# Role of gravitational decoupling on isotropization and complexity of self-
gravitating system under complete geometric deformation approach

S. K. Maurya<EMAIL_ADDRESS>Department of Mathematical and Physical
Sciences, College of Arts and Sciences, University of Nizwa, Nizwa, Sultanate
of Oman  Riju Nag<EMAIL_ADDRESS>Department of Mathematical and Physical
Sciences, College of Arts and Sciences, University of Nizwa, Nizwa, Sultanate
of Omana

###### Abstract

In the present paper, we discuss the role of gravitational decoupling to
isotropize the anisotropic solution of Einstein’s field equations in the
context of the complete geometric deformation (CGD) approach and its influence
on the complexity factor introduced by L. Herrera (Phys. Rev. D 97, 044010
(2018)) in the static self-gravitating system. Moreover, we also proposed a
simple and effective technique to generate new solutions for self-gravitating
objects via CGD approach by using two systems with the same complexity factor
and vanishing complexity factor proposed by Casadio et al. Eur. Phys. J. C 79,
826 (2019). The effect of decoupling constant and the compactness on the
complexity factor have been also analyzed for the obtained solutions.

###### pacs:

04.20.Jb, 04.40.Nr, 04.70.Bw

##  I Introduction

Quantifying the term ”complexity” has been quite a fascinating challenge among
researchers. Depending on various physical problems, the term complexity
changes its notion. For example, a perfect crystal can be thought of as a
system with zero complexity. Here, zero complexity means the crystal structure
is perfectly ordered and periodic. In contrast, an isolated ideal gas is fully
disordered and it contains maximum information as the system can be obtained
in any accessible state with equal probability. Now, if we consider the
concept of ”disequilibrium” i.e. how a system deviates from the equilibrium,
we find that the ideal gas has minimum disequilibrium while the perfect
crystal has maximum disequilibrium. So the contrasting views of complexity
while considering ”information” and ”disequilibrium” can be addressed by
defining complexity as a product of these concepts c7 . So in this way,
complexity is zero for both perfect crystal and ideal gas, as it should be.
The work of Lopez-Ruiz and collaborators c7 ; c10  about complexity has been
extended to self gravitating systems c13 ; c14 ; c15 ; c16 ; c17 ; c18 .
Recently, Carrasco-Hidalho and Contreras c20  proposed a polynomial complexity
factor containing gravastar model c19  as a special case. Contreras and
Fuenmayor c21  considered complexity factor as a physical quantity containing
anisotropy and gradients in the density. L. Herrera c22  proposed a new
definition of complexity for self-gravitating, spherically symmetric systems,
based on a specific parameter that arises in the orthogonal splitting of
Riemann tensor. Some more recent works in this regard can be found in c23 ;
c24 ; c25 ; c26 .
It can be said, that fixing some value of the complexity factor for a specific
scenario (example: a system having vanishing complexity) can act as an
equation of state which may lead close approximation to Einstein’s field
equations. But for these equations, it is very difficult to obtain analytical
solutions. In this scenario, a recent well-known tool called Gravitational
Decoupling (GD) by means of the minimal geometric deformation (MGD) o1  and
its extension known as complete geometric deformation m61  formalism works
perfectly to convert the isotropic solutions into anisotropic domain or can be
even used to obtain new solutions. Moreover, it is well-known that the MGD
tool is a transformation that is performed on the metric potential along with
the radial component of the line element by introducing a decoupler function.
As a consequence, the original system splits into two relatively simpler sets
of the equations. Another great advantage of MGD is that it can extend a
simple solution to more generalized and complex cases by the addition of an
extra source ($\Theta^{i}_{j}$) with the original energy-momentum tensor via
coupled with a dimensionless parameter. Later on, Ovalle and his collaborators
found a drawback in the MGD approach, such as considering only radial
transformation can’t explain a stable black hole with a well-defined horizon.
In this regard, the MGD was extended to deform both radial and temporal metric
functions m61 . Mathematically, in the extended case of MGD, the deformation
acts in the following way:

$\displaystyle\nu(r)\mapsto\xi(r)+\alpha
h(r),~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\lambda(r)\mapsto-\ln[\mu(r)+\alpha
f(r)]$

It is noted here that the extended gravitational decoupling (EGD) demands the
supposition of a seed solution, which allows reducing the number of free
variables (or number of degrees of freedom). Due to this, we need only two
extra conditions in order to close the system. There are many ways to solve
the equation despite the appearance of the new degree of freedom $h(r)$ in the
$\theta$-sector, such as implementing the mimick approach with particular form
of $h(r)$, or using EoS approach together with the mimick approach sunilepjc1
; sunilepjc2 ; sunilepjc3  to solve the $\Theta$-sector. On the other hand, it
must be noted that the hydrostatic balance gets severely modified because of
the deformation. Therefore, it is very important to check the hydrostatic
balance in order to assess the viability of the solution. Recently some
interior solutions were generalized into the anisotropic domain using
gravitational decoupling via MGD and CGD approaches in different contexts MGD0
; MGD1 ; MGD2 ; MGD3 ; MGD4 ; MGD5 ; MGD6 ; MGD7 ; MGD8 ; MGD9 ; MGD10 ; MGD11
; MGD12 ; MGD13 ; MGD14 ; MGD15 ; MGD16 ; MGD17 ; MGD18 ; MGD21 ; MGD22 .
Often for some cases like extremely dense compact star solutions, the
anisotropic scenario gives more realistic results, whereas, in some other
cases, the isotropic solution is more necessary. In compact stars, due to
extreme internal density and strong gravity, the pressure components break
down into two components, i.e. radial and tangential pressure. For the matter
to be isotropic in nature, these two components of pressure have to be of the
same value. In this connection, Ruderman rud  showed that for densities higher
than $10^{15}g/cm^{3}$, the two components of pressure don’t have the same
magnitude and nuclear matter transforms into anisotropic distribution.
On the other hand, the finding of the new physical viable anisotropic solution
for a static self-gravitating system is easier than the isotropic solution of
Einstein’s field equations. Till now, the researchers have obtained around 130
interior solutions of Einstein’s field equations (EFE) for perfect fluid
matter distributions but only few of them are well-behaved that can be used
for modeling of the self-gravitating compact objects Delgaty . This is why it
is still a challenge to obtain the new well-behaved isotropic solution of the
Einstein field equations. Therefore, recently Casadio and his collaborators
m79  have proposed is a very powerful methodology, known as isotropization
techniques via gravitational decoupling using the MGD approach to find the new
isotropic solutions for any known seed spacetime geometry corresponding to
anisotropic matter distribution. In this work, they have investigated a new
isotropic solution using the MGD approach as well as two other gravitationally
decoupled anisotropic solutions corresponding to two systems with the same
complexity factor, and zero complexity factor using Tolman IV solution. Some
recent solutions on anisotropic star with different complexity factor can be
seen in the following works GDC1 ; GDC2 ; GDC3 ; GDC4 ; GDC5 .
In the current article, we develop an isotropization technique using
gravitational decoupling in the framework of a complete geometric deformation
(CGD) approach to find the new isotropic solutions from a known spacetime
geometry for the anisotropic matter distribution. An example has been
presented to validate this methodology. Moreover, we also discussed the
complexity factor and the effect of the decoupling constant on complexity
factor for this isotropic solution. The present simple methodology has also
been utilized to obtain the new anisotropic solutions by taking two systems
with the same complexity factor as well as for zero complexity factor using
the Karori-Barua solution.

The article is arranged as follows: The section I is the introduction, while
in the section II, the Einstein field equation for two sources by
gravitational decoupling has been discussed. The section III consists of the
method of isotropization of the gravitationally decoupled system and new
solution obtained by taking Tolman-Kuchowicz spacetime for seed solution. The
complexity by gravitational decoupling has been analyzed in the section IV. In
this section, we also discussed the complexity factor for isotropic solution
obtained in section III. In section V, we have investigated new anisotropic
EGD solutions generated by two systems with same complexity factors and
vanishing complexity factor using Karori-Barua seed solution, which are
presented in subsections A and B. The last section VI contains the discussions
and conclusions of the article.

##  II Einstein’s field equation for two sources introduced by gravitational
decoupling

We propose the brief review of Einstein’s field equations with two different
sources,

$\displaystyle R_{ij}-\frac{1}{2}\,g_{ij}\,R=-8\pi(T_{ij}+\beta\,\Theta_{ij})$
(1)

Now with $G=c=1$, the relativistic units are considered to express the field
equations for the Ricci tensor denoted by $R_{ij}$, and $R$ is contracted
Ricci scalars, and $\beta$ is decoupling constant. Here, $T_{ij}$ denote the
energy-momentum tensor and the source $\theta_{ij}$ may contain new fields,
like scalar, vector and tensor fields. Since the Einstein tensor (1) satisfy
the Bianchi identity, therefore the effective energy momentum tensor
$\hat{T}_{ij}=T_{ij}+\beta\,\Theta_{ij}$ must be conserved, that is,

$\displaystyle{\nabla_{i}}\hat{T}^{ij}=0,$ (2)

The following static spherically symmetric line element is taken for
describing the space-time of interior region of the stellar system as,

$\displaystyle
ds^{2}=-e^{\lambda(r)}dr^{2}-r^{2}\big{(}d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}\big{)}~{}+e^{\nu(r)}dt^{2},$
(3)

where the metric potentials $\nu$ and $\lambda$ are only radially dependent.
However, the effective energy momentum tensor $\hat{T}_{ij}$ is considered for
anisotropic matter distribution,

$\displaystyle\hat{T}_{ij}=\left({\epsilon}+{P}_{r}\right)u_{i}u_{j}+{P}_{\perp}g_{ij}+({P}_{r}-{P}_{\perp})\chi_{i}\,\chi_{j},$
(4)

where ${P}_{r}$ and ${P}_{\perp}$ denote the radial and tangential pressures,
respectively while ${\epsilon}$ is the energy density of matter. Moreover,
$u^{i}$ denotes a contravariant $4$-velocity and
$\chi^{i}=\sqrt{1/g_{rr}}\,\delta^{i}_{1}$ is a unit space-like vector in the
radial direction. Then under the line element (3), the Einstein field equation
(1) with Eq.(4) provides the following differential equations,

$\displaystyle
8\pi{\epsilon}=T^{0}_{0}+\beta\,\Theta^{0}_{0}=\frac{1}{r^{2}}-e^{-\lambda}\left(\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right),$
(5) $\displaystyle
8\pi{P}_{r}=-T^{1}_{1}-\beta\,\Theta^{1}_{1}=-\frac{1}{r^{2}}+e^{-\lambda}\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right),$
(6) $\displaystyle
8\pi{P}_{\perp}=-T^{2}_{2}-\beta\,\Theta^{2}_{2}=\frac{e^{-\lambda}}{4}\left(2\nu^{\prime\prime}+\nu^{\prime
2}-\lambda^{\prime}\nu^{\prime}+2\frac{\nu^{\prime}-\lambda^{\prime}}{r}\right),~{}~{}~{}$
(7)

and, the conservation equation for system (5) - (7) will become,

$\displaystyle(P_{r})^{\prime}+\frac{\nu^{\prime}}{2}\big{(}\epsilon+P_{r}\big{)}-\frac{2}{r}(P_{\perp}-P_{r})=-(T^{1}_{1})^{\prime}$
$\displaystyle+\frac{\nu^{\prime}}{2}\left(T^{0}_{0}-T^{1}_{1}\right)+\frac{2(T^{2}_{2}-T^{1}_{1})}{r}-\beta\,L(\Theta_{i}^{i})=0,~{}~{}$
(8)

where the function $L(\theta_{i}^{i})$ is given by

$L(\Theta_{i}^{i})\equiv\left(\Theta^{1}_{1}\right)^{\prime}+\frac{\nu^{\prime}}{2}\left(\Theta^{0}_{0}-\Theta^{1}_{1}\right)+\frac{2}{r}\left(\Theta^{1}_{1}-\Theta^{2}_{2}\right).$
(9)

Now it is important to mention here that the source $T_{ij}$ can describe
perfect fluid or anisotropic fluid matter distribution. Suppose It describes
an anisotropic fluid matter distribution then, the effective density and
effective pressures can be read as,

$\displaystyle\epsilon=T^{0}_{0}+\beta\,\Theta^{0}_{0}=\rho+\beta\,\Theta^{0}_{0},$
(10) $\displaystyle
P_{r}=-T^{1}_{1}-\beta\,\Theta^{1}_{1},=p_{r}-\beta\,\Theta^{1}_{1}~{}$ (11)
$\displaystyle
P_{\perp}=-T^{2}_{2}-\beta\,\Theta^{2}_{2}=p_{t}-\beta\,\Theta^{2}_{2}.$ (12)

where, $\rho$, $p_{r}$ and $p_{t}$ denote the energy density, radial pressure
and tangential pressure, respectively. Then the effective anisotropy can be
given as,

$\displaystyle\hat{\Pi}=P_{\perp}-P_{r}=\Pi+\beta\,\Pi_{\Theta}$ (13)

where,

$\displaystyle\Pi=p_{t}-p_{r}~{}~{}~{}\text{and}~{}~{}~{}\Pi_{\Theta}=(\Theta^{1}_{1}-\Theta^{2}_{2}).$
(14)

Here the anisotropy $\Pi_{\Theta}$ is generated by second source
$\Theta_{ij}$, and Misner-Sharp mass function $m(r)$ for the effective system
can be calculated by the formula,

$\displaystyle m(r)$ $\displaystyle=$
$\displaystyle\frac{r}{2}[1-e^{-\lambda(r)}]=4\pi\int^{r}_{0}x^{2}\epsilon(x)dx$
(15) $\displaystyle=$
$\displaystyle\underbrace{4\pi\int^{r}_{0}x^{2}\rho(x)dx}_{m_{GR}}+\underbrace{4\pi\beta\int^{r}_{0}x^{2}\Theta^{0}_{0}(x)dx}_{m_{\Theta}}.~{}~{}~{}~{}~{}~{}~{}$

The $m_{GR}$ and $m_{\Theta}$ represent the mass function due matter
distribution $T_{ij}$ and $\Theta_{ij}$, respectively.
Also, there is an another definition in order describe the energy content
inside a fluid sphere which was proposed by Tolman many years before. The
Tolman mass ($m_{T}$) for the spherically symmetric static spacetime (3) and
energy-momentum tensor $\hat{T}_{ij}$ can be given by the formula [19]

$\displaystyle
m_{T}=4\pi\,\int^{r}_{0}x^{2}\,e^{(\lambda+\nu)/2}(\rho+P_{r}+2P_{\perp})dx,$
(16)

The above formula was proposed in order a measure of the energy contained
inside a fluid sphere of radius $r$. However, Tolman mass function $m_{T}$
using the field equations (5)-(7) under the spacetime (3) can be written as,

$\displaystyle m_{T}=\frac{r^{2}\,\nu^{\prime}}{2}\,e^{(\nu-\lambda)/2},$ (17)

The above formula states about the physical meaning of $m_{T}$ as the active
gravitational mass. Since instantaneously at rest in a static gravitational
field, the gravitational acceleration of a test particle is given by (see c22
, for more details)

$\displaystyle
a=\frac{\nu^{\prime}\,e^{-\lambda}}{2}=\frac{e^{-\nu/2}\,m_{T}}{r^{2}}$ (18)

Now we apply the extended gravitational decoupling by means of a complete
geometric deformation (CGD) approach in order to see the general effects of
the extra source $\Theta$ on the energy-momentum tensor $T_{ij}$. Under this,
the metric functions $e^{\lambda}$ and $e^{\nu}$ undergone by the following
transformation Ovalle m61  as,

$\displaystyle\xi(r)\mapsto\nu(r)=\xi(r)+\alpha~{}h(r),$ (19)
$\displaystyle\mu(r)\mapsto e^{-\lambda(r)}=\mu(r)+\alpha~{}f(r).$ (20)

where, $f(r)$ and $h(r)$ denote the geometric deformation functions for the
radial and temporal metric components, respectively. Since here we are
considering the extended case therefore we need to set $f(r)\neq 0$ and
$h(r)\neq 0$. Then the transformations (19) and (20) allow us to split the
field equations (5)-(7) into two sets of equations: (i) the standard Einstein
field equations corresponding to energy-momentum tensor $T_{ij}$ (same as at
$\alpha$ = 0) as

$\displaystyle 8\pi{\rho}=\frac{1-\mu}{r^{2}}-\frac{\mu^{\prime}}{r},$ (21)
$\displaystyle 8\pi
p_{r}=\frac{\mu-1}{r^{2}}-\frac{\mu~{}~{}\xi^{\prime}}{r},$ (22)
$\displaystyle 8\pi
p_{t}=\mu\bigg{(}\frac{\xi^{\prime\prime}}{2}+\frac{\xi^{\prime
2}}{4}+\frac{\xi^{\prime}}{2r}\bigg{)}+\bigg{(}\frac{\xi^{\prime}\mu^{\prime}}{4}+\frac{\mu^{\prime}}{2r}\bigg{)},~{}~{}~{}$
(23)

with the conservation equation,

$\displaystyle(p_{r})^{\prime}+\frac{\xi^{\prime}}{2}\big{(}\rho+p_{r}\big{)}=\frac{2\Pi}{r}.$
(24)

and the solution of this system can be described by the following spacetime,

$\displaystyle ds^{2}=-\mu dr^{2}-r^{2}\big{(}d\theta^{2}-\sin^{2}\theta
d\phi^{2}\big{)}+e^{\xi}dt^{2},$ (25)

with

$\displaystyle\mu=1-\frac{2m_{GR}}{r}=4\pi\int^{r}_{0}x^{2}\rho(x)dx.$ (26)

(ii) Now the second set of equations for the extra source is determined by
turning on $\beta$ as,

$\displaystyle
8\pi\Theta^{0}_{0}=-~{}~{}\bigg{(}\frac{f^{\prime}}{r}+\frac{f}{r^{2}}\bigg{)},$
(27) $\displaystyle
8\pi\Theta^{1}_{1}=-f~{}~{}\bigg{(}\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\bigg{)}-\frac{\mu
h^{\prime}}{r},$ (28) $\displaystyle
8\pi\Theta^{2}_{2}=-\frac{f}{2}~{}~{}\bigg{(}\nu^{\prime\prime}+\frac{\nu^{\prime
2}}{2}+\frac{\nu^{\prime}}{r}\bigg{)}-\frac{\alpha
f^{\prime}}{2}~{}~{}\bigg{(}\frac{\nu^{\prime}}{2}+\frac{1}{r}\bigg{)}$
$\displaystyle\hskip 34.14322pt-\frac{\mu}{4}\big{(}2h^{\prime\prime}+\alpha
h^{\prime
2}+\frac{2~{}h^{\prime}}{r}+2\xi^{\prime}h^{\prime}\big{)}-\frac{\mu^{\prime}~{}h^{\prime}}{4}.~{}~{}~{}~{}~{}~{}$
(29)

whose conservation equation read as,

$\displaystyle-\frac{\nu^{\prime}}{2}\big{(}\theta^{0}_{0}-\theta^{1}_{1}\big{)}+\big{(}\theta^{1}_{1}\big{)}^{\prime}-\frac{h^{\prime}}{2}\big{(}p_{r}+\rho\big{)}=\frac{2\Pi_{\Theta}}{r}.~{}~{}~{}~{}~{}~{}~{}~{}$
(30)

##  III Isotropization of gravitationally decoupled system

In this section, we will adopt the systematic approach proposed by Casadio and
his collaborators m79  to isotropize the decoupled system (5)-(7) under EGD
scenario. As we discussed previously, the effective anisotropy $\hat{\Pi}$
given by (13) may not be same as the anisotropy $\Pi$ due to extra
contribution $\beta\,\Pi_{\Theta}$. Here, our aim is to isotropize the
effective system, which can be obtained by setting $\hat{\Pi}=0$ with assuming
$\Pi\neq 0$ [see Ref.m79  for more details]. Therefore,
$\hat{\Pi}=\Pi+\beta\,\Pi_{\Theta}=0$ leads

$\displaystyle\Pi=-\beta\,\Pi_{\Theta}~{}~{}\Longrightarrow~{}~{}\Pi=-\beta(\Theta^{1}_{1}-\Theta^{2}_{2}).$
(31)

Now by plugging the Eqs.(28) and (II) in Eq.(31), we get the following non-
linear differential equation,

$\displaystyle
f^{\prime}(2+\nu^{\prime}r)+f(-4-2\nu^{\prime}r+2\nu^{\prime\prime}r^{2}+\nu^{\prime
2}r^{2})+r(-2h^{\prime}\mu$
$\displaystyle+h^{\prime}\mu^{\prime}r+2h^{\prime\prime}\mu r+bh^{\prime 2}\mu
r+2h^{\prime}\nu^{\prime}\mu r)+\frac{4r^{2}}{\beta}\,\Pi=0.~{}~{}~{}$ (32)

As we can see that the above equation (32) is a first order linear ODE in
$f(r)$ while it is a second order non-linear in $h(r)$. Therefore, we solve
the above differential for $f(r)$ due to simplicity. Now we assume a spacetime
corresponding energy-momentum tensor $T_{ij}$ generated by Tolman-Kuchowicz
metric functions $\\{\xi,\,\mu\\}$

$\displaystyle
ds^{2}=-(1+Kr^{2}+Lr^{4})\,dr^{2}-r^{2}d\Omega_{2}^{2}~{}+e^{Ar^{2}+B}dt^{2}.~{}~{}~{}$
(33)

together with temporal deformation function $h(r)=Cr^{2}$ in order to
isotropize the gravitationally decoupled system (5)-(7). Then, the metric
functions $\mu(r)=1/(1+Kr^{2}+Lr^{4})$ and $e^{\xi(r)}=e^{Ar^{2}+B}$ describe
the anisotropic solution for the system (21)-(23). The constant parameters
$K$, $L$, $A$, and $B$ will be determined by matching of the seed spacetime
(33) with exterior vacuum spacetime at surface $r=R$, If we consider exterior
vacuum spacetime is described by exterior Schwarzschild solution, then

$\displaystyle 1-\frac{2M_{s}}{R}=e^{\xi(R)},~{}~{}~{}$ (34) $\displaystyle
1-\frac{2M_{s}}{R}=\mu(R),~{}~{}~{}$ (35) $\displaystyle p_{r}(R)=0,~{}~{}~{}$
(36)

where $m_{GR}(R)=M_{s}$ is the total mass of the seed spacetime (33) related
to the energy-momentum tensor $T_{ij}$. Using the conditions (34)-(36), we
find the constants $A$, $B$, and $K$

$\displaystyle A=\frac{M_{s}}{R^{2}(R-2M_{s})},$ (37) $\displaystyle
B=\frac{-4M_{s}^{2}+5M_{s}R-R^{2}}{(2M_{s}-R)R},$ (38) $\displaystyle
K=\frac{-2M_{s}-2LM_{s}R^{4}+LR^{5}}{(2M_{s}-R)R^{2}}.$ (39)

By plugging of spacetime geometry (33) into Eq.(32) and using $h=Cr^{2}$, we
obtain the deformation function $f(r)$ as

$\displaystyle
f(r)=\frac{r^{2}e^{-\left(r^{2}(A+bC)\right)}}{\beta\sqrt{K^{2}-4L}\left(Kr^{2}+Lr^{4}+1\right)\left(A^{2}+2A\beta
C-AK+\beta^{2}C^{2}-\beta CK+L\right)}\Bigg{[}e^{-\frac{AK+\beta
CK+L}{L}}\Bigg{(}e\Bigg{\\{}\sqrt{K^{2}-4L}\left(K+Lr^{2}\right)$
$\displaystyle\times\big{(}A^{2}+2A\beta C-AK+\beta^{2}C^{2}-\beta
CK+L\big{)}e^{\frac{(A+\beta
C)\left(K+Lr^{2}\right)}{L}}+\beta^{2}C^{2}f_{2}(r)\left(Kr^{2}+Lr^{4}+1\right)\chi_{1}+\beta^{2}C^{2}f_{3}(r)\,\big{(}Kr^{2}+Lr^{4}$
$\displaystyle+1\big{)}\chi_{2}\Bigg{\\}}-f_{1}(r)\sqrt{K^{2}-4L}(A+\beta
C)\left(Kr^{2}+Lr^{4}+1\right)e^{\frac{K(A+\beta C)}{L}}\left(A^{2}+2A\beta
C-AK+3\beta^{2}C^{2}-\beta CK+L\right)\Bigg{)}+F\Bigg{]},~{}~{}~{}~{}~{}$ (40)

where, $F$ is a constant of integration, and then the solution of the system
(5)-(7) can be described by the spacetime

$\displaystyle
ds^{2}=-\frac{(1+Kr^{2}+Lr^{4})}{1+\beta\,(1+Kr^{2}+Lr^{4})\,f(r)}dr^{2}-r^{2}d\Omega_{2}^{2}+\big{(}e^{Ar^{2}+B+\beta
Cr^{2}}\big{)}dt^{2}.$ (41)

However, the effective energy density and effective pressures can be given as,

$\displaystyle P_{r}(r,\,\beta)=\frac{1}{Kr^{4}+Lr^{6}+r^{2}}\Big{[}\beta
f(r)\left(2Ar^{2}+1\right)\left(Kr^{2}+Lr^{4}+1\right)+r^{2}\left(2A-K-
Lr^{2}\right)+2\beta^{2}Cr^{2}f(r)\left(Kr^{2}+Lr^{4}+1\right)+2\beta
Cr^{2}\Big{]},~{}~{}~{}~{}$ (42) $\displaystyle
P_{\perp}(r,\,\beta)=\frac{1}{2r\left(Kr^{2}+Lr^{4}+1\right)^{2}}\Big{[}2r\left(A^{2}\left(Kr^{4}+Lr^{6}+r^{2}\right)+A\left(Kr^{2}+2\right)-K-2Lr^{2}\right)+\beta^{2}Cr\left(Kr^{2}+Lr^{4}+1\right)\Big{(}4f(r)\left(Ar^{2}+1\right)$
$\displaystyle\hskip
17.07182pt\times\left(Kr^{2}+Lr^{4}+1\right)+r\left(6Cr+\Omega(r)Kr^{2}+\Omega(r)Lr^{4}+\Omega(r)\right)\Big{)}+\beta\Big{\\{}2r\Big{(}C\left(2A\left(Kr^{4}+Lr^{6}+r^{2}\right)+Kr^{2}+2\right)+Af(r)$
$\displaystyle\hskip
17.07182pt\times\left(Ar^{2}+2\right)\left(Kr^{2}+Lr^{4}+1\right)^{2}\Big{)}+\Omega(r)\left(Ar^{2}+1\right)\left(Kr^{2}+Lr^{4}+1\right)^{2}\Big{\\}}+2\beta^{3}C^{2}f(r)r^{3}\left(Kr^{2}+Lr^{4}+1\right)^{2}\Big{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(43)
$\displaystyle\epsilon(r,\,\beta)=\frac{1}{\left(Kr^{3}+Lr^{5}+r\right)^{2}}\Big{[}-\beta(\Omega(r)r+f)\left(Kr^{2}+Lr^{4}+1\right)^{2}+K^{2}r^{4}+Kr^{2}\left(2Lr^{4}+3\right)+Lr^{4}\left(Lr^{4}+5\right)\big{]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(44)

The matching conditions (34)-(36) for new solution (41)-(44)

$\displaystyle
e^{-\lambda(R)}=\frac{1}{1+KR^{2}+LR^{4}}+\beta\,f(R)=1-\frac{2{M}}{R},~{}~{}~{}$
(45) $\displaystyle e^{\nu(R)}=e^{AR^{2}+B+\beta\,CR^{2}}=1-\frac{2{M}}{R},$
(46) $\displaystyle P_{r}(R)=0,$ (47)

determine the constant $B$, total mass $M$, and integration constant $F$ as

$\displaystyle B=\ln\Big{[}1-\frac{2M}{R}\Big{]}-AR^{2}-\beta\,CR^{2},$ (48)
$\displaystyle M=M_{s}-\frac{\beta\,R}{2}\,f(R)$ (49)

where $m(R)=M$ is the total mass of the deformed compact object corresponding
to energy-momentum tensor $\hat{T}_{ij}$ given by equation (4). However, we
avoid to write expression for $F$ due to lengthy expression.

Figure 1: The behavior of radial pressure ($P_{r}\times 10^{4}$)-top left,
tangential pressures ($P_{\perp}\times 10^{4}$)-top right, energy density
($\epsilon\times 10^{4}$)-bottom left and anisotropy ($\hat{\Pi}\times
10^{4}$)-bottom right versus radial coordinate $r/R$ for different coupling
constant $\beta$ with compactification factor $\frac{M_{s}}{R}=0.2$ with
$C=0.0002$, and $L=1.5\times 10^{-8}$. The description of the curves in the
figures is as follows: i. Solid-dark yellow for $\beta=0$, ii. dashed-blue for
$\beta=0.2$, iii. Dashed-dark red for $\beta=0.4$, and Dotted dash-green for
$\beta=0.6$. The figures are plotted corresponding to the isotropic solution
discussed in section III.

It is important to mention here that the expressions (42) and (43) given by
$P_{r}$ and $P_{\perp}$ are the same at each point within the compact object
for all $\beta$ i.e. the effective anisotropy $\hat{\Pi}=0$ (see Fig.1), which
implies that the solution given by spacetime geometry (41) represents an
isotropic solution of the decoupled system (5)-(7). Therefore, the
gravitational decoupling not only extends the isotropic solution to
anisotropic domain but it also plays an important role to convert anisotropic
solution in isotropic domain.

##  IV Complexity by gravitational decoupling

The definition of the complexity factor in static and spherically symmetric
self-gravitating systems was initially proposed by Herrera, which is a scalar
function denoted by $Y_{TF}$ and it can be measured by anisotropy $\Pi$ and
energy density gradient $\rho^{\prime}$. Later on Herrara and his
collaborators extended this complexity in the context of dynamical spherically
symmetric dissipative self-gravitating fluid distributions. Based on the
Herrera definition, we denote $\hat{Y}_{TF}$ as a complexity factor for the
spherically symmetric static self-gravitating systems (5)-(7) which is given
by,

$\displaystyle\hat{Y}_{TF}=8\pi\hat{\Pi}-\frac{4\pi}{r^{3}}\int^{r}_{0}x^{3}\epsilon^{\prime}(x)dx.$
(50)

As it is mentioned by Herrera that the complexity factor $\hat{Y}_{TF}$
represents the influence of local anisotropy of pressure and density
inhomogeneity on the Tolman mass ($m_{T}$) Or, how the Tolman mass is changed
by the above two factors defined in $Y_{TF}$. In order to see the influence of
$\hat{Y}_{TF}$ on the Tolman $m_{T}$, we write the Eq.(16) in terms of
complexity factor as,

$\displaystyle
m_{T}=M_{T}\Big{(}\frac{r}{R}\Big{)}^{2}+r^{3}\,\int^{R}_{r}\frac{e^{(\nu+\lambda)/2}}{x}\,Y_{TF}dx.$
(51)

Here, $M_{T}$ denote the total Tolman mass of the fluid sphere of radius $R$.

According to Herrera c22  observations, it is worthwhile noting that
(i) The complexity factor vanishes for not only isotropic fluid but also for
all other configurations where both the terms in (50) identically vanish.
(ii) From the abovementioned criteria, it is evident that there are plenty of
configurations with vanishing complexity factors.
(iii) It must also be noted that although the contribution of pressure
anisotropy to $Y_{TF}$ is local in nature, this is not the case for density
energy inhomogeneity.

In the context of MGD, Casadio et al. argued that the complexity factor
satisfies the additive property and then the complexity factor for
gravitationally decoupled system will be the sum of two existing complexity
factors generated by the sources $T_{ij}$ and $\theta_{ij}$. Therefore, using
the above facts, the complexity factor $\hat{Y}_{TF}$ given by Eq.(50) can be
also written into sum of two complexity factors corresponding to the source
$T_{ij}$ and $\Theta_{ij}$ as,

$\displaystyle\hat{Y}_{TF}$ $\displaystyle=$ $\displaystyle
8\pi\,\hat{\Pi}-\frac{4\pi}{r^{3}}\int^{r}_{0}x^{3}\epsilon^{\prime}(x)dx,$
(52) $\displaystyle=$ $\displaystyle
8\pi\,\Pi-\frac{4\pi}{r^{3}}\int^{r}_{0}x^{3}\rho^{\prime}(x)dx$
$\displaystyle+8\pi\,\beta\,\Pi_{\Theta}-\frac{4\pi\,\beta}{r^{3}}\int^{r}_{0}x^{3}\left[\Theta^{0}_{0}(x)\right]^{\prime}dx,$

which is denoted as,

$\displaystyle\hat{Y}_{TF}=Y_{TF}+Y^{\Theta}_{TF}.$ (53)

Here, we denote the $Y_{TF}$ is the complexity factor for the system (21)-(23)
while $Y^{\Theta}_{TF}$ for (27)-(II) corresponding to the sources $T_{ij}$
and $\Theta_{ij}$, respectively.
Now we will discuss two following cases: (A). Complexity factor generated by
the isotropic solution (40)-(44) for the energy-momentum tensor
$\hat{T}_{ij}$, and (B). Some new solutions generated EGD approach for the
systems having same or vanishing complexity factors.

###  IV.1 Complexity factor generated by isotropic solution (41)-(44)

The complexity factor for the systems (5)-(7) corresponding to the energy-
momentum tensor $\hat{T}_{ij}$ is

$\displaystyle\hat{Y}_{TF}$ $\displaystyle=$ $\displaystyle
8\pi\,\hat{\Pi}-\frac{4\pi}{r^{3}}\int^{r}_{0}x^{3}\epsilon^{\prime}(x)dx,$
(54)

Since the solution (41)-(44) is isotropic, then the effective anisotropy will
be zero i.e. $\hat{\Pi}=0$, yields

$\displaystyle\hat{Y}_{TF}$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{r^{3}}\int^{r}_{0}x^{3}\epsilon^{\prime}(x)dx,$
(55)

Using Eq.(5), we get

$\displaystyle\hat{Y}_{TF}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}-\frac{e^{-\lambda}}{r^{2}}-\frac{\lambda^{\prime}\,e^{-\lambda}}{2r},$
(56)

Now using the solution(41), we find expression for complexity factor
$\hat{Y}_{TF}$

$\displaystyle\hat{Y}_{TF}=\frac{r^{2}\left(K^{2}+2KLr^{2}+L\left(Lr^{4}-1\right)\right)}{\left(Kr^{2}+Lr^{4}+1\right)^{2}}+\frac{\beta}{2r^{2}}\big{[}r\,\Omega(r)$
$\displaystyle-2f(r)\big{]}.~{}~{}~{}~{}$ (57)

Figure 2: The behavior complexity factor $(\hat{Y}_{TF}\times 10^{4})$ versus
radial coordinate $r/R$ for different coupling constant $\beta$ with
compactness factor $\frac{M_{s}}{R}=0.2$ with $C=0.0002$, and $L=1.5\times
10^{-8}$. The description of the curves in the figures is as follows: i.
Solid-dark yellow for $\beta=0$, ii. dashed-blue for $\beta=0.2$, iii. Dashed-
dark red for $\beta=0.4$, and Dot dashed-green for $\beta=0.6$. This
complexity figure is plotted for the isotropic solution obtained in section
III.

It is noticed from the Fig.2, the decoupling constant $\beta$ is influencing
the complexity factor $\hat{Y}_{TF}$. The $\hat{Y}_{TF}$ increases when
$\beta$ increase, which implies that gravitational decoupling enhances the
complexity of the self-gravitating isotropic models.

##  V Some new solutions generated EGD approach for the systems having same or
vanishing complexity factors

###  V.1 EGD solution for two systems with same complexity factor

In this section, we will consider the situation where the complexity factor
$Y_{TF}$ related to energy-momentum tensor $T_{ij}$ remains same after using
gravitational decoupling via CGD, that is $\hat{Y}_{TF}=Y_{TF}$, which implies
$Y^{\Theta}_{TF}=0$ or

$\displaystyle\Pi_{\Theta}=\frac{1}{2r^{3}}\int^{r}_{0}x^{3}\,\big{[}\Theta^{0}_{0}(x)\big{]}^{\prime}\,dx,$
(58)

where,

$\displaystyle\int^{r}_{0}x^{3}\left[\Theta^{0}_{0}(x)\right]^{\prime}dx=\frac{r^{3}}{4\pi}\Bigg{(}\frac{f}{r^{2}}-\frac{f^{\prime}}{2r}\Bigg{)}.$
(59)

Now using the Eqs. (27)-(II), the equation (58) yields,

$\displaystyle
f^{\prime}(4r+\nu^{\prime}r^{2})+f(2\nu^{\prime\prime}r^{2}+\nu^{\prime
2}r^{2}-8-2\nu^{\prime}r)+r(-2h^{\prime}\mu$
$\displaystyle+h^{\prime}\mu^{\prime}r+2h^{\prime\prime}\mu r+\beta h^{\prime
2}\mu r+2h^{\prime}\nu^{\prime}\mu r)=0,~{}~{}$ (60)

It is clear that new source $\Theta_{ij}$ can be determined by any solution of
the Eq.(60) which can be obtained through any known solution of the system
(21)-(23) described by the metric functions $\xi$ and $\mu$ together with
imposing any viable form of deformation function $f(r)$ or $h(r)$. For this
purpose, we consider a well-known spacetime geometry proposed by Karori-Barau,

$\displaystyle
e^{\xi(r)}=e^{Ar^{2}+B},~{}~{}~{}~{}\text{and}~{}~{}~{}\mu(r)=e^{-Dr^{2}}.$
(61)

where $A$, $B$, and $D$ are constant parameters. Using above $\xi$ and $\mu$,
the system (21)-(23) provides the energy density and pressures expressions for
the energy-momentum tensor $T_{ij}$ as,

$\displaystyle
p_{r}=\frac{e^{-Dr^{2}}\left(1-e^{Dr^{2}}+2Ar^{2}\right)}{r^{2}},~{}~{}~{}$
(62) $\displaystyle
p_{t}=e^{-Dr^{2}}\left[A^{2}r^{2}+A\left(2-Dr^{2}\right)-D\right],~{}~{}~{}$
(63)
$\displaystyle\rho=\frac{e^{-Dr^{2}}\left(-1+e^{Dr^{2}}+2Dr^{2}\right)}{r^{2}}.~{}~{}~{}$
(64)

The constants involved in the solution are determined by the same matching
conditions (34)-(36) for metric functions (61), which yields

$\displaystyle D=\frac{\ln R-\ln\left[R-2M_{s}\right]}{R^{2}},~{}~{}~{}$ (65)
$\displaystyle A=\frac{M_{s}}{(R-2M_{s})R^{2}},~{}~{}~{}$ (66) $\displaystyle
B=-\frac{M_{s}}{R-2M_{s}}+\ln\left[1-\frac{2M_{s}}{R}\right].~{}~{}~{}$ (67)

where $m_{GR}(R)=M_{s}$ is the total mass of the object. Now we find the
complexity factor $Y_{TF}$ by using the definition (50) as,

$\displaystyle
Y_{TF}=\frac{e^{-Dr^{2}}\left(-2+2e^{Dr^{2}}+A^{2}r^{4}-Dr^{2}\left(2+Ar^{2}\right)\right)}{r^{2}},~{}~{}~{}$
(68)

Using the spacetime geometry (61) together with the same form of temporal
deformation function $h(r)=Cr^{2}$ as used in previous section (III), we find
the radial deformation function by solving of the equation (60) as,

$\displaystyle f(r)=-\frac{C(2A+3\beta C-D)r^{2}f_{8}(r)}{(A+\beta
C)^{2}\,e^{\frac{2(A+\beta C-D)}{A+\beta C}+(A+\beta C)r^{2}}}$
$\displaystyle\hskip 42.67912pt+\frac{\left(2+Ar^{2}+\beta
Cr^{2}\right)\,r^{2}\,F}{e^{(A+\beta C)r^{2}}\,},$ (69)

where $F$ is a constant of integration with dimension $length^{-2}$ and

$\displaystyle f_{8}(r)=-(A+\beta C)e^{\frac{(A+\beta C-D)\left(2+Ar^{2}+\beta
Cr^{2}\right)}{A+\beta C}}+(A+\beta C-D)$ $\displaystyle\hskip
28.45274pt\times\left(2+Ar^{2}+\beta Cr^{2}\right)f_{9}(r),$ $\displaystyle
f_{9}(r)=\text{ExpIntegralEi}\left[\frac{(A+\beta C-D)\left(2+Ar^{2}+\beta
Cr^{2}\right)}{A+\beta C}\right],$

The deformation function $f(r)$ given in Eq.(69) leads the same complexity
factor $\hat{Y}_{TF}=Y_{TF}$ i.e. $Y^{\Theta}_{TF}=0$ for all $\beta$. Then
the deformed metric functions can be read as,

$\displaystyle e^{-\lambda(r)}=e^{-Dr^{2}}+\beta\,\Bigg{[}-\frac{C(2A+3\beta
C-D)r^{2}f_{8}(r)}{(A+\beta C)^{2}\,e^{\frac{2(A+\beta C-D)}{A+\beta
C}+(A+\beta C)r^{2}}}$ $\displaystyle\hskip
28.45274pt+\frac{\left(2+Ar^{2}+\beta Cr^{2}\right)\,r^{2}\,F}{e^{(A+\beta
C)r^{2}}\,}\Bigg{]},~{}~{}~{}$ (70) $\displaystyle
e^{\nu(r)}=e^{Ar^{2}+B+\beta\,Cr^{2}}.~{}~{}~{}$ (71)

Figure 3: The behavior of radial pressure ($P_{r}\times 10^{4}$)-top left,
tangential pressures ($P_{\perp}\times 10^{4}$)-top right, energy density
($\epsilon\times 10^{4}$)-bottom left and anisotropy ($\hat{\Pi}\times
10^{4}$)-bottom right versus radial coordinate $r/R$ for different coupling
constant $\beta$ with compactification factor $\frac{M_{s}}{R}=0.2$ with
$C=0.0002$. The description of the curves in the figures are as follows: i.
Solid-dark yellow for $\beta=0$, ii. dashed-blue for $\beta=0.2$, iii. Dashed-
dark red for $\beta=0.4$, and Dotted dash-green for $\beta=0.6$. The above
figures are plotted for EGD solution for the two systems with same complexity
factor obtained in Sec. V (A).

The expressions for effective pressure and energy density,

$\displaystyle
P_{r}(r,\,\beta)=\frac{e^{-Dr^{2}}}{r^{2}}\,\Big{[}1+2Ar^{2}+2\beta
Cr^{2}+e^{Dr^{2}}\big{(}-1+2\beta^{2}C$ $\displaystyle\hskip 28.45274pt\times
f(r)r^{2}+\beta\,\big{\\{}f(r)+2Ar^{2}\,f(r)\big{\\}}\big{)}\Big{]},~{}~{}~{}$
(72) $\displaystyle
P_{\perp}(r,\,\beta)=\frac{e^{-Dr^{2}}}{r^{2}}\Big{[}2+4\beta
Cr^{2}+6\beta^{2}C^{2}r^{4}-2\beta CDr^{4}+4A$ $\displaystyle\hskip
28.45274pt(r^{2}+\beta
Cr^{4})+e^{Dr^{2}}\big{(}-2+2\beta^{3}C^{2}f(r)\,r^{4}+\beta^{2}Cr^{2}$
$\displaystyle\hskip
28.45274pt\times\big{[}r\,\Psi(r)+4f(r)\,\left(1+Ar^{2}\right)\big{]}+\beta
r[\Psi(r)+A\Psi(r)r^{2}$ $\displaystyle\hskip
28.45274pt+2Af(r)\,r\times\left(2+Ar^{2}\right)]\big{)}\Big{]},~{}~{}~{}$ (73)
$\displaystyle\epsilon(r,\,\beta)=\frac{e^{-Dr^{2}}}{r^{2}}\left(2D\,r^{2}-1+\frac{\big{(}1-\beta\,[f(r)+\Psi(r)\,r]\big{)}}{e^{-Dr^{2}}}\right),~{}~{}~{}$
(74)

and effective anisotropy $\hat{\Pi}=P_{\perp}-P_{r}$ is

$\displaystyle\hat{\Pi}(r,\,\beta)=\frac{e^{-Dr^{2}}}{2r^{2}}\Big{[}e^{Dr^{2}}\Big{(}2+2\beta^{3}C^{2}f(r)\,r^{4}+\beta^{2}\,C\,r^{3}\big{(}\Psi(r)\,$
$\displaystyle\hskip
34.14322pt+4Af(r)\,r\big{)}+\beta\left(1+Ar^{2}\right)\big{\\{}-2f(r)+\Psi(r)r$
$\displaystyle\hskip
28.45274pt+2Af(r)r^{2}\big{\\}}\Big{)}-2\big{(}1-A^{2}r^{4}-2A\beta
Cr^{4}-3\beta^{2}C^{2}$ $\displaystyle\hskip
28.45274pt\times\,r^{4}+D\left(r^{2}+(A+\beta
C)r^{4}\right)\big{)}\Big{]}.~{}~{}~{}~{}~{}$ (75)

where the expression $\Psi(r)$ is mentioned in the Appendix. The metric
functions (70) and (71) together with the Eqs.(72)-(75) represent the complete
exact solution of the Einstein field equations (5)-(7), which is new a
anisotropic form of Karori-Barua solution whose complexity factor
$\hat{Y}_{TF}$ is same form as the complexity factor $Y_{TF}$ given by
Eq.(68). However, we impose the matching conditions (34)-(36) under the new
solution (69)-(75) in order to determine the constant parameters $F$ and $B$,
and $M$,

$\displaystyle F=\Bigg{[}\frac{\beta C(2A+3\beta
C-D)\,f_{8}(R)\,(2AR^{2}+2\beta CR^{2}+1)}{(A+\beta C)^{2}e^{\frac{2(A+\beta
C-D)}{A+\beta C}+(A+\beta C)R^{2}}}$ $\displaystyle\hskip
19.91684pt+\frac{\left(-1+e^{DR^{2}}-2AR^{2}\right)}{R^{2}e^{DR^{2}}}-2\beta
Ce^{-DR^{2}}\Bigg{]}F_{1}(R),$ (76) $\displaystyle
B=\frac{e^{-DR^{2}}+\beta\,f(R)}{e^{AR^{2}+\beta\,CR^{2}}},$ (77)
$\displaystyle M=M_{s}-\frac{\beta\,R}{2}f(R).$ (78)

where $M_{s}=\frac{R}{2}(1-e^{-Dr^{2}})$, while $F_{1}(R)$ is mentioned in the
Appendix. The Fig. 3 shows the behaviors of radial and tangential pressures,
energy density, and anisotropy inside the self-gravitating anisotropic compact
object. it can be observed that all the physical parameters $P_{r}$,
$P_{\perp}$, and $\epsilon$, and $\hat{\Pi}$ are satisfying the condition for
a viable compact object, which implies that the CGD approach is also a very
powerful technique to discover new physical viable GD solution for two systems
with same complexity factor.
The complexity factor $\hat{Y}_{TF}$ takes the form

$\displaystyle\hat{Y}_{TF}=\frac{\big{[}2e^{Dr^{2}}-2+A^{2}r^{4}-D\,r^{2}\left(2+A\,r^{2}\right)\big{]}}{r^{2}\,e^{Dr^{2}}}.$
(79)

Here we can generate the family of complexity factors by taking the different
values of the compactness factor $\frac{M_{s}}{R}$.

Figure 4: The behavior complexity factor $\hat{Y}_{TF}$ versus radial
coordinate $r/R$ for different compactification factor $\frac{M_{s}}{R}$ with
$C=0.0002$. The description of the curves in the figures are as follows: i.
Solid-dark yellow for $\frac{M_{s}}{R}=0.10$, ii. dashed-blue for
$\frac{M_{s}}{R}=0.15$, iii. Dashed-dark red for $\frac{M_{s}}{R}=0.20$, and
Dot dashed-green for $\frac{M_{s}}{R}=0.25$. The above complexity figure is
plotted corresponding to solution determined in Sec. V (A).

Since the complexity factor for new anisotropic solution is same as the seed
solution. Therefore $\beta$ will not show any direct effect on the complexity.
Therefore, we show the influence of the compactness on the complexity factor
$\hat{Y}_{TF}$. As we can see from the Fig. 4, the complexity is increasing
when the compactness factor $\frac{M_{s}}{R}$ increases.

###  V.2 EGD solution generated by zero complexity factor

In this section, we discuss the gravitational decoupling solution via complete
geometric deformation approach when the complexity factor is zero i.e.
$\hat{Y}_{TF}=0$ with the condition $Y_{TF}\neq 0$. Therefore, based on the
Eq.(52), we can write

$\displaystyle\hat{Y}_{TF}=Y_{TF}+Y^{\Theta}_{TF}=0$
$\displaystyle~{}\Longrightarrow~{}~{}Y_{TF}=-8\pi\,\beta\,\Pi_{\Theta}+\frac{4\pi\,\beta}{r^{3}}\int^{r}_{0}x^{3}\Theta^{0}_{0}(x)dx,~{}~{}~{}~{}$
(80)

Plugging of the Eqs.(27)-(II) in condition (80), we determine the following
differential equation in geometric deformation functions $f(r)$ and $h(r)$,

$\displaystyle(4+\nu^{\prime}r)r\,f^{\prime}-\big{[}2\nu^{\prime}r+8r-2\nu^{\prime\prime}r^{2}-\nu^{\prime
2}r^{2}\big{]}\,f+\big{[}r\,\beta h^{\prime 2}\mu r$
$\displaystyle-2h^{\prime}\mu
r+h^{\prime}\mu^{\prime}r^{2}+2h^{\prime\prime}\mu
r^{2}+2h^{\prime}\nu^{\prime}\mu
r^{2}\big{]}+\frac{4r^{2}}{\beta}Y_{TF}=0,~{}~{}$ (81)

Now by considering again the Karori-Barua solution and the complexity factor
given by Eqs. (61) and (68), respectively together with deformation function
$h(r)=Cr^{2}$, we get the following solution of the above differential
equation,

$\displaystyle f(r)=\frac{r^{2}\,\left(2+Ar^{2}+\beta
Cr^{2}\right)}{e^{(A+\beta
C)r^{2}}}\Bigg{[}F-\frac{f_{12}(r)}{2\beta}-f_{13}(r)\Bigg{]},~{}~{}~{}~{}~{}$
(82)

where $F$ is a constant of integration and this above deformation $f(r)$
provides the vanishing complexity factor i.e. $\hat{Y}_{TF}=0$. The
expressions for effective radial and tangential pressures with effective
energy density are given as,

$\displaystyle
P_{r}(r,\,\beta)=\frac{\left(1-e^{Dr^{2}}+2Ar^{2}\right)}{r^{2}\,e^{Dr^{2}}}-\beta\Bigg{[}\frac{\left(2Ar^{2}+2\beta
Cr^{2}+1\right)}{e^{(A+\beta C)r^{2}}}$ $\displaystyle\hskip
17.07182pt\times\left(2+Ar^{2}+\beta
Cr^{2}\right)\left(F-\frac{f_{12}(r)}{2\beta}-f_{13}(r)\right)+\frac{2C}{e^{Dr^{2}}}\Bigg{]},~{}~{}~{}~{}$
(83) $\displaystyle
P_{\perp}(r,\,\beta)=\frac{\left(1-e^{Dr^{2}}+2Ar^{2}\right)}{r^{2}\,e^{Dr^{2}}}+\beta\,\Bigg{[}\frac{\zeta(r)\left(1+Ar^{2}+\beta
Cr^{2}\right)}{2r}$ $\displaystyle\hskip
17.07182pt+\frac{CD\,r^{2}}{e^{Dr^{2}}}+\frac{\left(2+2Ar^{2}+3\beta
Cr^{2}\right)\,C}{e^{Dr^{2}}}+e^{-(A+\beta C)r^{2}}r^{2}$ $\displaystyle\hskip
17.07182pt\times\left(2+Ar^{2}+\beta
Cr^{2}\right)\Big{\\{}A^{2}r^{2}+2A\left(1+\beta Cr^{2}\right)+\beta C$
$\displaystyle\hskip 17.07182pt\times\left(2+\beta
Cr^{2}\right)\Big{\\}}\left(F-\frac{f_{12}(r)}{2\beta}-f_{13}(r)\right)\Bigg{]},$
(84)
$\displaystyle\epsilon(r,\,\beta)=\frac{\left(e^{Dr^{2}}+2Dr^{2}-1\right)}{r^{2}\,e^{Dr^{2}}}-\beta\,\Bigg{[}\frac{\zeta(r)}{r}+\frac{\left(2+Ar^{2}+bCr^{2}\right)}{e^{(A+bC)r^{2}}}$
$\displaystyle\hskip
17.07182pt\times\left(F-\frac{f_{12}(r)}{2\beta}-f_{13}(r)\right)\Bigg{]}.$
(85)

where, $\zeta(r)$ is given in the Appendix and the effective anisotropy factor
read as,

$\displaystyle\hat{\Pi}(r,\,\beta)=\frac{e^{-Dr^{2}}\left(-1+e^{Dr^{2}}-2Ar^{2}\right)}{r^{2}}+\beta\Bigg{(}-2Ce^{-Dr^{2}}$
$\displaystyle\hskip
17.07182pt-\frac{\big{[}2\beta(F-f_{13}(r))-f_{12}(r)\big{]}(2Ar^{2}+2\beta
Cr^{2}+1)}{2\beta e^{(A+\beta C)r^{2}}(2+Ar^{2}+\beta Cr^{2})^{-1}}\Bigg{)}$
$\displaystyle\hskip
17.07182pt-\beta\Bigg{[}\frac{CDr^{2}}{e^{Dr^{2}}}-\frac{\zeta(r)\left(1+Ar^{2}+\beta
Cr^{2}\right)}{2r}-\frac{(A+\beta C)}{e^{(A+\beta C)r^{2}}}$
$\displaystyle\hskip
17.07182pt\times\frac{\big{[}2\beta(F-f_{13}(r))-f_{12}(r)\big{]}r^{2}\left(2+Ar^{2}+\beta
Cr^{2}\right)^{2}}{2\beta}$ $\displaystyle\hskip
17.07182pt-\frac{C\left(2+2Ar^{2}+3\beta
Cr^{2}\right)}{e^{Dr^{2}}}\Bigg{]}-\frac{\left[D-A^{2}r^{2}+A(Dr^{2}-2)\right]}{e^{Dr^{2}}}.~{}~{}~{}$
(86)

Figure 5: The behavior of radial pressure ($P_{r}\times 10^{4}$)-top left,
tangential pressures ($P_{\perp}\times 10^{4}$)-top right, energy density
($\epsilon\times 10^{4}$)-bottom left and anisotropy ($\hat{\Pi}\times
10^{4}$)-bottom right versus radial coordinate $r/R$ for different coupling
constant $\beta$ with compactification factor $\frac{M_{s}}{R}=0.2$ with
$C=0.0002$. The description of the curves in the figures are as follows: i.
Solid-dark yellow for $\beta=0$, ii. dashed-blue for $\beta=0.1$, iii. Dashed-
dark red for $\beta=0.2$, and Dotted dash-green for $\beta=0.3$. The above
figures are plotted corresponding to EGD solution generated by zero complexity
factor presented in in Sec. V (B).

Now again use the boundary conditions (34)-(36) for present solution
(82)-(86), we find the $F$, $B$ and total mass $M$,

$\displaystyle F=\frac{e^{R^{2}(A+\beta C)}}{\beta\left(2AR^{2}+2\beta
CR^{2}+1\right)\left(AR^{2}+\beta CR^{2}+2\right)}\Bigg{[}\frac{f_{8}(R)}{2}$
$\displaystyle\times\frac{\left(AR^{2}+\beta CR^{2}+2\right)}{e^{(A+\beta
C)R^{2}}}\left(2AR^{2}+2\beta CR^{2}+1\right)+\frac{f_{13}(R)}{2}$
$\displaystyle\times\frac{\left(2AR^{2}+2\beta
CR^{2}+1\right)}{R^{2}\,e^{\left(R^{2}(A+\beta
C)\right)+2}}+\frac{\left(e^{DR^{2}}-2AR^{2}-1\right)}{R^{2}e^{DR^{2}}}-\frac{2\beta
C}{e^{DR^{2}}}\Bigg{]},~{}~{}~{}~{}~{}$ (87) $\displaystyle
B=\frac{e^{-DR^{2}}+\beta\,f(R)}{e^{AR^{2}+\beta\,CR^{2}}},$ (88)
$\displaystyle M=M_{s}-\frac{\beta\,R}{2}f(R).$ (89)

where $f(R)$ can be determined by Eq.(82) at $r=R$ while the constant $A$ and
$D$ will be same as given by Eqs. (65) and (66). The Fig.5 has been plotted
against the gravitationally decoupled solution obtained in the context of the
zero complexity factor. It is observed that $P_{r}$, $P_{\perp}$, and
$\epsilon$ are monotonically decreasing towards the surface but the tangential
pressure ($P_{\perp}$) is negative near the boundary. This happens due to
stronger attractive force generated by the anisotropy, known as anisotropic
force ($F_{a}$), near the surface of the object i.e.
$F_{a}=\frac{2\hat{\Pi}}{r}<0$. The same features also appear under the MGD
scenario for Tolman IV solution as discussed by Casadio et al. m79 .
Therefore, we can conclude that the gravitationally decoupled solution under
zero complexity factor may not be suitable for modelling of the self-
gravitating compact objects.

##  VI Discussions and Conclusions

In the present article, we have used GD via complete geometric deformation
approach to isotropize the self-gravitating anisotropic matter distribution
and discussed the complexity of this isotropic solution together with the
effect of the decoupling parameter on the complexity. Furthermore, we also
investigated two new gravitationally decoupled anisotropic solutions by
imposing the condition of two system with same complexity factor as well as
systems with zero complexity factor. As we know that most of the previous
works, the new solutions were investigated by taking some particular
procedures such as equations of state (EoS) for the extra sources added in the
original energy-momentum tensor, mimick approaches, and particular ansatz of
deformation functions, etc. However, in this article we have adopted some
different approaches to solve the systems by introducing the gravitational
decoupling in the context of CGD. For simplicity, first we started with two
energy-momentum tensors in which the first energy-momentum tensor
corresponding to anisotropic matter distribution while second one is an
unknown source. As usual, the decoupled system is divided into two sets of
equations through the CGD approach by introducing two unknown deformation
functions $f(r)$ and $h(r)$ along the radial and temporal component of the
metric function of the line element, respectively. After splitting the field
equations, we have considered the following cases:

In the section III, we have investigated the isotropic solution for the
gravitationally decoupled system. For this purpose, first we consider the
spactime geometry for seed system corresponding the Tolman-Kuchowicz metric,
which is necessary for GD system. After that we find the isotropic condition
of gravitationally decoupled system by employing the effective anisotropy
$\hat{\Pi}$ to be zero. In this way we get a differential equation containing
two unknowns $f(r)$ and $h(r)$ and solved this equation for $f(r)$ by assuming
a particular viable form of $h(r)=Cr^{2}$. The obtained solution for $f(r)$
gives the vanishing effective anisotropy throughout the star which implies an
isotropic solution of the gravitationally decoupled system.

In the section IV, we extended the definition of complexity proposed by
Herrera c22  under gravitational decoupling. This section contains full
details about the complexity for the gravitationally decoupled systems.
Moreover, we also discussed the complexity factor and impact of decoupling
constant $\beta$ on complexity for the obtained isotropic solution in section
III. We observe that the complexity is increasing when $\beta$ increases.

The section V contains the some new solutions generated EGD approach for the
systems having same or vanishing complexity factors which is divided into two
subsections A and B:
A). In the section A, we discover the anisotropic solution by imposing the
condition of two systems with same complexity factor by using Krori-Barua seed
solution. This said condition leads a vanishing complexity factor for extra
source i.e. $Y^{\Theta}_{TF}=0$, which governs a differential equation in
$f(r)$ and $g(r)$. This differential equation has been again solved for $f(r)$
by using the same ansatz $h(r)=Cr^{2}$. Furthermore, we also discussed the
complexity factor and the influence of the compactness on the complexity
within the compact objects for the obtained GD anisotropic solution.
B). The third section C contains the GD anisotropic solution which is obtained
by setting zero complexity factor corresponding to the gravitationally
decoupled systems i.e. $\hat{Y}_{TF}=0$. The solution obtained in this section
shows some drawbacks such that it gives a negative anisotropy throughout the
configuration. Since this behavior of the anisotropy leads to an attractive
force which may not much suitable for modeling the viable self-gravitating
compact objects.
Finally, we would like to mention here that the gravitational decoupling via
CGD approach is a very powerful and effective technique to generate new
physically viable isotropoic solution from an anisotropic matter distribution.
Moreover, this methodology is also useful to generate well-behaved anisotropic
solution by assuming of two systems with same complexity factors. In the
future projects, we will try to investigate some more conditions on complexity
factor for generating the new solutions of the Einstein’s field equations for
static self-gravitating system by using the gravitational decoupling
technique.

## Acknowledgement

The author SKM acknowledges 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.

## Appendix

$\displaystyle f_{1}(r)=\text{ExpIntegralEi}[1+(A+\beta
C)r^{2}],~{}~{}~{}f_{2}(r)=\text{ExpIntegralEi}\left[\frac{(A+\beta
C)\left(K-\sqrt{K^{2}-4L}+2Lr^{2}\right)}{2L}\right],$ $\displaystyle
f_{3}(r)=\text{ExpIntegralEi}\left[\frac{(A+\beta
C)\left(K+\sqrt{K^{2}-4L}+2Lr^{2}\right)}{2L}\right],~{}~{}\chi_{1}=\frac{\left(A\left(\sqrt{K^{2}-4L}+K\right)+\beta
C\left(\sqrt{K^{2}-4L}+K\right)-2L\right)}{e^{-\frac{\left(\sqrt{K^{2}-4L}+K\right)(A+\beta
C)}{2L}}},$
$\displaystyle\chi_{2}=\frac{\left(A\left(\sqrt{K^{2}-4L}-K\right)+\beta
C\left(\sqrt{K^{2}-4L}-K\right)+2L\right)}{e^{-\frac{\left(K-\sqrt{K^{2}-4L}\right)(A+\beta
C)}{2L}}},~{}~{}~{}\chi_{3}=\sqrt{K^{2}-4L}\left(A^{2}+2A\beta
C-AK+3\beta^{2}C^{2}-\beta\,CK+L\right),$
$\displaystyle\Omega(r)=\frac{r}{e^{\left(r^{2}(A+\beta
C)\right)}}\left(\frac{r\left[2f_{7}(r)r\left(Kr^{2}+Lr^{4}+1\right)-2r\,f_{5}(r)\left(K+2Lr^{2}\right)\right]}{\beta\,\chi_{3}\left(Kr^{2}+Lr^{4}+1\right)^{2}\,e^{\frac{AK+\beta
CK+L}{L}}}-2f_{4}(r)r^{2}(A+\beta C)+2f_{4}(r)\right)$ $\displaystyle
f_{4}(r)=F+\frac{1}{{\beta\chi_{3}\left(Kr^{2}+Lr^{4}+1\right)}}\Bigg{[}e^{-\frac{AK+\beta
CK+L}{L}}\Bigg{(}e\Bigg{\\{}\chi_{3}\left(K+Lr^{2}\right)e^{\frac{(A+\beta
C)\left(K+Lr^{2}\right)}{L}}+\beta^{2}C^{2}\chi_{1}f_{2}(r)\left(Kr^{2}+Lr^{4}+1\right)+\beta^{2}C^{2}$
$\displaystyle\hskip
19.91684pt\times\chi_{2}f_{3}(r)\left(Kr^{2}+Lr^{4}+1\right)\Bigg{\\}}-\chi_{3}f_{1}(r)(A+\beta
C)\left(Kr^{2}+Lr^{4}+1\right)e^{\frac{K(A+\beta C)}{L}}\Bigg{)}\Bigg{]}$
$\displaystyle
f_{5}(r)=e\Bigg{[}\frac{\chi_{3}\left(K+Lr^{2}\right)}{e^{-\frac{(A+\beta
C)\,(K+Lr^{2})}{L}}}+\beta^{2}C^{2}\Big{(}\chi_{1}f_{2}(r)+\chi_{2}f_{3}(r)\Big{)}\left(Kr^{2}+Lr^{4}+1\right)\Bigg{]}-\frac{\chi_{3}f_{1}(r)(A+\beta
C)\left(Kr^{2}+Lr^{4}+1\right)}{e^{-\frac{K(A+\beta C)}{L}}}$ $\displaystyle
f_{6}(r)=\frac{\chi_{3}L+\chi_{3}(A+\beta
C)\left(K+Lr^{2}\right)}{e^{-\frac{(A+\beta
C)\left(K+Lr^{2}\right)}{L}}}+\beta^{2}C^{2}\chi_{1}f_{2}(r)\left(K+2Lr^{2}\right)+\beta^{2}C^{2}\chi_{2}f_{3}(r)\left(K+2Lr^{2}\right)+f_{11}(r)+f_{12}(r),$
$\displaystyle
f_{7}(r)=-\chi_{3}\,f_{1}(r)(A+bC)\left(K+2Lr^{2}\right)e^{\frac{K(A+bC)}{L}}-\frac{\chi_{3}(A+bC)^{2}\left(Kr^{2}+Lr^{4}+1\right)e^{\frac{K(A+bC)}{L}+r^{2}(A+bC)+1}}{r^{2}(A+bC)+1}+e\,f_{6}(r),$
$\displaystyle\Psi(r)=e^{-(A+\beta C)r^{2}}r\,\bigg{[}2(A+\beta
C)Fr^{2}+2F\left(2+Ar^{2}+\beta Cr^{2}\right)-2(A+\beta
C)F\,r^{2}\left(2+Ar^{2}+\beta
Cr^{2}\right)-\frac{f_{10}(r)+f_{11}(r)}{A+\beta C}-\frac{f_{11}(r)}{(A+\beta
C)^{2}}\bigg{]},$ $\displaystyle f_{10}(r)=2C(A+\beta C-D)(2A+3\beta
C-D)e^{-\frac{2(A+\beta C-D)}{A+\beta
C}}r^{2}f_{9}(r),~{}~{}~{}~{}f_{11}(r)=2C(2A+3\beta C-D)e^{-\frac{2(A+\beta
C-D)}{A+\beta C}}r^{2}f_{8}(r),$ $\displaystyle f_{12}(r)=-(A+\beta
C)\text{ExpIntegralEi}\left[(A+\beta C)r^{2}\right]+\frac{(A+\beta
C)\text{ExpIntegralEi}\left[2+Ar^{2}+\beta
Cr^{2}\right]}{e^{2}}+\frac{2}{(A+\beta
C)^{2}}2\Bigg{[}2\beta^{2}C^{2}(A+\beta C$ $\displaystyle\hskip
19.91684pt-D)e^{-\frac{2(A+\beta C-D)}{A+\beta C}}f_{9}(r)+\frac{(A+\beta
C)e^{(A+\beta C-D)r^{2}}\left(A+\beta C\left(1-2\beta
Cr^{2}\right)\right)}{r^{2}\left[2+Ar^{2}+\beta Cr^{2}\right]}\Bigg{]},,$
$\displaystyle f_{13}(r)=\frac{1}{2\beta e^{2}r^{2}\left(2+Ar^{2}+\beta
Cr^{2}\right)}\bigg{(}-2e^{2+Ar^{2}+\beta Cr^{2}}\left(1+Ar^{2}+\beta
Cr^{2}\right)+e^{2}r^{2}\left(A^{2}r^{2}+2A\left(1+\beta Cr^{2}\right)+\beta
C\left(2+\beta Cr^{2}\right)\right)$ $\displaystyle\hskip
14.22636pt\times\text{ExpIntegralEi}\left[(A+\beta
C)r^{2}\right]+r^{2}\Big{\\{}A^{2}r^{2}+2A\left(1+\beta Cr^{2}\right)+\beta
C\left(2+\beta Cr^{2}\right)\Big{\\}}\text{ExpIntegralEi}\left[2+(A+\beta
C)r^{2}\right]\bigg{)},$ $\displaystyle F_{1}(R)=\frac{e^{(A+\beta
C)R^{2}}}{\beta(2AR^{2}+2\beta CR^{2}+1)\left(2+AR^{2}+\beta
CR^{2}\right)},~{}~{}~{}\zeta(r)=-\frac{1}{\beta(A+\beta
C)^{2}}e^{-2-(2A+2\beta C+D)r^{2}}r(\text{f14}-(A+\beta C)(\text{f15})),$
$\displaystyle f_{14}(r)=-(A+\beta C)^{3}e^{(A+\beta
C+D)r^{2}}\left(-2+A^{2}r^{4}+2A\beta
Cr^{4}+\beta^{2}C^{2}r^{4}\right)\text{ExpIntegralEi}\left[2+Ar^{2}+\beta
Cr^{2}\right]-4\beta^{2}C^{2}(A+\beta C-D)$ $\displaystyle\hskip
25.6073pt\times e^{(A+\beta C)r^{2}+D\left(\frac{2}{A+\beta
C}+r^{2}\right)}\left(-2+A^{2}r^{4}+2A\beta
Cr^{4}+\beta^{2}C^{2}r^{4}\right)\text{f9},$ $\displaystyle
f_{15}(r)=-2e^{2}\Bigg{[}\beta^{4}C^{3}e^{(A+\beta
C+D)r^{2}}Fr^{4}+\beta^{3}C^{2}r^{2}\left(Ce^{2(A+\beta
C)r^{2}}\left(2+e^{Dr^{2}}\right)+3Ae^{(A+\beta
C+D)r^{2}}Fr^{2}\right)+Ae^{2(A+\beta C)r^{2}}\big{(}-D+Ae^{Dr^{2}}$
$\displaystyle\hskip
14.22636pt\times\left(-1+Ar^{2}\right)\big{)}+\beta\left(Ae^{(A+\beta
C+D)r^{2}}F\left(-2+A^{2}r^{4}\right)-Ce^{2(A+\beta
C)r^{2}}\left(D-Ae^{Dr^{2}}\left(-2+3Ar^{2}\right)\right)\right)+\beta^{2}C\Big{(}e^{(A+\beta
C+D)r^{2}}F$ $\displaystyle\hskip
14.22636pt\times\left(-2+3A^{2}r^{4}\right)+Ce^{2(A+\beta
C)r^{2}}\left(-2+2Ar^{2}+e^{Dr^{2}}\left(-1+3Ar^{2}\right)\right)\Big{)}\Bigg{]}+(A+\beta
C)^{2}e^{(A+\beta C+D)r^{2}}\Big{(}-2+A^{2}r^{4}+2A\beta Cr^{4}$
$\displaystyle\hskip
14.22636pt+\beta^{2}C^{2}r^{4}\Big{)}\text{ExpIntegralEi}\left[2+(A+\beta
C)r^{2}\right].$

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