}{4}\left(\eta^{\prime}+\frac{2}{r}\right)+\Psi(r)\bigg{]},~{}~{}~{}~{}$
(3.16)

with

$\displaystyle\Psi(r)=\frac{\mu^{\prime}h^{\prime}}{4}+\frac{\mu}{4}\Big{(}2h^{\prime\prime}~{}+\beta
h^{\prime 2}~{}+\frac{2h^{\prime}}{r}~{}+2\xi^{\prime}h^{\prime}\Big{)}.$

Hence, in view of Eq. (2.16), we obtain $\nabla^{\mu}\theta_{\mu\nu}=0$ in the
following conservation equation for system of Eqs. (3.14)–(3.16)

$\displaystyle\left(\theta^{1}_{\
1}\right)^{\prime}-\frac{y^{\prime}}{2}\left(\theta^{0}_{\ 0}-\theta^{1}_{\
1}\right)-\frac{2}{r}\left(\theta^{2}_{\ 2}-\theta^{1}_{\
1}\right)=\frac{g^{\prime}}{2}(p_{r}+\rho).~{}~{}$ (3.17)

In addition to this, now we use the pressure anisotropy condition in Eq. (3.4)
and (3.1), i.e. $G^{1}_{1}=G^{2}_{2}$, which provides

$\displaystyle{\xi}\left(\frac{\eta^{\prime\prime}}{2}+\frac{\eta^{\prime
2}}{4}-\frac{\eta^{\prime}}{2\,r}-\frac{1}{r^{2}}\right)+\frac{\xi^{\prime}\eta^{\prime}}{4}+\frac{2\,\xi^{\prime}}{r}-\frac{1}{r^{2}}=\hat{\Delta},$
(3.18)

where $\hat{\Delta}={(8\pi+2\chi)}\,(p_{t}-p_{r})$.

One can observe here that condition (3.18) is same as the anisotropic
condition in GR. Since $f(R,T)$ gravity is an extended form of GR for the
linear choice of $f(R,T)$, we can say that the solution of the field equations
in $f(R,T)$ theory can be obtained by the known solution of GR. The coupling
parameter $\chi$ will only affect the matter variable. Using Eqs.
(2.21)–(2.23), we can obtain $\rho$, $p_{r}$ and $p_{t}$ form Eqs. (3.8)–(3.9)
with respect to the radial coordinate as

$\displaystyle
8\pi\rho=\frac{1}{48(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}}\Big{[}48\pi(1-\xi^{\prime}r-\xi)+\chi\big{\\{}16$
$\displaystyle\hskip
8.5359pt+\xi^{\prime}r(\eta^{\prime}r-16)+(4\eta^{\prime}r+2\eta^{\prime\prime}r^{2}+\eta^{\prime
2}r^{2}-16)\xi\big{\\}}\Big{]},~{}~{}~{}~{}~{}$ (3.19) $\displaystyle 8\pi
p_{r}=\frac{1}{48(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}}\Big{[}48\pi(\xi+\eta^{\prime}r\xi-1)-\chi\big{\\{}16$
$\displaystyle\hskip
8.5359pt+\xi^{\prime}r(8+\eta^{\prime}r)-(16+20\eta^{\prime}r-2\eta^{\prime\prime}r^{2}-\eta^{\prime
2}r^{2})\xi\big{\\}}\Big{]},~{}~{}$ (3.20) $\displaystyle 8\pi
p_{t}=\frac{1}{48(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}}\Big{[}12\pi
r(\xi^{\prime}(2+\eta^{\prime}r)+(2\eta^{\prime}$ $\displaystyle\hskip
8.5359pt+2\eta^{\prime\prime}r+\eta^{\prime
2}r)\xi)+\chi\big{\\{}8+\xi^{\prime}r(4+5\eta^{\prime}r)+(-8+8\eta^{\prime}r$
$\displaystyle\hskip 8.5359pt+10\eta^{\prime\prime}r^{2}+5\eta^{\prime
2}r^{2})\xi\big{\\}}\Big{]}.$ (3.21)

To find the contribution of $f(R,T)$, it is required to separate out $\rho$,
$p_{r}$ and $p_{t}$, because the term $\hat{T}$ in Eqs. (3.8)–(3.10) is
appearing with $\rho$, $p_{r}$ and $p_{t}$ respectively in Eqs. (3.8)–(3.10).
If we interpret the term $\rho+\hat{T}^{0}_{\ 0}$ in Eq. (3.8) as effective
seed density (similarly $p_{r}-\hat{T}^{1}_{1}$ in (3.9) and
$p_{r}-\hat{T}^{2}_{2}$ in 3.10 as the effective seed pressures), then the
system of the equations can be treated as the set of Einstein’s field
equations. The $f(R,T)$ contribution is hidden within the redefined
thermodynamic quantities as mentioned in Eqs. (3.1)–(3.1). We would like to
mention here that no substantial effect has been noticed in Eqs. (3.1)–(3.1)
$f(R,T)$ since $\theta$-sector is separated by applying the CGD. Hence, in
order to close the problem at least mathematically and to check the physical
viability, we need to find a solution for the system
$\\{\theta_{\mu\nu},h,g\\}$.

Let us now define the following new physical parameters:

$\displaystyle\hskip 8.5359pt\rho^{\text{eff}}=\rho+\beta\,\theta^{0}_{0},$
(3.22) $\displaystyle\hskip
8.5359ptp^{\text{eff}}_{r}=p_{r}-\beta\,\theta^{1}_{1},$ (3.23)
$\displaystyle\hskip 8.5359ptp^{\text{eff}}_{t}=p_{t}-\beta\,\theta^{2}_{2},$
(3.24)

where the effective thermodynamic variables characterizing the matter
distribution of the model.

Here we can observe the effects of CGD on the mass function $m(r)$ from
$m_{CGD}(r)$. Moreover, the effective anisotropy factor $\Delta$ can be given
as

$\Delta^{\text{eff}}~{}=~{}\underbrace{(p_{t}-p_{r})}_{\Delta_{FRT}}~{}+~{}\underbrace{\beta\,\,(\theta^{1}_{1}-\theta^{2}_{2})}_{\Delta_{CGD}}.$
(3.25)

Here we would like to highlight an important point that the CGD induced an
extra contribution $\Delta_{CGD}$ in the seed anisotropy $\Delta_{FRT}$ which
may enhance the anisotropy within the matter distribution. This contribution
may also improve the equilibrium mechanism of the stellar system via the
anisotropic force.

###  3.2 Embedding Class I solution in $f(R,T)$ gravity for the seed system

A literature survey shows that the temporal and radial components of the
chosen seed spacetime corresponds to the well-known Adler [100] and Finch-Skea
[101], respectively. Furthermore, this seed Class I spacetime was previously
discussed in $f(R,T)$ gravity in the context of MGD approach [71]. Therefore,
hybridization which we are considering here is reasonable. Now, we consider
the seed spacetime in $f(R,T)$ gravity, which satisfies the Karmarker
condition (2.31), is given by

$\displaystyle
ds^{2}=(X+B\,r^{2})^{2}\,dt^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})-(1+Yr^{2})\,dr^{2},~{}~{}~{}$
(3.26)

where $X$ and $B$ are constant and $Y=16\,A\,B^{2}$.

So, the density and the pressures in $f(R,T)$ scenario [using Eqs. (3.1) –
(3.1)] characterizing this model can be given as

$\displaystyle 8\pi\rho=\frac{2(\chi+3\pi)Y(3+Yr^{2})+C[6\pi
Yr^{2}(3+Yr^{2})+\psi_{1}(r)]}{6(\chi^{2}+6\chi\pi+8\pi^{2})(1+Cr^{2})(1+Yr^{2})^{2}},~{}~{}~{}~{}$
(3.27) $\displaystyle 8\pi
p_{r}=\frac{-2Y\,\psi_{3}(r)+C[\chi(9+10Yr^{2}-2Y^{2}r^{4})+\psi_{2}(r)]}{6(\chi^{2}+6\chi\pi+8\pi^{2})(1+Cr^{2})(1+Yr^{2})^{2}},$
(3.28) $\displaystyle 8\pi p_{t}=\frac{Y(\chi
Yr^{2}-6\pi)+C[6\pi(4+Yr^{2})+\chi\psi_{4}(4)]}{6(\chi^{2}+6\chi\pi+8\pi^{2})(1+Cr^{2})(1+Yr^{2})^{2}},~{}~{}~{}$
(3.29)

where we considered here $C=B/X$ for simplicity
$\psi_{1}(r)=\chi(3+8Yr^{2}+2r^{4}Y^{2})$,
$\psi_{2}(r)=6\pi(4+3Yr^{2}-r^{4}Y^{2})$,
$\psi_{3}(r)=[\chi\,Yr^{2}+3\pi(1+Yr^{2})]$,
$\psi_{4}(r)=(9+4Yr^{2}+Y^{2}r^{4})$.

Now we have already specified the $\xi$ and $\eta$, so only we need to
determine the deformation functions $h(r)$ and $g(r)$ to close the
$\theta$-sector completely. Therefore, we will apply two different approach to
find deformation functions $h(r)$ and $g(r)$ as discussed below:

####  3.2.1 Mimic constraints for the density approach for determining the
deformation function $h(r)$

In this case we will consider the mimic constraints for density

$\displaystyle\theta^{0}_{0}(r)=\rho(r),$ (3.30)

which leads

$\displaystyle
h^{\prime}+\frac{h}{r}=-8\pi\,r\,\rho~{}~{}\Longrightarrow~{}~{}h=-\frac{8\pi}{r}\,\int{\rho\,r^{2}\,dr}+\frac{F}{r},$
(3.31)

where $F$ is a constant of integration.

Now, using the Eqs. (3.27) and (3.31), we find the deformation function $h(r)$

$\displaystyle
h(r)=\frac{h_{1}(r)+h_{2}(r)\,\tan^{-1}[\sqrt{C}\,r]-h_{3}(r)\,\tan^{-1}[\sqrt{Y}\,r]}{12(\chi^{2}+6\chi\pi+8\pi^{2})\,r\,(C-Y)^{2}\sqrt{Y}(1+Yr^{2})}.~{}~{}$
(3.32)

Here, we take the arbitrary constant $F=0$ to avoid the singularity at $r=0$.
The coefficient used in $h(r)$ can be provided as

$\displaystyle
h_{1}(r)=-r\,(C-Y)\,\sqrt{Y}\,[-4(\chi+3\pi)Y^{2}r^{2}+C\,(\chi+4\chi
Yr^{2}+12\pi Yr^{2})],$ $\displaystyle
h_{2}(r)=2\sqrt{C}\,\chi\,(3C-2Y)\,\sqrt{Y}\,(1+Yr^{2}),$ $\displaystyle
h_{3}(r)=C\chi(5C-3Y)(1+Yr^{2}).$

We would like to mention that all the calculations have been done by taking
the following assumption to avoid the singularity in the expression of $f(r)$
as: (i) by expanding the $\tan^{-1}(x)$, where
$x=\sqrt{C}\,r~{}~{}or~{}~{}\sqrt{Y}\,r$, upto linear term using Taylor series
expansion around $x=0$ and (ii) the integration constant involve in the
solution is taken to be zero. While the other deformation function $g(r)$ will
be obtained by taking a linear equation of state (EOS) between the
$\theta$-components which we discuss in the next section.

####  3.2.2 Equation of state approach for determining $g(r)$

We consider the following linear EOS in $\theta$-components to determine the
function $h(r)$ as

$\displaystyle~{}~{}\theta^{1}_{1}=\alpha\,\theta^{0}_{0}+\gamma,$ (3.33)

where $\alpha$ and $\gamma$ are the constants.

This EOS will provide the first order linear differential equation whose
solution yields the expression for $h(r)$ as

$\displaystyle
g(r)=\frac{-4\,Y\,g_{1}(r)\,\ln(1+Cr^{2})+C[\,g_{2}(r)+g_{3}(r)\,]}{24\,C\,(\chi^{2}+6\chi\pi+8\pi^{2})\,(C-Y)},~{}~{}~{}~{}$
(3.34)

where

$\displaystyle g_{1}(r)=3C(\chi+4\pi)-4(\chi+3\pi)Y,$ $\displaystyle
g_{2}(r)=r^{2}\big{[}3C(\chi+4\pi)(-(-5+\alpha)Y+2\chi\gamma(2+Yr^{2})$
$\displaystyle\hskip
28.45274pt+4\gamma\pi(2+Yr^{2}))-2Y\big{\\{}\,3\chi^{2}\gamma(2+Yr^{2})$
$\displaystyle\hskip
28.45274pt+6\pi((5-\alpha)Y+4\gamma\pi(2+Yr^{2}))+2\chi((5-\alpha)Y$
$\displaystyle\hskip 28.45274pt+9\gamma\pi(2+Yr^{2}))\,\big{\\}}\big{]}$
$\displaystyle g_{3}(r)=2\alpha(-3C(\chi+4\pi)+4(\chi+3\pi)Y)\ln(1+Yr^{2}).$

Now the $\theta$ components can be given as

$\displaystyle 8\pi\theta^{0}_{0}=\frac{2(\chi+3\pi)Y(3+Yr^{2})+C[6\pi
Yr^{2}(3+Yr^{2})+\psi_{1}(r)]}{6(\chi^{2}+6\chi\pi+8\pi^{2})(1+Cr^{2})(1+Yr^{2})^{2}},~{}~{}~{}~{}~{}$
(3.35) $\displaystyle
8\pi\theta^{1}_{1}=\frac{-3C(\chi+4\pi)[\theta_{11}(r)-\alpha
Y(3+r^{2}Y)]+\theta_{12}(r)}{12\,(\chi^{2}+6\,\chi\,\pi+8\pi^{2})(C-Y)(1+Yr^{2})^{2}},$
(3.36) $\displaystyle
8\pi\theta^{2}_{2}=\frac{\theta_{21}(r)+\Psi_{22}(r)}{2304(\chi^{2}+6\chi\pi+8\pi^{2})^{2}(C-Y)^{2}(1+r^{2}Y)^{2}}.~{}~{}~{}~{}$
(3.37)

The coefficients used in the expressions (3.35)–(3.37) have been mentioned in
the Appendix.

Then the deformed spacetime can be given by

$\displaystyle
ds^{2}=(X+Br^{2})^{2}\,e^{\beta\,g(r)}\,dt^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})-\frac{(1+Yr^{2})}{1+\beta\,h(r)(1+Yr^{2})}\,dr^{2},$
(3.38)

where $h(r)$ and $g(r)$ is given in Eqs. (3.32) and (3.34).

Thus, the effective quantities are given as

$\displaystyle\hskip
8.5359pt\rho^{\textrm{eff}}(r)=\rho(r)+\beta\,\theta^{0}_{\
0}=(1+\beta)\,\rho(r),$ (3.39) $\displaystyle\hskip
8.5359ptp_{r}^{\textrm{eff}}(r)=p_{r}(r)-\beta\,\theta^{1}_{\ 1},$ (3.40)
$\displaystyle\hskip
8.5359ptp_{t}^{\textrm{eff}}(r)=p_{t}(r)-\beta\,\theta^{2}_{\ 2}.$ (3.41)

##  4 Matching condition for the astrophysical system

In the astrophysical system, the matching of space time geometries leads to
its physical viability. Under the purview of general relativity, any
spherically symmetric stellar object provides strict limiting case. It is must
that the stellar distribution at the surface of the star ($r=R$) between the
interior ($r<R$) and exterior ($r>R$) solution be smooth and continuous. The
$f(R,T)$ theory of gravity can describe a non-minimal coupling between the
geometry and the matter. The non-trivial coupling between the gravity and
matter sectors can contribute the matter content of the outer space time that
surrounds the compact structure. Also, the junction conditions in $f(R,T)$
gravity needs to be established. To establish this, the field equations of
$f(R,T)$ gravity (2.1) for the functional $f(R,T)=R+2\chi T$ required to be
expressed in terms of the Ricci tensor and matter distribution. Also to
analyze the contribution of the extended gravity on the outer space time
matter distribution, we shall ignore $\theta_{\mu\nu}$ term. Now, we can
establish a relation between the trace and Ricci scalar from (2.1) as

$R=-\left(8\pi+6\chi\right)T-8\chi\mathcal{P}.$ (4.1)

Subsequently, we can obtain the field equations as

$R_{\mu\nu}=2\left(\chi+4\pi\right)T_{\mu\nu}-2\left(2\pi+\chi\right)T\,g_{\mu\nu}-2\chi\,\mathcal{P}\,g_{\mu\nu}.$
(4.2)

Now, to get an idea on the matching condition, it is informative to know the
contribution from $T_{\mu\nu}$ and $f(R,T)$ gravity sector inside the compact
object. For brevity, we consider $T_{\mu\nu}=0$ so that the trace, energy
density and the pressures terms vanish. So, without loss of generality, we
consider the geometry of outer space time as the Schwarzschild exterior
metric. This is possible because of the linear relation between the Ricci
scalar and trace of the energy momentum tensor(2.13). If the gravity-matter
coupling were represented by the functional $f(R,T)$ such that non-linear
terms would exist between them and at a result obtaining $R$ and $T$
explicitly become difficult. The minimal coupling further allows the
decoupling of $\theta_{\mu\nu}$ and seed source. would violate. Hence, the
Schwarzschild metric as the exterior space time that surrounds the collapsed
configuration. As usual this solution is given by

$\displaystyle
ds^{2}_{+}=-\bigg{(}1-\frac{2{{M}}}{r}\bigg{)}^{-1}dr^{2}-r^{2}(d\theta^{2}-\sin^{2}\theta\,d\phi^{2})+\bigg{(}1-\frac{2{{M}}}{r}\bigg{)}\,dt^{2}.~{}$
(4.3)

In this connection, the deformed solution can be given by the most general
interior spacetime as

$\displaystyle ds^{2}_{-}=-\left[\xi(r)+\beta
h(r)\right]^{-1}dr^{2}-r^{2}(d\theta^{2}-\sin^{2}\theta\,d\phi^{2})+e^{\eta(r)+\beta
g(r)}dt^{2}.$ (4.4)

Now, at the boundary $\Sigma$, the matching of the geometries can be performed
smoothly between the the outer manifold $ds^{2}_{+}$ (4.3) and the inner
manifold $ds^{2}_{-}$ (4.4) as per the junction conditions. Joining both the
geometries at the boundary is the continuity equation. This will provide the
first and second fundamental forms across the surface $\Sigma$. In the first
fundamental form, the inner geometry described by the metric tensor
$g_{\mu\nu}$ induced by $ds^{2}_{-}$ and $ds^{2}_{+}$ on the interface, which
can be written as

$\displaystyle
g^{-}_{00}|_{r=r_{b}}=g^{+}_{00}|_{r=r_{b}}~{}~{}\text{and}~{}~{}g^{-}_{33}|_{r=r_{b}}=g^{+}_{33}|_{r=r_{b}}.$
(4.5)

Explicitly it reads

$\displaystyle\xi(r_{b})+\alpha\,h(r_{b})$ $\displaystyle=$
$\displaystyle\bigg{(}1-{\frac{2{{M}}}{r_{b}}}\bigg{)},~{}$ (4.6)
$\displaystyle e^{\eta(r_{b})+\beta g(r_{b})}$ $\displaystyle=$
$\displaystyle\bigg{(}1-{\frac{2{{M}}}{r_{b}}}\bigg{)},~{}$ (4.7)

where

$\displaystyle
M=m(r_{b})=m_{0}(r_{b})+4\pi\,\beta\,\int_{0}^{r}{\theta^{0}_{0}(x)\,x^{2}\,dx}.$
(4.8)

Furthermore, the second fundamental form is associated with the continuity of
the extrinsic curvature $K_{\mu\nu}$ through the $\mathcal{M}^{-}$ and
$\mathcal{M}^{+}$ on $\Sigma$. Then by matching of interior
($\mathcal{M}^{-}$) and outer ($\mathcal{M}^{+}$) manifold at the $\Sigma$
gives

$\left[p^{(\text{eff})}_{r}(r)\right]_{\Sigma}=\left[p(r)-\alpha\theta^{1}_{1}(r)\right]_{\Sigma}=0.$
(4.9)

Here, it is noted that the $\theta$-sector could in principle introduce some
modifications on the outer spacetime and matter content. Due to above fact,
the second fundamental form (4.9) can be written in the following from as

$\begin{split}p(r_{b})-\alpha\,[\theta^{1}_{\
1}(r_{b}]^{-}=-\alpha\,[\theta^{1}_{\ 1}(r_{b})]^{+},~{}\end{split}$ (4.10)

where $p_{r}(r_{b})=p_{r}^{-}(r_{b})$.

After substituting the value of $(\theta^{1}_{\ 1})^{-}(r_{b})$ from Eq.
(3.15), the above expression reads

$\displaystyle{p}_{r}(r_{b})+\beta\,\bigg{[}\frac{h}{8\pi}\,\bigg{(}\frac{y^{\prime}}{r}+\frac{1}{r^{2}}\bigg{)}+\frac{\mu\,h^{\prime}}{r}\bigg{]}_{r=r_{b}}=-\beta\,(\theta^{1}_{1})^{+}(r_{b}),~{}~{}~{}$
(4.11)

where $y^{\prime}\equiv\partial_{r}\,y^{-}$.

In order to find out $(\theta^{1}_{1})^{+}(r_{b})$ in (2.24), we employ
Eqs.(3.15), (4.3) and (4.7), obtaining

$\displaystyle{p}_{r}(r_{b})+\beta\,\bigg{[}\frac{h(r_{b})}{8\pi}\,\bigg{(}\frac{y^{\prime}(R)}{R}+\frac{1}{R^{2}}\bigg{)}+\frac{\xi(r_{b})\,g^{\prime}(r_{b})}{8\pi
R}\bigg{]}$
$\displaystyle=\frac{\beta\,h^{\ast}(r_{b})}{8\pi}\Bigg{[}\frac{2M}{R^{2}\,\big{(}R-2M\big{)}}+\frac{1}{R^{2}}\Bigg{]}+\beta\,\frac{\big{[}g^{\ast}(r_{b})\big{]}^{\prime}}{8\pi}\,\bigg{(}\frac{R-2M}{R^{2}}\bigg{)},~{}~{}~{}$
(4.12)

where $h^{\ast}(r_{b})$ and $g^{\ast}(r_{b})$ denote the deformation functions
for exterior solution under the extra source $\theta_{\mu\nu}$, which can be
given by following in the exterior metric as

$\displaystyle
ds^{2}_{+}=-\bigg{(}1-\frac{2{{M}}}{r}+\beta\,g^{\ast}\bigg{)}dt^{2}+r^{2}(d\theta^{2}-\sin^{2}\theta\,d\phi^{2})+\bigg{(}1-\frac{2{{M}}}{r}+\beta\,h^{\ast}\bigg{)}^{-1}\,dr^{2}.~{}~{}$
(4.13)

If this the exterior solution (4.13) is given by the Schwarzschild solution
(4.3), then we must put $h^{\ast}(r_{b})=0$ and $g^{\ast}(r_{b})=0$ in (4.13).
Then the Eq. (4.12) yields the following

$\displaystyle
p(r_{b})+\frac{h(r_{b})}{8\pi}\,\bigg{(}\frac{\eta^{\prime}(r_{b})}{R}+\frac{1}{R^{2}}\bigg{)}-\frac{\xi(r_{b})g^{\prime}(r_{b})}{8\pi
r_{b}}=0.~{}~{}~{}$ (4.14)

The above condition (4.14) can be also written as

$\displaystyle p(r_{b})-\alpha\,\big{(}\theta^{1}_{1}(r_{b})\big{)}^{-}=0.$
(4.15)

The constants involves in the solutions will be determined by the necessary
and sufficient conditions (4.6), (4.7) and (4.14). So, by using the boundary
conditions we have obtained the values of the constants

$\displaystyle
C=\frac{-2Y\big{[}\chi\,Y\,r_{b}^{2}+3\pi(1+r_{b}^{2}Y)\big{]}+2\beta\,C_{11}(r_{b})}{-6\beta\chi^{2}\gamma(r_{b}+r_{b}^{3}Y)^{2}+C_{22}(r_{b})},$
(4.16) $\displaystyle
M=M_{0}-\frac{\beta\,r}{2}\,h(r_{b})=\frac{YR^{3}}{2(1+YR^{2})}-\frac{\beta\,R\,h(R)}{2},$
(4.17)

where $M_{0}=m_{0}(r_{b})$. Also, we are avoiding to write the expression for
the constant $B$ due to its long cumbersome form.

##  5 Physical analysis of the gravitational decoupling solution for $f(R,T)$
gravity

###  5.1 Regularity conditions

It is to note that the regular behavior of the solution depends on the
deformed metric functions $e^{\lambda}$ and $e^{y}$ which are
$e^{\lambda(0)}=1$ and $e^{y(0)}>0$ together with the monotonic increasing
function of $r$ to describe the realistic objects. In the present case we see
that $e^{\lambda(r)}$ and $e^{y}$ are solely dependent on the parameters
$\xi,~{}\eta,~{}h,~{}g$. Since $\xi$ and $\eta$ are metric function
corresponding to the seed space-time which are already well behaved, then only
testing of the physical validity of the deformation functions $h(r)$ and
$g(r)$ are required. At the centre $r=0$, $f(r)$ and $g(r)$ must be freed from
singularity and $f(r)$ must be vanished to preserve $e^{\lambda(0)}=1$.

In addition to the above, the following conditions also must be satisfied:

#### Case I: For $\beta>0$

  1. 1.

if $h(r)\geq 0$, $g(r)\geq 0$ and for all $r\in[0,R]$, both are increasing,
then the deformed metric function $e^{\lambda(r)}>0$, $e^{y(r)}>0$, mass
function $m(r)>0$ are also increasing when the growth of $\xi(r)$ is faster
than $h(r)$.

  2. 2.

if $h(r)\leq 0$, $g(r)\leq 0$ and for all $r\in[0,R]$, both are decreasing,
then the deformed metric function $e^{\lambda(r)}>0$, $e^{y(r)}>0$, mass
function $m(r)>0$ are increasing when the growth of $\eta(r)$ is faster than
$g(r)$.

  3. 3.

if $h(r)\leq 0$, $g(r)\geq 0$ for all $r\in[0,R]$ then the deformed metric
function $e^{\lambda(r)}>0$, $e^{y(r)}>0$, mass function $m(r)>0$ and
increasing automatically.

  4. 4.

if $h(r)\geq 0$, $g(r)\leq 0$ for all $r\in[0,R]$, then the growth of $\xi(r)$
and $\eta(r)$ must be respectively faster than $h(r)$ and $g(r)$, to maintain
the deformed metric function $e^{\lambda(r)}>0$, $e^{y(r)}>0$ and the mass
function $m(r)>$ and its increasing behaviour.

Figure 1: Variation of the radial deformation function $f(r)$ and temporal
deformation function $h(r)$ with respect to the radial coordinate $r/r_{b}$.
For plotting of this figure, we use the numerical values of the constants as
$\alpha=1.4$, $\gamma=-0.002$, $\frac{M_{0}}{R}=0.2$, and $Y=0.005$.
Henceforth we shall use this same data set for plotting other figures.

Figure 2: Variation of the metric functions $e^{\lambda}$ and $e^{\nu}$ with
respect to the radial coordinate $r/r_{b}$.

Figure 3: The left panel shows the effective radial pressure
($p^{\text{eff}}_{r}$) and the right panel shows the effective tangential
pressure ($p^{\text{eff}}_{t}$) with respect to the radial coordinate
$r/r_{b}$.

Figure 4: The left panel shows the effective density $\rho^{\text{eff}}_{r}$
for Case I and the right panel shows the effective density
$\rho^{\text{eff}}_{r}$ for Case II with respect to the radial coordinate
$r/r_{b}$.

Figure 5: The left panel shows the anisotropy for the CGD contribution
($\Delta_{CGD}$) and the right panel shows the effective anisotropy
($\Delta^{\text{eff}}$) with respect to $r/r_{b}$.

Figure 6: Variation of the radial deformation function, $f(r)$, and the
temporal deformation function, $h(r)$ with respect to the radial coordinate
$r/r_{b}$. For plotting of this figure, we use the numerical values of the
constants as $\alpha=1.4$, $\gamma=-0.002$, $\frac{M_{0}}{R}=0.2$, and
$Y=0.005$. Henceforth we shall use this same data set for plotting other
figures.

Figure 7: Variation of the metric functions $e^{\lambda}$ and $e^{\nu}$ with
respect to the radial coordinate $r/r_{b}$.

Figure 8: The left panel shows the effective radial pressure
($p^{\text{eff}}_{r}$) and the right panel shows the effective tangential
pressure ($p^{\text{eff}}_{t}$) with respect to the radial coordinate
$r/r_{b}$.

Figure 9: The left panel shows the anisotropy for the CGD contribution
($\Delta_{CGD}$) and the right panel shows the effective anisotropy
($\Delta^{\text{eff}}$) with respect to $r/r_{b}$.

#### Case II: For $\beta<0$

  1. 1.

For non-negative and increasing $h(r)$ and $g(r)$, $\forall$ $r\in[0,R]$, the
deformed metric function $e^{\lambda(r)}$, $e^{y(r)}$ and the mass function
$m(r)$ will be positive and increasing, when the growth of $\eta(r)$ is higher
than $g(r)$.

  2. 2.

if $h(r)$ and $g(r)$ are non-positive and decreasing for all $r\in[0,R]$ then
growth of $\xi(r)$ must higher than $h(r)$ in order to preserve the increasing
and positive behavior of $e^{\lambda(r)}$, $e^{y(r)}$ and the mass function
$m(r)$.

  3. 3.

if $h(r)\leq 0$ and $g(r)\geq 0$ , $\forall$ $r\in[0,R]$ then the deformed
metric function $e^{\lambda(r)}$, $e^{y(r)}$ and the mass function $m(r)$ will
be positive and increasing, if the growth of $\xi(r)$ and $\eta(r)$ are higher
than $h(r)$ and $g(r)$, respectively.

  4. 4.

if $h(r)\geq 0$ and $g(r)\leq 0$, $\forall$ $r\in[0,R]$, then it yields
positive and increasing behavior of $e^{\lambda(r)}$, $e^{y(r)}$ and $m(r)$,
automatically.

###  5.2 Adiabatic Index

The adiabatic index $\Gamma$ is defined as the ratio of two specific heats
[102] as follows:

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\rho+p_{r}}{p_{r}}\left[\frac{dp_{r}}{d\rho}\right].$
(5.1)

This provides a tool to study the density profile as well as the stiffness of
EOS of a spherically symmetric system [103, 104, 105, 106, 107, 108]. It is
argued that $\Gamma$ can play a key to explain the dynamical stability of a
stellar system under the application of an infinitesimal radial adiabatic
perturbation [110, 111, 112, 113, 114, 115, 116, 117, 118, 119]. It is worthy
to mention that $\Gamma>\frac{4}{3}$ is prescribed for a stable Newtonian
sphere, $\Gamma=\frac{4}{3}$ gives rise to a neutral equilibrium [109]. In the
scenario of relativistic fluid distribution, some extra term will involve
which may produce some correction in previous bound, which can be given as
[120, 121]

$\Gamma<\frac{4}{3}+\left[\frac{1}{3}\kappa\frac{\rho_{0}p_{r0}}{|p^{\prime}_{r0}|}r+\frac{4}{3}\frac{\left(p_{t0}-p_{r0}\right)}{|p^{\prime}_{r0}|r}\right]_{max},$
(5.2)

where $\rho_{0}$, $p_{r0}$ and $p_{t0}$ are called the initial density, radial
and tangential pressure when the matter distribution is in static equilibrium.
The first term in bracket quantity in above inequality describe the
relativistic corrections to the Newtonian perfect fluid while the last term
present in the bracket due to anisotropy.

Figure 10: Variation of the adiabatic index ($\Gamma$) with respect to the
radial coordinate $r/r_{b}$.

Figure 11: Variation of different forces as follows: $F_{g}$-short dash
curves, $F_{h}$-long dashed curves, $F_{a}$-Solid curves, and $F_{\chi}$-long
dot dashed curves with respect to the radial coordinate $r/r_{b}$ for Case I
(left panel) and Case II (right panel).

However, for an anisotropic, stable and relativistic dynamical system
$\Gamma>\frac{4}{3}$ [121, 122, 102], since positive anisotropy factor may
slow down the growth of instability. The relativistic correction to the
adiabatic index $\Gamma$ could introduce some instabilities inside the star
[123, 124]. In this connection, Moustakidis [125] was proposed a more strict
condition on the adiabatic index $\Gamma$ to solve this issue and achive the
stable steallr structure. This condition leads the existence of a critical
value for the adiabatic index $\Gamma$, denoted by $\Gamma_{crit}$, which
depends on the amplitude of the Lagrangian displacement from equilibrium and
the compactness factor $u=M/R$ (where $M$ and $R$ being the total mass and
radius of the spherical system). Specifically in the present situation, the
critical relativistic adiabatic index under in $f(R,T)$ theory can be given as

$\displaystyle\Gamma_{crit}$ $\displaystyle=$
$\displaystyle\frac{4}{3}+\frac{19}{21}u.$ (5.3)

The variation of the adiabatic index ($\Gamma$) with respect to the radial
coordinate $r/r_{b}$ is shown in Fig. 10 which is physically satisfactory.

Table 1: The numerical values of physical parameters effective central density ($\rho^{\text{eff}}_{0}$), effective surface density ($\rho^{\text{eff}}_{s}$), and effective central pressure ($p^{\text{eff}}_{0}$) for $\beta=0.004$, $\alpha=1.4$, $\gamma=-0.002$, and $Y=0.005$.  $\chi$ and $\beta$ | $\rho^{\text{eff}}_{0}\times 10^{13}$ | $\rho^{\text{eff}}_{s}\times 10^{13}$ | $p^{\text{eff}}_{0}\times 10^{33}$
---|---|---|---
| ($gm/cm^{3}$) | ($gm/cm^{3}$) | ($dyne/cm^{2}$)
$\chi$ = 0.4, $\beta$ = 0 | 3.06204 | 1.34680 | 1.84404
$\chi$ = -0.4, $\beta$ = 0 | 3.36193 | 1.47900 | 2.02227
$\chi$ = 0.4, $\beta$ = 0.004 | 3.07614 | 1.35285 | 2.39666
$\chi$ = -0.4, $\beta$ = 0.004 | 3.37324 | 1.48417 | 2.60919
$\chi$ = 0.4, $\beta$ = 0.007 | 3.08677 | 1.35739 | 2.83532
Table 2: The numerical values of physical parameters ($\rho^{\text{eff}}_{0}$), effective surface density ($\rho^{\text{eff}}_{s}$), and effective central pressure ($p^{\text{eff}}_{0}$) for $\beta=0.004$, $\alpha=1.4$, $\gamma=-0.002$, and $Y=0.005$.  $\chi$ and $\beta$ | $\rho^{\text{eff}}_{0}\times 10^{13}$ | $\rho^{\text{eff}}_{s}\times 10^{13}$ | $p^{\text{eff}}_{0}\times 10^{33}$
---|---|---|---
| ($gm/cm^{3}$) | ($gm/cm^{3}$) | ($dyne/cm^{2}$)
$\chi$ = 0.4, $\beta$ = 0 | 3.06204 | 1.34680 | 1.84404
$\chi$ = -0.4, $\beta$ = 0 | 3.36193 | 1.47900 | 2.02227
$\chi$ = 0.4, $\beta$ = -0.001 | 3.05853 | 1.34528 | 1.71146
$\chi$ = -0.4, $\beta$ = -0.001 | 3.35909 | 1.47771 | 1.88101
$\chi$ = 0.4, $\beta$ = -0.002 | 3.05502 | 1.34377 | 1.58106
Table 3: The numerical values of physical parameters mass-radius ratio $\big{(}\frac{M}{R}\big{)}$, surface red-shift ($z_{s}$), central adiabatic index ($\Gamma_{0}$), and $\Gamma_{crit}$ for different $\beta$ with $\alpha=1.4$, $\gamma=-0.002$, $Y=0.005$ and $\chi=0.4$.  $\beta$ | $u=\frac{M}{R}$ | $z_{s}$ | $\Gamma_{0}$ | $\Gamma_{crit}$
---|---|---|---|---
$\beta$ = 0 | 0.2 | 0.29113 | 1.81430 | 1.51434
$\beta$ = 0.004 | 0.20009 | 0.29119 | 1.82829 | 1.51437
$\beta$ = 0.007 | 0.20012 | 0.29124 | 1.84826 | 1.51439
Table 4: The numerical values of physical parameters mass-radius ratio $\big{(}\frac{M}{R}\big{)}$, surface red-shift ($z_{s}$), central adiabatic index ($\Gamma_{0}$), and $\Gamma_{crit}$ for different $\beta$ with $\alpha=1.4$, $\gamma=-0.002$, $Y=0.005$ and $\chi=0.4$.  $\beta$ | $u=\frac{M}{R}$ | $z_{s}$ | $\Gamma_{0}$ | $\Gamma_{crit}$
---|---|---|---|---
$\beta$ = 0 | 0.2 | 0.29113 | 1.81430 | 1.51434
$\beta$ = -0.001 | 0.200055 | 0.291112 | 1.81400 | 1.514335
$\beta$ = -0.002 | 0.200047 | 0.291095 | 1.81547 | 1.514328

###  5.3 Equilibrium condition

Any stellar configuration will be in stable equilibrium state only if the
combined effect of all the forces related to the system becomes zero. In the
present case, the obtained modified form of Tolman [98] and Oppenheimer-
Volkoff (TOV) [98, 99] equation which comply with the hydrostatic equilibrium
of the stellar structure can be written as

$\displaystyle
p^{\prime}_{r}+\frac{\eta^{\prime}}{2}\,(\rho+p_{r})-\frac{2}{r}\,(p_{t}-p_{r})-\beta\,\Big{[}\left(\theta^{1}_{\
1}\right)^{\prime}-\frac{y^{\prime}}{2}\left(\theta^{0}_{\ 0}-\theta^{1}_{\
1}\right)$ $\displaystyle-\frac{2}{r}\left(\theta^{2}_{\ 2}-\theta^{1}_{\
1}\right)\Big{]}-\frac{\chi\,(3\rho^{\prime}-p^{\prime}_{r}-2p^{\prime}_{t})}{6\,(4\,\pi+\chi)}+\frac{\beta\,g^{\prime}}{2}(p_{r}+\rho)=0.$
(5.4)

The above modified TOV equation can be written in the form of different forces
components as hydrostatic force
$F_{h}=-\big{[}p^{\prime}_{r}-\beta\,\left(\theta^{1}_{\ 1}\right)^{\prime}$],
gravitational gradient force
$F_{g}=-\big{[}\frac{\eta^{\prime}}{2}\,(\rho+p_{r})+\frac{\beta\,g^{\prime}}{2}(p_{r}+\rho)+\beta\,\frac{y^{\prime}}{2}\left(\theta^{0}_{\
0}-\theta^{1}_{\ 1}\right)\big{]}$, anisotropic force
$F_{a}=\Big{[}\frac{2}{r}\,(p_{t}-p_{r})-\frac{2}{r}\left(\theta^{2}_{\
2}-\theta^{1}_{\ 1}\right)\big{]}$, and coupling force due to $f(R,T)$-gravity
$F_{\chi}=\frac{\chi\,(3\rho^{\prime}-p^{\prime}_{r}-2p^{\prime}_{t})}{6\,(4\,\pi+\chi)}$
such that $F_{h}+F_{g}+F_{a}+F_{\chi}=0$.

We have shown the features of all the forces for different values of the
parameters in Fig. 11. From this figure it can be observed that the fluid
distribution possesses a stable equilibrium under the combined effect of all
these forces involved in Eq. (5.3).

##  6 Discussion and conclusion

In the present investigation we deal with the decoupling gravitational sources
in $f(R,T)$ gravity under anisotropic matter distribution. However, the
energy-momentum tensor $T_{\mu\nu}$ used here under the CGD is not the usual
one as in Eq. (2.8) rather for the present study we consider the matter
Lagrangian to be ${L}_{m}=-\mathcal{P}$, where
$\mathcal{P}=-\frac{1}{3}\big{(}p_{r}+2\,p_{t})$. The basis of this particular
choice is not a general one as the matter Lagrangian enters explicitly in the
field equations (2.1) and different choices lead to different equations of
motion. The adopted technique of the complete geometric deformation (CGD) for
solving the system of equations seems a unique pathway. This technique
provides a systematic approach which are as follows: (1) split the decoupled
system into two subsystems, and (2) solve these system individually. A detail
discussion on this aspect has been provided in the preliminary Sec. 5 of the
present manuscript. Based on the above approach we have basically an extended
case of previously studied all such investigations. The results of the present
study is overall interesting, distinctive and satisfactory as far as physical
viability is concerned.

Some salient features of the present investigation can be discussed as
follows:

(i) We have shown variation of the radial deformation function $f(r)$ and
temporal deformation function $h(r)$ with respect to the radial coordinate
$r/r_{b}$ in Fig. 1. From this figure, for $f(r)$ and $h(r)$, we note that
both are negative and decreasing function $\forall$ $r\in(r,R]$ which give the
scenario of point 2 in Case I.

(ii) In Fig. 2, the metric functions $\lambda(r)=-\ln[\xi(r)+\alpha\,f(r)]$
and $\nu(r)=\eta(r)+\alpha\,h(r)$ are both increasing and positive throughout
the star, which imply that the growth of $\eta(r)$ is faster than the
deformation function $h(r)$.

(iii) In Fig. 3, the left panel shows the effective radial pressure
($p^{\text{eff}}_{r}$) and the right panel shows the effective tangential
pressure ($p^{\text{eff}}_{t}$) with respect to the radial coordinate
$r/r_{b}$. Since the constants $\chi$ and $\beta$ influence the radial and
tangential pressures, therefore the following observations can be made from
Fig. 3 which are:
For fixing $\chi$: when we increase $\beta$, both $p^{\text{eff}}_{r}$ and
$p^{\text{eff}}_{t}$ increase at the core of compact object;
For fixing $\beta$: $p^{\text{eff}}_{r}$ and $p^{\text{eff}}_{t}$ show
decreasing value at core of the star when we move $\chi$ from negative to
positive value.

(iv) Variation of the effective energy density ($\rho^{\text{eff}}$) with
respect to $r/r_{b}$ are shown in Fig. 9. We set the same numerical values as
used in Fig. 1. Since effective energy density is
$\rho^{\text{eff}}=(1+\beta)\,\rho$, therefore value of the effective energy
density at the core and boundary of the stellar object will be decreasing for
the decreasing value of $\beta$. Obviously values of the effective energy
density of Case I is higher than the Case II at each point of the stellar
model.

(v) The left panel of Fig. 5 shows the anisotropy for the CGD contribution
($\Delta_{CGD}$) and the right panel shows the effective anisotropy
($\Delta^{\text{eff}}$) with respect to $r/r_{b}$. We set the same numerical
values as used in Fig. 1. One can observe from the Fig. 5, the anisotropic
contribution $\Delta_{CGD}$ due to CGD shows the negative and decreasing
behavior within the stellar model but the effective anisotropy is still
increasing when $\beta\geq 0$, which shows that the growth of seed anisotropy
($\Delta_{FRT}$) is faster than $\Delta_{CGD}$. On the other hand, we would
like to mention here that when $\beta$ is positive then the effective
anisotropic force $F_{a}=\frac{2\,\Delta^{\text{eff}}}{r}$ will introduce less
effect to balance the system in order to achieve the hydrostatic Equilibrium
near the surface.

(vi) Variation of the radial deformation function $f(r)$ and the temporal
deformation function $h(r)$ with respect to the radial coordinate $r/r_{b}$
are depicted in Fig. 6. For plotting of this figure, we use the numerical
values of the constants as $\beta=0.004$, $\alpha=1.4$, $\gamma=-0.002$,
$\frac{M_{0}}{R}=0.2$, and $Y=0.005$. In this case II: The deformation
functions $f(r)$ and $h(r)$ are also negative and decreasing function
$\forall$ $r\in(r,R]$, see Fig. 6 as happens in Case I.

(vii) Variation of the metric functions $e^{\lambda}$ and $e^{\nu}$ with
respect to the radial coordinate $r/r_{b}$ are featured in Fig. 7. For
plotting of this figure, we use the same numerical values of constants as used
in Fig. 1. It can be observed that the deformation functions $f(r)$ and $h(r)$
are negative and decreasing function throughout the stellar model and both
metric functions $\lambda(r)=-\ln[\xi(r)+\alpha\,f(r)]$ and
$\nu(r)=\eta(r)+\alpha\,h(r)$ are increasing as well as positive $\forall$
$r\in[0,R]$. This clearly shows the growth of $\eta(r)$ is faster than the
deformation function $h(r)$ in Case II also.

(viii) In Fig. 8, the left panel shows the effective radial pressure
($p^{\text{eff}}_{r}$) and the right panel shows the effective tangential
pressure ($p^{\text{eff}}_{t}$) with respect to the radial coordinate
$r/r_{b}$. We set the same numerical values as used in Fig. 1. Here the
constants $\chi$ and $\beta$ also influence the radial and tangential
pressures ($p^{\text{eff}}_{r}$ and $p^{\text{eff}}_{t}$), and we observe the
following important points from Fig. 3 as: When we fix $\beta$ and $\chi$ move
from positive to negative, the values of the pressures $p^{\text{eff}}_{r}$
and $p^{\text{eff}}_{t}$ at the core of stellar object increase, but when
$\beta$ decreasing then $p^{\text{eff}}_{r}$ and $p^{\text{eff}}_{t}$ at core
also decrease for fixing $\chi$. On the other hand, if we compare the Case I
($\beta>0$) and Case II ($\beta<0$), the pressure at core in Case I is higher
than the Case II (see Table 1 and 2 as well as Figs. 3 and 8).

(ix) In Fig. 9, the left panel shows the anisotropy for the CGD contribution
($\Delta_{CGD}$) and the right panel shows the effective anisotropy
($\Delta^{\text{eff}}$) with respect to $r/r_{b}$. We set the same numerical
values as used in Fig. 6. Here we find an interesting observations that when
we look at the left panel of Fig. 9, the anisotropic contribution
$\Delta_{CGD}$ is increasing throughout the compact star model which shows
that CGD approach can also introduce a stronger anisotropy within the object
since the effective anisotropy $\Delta^{\text{eff}}=\Delta_{FRT}+\Delta_{CGD}$
will be higher than seed anisotropy $\Delta_{FRT}$. Due to this, the effective
anisotropic force $F_{a}=\frac{2\,\Delta^{\text{eff}}}{2}$ will produce
stronger effects to balance the system in order to achieve the hydrostatic
equilibrium near the surface.

(x) Variation of the adiabatic index ($\Gamma$) with respect to the radial
coordinate $r/r_{b}$ are featured in Fig. 10. Here, we set the numerical
values of constants $\beta=0.004$, $\alpha=1.4$, $\gamma=-0.002$,
$\frac{M_{0}}{R}=0.2$, and $Y=0.005$. We have mentioned earlier that the
anisotropy may improve the stability of the model. Here we can note from Fig.
10 that value of the adiabatic index $\Gamma$ for both Cases I and II is
higher in the presence of gravitational decoupling. On the other hand, we
notice from Fig. 11 that the adiabatic index $\Gamma$ is monotonic increasing
towards the surface. Tables 3 and 4 show that the adiabatic index
$\Gamma_{0}>\Gamma_{crit}$ for both the Cases I and II and hence are stable
against the radial adiabatic infinitesimal perturbations.

(xi) As earlier mention that anisotropic contribution $\Delta_{CGD}$ due
gravitational decoupling is negative and decreasing, then anisotropic force
will show less impact as compare to hydrostatic force for balancing the system
of Case I which can be seen from Fig. 11 (left panel), while in Case II the
anisotropic force is stronger than the hydrostatic force near the surface. On
the other hand, this force in increasing throughout the star which implies
that the anisotropic force in Case 2 will introduce a strong repulsive force
to avoid the gravitational collapse.

As a final comment we would like to add here that though the present work is
an extension of the work of Maurya et al. [71], the obtained results widely
differ from that due to particular CGD technique as can be observed from all
the case studies in connection to the above descriptions on Figs. 1–11. As far
as literature survey reveals that this is the first ever approach to solve
filed equations in $f(R,T)$ gravity under the extended gravitational
decoupling approach, where CGD has been employed and successfully generated
physically viable solutions for anisotropic system.

## Appendix

$\displaystyle\theta_{11}(r)=4\chi\gamma(1+r^{2}Y)^{2}+8\gamma\pi(1+r^{2}Y)^{2},$
$\displaystyle\theta_{12}(r)=4Y[3\gamma(\chi+\chi
r^{2}Y)^{2}+3\pi(8\gamma\pi(1+r^{2}Y)^{2}-\alpha
Y(3+r^{2}Y))+\chi(18\gamma\pi(1+r^{2}Y)^{2}-\alpha Y(3+r^{2}Y))],$
$\displaystyle\theta_{21}(r)=\frac{192(\chi^{2}+6\chi\pi+8\pi^{2})(1+3Cr^{2})(C-Y)YL_{1}}{(1+Cr^{2})}+\frac{768C(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}(C-Y)YL_{1}(1+r^{2}Y)}{(1+Cr^{2})}$
$\displaystyle\hskip
22.76228pt+96(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}(C-Y)Y\Psi_{21}(r)$
$\displaystyle\theta_{22}(r)=-(1+r^{2}Y)\Big{[}96(\chi^{2}+6\chi\pi+8\pi^{2})(C-Y)\Psi_{21}(r)+\frac{384C(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}(C-Y)\Psi_{21}(r)}{(1+Cr^{2})}$
$\displaystyle\hskip 22.76228pt+4\alpha
r^{2}\Psi^{2}_{21}(r)+96(\chi^{2}+6\chi\pi+8\pi^{2})(C-Y)\Big{\\{}30\gamma(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}(C-Y)Y\frac{8Cr^{2}YL_{1}}{(1+Cr^{2})^{2}}$
$\displaystyle\hskip 22.76228pt-\frac{4YL_{1}}{1+Cr^{2}}+\frac{4\alpha
r^{2}Y^{2}L_{1}}{(1+r^{2}Y)^{2}}-\frac{2\alpha
YL_{1}}{1+r^{2}Y}L_{4}-2Y\big{(}3\gamma\,(2+r^{2}Y)+L_{2}+L_{3}\big{)}\Big{\\}}\Big{]},$

where

$\displaystyle\Psi_{21}(r)=L_{5}-\frac{4YL_{1}}{(1+Cr^{2})}-\frac{-2\alpha
Y\,L_{1}}{(1+r^{2}Y)}+L_{4}-2Y\,\big{[}3\chi^{2}\gamma(2+r^{2}Y)+L_{2}+L_{3}\,\big{]},$
$\displaystyle
L_{1}=3C(\chi+4\pi)-4(\chi+3\pi)Y,~{}~{}~{}L_{2}=6\pi[(5-\alpha)Y+4\gamma\pi(2+r^{2}Y)],~{}~{}L_{3}=2\chi[(5-\alpha)Y+9\gamma\pi(2+r^{2}Y)],$
$\displaystyle
L_{4}=3C(\chi+4\pi)[(5-\alpha)Y+2\chi\gamma(2+r^{2}Y)+4\gamma\pi(2+r^{2}Y)],~{}~{}L_{5}=6\gamma(\chi^{2}+6\chi\pi+8\pi^{2})r^{2}(C-Y)Y.$
$\displaystyle C_{11}(r_{b})=3\gamma(\chi+\chi
r_{b}^{2}Y)^{2}+3\pi\big{(}8\gamma\pi(1+r_{b}^{2}Y)^{2}-\alpha
Y(3+r_{b}^{2}Y)\big{)}+\chi\big{[}18\gamma\pi(1+r_{b}^{2}Y)^{2}-\alpha
Y(3+r_{b}^{2}Y)\big{]},$ $\displaystyle
C_{22}(r_{b})=6\pi\big{[}-4-3r_{b}^{2}Y+r_{b}^{4}Y^{2}+\beta
r_{b}^{2}(-8\gamma\pi(1+r_{b}^{2}Y)^{2}+\alpha
Y(3+r_{b}^{2}Y))\big{]}+\chi\big{[}-9-10r_{b}^{2}Y+2r_{b}^{4}Y^{2}$
$\displaystyle\hskip
25.6073pt+\beta\big{(}-36\gamma\pi(r_{b}+r_{b}^{3}Y)^{2}+\alpha(3+8r_{b}^{2}Y+2r_{b}^{4}Y^{2})\big{)}\big{]}$

## Acknowledgement

SKM and SR acknowledge that this work is carried out under TRC Project (Grant
No. BFP/RGP/CBS-/19/099), the Sultanate of Oman. SKM is thankful for
continuous support and encouragement from the administration of University of
Nizwa.

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