# Existence of Wormholes in $f(\mathcal{G})$ Gravity using Symmetries

Tayyaba Naz<EMAIL_ADDRESS>National University of Computer and
Emerging Sciences,
Lahore Campus, Pakistan.  G. Mustafa<EMAIL_ADDRESS>Department of
Physics, Zhejiang Normal University, Jinhua, 321004, China.  M. Farasat Shamir
<EMAIL_ADDRESS>
###### Abstract

The current study examines the geometry of static wormholes with anisotropic
matter distribution in context of modified $f(\mathcal{G})$ gravity. We
consider the well known Noether and conformal symmetries, which help in
investigating wormholes in $f(\mathcal{G})$ gravity. For this purpose, we
develop symmetry generators associated with conserved quantities by taking
into consideration the $f(\mathcal{G})$ gravity model. Moreover, we use the
conservation relationship gained from the classical Noether method and
conformal Killing symmetries to develop the metric potential. These symmetries
provide a strong mathematical background to investigate wormhole solutions by
incorporating some suitable initial conditions. The obtained conserved
quantity performs a significant role in defining the essential physical
characteristics of the shape-function and energy conditions. Further, we also
describe the stability of obtained wormholes solutions by employing the
equilibrium condition in modified $f(\mathcal{G})$ gravity. It is observed
from graphical representation of obtained wormhole solutions that Noether and
conformal Killing symmetries provide the results with physically accepted
patterns.

Keywords: Wormhole; $f(\mathcal{G})$ gravity; Noether symmetries; Conformal
motion; Conserved quantities.
PACS: 04.20.Jb; 98.80.Jk; 98.80.-k.

##  I Introduction

The approximation symmetry method played a significant role in evaluating the
precise solutions of the differential equations. Such approximations
dynamically reduce the complexity of the non-linear equation involved in a
scheme by seeking the unknown parameter of equations. The Noether symmetries,
in particular, are not only a mechanism for dealing with the dynamics
solution, but their presence also provides suitable conditions so that one can
specify the universe models physically and analytically according to our
measured observations. In addition to this, Noether symmetry technique is
believed to be a suitable mathematical approach, which often investigates the
exact solutions and computes the associated conserved quantities. This method
plays a central role in reducing the nonlinear equation system to a linear
equation system. The numerous conservation principles, such as conservation of
energy and angular momentum etc., are specifically linked to the symmetries of
a specified dynamic system and provide the conserved quantities, which seem to
be the consequence of certain type of symmetry being present in that
mechanism. Moreover, conserved quantities can be determined by applying the
Noether symmetry technique, asking for the Lagrangian symmetry. The presence
of any specific type of symmetry for the Euler-Lagrange equations of motion,
along with the Lagrangian, will precisely be related to the Noether’s
symmetry. Whereas, no particular theory is endorsed by the technique of
Noether symmetry, the literature studies have indicated that the existence of
Noether symmetries is capable of selecting suitable theory and then
integrating dynamics through the first integrals referring to Noether
symmetries 1 -5 . In fact, it should be noticed that the Noether symmetries
are not just a mathematical method for solving or reducing dynamics, yet their
presence even enables to choice of observable universes/wormholes/black holes,
etc. and the collection of analytical models relevant to observations 6 .
Recently, we have proposed some compact star solutions incorporating Noether
symmetry in frame of the modified $f(\mathcal{G})$ gravity fartay . The
Noether Symmetry approach for $f(\mathcal{G})$ cosmology in $n$ dimensions has
been discussed in BC1 . Moreover, in spherically symmetric context, the
$f(\mathcal{G})$ theory of gravity can be employed to address general
relativity ($\mathcal{GR}$) inconsistencies BC2 . Further, a detailed overview
of the Noether symmetry technique to investigate a variety of cosmic
scenarios, including viable mimetic $f(R)$ and $f(R,T)$ theories is given in
Momen -CAPO . In this regard, this approach has successively used to cope with
cosmologies generated from various theories of gravity BC3 -BC5 .

Our universe often exhibits eye-opening challenges for cosmologists, regrading
their fascinating and enigmatic existence. The presence of hypothetical
geometries are perceived to be the most contentious topic leading to wormhole
geometry. A debate regarding the existence of the wormhole and the
construction of its solutions is among the most interesting challenges in
modern astrophysics. A wormhole is a path or tunnel that connects two separate
regions of the same or two different type of universes. Flamm 7  used the term
bridge for the very first time in 1916. In 1935, Einstein and Rosen
mathematically described such bridge as structures renowned as the Einstein-
Rosen bridges 8 . In addition, Morris and Throne 9  established wormholes by
taking into account exotic matter. Exotic matter is regarded as the necessary
component for the formation of these wormholes. The existence of exotic matter
by using various techniques have been addressed by many authors 10 -12 .
Moreover, extra geometric terms are thought to be the cause of these exotic
matter in modified theories of gravity 13 -17 . Recently, Sharif and Nawazish
Sharif1 ; Sharif2  have explored static wormhole solutions utilizing the
Noether symmetry methodology in modified gravity and they noticed the stable
structure of red-shift functions for various cases. Furthermore, Sharif and
Hussain Sharif3  used the same technique to explore wormhole physical presence
in frame of $f(\mathcal{G},T)$ gravity and investigate its properties of fluid
distributions for both dust and non-dust case.

The current cosmic accelerated expansion has always been considered to be the
most revolutionizing reality on the landscape of theoretical and observational
modern cosmology. In order to take into account the late-time accelerated
expansion, two main approaches have been proposed. The first effective way to
describe the idea of accelerated cosmos expansion in context of $\mathcal{GR}$
is the existence of dark energy, which exhibits strong negative pressure. The
second innovative approach to ponder this concept of universe expansion is to
modify the Einstein-Hilbert action at large scales. These modifications of the
$\mathcal{GR}$ play an influential role in revealing the intriguing dynamics
behind the expansion of the universe. Among the various gravitational
theories, the theory that has acquired the prominence in the last few years is
modified $f(\mathcal{G})$ gravity Noj1 . This modified gravity was obtained by
incorporating the function $f(\mathcal{G})$ in Einstein Hilbert action. The
Gauss-Bonnet term is of great importance as it facilitates the regularization
of gravitational action and can serve to avoid ghost contributions Chiba . It
is believed that $f(\mathcal{G})$ gravity is also quite helpful in explaining
late cosmic acceleration and reconstructing some form of cosmological
solution. Indeed, this theory was used as a significant approach for revealing
the mystical nature of the cosmos Noj2 .

Several important studies from literature have shown that anisotropic stars
can be modeled utilizing solutions that endorse a single parameter group of
conformal motion. Herrera and his colleagues Her1 -Her3  were among the
pioneers who provided the general treatment of the spheres that accepted a
single parameter category of conformal motions. Some significant findings
employing conformal Killing vectors (CKVs) have been presented in literature
Rah2 -Aktas . Nevertheless, CKVs approach is helpful to make the governing
system easier to analyze by reducing the nonlinear structure of partial
differential equations (PDE’s) into the ordinary differential equations. In
the spacetime, due to conformal symmetry, some constraints on the
gravitational potential are imposed. However, the idea of CKVs was considered
in literature to investigate the presence of spherically symmetric wormholes,
as the static symmetric spacetime presents a limited category of conformal
motions. Kuhfitting Kuh  has recently investigated the stable wormholes
solutions through CKVs and non-commutational distribution. Moreover, Rahaman
Rahaman  used the CKVs technique to construct the wormhole solutions in the
context of non-commutative geometry.

Motivated by the aforementioned literature, our focus is to construct the
wormhole solutions admitting symmetries in the frame of conformal motion. To
the best of our understanding, no attempt has yet been made to explore the
wormhole solutions using the Noether symmetry technique under conformal motion
by considering the $f(\mathcal{G})=\alpha\mathcal{G}^{n}$ gravity model Cog2 ,
where $n=2$. For this aim, we extended the idea of Shamir and Tayyaba fartay
and use the conservation relationship gained from Noether method by
incorporating some peculiar initial conditions to construct the relation of
metric potential to address the formulation of wormhole. The manuscript is
organized as follows. In section $2$, we discuss some basics formulism of
$f(\mathcal{G})$ gravity in frame of anisotropic matter distributions. In
section 3, the geometry of wormhole by employing the Noether and conformal
motion has been discussed. Conclusive remarks are presented in Section $4$.

##  II Some Formulation of Modified $f(\mathcal{G})$ Gravity

The action for modified $f(\mathcal{G})$ gravity is expressed as Noj1

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{\mathcal{R}}{2\kappa^{2}}+f(\mathcal{G})+L_{m}\Bigg{]},$
(1)

here $L_{m}$ shows matter Lagrangian, $\mathcal{R}$ being the Ricci scalar,
$\kappa^{2}={8\pi G}$ represents the coupling constant term and
$f(\mathcal{G})$ is an arbitrary function of the Gauss-Bonnet invariant term
represented as

$\mathcal{G}=\mathcal{R}^{2}-4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}_{\mu\nu\sigma\rho}\mathcal{R}^{\mu\nu\sigma\rho},$
(2)

here $\mathcal{R}_{\mu\nu}$ and $\mathcal{R}_{\mu\nu\rho\sigma}$ specify the
Ricci and Riemann tensors, respectively. The variation of above action with
respect to metric tensor yield the following field equations

$\displaystyle
G_{\xi\eta}+8\big{[}\mathcal{R}_{\xi\rho\eta\sigma}+\mathcal{R}_{\rho\eta}g_{\sigma\xi}-\mathcal{R}_{\rho\sigma}g_{\eta\xi}-\mathcal{R}_{\xi\eta}g_{\sigma\rho}+\mathcal{R}_{\xi\sigma}g_{\eta\rho}\frac{\mathcal{R}}{2}(g_{\xi\eta}g_{\sigma\rho}-g_{\xi\sigma}g_{\eta\rho})\big{]}\nabla^{\rho}\nabla^{\sigma}f_{\mathcal{G}}+(\mathcal{G}f_{\mathcal{G}}-f)g_{\xi\eta}=\kappa^{2}T_{\xi\eta}.$
(3)

An alternate representation of above aforementioned field equations (3), which
are familiar with $\mathcal{GR}$ may be described as

$\displaystyle G_{\xi\eta}=\kappa^{2}T_{\xi\eta}^{eff},$ (4)

the effective stress-energy tensor $T_{\xi\eta}^{eff}$ is given by

$\displaystyle
T_{\xi\eta}^{eff}=T_{\xi\eta}-\frac{8}{\kappa^{2}}\big{[}\mathcal{R}_{\xi\rho\eta\sigma}+\mathcal{R}_{\rho\eta}g_{\sigma\xi}-\mathcal{R}_{\rho\sigma}g_{\eta\xi}-\mathcal{R}_{\xi\eta}g_{\sigma\rho}+\mathcal{R}_{\xi\sigma}g_{\eta\rho}+\frac{\mathcal{R}}{2}(g_{\xi\eta}g_{\sigma\rho}-g_{\xi\sigma}g_{\eta\rho})\big{]}\nabla^{\rho}\nabla^{\sigma}f_{\mathcal{G}}-(\mathcal{G}f_{\mathcal{G}}-f)g_{\xi\eta}.$
(5)

We consider the static, spherically symmetric spacetime Krori

$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\Phi^{2}).$ (6)

The source of the configuration of matter presumed in this study is
anisotropic in nature, represented as

$\mathcal{T}_{\chi\gamma}=(\rho+p_{t})\upsilon_{\chi}\upsilon_{\gamma}-p_{t}g_{\chi\gamma}+(p_{r}-p_{t})\xi_{\chi}\xi_{\gamma},$
(7)

here $\rho$, $p_{r}$ and $p_{t}$ indicate energy density, radial and
tangential pressures respectively. The four velocity and radial vector are
symbolized by $\upsilon_{\chi}$ and $\xi_{\chi}$ respectively, which are
satisfying the following condition

$\upsilon^{\alpha}=e^{\frac{-\nu}{2}}\delta^{\alpha}_{0},~{}~{}~{}\upsilon^{\alpha}\upsilon_{\alpha}=1,~{}~{}~{}\xi^{\alpha}=e^{\frac{-\lambda}{2}}\delta^{\alpha}_{1},~{}~{}~{}\xi^{\alpha}\xi_{\alpha}=-1.$

Using equations (5), (6) and (7), we obtain

$\displaystyle\rho^{eff}$
$\displaystyle=~{}~{}\rho-8e^{-2\lambda}(f_{\mathcal{GGG}}\mathcal{G}^{\prime
2}+f_{\mathcal{GG}}\mathcal{G}^{\prime\prime})(\frac{e^{\lambda}-1}{r^{2}})+4e^{-2\lambda}\lambda^{\prime}\mathcal{G}^{\prime}f_{\mathcal{GG}}(\frac{e^{\lambda}-3}{r^{2}})-(\mathcal{G}f_{\mathcal{G}}-f),$
(8) $\displaystyle p_{r}^{eff}$
$\displaystyle=~{}~{}~{}p_{r}-4e^{-2\lambda}\nu^{\prime}\mathcal{G}^{\prime}f_{\mathcal{GG}}(\frac{e^{\lambda}-3}{r^{2}})+(\mathcal{G}f_{\mathcal{G}}-f),\quad\quad\quad\quad\quad$
(9) $\displaystyle p_{t}^{eff}$
$\displaystyle=~{}~{}p_{t}-\frac{4e^{-2\lambda}\nu^{\prime}}{r}(f_{\mathcal{GGG}}\mathcal{G}^{\prime
2}+f_{\mathcal{GG}}\mathcal{G}^{\prime\prime})-\frac{2e^{-2\lambda}{\nu^{\prime}}^{2}f_{\mathcal{GG}}\mathcal{G}^{\prime}}{r}-\frac{2e^{-2\lambda}f_{\mathcal{GG}}\mathcal{G}^{\prime}}{r}(2\nu^{\prime\prime}-3\nu^{\prime}\lambda^{\prime})+(\mathcal{G}f_{\mathcal{G}}-f).$
(10)

Here $\rho$, $p_{r}$ and $p_{t}$ are usual energy density, radial pressure and
transverse pressure respectively. The Gauss-Bonnet invariant term for the
spherically symmetric space time (6) appears as

$\mathcal{G}=\frac{2e^{-\lambda}}{r^{2}}(\nu^{\prime}\lambda^{\prime}+{\nu^{\prime}}^{2}e^{-\lambda}-3\nu^{\prime}\lambda^{\prime}e^{-\lambda}-2\nu^{\prime\prime}-{\nu^{\prime}}^{2}+2\nu^{\prime\prime}e^{-\lambda}).$
(11)

. To solve the field equations (8-10), which are extremely nonlinear,
complicated and involve many unknowns, we need some suitable mathematical
method. For this purpose, we use a special class of Lie point symmetries
namely Noether symmetry approach.

In the current study, we consider the following integral of motion as
discussed fartay , i.e.,

$\displaystyle\mathcal{I}_{1}=e^{\frac{\nu-3\lambda}{2}}\alpha[-\mathcal{G}\\{e^{2\lambda}\mathcal{G}r^{3}+32(e^{\lambda}-1)\nu^{\prime}\\}-8(e^{\lambda}-1)(r\nu^{\prime}-10)\mathcal{G}^{\prime}].$
(12)

Here, we use another interesting approach in the context of symmetries, is the
use of Killing vectors K1 . It has been argued that Killing symmetries form a
subalgebra of Noether symmetries. Moreover, Noether equations may be termed as
generalized Killing equations for some special cases K2 ; K3 . Now we discuss
CKVs in to connect wormhole geometry with Noether symmetry for the metric (6).
In a given space time with manifold $\mathcal{M}$, the field for conformal
vectors $\gamma$ is defined as

$\mathfrak{L}_{\gamma}g_{\mu\nu}=g_{\eta\nu}\gamma^{\eta}_{;\lambda}+g_{\mu\eta}\gamma^{\eta}_{;\nu}=\Theta_{f}(r)g_{\mu\nu},$
(13)

where $\mathfrak{L}$ represents the Lie derivative. The expressions
$\gamma^{\eta}$ and $\Theta_{f}(r)$ denote the conformal functions. Among all
the symmetries, conformal symmetry and Noether symmetry can approach some
results with goodness as both the symmetry yield a more profound knowledge
into the geometry of spacetime. By plugging the spacetime from Eq. (6) in Eq.
(13) the following relations can be easily obtained

$\displaystyle\gamma^{1}\nu^{{}^{\prime}}(r)=\Theta_{f}(r),\quad\gamma^{1}=\frac{r\Theta_{f}(r)}{2},\quad\gamma^{1}\lambda^{{}^{\prime}}(r)+2\gamma^{1}_{,1}=\Theta_{f}(r).$

The above results further lead to

$\displaystyle e^{\nu(r)}=\Lambda_{1}^{2}r^{2},\quad\quad\quad\quad\quad
e^{\lambda(r)}=\left(\frac{\Lambda_{2}}{\Theta_{f}(r)}\right)^{2},$ (14)

where $\Lambda_{1}$ and $\Lambda_{2}$ represent the constants of integration.

##  III Wormholes in $f(\mathcal{G})$ Gravity Admitting Noether and Conformal
Symmetries

We shall describe the wormhole solutions in the background of conformal and
Noether symmetries in this section. The standard spacetime for wormhole
geometry is defined as

$ds^{2}=e^{2\Omega(r)}dt^{2}-\left(1-\frac{\mathbb{S}_{f}(r)}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\Phi^{2}),$ (15)

where $\Omega(r)$ and $\mathbb{S}_{f}(r)$ denote the redshift function and
shape-function respectively. A certain significant criteria for wormhole
physics to be satisfied by the shape function and the red-shift function are
summarized here. The value of the red-shift function $\Omega(r)$ must be
finite within the configuration. There is no horizon restriction on
$\Omega(r)$, for the wormhole to be traversable. The appropriate radial
distance by enforcing the constraint in the shape function relation
$\mathbb{S}_{f}(r)$,
$\mathcal{L}(r)=\pm\int^{r}_{r_{0}}(1+\frac{\mathbb{S}_{f}(r)}{r})^{-1/2}dr$,
with $r>r_{0}$, should be finite every where in the spacetime geometry. Here,
the $\pm$ incorporates the two different parts of the spacetime geometry,
interconnected by the wormhole configuration. The upper segment of the
wormhole decreases and hits its lowest at the position of wormhole of the
throat, and then rises to the lower part. The $\mathbb{S}_{f}(r)$ needs to
satisfy the inequality given by
$(\mathbb{S}_{f}(r)-\mathbb{S}_{f}(r)^{{}^{\prime}}r)/\mathbb{S}_{f}(r)^{2}>0$
and the equality $\mathbb{S}_{f}(r_{0})=r_{0}$. The $\mathbb{S}_{f}(r)$ should
also satisfy the condition $\mathbb{S}_{f}^{{}^{\prime}}(r_{0})<1$. By
equating the spherically symmetric spacetime (6) with Eq. (15), we get the
following relations

$g_{tt}=e^{\nu(r)}=e^{2\Omega(r)},\quad\quad\quad\quad\quad
g_{rr}=e^{\lambda(r)}=\left(1-\frac{\mathbb{S}_{f}(r)}{r}\right)^{-1}=\left(\frac{\Lambda_{2}}{\Theta_{f}(r)}\right)^{2}.$
(16)

Using Eq. (16), (8-10) and considering $\Lambda_{1}^{2}=\Lambda_{3}$,
$\Lambda_{2}^{2}=\Lambda_{4}$, $(\Theta_{f}(r))^{2}=\Theta_{f_{0}}(r)$, we get
the simplified system of field equations

$\displaystyle\rho^{eff}$
$\displaystyle=\frac{1}{\Lambda_{3}^{4}r^{15}\Theta_{f_{0}}(r)^{4}}\bigg{(}540\alpha
r^{3}\left(\Lambda_{3}r^{2}-1\right){}^{2}\Theta_{f_{0}}^{\prime}(r)^{4}-16\alpha
r^{2}\Theta_{f_{0}}(r)\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime}(r)^{2}\left(65r\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime\prime}(r)\right.$
(17)
$\displaystyle\left.+\left(195-113\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime}(r)\right)+16\alpha
r\Theta_{f_{0}}(r)^{2}\left(15r^{2}\left(\Lambda_{3}r^{2}-1\right){}^{2}\Theta_{f_{0}}^{\prime\prime}(r)^{2}+\left(\Lambda_{3}r^{2}\left(187\Lambda_{3}r^{2}-706\right)+555\right)\right.$
$\displaystyle\times\left.\Theta_{f_{0}}^{\prime}(r)^{2}+2r\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime}(r)\left(10r\Theta_{f_{0}}^{(3)}(r)\left(\Lambda_{3}r^{2}-1\right)+\left(135-77\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime\prime}(r)\right)\right)+64\alpha\Theta_{f_{0}}(r)^{3}$
$\displaystyle\times\left(\left(\Lambda_{3}r^{2}\left(35\Lambda_{3}r^{2}-234\right)+231\right)\Theta_{f_{0}}^{\prime}(r)+r\left(\left(\Lambda_{3}r^{2}\left(138-35\Lambda_{3}r^{2}\right)-111\right)\Theta_{f_{0}}^{\prime\prime}(r)+r\left(\Lambda_{3}r^{2}-1\right)\right.\right.$
$\displaystyle\times\left.\left.\left(r\Theta_{f_{0}}^{(4)}(r)\left(1-\Lambda_{3}r^{2}\right)+2\Theta_{f_{0}}^{(3)}(r)\left(5\Lambda_{3}r^{2}-9\right)\right)\right)\right)+\Lambda_{3}^{4}\rho
r^{15}\Theta_{f_{0}}(r)^{4}\bigg{)},$ $\displaystyle p^{eff}_{r}$
$\displaystyle=\frac{1}{\Lambda_{3}^{4}r^{14}\Theta_{f_{0}}(r)^{4}}\bigg{(}-12\alpha
r^{2}\left(\Lambda_{3}r^{2}-1\right)\left(5\Lambda_{3}r^{2}-21\right)\Theta_{f_{0}}^{\prime}(r)^{4}+16\alpha
r\Theta_{f_{0}}(r)\Theta_{f_{0}}^{\prime}(r)^{2}\left(r\left(\Lambda_{3}r^{2}-1\right)\left(5\Lambda_{3}r^{2}-21\right)\right.$
(18)
$\displaystyle\times\left.\Theta_{f_{0}}^{\prime\prime}(r)+\left(\Lambda_{3}r^{2}\left(54-11\Lambda_{3}r^{2}\right)-63\right)\Theta_{f_{0}}^{\prime}(r)\right)+16\alpha\Theta_{f_{0}}(r)^{2}\left(r^{2}\left(\Lambda_{3}r^{2}-1\right){}^{2}\Theta_{f_{0}}^{\prime\prime}(r)^{2}-3\left(\Lambda_{3}r^{2}-3\right)\right.$
$\displaystyle\times\left.\left(3\Lambda_{3}r^{2}-13\right)\Theta_{f_{0}}^{\prime}(r)^{2}-2r\left(\Lambda_{3}r^{2}-3\right)\Theta_{f_{0}}^{\prime}(r)\left(r\Theta_{f_{0}}^{(3)}(r)\left(\Lambda_{3}r^{2}-1\right)+\left(8-4\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime\prime}(r)\right)\right)$
$\displaystyle+\Lambda_{3}^{4}p_{r}r^{14}\Theta_{f_{0}}(r)^{4}\bigg{)},$
$\displaystyle p^{eff}_{t}$
$\displaystyle=\frac{1}{\Lambda_{3}^{4}r^{14}\Theta_{f_{0}}(r)^{5}}\bigg{(}-432\alpha
r^{3}\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime}(r)^{5}+4\alpha
r^{2}\Theta_{f_{0}}(r)\Theta_{f_{0}}^{\prime}(r)^{3}\left(208r\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime\prime}(r)+\left(\Lambda_{3}r^{2}\right.\right.$
(19)
$\displaystyle\times\left.\left.\left(9\Lambda_{3}r^{2}-382\right)+561\right)\Theta_{f_{0}}^{\prime}(r)\right)-16\alpha
r\Theta_{f_{0}}(r)^{2}\Theta_{f_{0}}^{\prime}(r)\left(16r^{2}\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime\prime}(r)^{2}+\left(\Lambda_{3}r^{2}\left(149-3\Lambda_{3}r^{2}\right)\right.\right.$
$\displaystyle-\left.\left.354\right)\Theta_{f_{0}}^{\prime}(r)^{2}+r\Theta_{f_{0}}^{\prime}(r)\left(13r\Theta_{f_{0}}^{(3)}(r)\left(\Lambda_{3}r^{2}-1\right)+3\left(\Lambda_{3}r^{2}\left(\Lambda_{3}r^{2}-43\right)+64\right)\Theta_{f_{0}}^{\prime\prime}(r)\right)\right)+16\alpha\Theta_{f_{0}}(r)^{3}$
$\displaystyle\times\left(\left(\Lambda_{3}r^{2}\left(\Lambda_{3}r^{2}-96\right)+471\right)\Theta_{f_{0}}^{\prime}(r)^{2}+r^{2}\Theta_{f_{0}}^{\prime\prime}(r)\left(2r\Theta_{f_{0}}^{(3)}(r)\left(\Lambda_{3}r^{2}-1\right)+\left(\Lambda_{3}r^{2}\left(\Lambda_{3}r^{2}-12\right)+19\right)\right.\right.$
$\displaystyle\times\left.\left.\Theta_{f_{0}}^{\prime\prime}(r)\right)-2r\Theta_{f_{0}}^{\prime}(r)\left(\left(\Lambda_{3}r^{2}\left(\Lambda_{3}r^{2}-54\right)+135\right)\Theta_{f_{0}}^{\prime\prime}(r)+r\left(r\Theta_{f_{0}}^{(4)}(r)\left(1-\Lambda_{3}r^{2}\right)\right.\right.\right.$
$\displaystyle+\left.\left.\left.6\Theta_{f_{0}}^{(3)}(r)\left(2\Lambda_{3}r^{2}-3\right)\right)\right)\right)+\Lambda_{3}^{4}p_{t}r^{14}\Theta_{f_{0}}(r)^{5}\bigg{)}.$

It is interesting to notice that Eqs. (17-19) now involve only one unknown
$\Theta_{f_{0}}(r)$. By using the Eq. (16) in Eq. (12), we have the following
differential equation

$\displaystyle\frac{4\sqrt{\frac{\Lambda_{2}^{2}}{\Theta_{f_{0}}(r)}}}{r^{4}\Theta_{f_{0}}^{4}(r)\left(\Lambda_{3}r^{2}\right){}^{7/2}}\bigg{(}-4r^{2}\Theta_{f_{0}}(r)\left(\Lambda_{3}r^{2}-1\right)(\Theta_{f_{0}}^{\prime}(r)^{2})\left(5r\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime\prime}(r)+\left(63-59\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime}(r)\right)$
$\displaystyle+\Theta(r)^{3}\left(\Lambda_{3}r^{2}-1\right)\left(\left(5\Lambda_{3}r^{2}-21\right)\Theta_{f_{0}}^{\prime}(r)+r\left(r\Theta_{f_{0}}^{(3)}(r)\left(\Lambda_{3}r^{2}-1\right)+\left(9-5\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime\prime}(r)\right)\right)+4r\Theta_{f_{0}}(r)^{2}$
$\displaystyle\times\left(-r^{2}\left(\Lambda_{3}r^{2}-1\right){}^{2}\Theta^{\prime\prime}(r)^{2}+\left(\Lambda_{3}r^{2}\left(141\Lambda_{3}r^{2}-394\right)+249\right)(\Theta_{f_{0}}^{\prime}(r))^{2}+2r\left(\Lambda_{3}r^{2}-1\right)\Theta_{f_{0}}^{\prime}(r)\left(r\Theta_{f_{0}}^{(3)}(r)\right.\right.$
$\displaystyle\times\left.\left.\left(\Lambda_{3}r^{2}-1\right)\left(42-40\Lambda_{3}r^{2}\right)\Theta_{f_{0}}^{\prime\prime}(r)\right)\right)+15r^{3}\left(\Lambda_{3}r^{2}-1\right){}^{2}(\Theta_{f_{0}}^{\prime}(r))^{4}\bigg{)}-\mathcal{I}_{1}=0.$
(20)

In the context of the current study, this differential equation is the most
relevant, since all physical effects depend on its solution. Since it is
extremely non-linear, we can solve it numerically by computing
$\Theta_{f{0}}(r)$ for some appropriate initial condition.

Figure 1:  is showing the behavior of $\mathbb{S}_{f}(r)$ and
$\mathbb{S}_{f}(r)-r$.

Figure 2:  is showing the behavior of $\mathbb{S}_{f}^{{}^{\prime}}(r)$ and
$\frac{\mathbb{S}_{f}}{(r)}$.

Some important results using numerical solution of Eq. (20) are itemized
below.

  * •

The positive and increasing behavior of $\mathbb{S}_{f}(r)$ can be verified by
the Fig. (1) of left panel, which justifies the existence of wormholes. The
right side of the Fig. (1) describe the difference of shape-function and
radial coordinate, i.e., $\mathbb{S}_{f}(r)-r$. Basically, this difference
indicates the wormhole throat location, i.e., $r_{0}$. In particular, it can
be seen from Fig. (1) that the difference cut the $x$ axis at $r=r_{0}=0.0420$
which provides the evidence for the opening of wormhole throat. The wormhole
throat opening properties are the fundamental requirement of the wormhole
geometry.

  * •

The derivative of shape function with respect to radial coordinate $r$ can be
perceived from the left panel of Fig. (2), which is seen less than one, i.e.,
$\frac{d\mathbb{S}_{f}}{dr}<1$ at $r_{0}=0.0420$. The condition
$\frac{d\mathbb{S}_{f}}{dr}<1$ provides the flaring out property of wormhole
physics. From Fig. (2), the ratio of shape function and radial coordinate,
i.e., $\frac{\mathbb{S}_{f}(r)}{r}$ is seen by right part. In this study, the
expression $\frac{\mathbb{S}_{f}(r)}{r}$ approaches to small value, i.e.,
$0.855$ nearing zero as $r$ approaches infinity, perhaps due to the use of
Noether and conformal symmetries.

###  III.1 Energy Bounds

The energy bounds are very important for the physical acceptability of
wormhole geometry. It would be an interesting task to derive modified energy
bounds in frame of $f(\mathcal{G})$ gravity. For present study, we may use the
$\mathcal{GR}$ energy condition with the following justification.

Theorem: For a solution of Eqs. (8)-(10), expressed by functions
$K_{1}=\big{\\{}\nu(r),\lambda(r),f(\mathcal{G})\big{\\}}$, if we have a
solution in $\mathcal{GR}$ defined by function
$K_{2}=\big{\\{}\nu(r),\lambda(r)\big{\\}}$, then the energy conditions would
be the same for $K_{1}$ and $K_{2}$ since $T^{eff}_{\xi\eta}$ in (4) performs
the significant role of stress energy tensor in $\mathcal{GR}$ M.V ; Shamir1 .

The energy conditions are described as null energy condition $($NEC$)$, weak
energy condition $($WEC$)$, dominant energy condition $($DEC$)$, and strong
energy condition $($SEC$)$, which are mentioned as Bamba .

$\displaystyle NEC:\;\;\;\forall i,\;\;\;\;\;\rho^{eff}+p^{eff}_{i}\geq
0.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;WEC:\;\;\rho^{eff}\geq 0\;\;\;\;\;\forall
i,\;\;\rho^{eff}+p^{eff}_{i}\geq 0.$ $\displaystyle DEC:\;\;\rho^{eff}\geq
0\;\;\;\forall i,\;\;\;\;\;\rho^{eff}\pm p^{eff}_{i}\geq
0.\;\;\;\;\;SEC:\;\;\rho^{eff}+p^{eff}_{i}\geq 0\;\;and\;\;\;\forall
i,\;\;\;\;\;\rho^{eff}+\sum p^{eff}_{i}\geq 0.$

The $NEC$ and $WEC$ violation lead to exotic matter. The exotic matter is the
main cause for the validity of the existence of wormhole geometry. The
pictorial representation of $NEC$ and $WEC$ can be seen from the Fig. (3) and
Fig. (4) respectively. We can see the violation of $NEC$ from the Fig. (3),
which describes the presence of exotic matter. Moreover, Figs. (5) and (6)
depict the evolution of radial and tangential equation of parameter. The
magnified views (right panel) show that the values goes in negative range near
the throat indicting existence of exotic matter. The presence of exotic matter
shows the goodness and superiority of our study based on the symmetries. Due
to presence of exotic matter wormhole throat should be open, which is
necessary for the physically acceptable wormhole geometry. However, a little
drawback is witnessed that the physical parameters are seen justified in a
narrow space due to the use of symmetry approach. It is evident from graphs
that fluctuated behavior is obtained for large scale, however, magnified views
show that physically viable wormholes are possible.

Figure 3:  is indicating the graphically behavior of $NEC$.

Figure 4:  is indicating the graphically behavior of $WEC$.

###  III.2 Equilibrium Condition

Figure 5:  shows the evolution of radial EoS parameter

Figure 6:  shows the evolution of tangential EoS parameter

Here, we provide the stability analysis of the wormhole solutions just
discussed in the previous section by incorporating the equilibrium limitations
under Noether symmetry framework with conformal symmetry, for the
$f(\mathcal{G})$ gravity models under investigation. Here, we will assume
Tolman-Oppenheimer-Volkoff equation, which is given as:

$-\frac{dp^{eff}_{r}}{dr}-\frac{\nu^{{}^{\prime}}(r)}{2}(\rho^{eff}+p^{eff}_{r})+\frac{2}{r}(p^{eff}_{t}-p^{eff}_{r})=0,$
(21)

where $\nu(r)=2\Omega(r)$. The forces namely, hydrostatic
$(\mathcal{F}_{\mathrm{h}})$, the gravitational ($\mathcal{F}_{\mathrm{g}})$
and anisotropic force $(\mathcal{F}_{\mathrm{a}})$ are represented by
following expressions

$\mathcal{F}_{\mathrm{h}}=-\frac{dp^{eff}_{r}}{dr},\;\;\;\;\;\;\;\;\mathcal{F}_{\mathrm{a}}=\frac{2}{r}(p^{eff}_{t}-p^{eff}_{r}),\;\;\;\;\;\;\;\;\mathcal{F}_{\mathrm{g}}=-\frac{\nu^{{}^{\prime}}}{2}(\rho^{eff}+p^{eff}_{r}),$
(22)

and thus Eq. (21) takes the form given by

$\mathcal{F}_{\mathrm{a}}+\mathcal{F}_{\mathrm{g}}+\mathcal{F}_{\mathrm{h}}=0.$

Figure 7:  indicates the balancing behavior of hydrostatic, the gravitational
and anisotropic forces.

Fig. (7) depicts the graphical behavior of these forces, which are shown
balance to each other by left part. In Fig. (7), the red, black, and green
lines represent the hydrostatic, gravitational, and anisotropic forces. The
balancing behavior of gravitational and anisotropic forces against hydrostatic
force shows the stability of wormhole existence via Tolman-Oppenheimer-Volkoff
equation in the background of Noether symmetry with conformal symmetry. The
right part of Fig. (7) shows the real impact of different forces in very small
interval of radial coordinate. The stability via Tolman-Oppenheimer-Volkoff
equation shows that the our inquired wormhole solutions are physically
realistic and acceptable in $f(\mathcal{G})$ gravity.

##  IV Conclusive Remarks

A basic technique for solving the dynamical equations is known as the Noether
symmetry technique. The Noether symmetries, in particular, are not only a
mechanism for dealing with the dynamics solution, but their presence also
provides suitable conditions so that one can specify the universe models
physically and analytically according to our measured observations. The
procedure of the Lagrange multiplier enables one to resolve some problems
related to the $f(\mathcal{G})$ gravity model and to minimize the Lagrangian
to a canonical form. It is perhaps quite useful to reduce the dynamics of the
system in order to identify the exact solutions. In this study, we examine the
existence of static wormhole through Noether and conformal symmetries in the
frame anisotropic matter distribution. For this purpose, we solve
overdetermined system of PDE’s and find the Noether symmetry generators along
with corresponding conserved quantity by incorporating the $f(\mathcal{G})$
gravity model. Moreover, a useful conserved quantity is gained from the
Noether symmetry of spherically symmetric spacetime. The presence of conserved
quantity plays a significant part in defining the possible existence of
wormhole solutions under Noether symmetry by utilizing conformal symmetry.
Moreover, we have also investigated stable condition of wormhole solutions via
modified equilibrium condition by considering the specific red-shift function.
Some essential findings and observations regarding the existence of wormhole
in modified $f(\mathcal{G})$ gravity are summarized below.

  * •

The shape-function $\mathbb{S}_{f}(r)$ remains positive and continues to
increase as shown in the left plot of Fig. (1). The positive behavior of shape
function shows that our obtained solutions are physically acceptable.

  * •

The condition $\mathbb{S}_{f}(r_{0})=r_{0}$ is justified at $r_{0}=0.0420$ as
depicted in Fig. (1), which shows that the wormhole throat should be opened.

  * •

The flaring out condition, i.e., $\mathbb{S}_{f}(r)^{{}^{\prime}}(r_{0})<1$,
is satisfied, as it can be noted from the right panel of Fig. (2).

  * •

The flatness condition also can be seen from the right panel of Fig. (2).

  * •

Referring to the energy bounds $NEC$ and $WEC$, it can be seen from the Fig.
(3) and Fig. (4). The violation of $NEC$ can be perceived from the Fig. (3),
which describes the presence of exotic matter. The presence of exotic matter
shows the goodness and superiority of our study based on the symmetries, i.e.,
Noether symmetry and conformal symmetry. Due to presence of exotic matter
wormhole throat should be open, which is necessary for the physically
acceptable wormhole geometry.

  * •

Fig. (7) represents the graphical analysis of three different forces, i.e.,
$\mathcal{F}_{\mathrm{a}},\mathcal{F}_{\mathrm{g}}$ and
$\mathcal{F}_{\mathrm{h}}$, which are shown balance to each other by left
part. In Fig. (7), the red, black, and green lines represent the hydrostatic,
gravitational, and anisotropic forces. The balancing behavior of gravitational
and anisotropic forces against hydrostatic force shows the stability of
wormhole existence via Tolman-Oppenheimer-Volkoff equation in the background
of Noether symmetry with conformal symmetry.

All above discussions suggest that wormholes exist in $f(\mathcal{G})$ gravity
using symmetry approach. However, a little drawback is witnessed that the
physical parameters are seen justified in a narrow space due to the use of
symmetry approach. It is evident from graphs that fluctuated behavior is
obtained for large scale, however, magnified views show that physically viable
wormholes are possible. Conclusively, it is worthy to mention here that
Noether and conformal symmetries are quite helpful in obtaining physically
realistic and acceptable wormhole solutions in $f(\mathcal{G})$ gravity.

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