#  Noncommutative wormhole in de Rham-Gabadadze-Tolley like massive gravity

Piyali Bhar<EMAIL_ADDRESS>Department of Mathematics,Government
General Degree College Singur, Hooghly, West Bengal 712 409, India  Allah
Ditta<EMAIL_ADDRESS>Department ofMathematics, Shanghai University
and Newtouch Center for Mathematics of Shanghai University, Shanghai,200444,PR
China  Abdelghani Errehymy<EMAIL_ADDRESS>Astrophysics
Research Centre, School of Mathematics, Statistics and Computer Science,
University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

###### Abstract

The wormhole solution in dRGT massive gravity is examined in this paper in the
background of non-commutative geometry. In order to derive the wormhole model,
along with the zero tidal force, we assume that the matter distribution is
given by the Gaussian and Lorentzian distributions. The shape function in both
models involves the massive gravity parameters $m^{2}c_{1}$ and $m^{2}c_{2}$.
But the spacetime loses its asymptotic flatness due to the action of the
massive gravity parameter. It is noticed that the asymptotic flatness is
affected by the repulsive effect induced in the massive gravitons that push
the spacetime geometry very strongly. We observed that each model violates the
null energy criteria, indicating the presence of exotic matter which is
necessary to sustain the wormholes. The exotic matter is measured using the
volume integral quantifier. Moreover, it is discovered that the model is
stable under the hydrostatic equilibrium condition by utilizing the TOV
equation. Finally, our research encompassed an exploration of the repulsive
influence exerted by gravity. Our findings demonstrated that the presence of
repulsive gravity results in a negative deflection angle for photons following
null geodesics. Remarkably, we consistently observed negative values for the
deflection angle across all values of $r_{0}$ in the two scenarios examined.
This consistent negativity unequivocally signifies the manifestation of the
repulsive gravity effect.

###### pacs:

95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.Kd

##  I Introduction

The general theory of relativity (GR) formulated by Einstein has been
remarkably successful in elucidating various phenomena in astrophysics and
cosmology. However, the discovery of the late-time acceleration of the
universe, attributed to dark energy, and the existence of dark matter have
presented challenges to the prevailing understanding based on GR
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns ;
Hanany:2000qf ; deBernardis:2001mjh ; Peebles:2002gy ; SDSS:2003tbn ;
Riess:2006fw ; Amanullah:2010vv ; WMAP:2010qai . These findings have raised
important questions and cast doubts on the continued success of Einstein’s
theory of gravitation. It is worth noting that one of the notable solutions to
Einstein’s field equations, known as the Schwarzschild solution, was derived
by K. Schwarzschild in 1916 Schwarzschild:1916uq . This solution describes a
static and spherically symmetric black hole. The confirmation of gravitational
waves (GWs) LIGOScientific:2016aoc  has provided compelling evidence for the
existence of stellar-mass black holes in the natural universe. Interestingly,
as far back as 1916, Ludwing Flamm Flamm:1916aoc  recognized that Einstein’s
equations have another solution, now known as a white hole. Unlike black
holes, white holes were initially believed to expel matter and light from
their event horizon. These two solutions could represent distinct regions in
spacetime that are connected by a conduit, which later became known as a
“bridge”. In 1935, Albert Einstein and Nathan Rosen used the principles of
general relativity to propose the existence of such “bridges” in spacetime
Einstein:1935tc . Several decades later, in 1957, the term wormhole was coined
by Misner and Wheeler Misner:1957mt , leading to a renewed focus on studying
these intriguing phenomena. However, it was later discovered that the original
version of wormholes is not traversable due to the rapid opening and closing
of their throat. This means that they are not practical for passage or
communication.

To avoid the rapid closure of the wormhole’s throat, a potential solution is
to introduce a scalar field that interacts with gravity. This additional
component gives rise to a broader category of wormholes, which were initially
proposed by Ellis Ellis:1973yv  and independently by Bronnikov
Bronnikov:1973fh . However, one of the major obstacles in the study of
wormholes is the need for exotic matter to maintain their stability and
structure. Exotic matter is characterized by violating the energy conditions
that are commonly observed in conventional physics. In 1988, Morris and Thorne
introduced certain conditions for wormholes to be traversable, which can be
found in Morris:1988cz . These solutions are derived by considering a peculiar
type of matter, known as exotic matter, which possesses unusual properties
necessary for maintaining the structure of the wormhole. Notably, this exotic
matter exhibits negative energy density, satisfying the flare-out condition
while violating the weak energy condition Morris:1988cz ; Morris:1988tu .

It has been established that modified theories of gravity, such as $f(R)$
theory, have a noteworthy contribution in the study of wormholes. Lobo and
Oliveira conducted a study Lobo:2009ip  on traversable wormholes within the
framework of $f(R)$ gravity. They examined the factors responsible for
violating the null energy condition and supporting the existence of wormholes.
They also investigated wormhole solutions for various shape functions. Another
work by Bronnikov et al. Bronnikov:2010tt  discussed the presence of wormholes
in scalar-tensor theory and $f(R)$ gravity. The null and weak energy
conditions for wormholes with constant shape and redshift functions in $f(R)$
gravity can be found in Saeidi:2011zz . S. Bahamonde et al. Bahamonde:2016ixz
developed dynamical wormholes in $f(R)$ gravity, while Kuhfittig
Kuhfittig:2018vdg  explored wormhole solutions using different types of shape
functions in the same framework. Lemos et al. Lemos:2003jb  explored the study
of static and spherically symmetric traversable wormholes in the presence of a
cosmological constant, while Errehymy et al. Errehymy:2023rnd ;
Errehymy:2024yey  addressed the topic of wormholes surrounded with and without
dark matter halos.

A diverse array of wormhole solutions can be found within modified theories of
gravity. For instance, wormholes have been studied in Einstein-Gauss-Bonnet
gravity Maeda:2008nz , $f(R,\Phi)$ gravity Zubair:2017oir , as well as
theories involving both $f(R)$ and $f(R,T)$ DeFalco:2023twb ; Errehymy:2024spg
. Additionally, wormhole solutions have been explored in Born-Infeld gravity
Shaikh:2018yku , Eddington-inspired Born-Infeld gravity Harko:2013aya , and
even in the context of noncommutative geometry Jamil:2013tva ; Rahaman:2012pg
. Mustafa et al. Mustafa:2022fxn  used observational data in the matter
coupling gravity formalism in $f(R,\,T)$ gravity to determine the potential of
generalized wormhole development due to dark matter in the galactic halo.
Using the Karmarkar condition in the f(Q) gravity formalism-where Q is the
nonmetricity scalar-authors in ref. Mustafa:2021ykn  achieved wormhole
solutions. They demonstrate how the combination of these components offers the
prospect of obtaining traversable wormholes that satisfy the energy
conditions. New wormhole solutions in the fourth-order modified Ricci inverse
gravity background are the subject of the research work by Mustafa
Mustafa:2023mls . By displaying the valid region for the majority of the
wormhole geometry under the influence of the relevant parameters, two new
classes of wormhole solutions are analyzed in this paper. In the extended
f(R,Lm) gravity, Mustafa et al. Mustafa:2024iwu  used an anisotropic matter
source and a particular type of energy density demonstrating cold dark matter
halo and quantum wave dark matter halo to calculate two different wormhole
solutions. The properties of the exotic matter within the wormhole geometry
and the matter contents via energy conditions are studied in detail in this
paper. Another interesting work on wormholes can be found in Mustafa:2021vqz .
Moreover, there are numerous other models of wormholes beyond those mentioned
above, as seen in works such as Boehmer:2012uyw ; KordZangeneh:2015dks ;
Mehdizadeh:2017tcf ; Jusufi:2018waj ; Ovgun:2018xys ; Golchin:2019qch ;
Dai:2019mse ; Ditta:2021uoe ; Mustafa:2024jsv ; Errehymy:2024cgy ;
Errehymy:2023rsm ; Waseem:2023ejh ; Mustafa:2023ojf ; Mustafa:2024ark . These
encompass the construction of traversable wormholes within various types of
modified gravity theories. Einstein Einstein:1936llh  was the first to explore
the field of GR, and one of its early and valuable discoveries was
gravitational lensing. This phenomenon occurs when an object that is very
massive bends incoming light in the same way as a lens, providing viewers with
more information about the source of the light. Following the observational
confirmation of light’s deflection, as suggested theoretically, interest in
the field of gravitational lensing grew Dyson:1920cwa ; Eddington:1920cwa .
This phenomenon provides opportunities to investigate exoplanets, dark matter,
and dark energy, reaching beyond the solar system. It was first identified
observationally as a double quasar system in 1979 Walsh:1979nx . Since then,
it has become one of the important research topics in gravitational physics.
The first successful astrometric microlensing measurement of a white dwarf’s
mass, Stein 2051 B, marked a significant milestone Sahu:2017ksz . One
fascinating feature of gravitational lensing is that, in situations with
intense lensing, it may bend light eternally in unstable light rings,
producing a multitude of relativistic pictures Virbhadra:1999nm ; Bozza:2001xd
; Bozza:2002zj . Strong and weak versions of this phenomena are useful
analytical tools for studying gravitational fields around different cosmic
objects, such as wormholes and black holes. Gravitational lensing has been
widely used in theoretical and astrophysical studies to investigate wormholes,
demonstrating the importance of this phenomenon in current research
Tsukamoto:2016qro ; Shaikh:2017zfl ; Jusufi:2017mav ; Ovgun:2018fnk ;
Javed:2019qyg ; Ovgun:2020yuv ; Javed:2022fsn .

In this study, we investigate a noncommutative wormhole in the context of de
Rham-Gabadadze-Tolley (dRGT)-like massive gravity. The investigation into
massive gravity commenced before the discovery of the accelerated expansion of
the cosmos. In 1939, the authors Fierz:1939r  introduced a linear theory of
massive gravity, which treated the graviton as a mass term. However, this
proposed model had some drawbacks Zakharov:1970cc ; vanDam:1970vg , as the
asymptotic-massless limits of the linear theory did not align with the
predictions of GR. In 1972, the author Vainshtein:1972sx  put forward the idea
that the nonlinear approach in the theory of massive gravity could potentially
address the problem. However, this approach gave rise to a new issue known as
the Boulware-Deser ghost Hinterbichler:2011tt . Subsequently, the Boulware-
Deser ghost was successfully eliminated through the introduction of a new
nonlinear version of massive gravity proposed by de Rham, Gabadadze, and
Tolley (dRGT) deRham:2010kj . Since then, the dRGT formulation has become a
notable framework for studying the universe on a cosmic scale. Furthermore,
there have been numerous intriguing articles exploring the applications of
dRGT massive gravity to exotic objects such as black holes Ghosh:2015cva ;
Boonserm:2017qcq .

The paper followed a structured organization, beginning with Section II
offering a concise overview of massive gravity. This was followed by Section
III, which delved into the exploration of wormhole solutions. In Section IV,
the study focused on analyzing the photon deflection angle on null geodesics
and investigating the deviation of photons’ paths. Finally, Section V
encompassed discussions and conclusions drawn from the findings presented in
the paper.

##  II Brief review of massive gravity

Here in this section, we describe the field equations for dRGT model of
massive gravity, by using the spherically symmetric space time. As stated
distinctly in the references 1 ; I21 , we here start with our study with the
massive gravity dynamics equivalent to the dRGT model, using the matter
Lagrangian $S_{\text{matter }}$:

$\displaystyle S_{G}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[R+m^{2}\sum_{i}^{4}c_{i}U_{i}(g,f)\right]+S_{\text{matter
}},$ (1)

in this instance, $R$ is the Ricci scalar curvature, $m$ is a graviton mass-
related variable, and $G$ is the Newton constant of gravitation. The constants
$c_{i}$, the metric tensor $g$, and a fixed symmetric tensor $f$ are examples
of integral components, and show the e dynamical and reference metrics,
respectively. The effective potential of a graviton, $U_{i}$, can be
determined in four dimensions of spacetime as follows:

$\displaystyle\begin{cases}U_{1}&=[\mathcal{K}],\\\
U_{2}&=[\mathcal{K}]^{2}-\left[\mathcal{K}^{2}\right],\\\
U_{3}&=[\mathcal{K}]^{3}-3[\mathcal{K}]\left[\mathcal{K}^{2}\right]+2\left[\mathcal{K}^{3}\right],\\\
U_{4}&=[\mathcal{K}]^{4}-6\left[\mathcal{K}^{2}\right][\mathcal{K}]^{2}+8\left[\mathcal{K}^{3}\right][\mathcal{K}]+3\left[\mathcal{K}^{2}\right]^{2}-6\left[\mathcal{K}^{4}\right],\end{cases}$
(2)

in this instance, $[\mathcal{K}]$ represents the trace of the metric
$\mathcal{K}_{v}^{\mu}$, and $\mathcal{K}v^{\mu}=\sqrt{g^{\mu\lambda}f{\lambda
v}}$. Changing the action (1) with regard to $g_{v}^{\mu}$ yields the equation
of motion controlling this gravitational system:

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+m^{2}\chi_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu},$ (3)

where

$\displaystyle\begin{cases}\chi_{\mu\nu}&=-\frac{c_{1}}{2}\left(U_{1}g_{\mu\nu}-\mathcal{K}_{\mu\nu}\right)-\frac{c_{2}}{2}[U_{2}g_{\mu\nu}-2U_{1}\mathcal{K}_{\mu\nu}\\\
&+2\mathcal{K}_{\mu\nu}^{2}]-\frac{c_{3}}{2}\big{[}U_{3}g_{\mu\nu}-3U_{1}\mathcal{K}_{\mu\nu}+6U_{1}\mathcal{K}_{\mu\nu}^{2}\\\
&-6\mathcal{K}_{\mu\nu}^{3}\big{]}-\frac{c_{4}}{2}\big{[}U_{4}g_{\mu\nu}-4U_{3}\mathcal{K}_{\mu\nu}+12U_{2}\mathcal{K}_{\mu\nu}^{2}\\\
&-24U_{1}\mathcal{K}_{\mu\nu}^{3}+24\mathcal{K}_{\mu\nu}^{4}\big{]}.\end{cases}$
(4)

In this case, we present a spherically symmetric metric as:

$\displaystyle
ds^{2}=-e^{2\phi(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin\theta^{2}d\phi^{2}\right),$
(5)

here, radial functions are denoted by $\lambda(r)$ and $\phi(r)$. The
particular form of the reference metric that we have chosen is as follows:

$\displaystyle
f_{\mu\nu}=\operatorname{diag}\left(0,0,C^{2},C^{2}\sin^{2}\theta\right),$ (6)

where $C$ denotes the positive constant. The distinct functional forms of
$U_{i}$ can be obtained using the metric ansatz (6) and are as follows:

$\displaystyle\begin{cases}U_{1}&=\frac{2C}{r},\quad
U_{2}=\frac{2C^{2}}{r^{2}},\\\ U_{i}&=0,\quad i>2.\end{cases}$ (7)

Moreover, the energy-momentum tensor used for an anisotropic fluid is 3 :

$\displaystyle
T_{\mu\nu}=\left(\rho+p_{t}\right)u_{\mu}u_{v}+p_{t}g_{\mu\nu}+\left(p_{r}-p_{t}\right)n_{\mu}n_{v},$
(8)

where $\rho$ denotes energy density and $p_{\mathrm{r}}$ and $p_{t}$ denote
radial and transverse pressure, respectively. The four-velocity is denoted by
$u_{\mu}$, and the four-radial vector by $n_{\mu}$. The field equations’ non-
zero components can be acquired by using the energy-momentum tensor from Eq.
(8) and the metric given in Eq. (5) as follows:

$\displaystyle\frac{8\pi G}{c^{2}}r^{2}\rho$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}\left(2r\lambda^{\prime}-1\right)+1+m^{2}C\left(c_{2}C+c_{1}r\right),$
(9) $\displaystyle\frac{8\pi G}{c^{4}}r^{2}p_{r}$ $\displaystyle=$
$\displaystyle
e^{-2\lambda}\left(2r\phi^{\prime}+1\right)-1-m^{2}C\left(c_{2}C+c_{1}r\right),$
(10) $\displaystyle\frac{8\pi G}{c^{4}}rp_{t}$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}\left(\phi^{\prime}+r\phi^{\prime
2}-\lambda^{\prime}-r\lambda^{\prime}\phi^{\prime}+r\phi^{\prime\prime}\right)-\frac{m^{2}c_{1}C}{2},$
(11)

where ′ denotes the derivative with respect to radial coordinate $r$. In the
next section, we are going to solve the eqns. (9)-(11) under certain
conditions to obtain the model of the wormhole in this modified gravity.

##  III Wormhole Solutions

In eqn. (5), $\phi(r)$ is referred to as the redshift function because of its
relationship to the gravitational redshift and is taken to be finite
everywhere to prevent event horizons which makes the wormhole (WH)
traversable. $e^{-2\lambda}=1-\frac{b(r)}{r}$, $b(r)$, on the other hand,
represents the shape function of the wormhole. The radial coordinate r extends
to $\infty$ from a minimum value $r_{0}$, where $r_{0}$ represents the throat
of the WH and it satisfies the formula $b(r_{0})=r_{0}$.
For the sake of physical admissibility, the wormhole shape function $b(r)$
needs to adhere to the following conditions:

  * •

Throat condition: At the throat $r_{0}$, $b(r_{0})=r_{0}$ and for $r>r_{0}$,
$1-\frac{b(r)}{r}>0$.

  * •

Flaring-out condition: At the throat, the radial differential of the shape
function $b^{\prime}(r)$ should be less than $1$.

  * •

Asymptotic Flatness condition: If $r\rightarrow\infty$, then
$\frac{b(r)}{r}\rightarrow 0$.

  * •

Proper radial distance function: In traversable wormhole the proper radial
distance function $l(r)=\pm\int(\frac{r-b(r)}{r})^{-\frac{1}{2}}dr$ must be
finite everywhere in the domain. Thus, $b(r)$ should be less than $r$ for all
$r$. The sign $\pm$ represents the upper and lower universes.

###  III.1 Model I with Gaussian distribution

In non-commutative geometry, smeared objects take the place of point-like
particles. Space-time is encoded in the commutator
$[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}$, $\theta^{\mu\nu}$ being the anti-
symmetric matrix. It causes the fundamental cell discretization of spacetime,
like how the phase space is discretized by the Planck constant $\hbar$. We
shall take into consideration the Gaussian distribution as a noncommutative
energy density profile to describe our current model in this subsection, which
has the following expression Nicolini:2005vd :

$\displaystyle\rho=\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}},$
(12)

where the source’s total mass is $M$ and $\theta$ is the non commutative
parameter. Mass $M$ of the source is not perfectly localized at the point, but
rather it is distributed throughout a region of linear size $\sqrt{\theta}$.
The coordinate commutator’s inherent uncertainty is responsible for this. This
results from the inherent uncertainty of the coordinate commutator.
Along with this, to solve the field equations, let us assume a constant
redshift function, i.e., $\phi$=constant.
Assuming $G=c=1$ and using the expression of $\rho$ given in (12), we solve
the equation (9) for $e^{-2\lambda}$ which is obtained as,

$\displaystyle e^{-2\lambda}$ $\displaystyle=$ $\displaystyle
1+Cm^{2}\left(Cc_{2}+\frac{c_{1}r}{2}\right)+\frac{2M}{\sqrt{\theta\pi}}e^{-\frac{r^{2}}{4\theta}}-\frac{2M}{r}\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right)+\frac{E_{1}}{r},$
(13)

where $E_{1}$ is the constant of integration. ‘erf(z)’ is the “error function”
that arises in integrating the normal distribution. It has the following
expression:

$\displaystyle erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt,$

and in Maclaurin series expansion, it can be written as,

$\displaystyle erf(z)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}\left[z-\frac{z^{3}}{3}+\frac{z^{5}}{10}-\frac{z^{7}}{42}+...\right].$

Now using the relationship $e^{-2\lambda}=1-\frac{b(r)}{r}$, the shape
function of the wormhole is obtained as,

$\displaystyle
b(r)=-\frac{1}{2}Cm^{2}r(2Cc_{2}+c_{1}r)-\frac{2Mr}{\sqrt{\pi\theta}}e^{-\frac{r^{2}}{4\theta}}+2M\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right)-E_{1}.$
(14)

The value of the integration constant can be obtained from the formula
$b(r_{0})=r_{0}$. The radial and transverse pressure are obtained as,

$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{M}{4r^{2}\pi^{\frac{3}{2}}\sqrt{\theta}}e^{-\frac{r^{2}}{4\theta}}+\frac{2E_{1}-c_{1}Cm^{2}r^{2}-4M\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right)}{16\pi
r^{3}},$ (15) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{M}{8\pi
r^{3}}\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right)-\frac{1}{32\pi
r}Cm^{2}c_{1}-\frac{M(r^{2}+2\theta)}{16r^{2}(\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}}-\frac{E_{1}}{16\pi
r^{3}}.$ (16)

Figure 1: Different properties of the shape function with Gaussian
distribution. For drawing the plots the values of the parameters are taken as
follows: $C=0.05,\,m^{2}c_{2}=-0.05,\,M=1.8,\,\theta=0.02$ and $E_{1}=-1.05$.

###  III.2 Model II with Lorentzian distribution

We draw inspiration for the second wormhole model from the work of Mehdipour
Mehdipour:2011mc . Consequently, we suppose that the particle-like
gravitational source has a Lorentzian distribution, and as a result, the
energy density profile is as follows:

$\displaystyle\rho=\frac{\mathcal{M}\sqrt{\psi}}{\pi^{2}(r^{2}+\psi)^{2}},$
(17)

where $\mathcal{M}$ and $\psi$ respectively denote the smeared mass
distribution and the noncommutativity parameter.
Now by assuming the constant redshift function, we solve the field equations
(9)-(11) with the help of the density function given in eqn. (17) which gives,

$\displaystyle e^{-2\lambda}$ $\displaystyle=$ $\displaystyle
1+\mathcal{M}^{2}C(c_{2}C+\frac{c_{1}}{2}r)-\frac{8\mathcal{M}\sqrt{\psi}}{r\pi}\left[-\frac{r}{2(r^{2}+\psi)}+\frac{1}{2\sqrt{\psi}}tan^{-1}\left(\frac{r}{\sqrt{\psi}}\right)\right]+\frac{E_{2}}{r},$
(18)

here, $E_{2}$ represents the constant of integration. Using the well-known
relationship between the shape function and the metric coefficient
$e^{-2\lambda}$, the shape function of the wormhole in this model is obtained
as,

$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{8\mathcal{M}\sqrt{\psi}}{\pi}\left[-\frac{r}{2(r^{2}+\psi)}+\frac{1}{2\sqrt{\psi}}tan^{-1}\left(\frac{r}{\sqrt{\psi}}\right)\right]-\frac{1}{2}C\mathcal{M}^{2}r(2Cc_{2}+c_{1}r)-E_{2},$
(19)

The integration constant $E_{2}$ can be obtained from the relationship
$b(r_{0})=r_{0}$.
We consequently obtain the radial and transverse pressure as follows:

$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{8\mathcal{M}r\sqrt{\psi}+2E_{2}\pi(r^{2}+\psi)-c_{1}C\mathcal{M}^{2}\pi
r^{2}(r^{2}+\psi)-8\mathcal{M}(r^{2}+\psi)tan^{-1}\left(\frac{r}{\sqrt{\psi}}\right)}{16\pi^{2}r^{3}(r^{2}+\psi)},$
(20) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{8\mathcal{M}(r^{2}+\psi)^{2}tan^{-1}\left(\frac{r}{\sqrt{\psi}}\right)-2E_{2}\pi(r^{2}+\psi)^{2}-r(c_{1}C\mathcal{M}^{2}\pi
r(r^{2}+\psi)^{2}-8\mathcal{M}\sqrt{\psi}(3r^{2}+\psi))}{32\pi^{2}r^{3}(r^{2}+\psi)^{2}}$
(21)

Figure 2: Different properties of the shape function with Lorentzian
distribution. For drawing the plots we have taken
$C=0.05,\,m^{2}c_{2}=-0.05,\,\mathcal{M}=1.8,\,\psi=0.02$ and $E_{2}=-1.05$

We are going to discuss the shape function that we have obtained and the
conditions that must be met for a wormhole to exist. To accomplish this, we
carefully choose the appropriate free parameters which are mentioned in Figs.
1-2. Assuming a constant redshift function, we examine how the shape functions
related to the Gaussian distribution as well as the Lorentzian distribution
behave. The shape function and the flaring out condition for three different
values of $m^{2}c_{1}$ are shown in Figs. 1-2. The shape function clearly
shows positive increasing behavior. Nevertheless, the shape function shows a
declining trend as the value of the model parameter $m^{2}c_{1}$ increases for
both the two cases. Moreover, the upper right panels of Figs. 1-2 show the
behavior of $b(r)-r$, where the graphs cut the ‘r-axis’ the position of the
throat located at that point. By assuming the suitable values of the model
parameters, the position of the throats for both the wormholes are listed in
Tables 1-2 for three different values of $m^{2}c_{1}$.
Furthermore, as can be seen from the bottom panels of Figs. 1-2, at the throat
of the wormhole, the flaring out requirement, $b^{{}^{\prime}}(r)<1$ for three
different values of $m^{2}c_{1}$, is met. Additionally, the asymptotic
behavior of the shape function over a range of $m^{2}c_{1}$ is shown in the
right-hand side of the bottom panels of Fig. 1-2. It shows that the ratio
$\frac{b(r)}{r}$ does not approach zero as $r~{}\rightarrow~{}\infty$, and
therefore the asymptotic flatness condition is violated for both cases.

Table 1: The position of the throat of the wormhole for Gaussian distribution with $C=0.05,\,m^{2}c_{2}=-0.05,\,M=1.8,\,\theta=0.02$ and $E_{1}=-1.05$. $m^{2}c_{1}$ |  |  $r_{0}$(in km.)
---|---|---
-0.1 |  | 4.706
-0.2 |  | 4.764
-0.3 |  | 4.825
Table 2: The position of the throat of the wormhole for Lorentzian distribution with $C=0.05,\,m^{2}c_{2}=-0.05,\,\mathcal{M}=1.8,\,\psi=0.02$ and $E_{2}=-1.05$ $m^{2}c_{1}$ |  |  $r_{0}$(in km.)
---|---|---
-0.1 |  | 4.56
-0.2 |  | 4.617
-0.3 |  | 4.676

###  III.3 Visualization of wormhole

We next look over the embedding surface diagram and determine which conditions
are necessary in order to represent the embedded wormhole configuration. Using
$\theta=2\pi$ and $t=$ const. in Eq. (5), we can take specified spherical
symmetric space-time with an equatorial slice, which in turn is as follows:

$\displaystyle
ds^{2}=\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}d\phi^{2},$ (22)

One can embed Eq. (22) into 3-dimensional cylindrically symmetric Euclidean
spacetime as follows:

$\displaystyle ds_{\Xi}^{2}=dz(r)^{2}+dr^{2}+r^{2}d\phi^{2}.$ (23)

The above Eq. (23) can be rewritten as

$\displaystyle
ds_{\Xi}^{2}=\left(1+\left(\frac{dz(r)}{dr}\right)^{2}\right)dr^{2}+r^{2}d\phi^{2}.$
(24)

Figure 3: The embedding diagrams of the wormholes are shown. The values of the
parameters are the same as Fig. 1-2.

Figure 4: The full visualization of the wormholes is shown. The values of the
parameters are the same as Fig. 1-2.

The relationship that results from matching Eqs. (22-24) is as follows:

$\displaystyle\frac{dz(r)}{dr}=\pm\left(\frac{r}{b(r)}-1\right)^{-1/2}.$ (25)

Fig. 3 displays the embedded surface diagram of the wormhole for three
different values of $m^{2}c_{1}$. In Fig. 4, the full visualization of the
wormhole is obtained by revolving the Fig. 3 with respect to ‘z-axis’.

###  III.4 Energy conditions

The energy conditions become important when examining the properties of matter
in the framework of modified gravity. The famous Raychaudhuri equations are
the source of these conditions ad1 ; ad2 ; ad3 . Moreover, the purpose of all
energy conditions is to investigate the geometry and composition of
spherically symmetric spacetime. For an anisotropic matter distribution, the
energy conditions are satisfied if the following inequalities hold:

$\displaystyle(\mathcal{NEC})$ $\displaystyle\Leftrightarrow$
$\displaystyle\rho+p_{t}\geq 0\;\text{and}\;\rho+p_{r}\geq 0.$
$\displaystyle(\mathcal{WEC})$ $\displaystyle\Leftrightarrow$
$\displaystyle\rho\geq 0,\rho+p_{t}\geq\;\text{and}\;\rho+p_{r}\geq 0.$
$\displaystyle(\mathcal{SEC})$ $\displaystyle\Leftrightarrow$
$\displaystyle\rho+p_{j}\geq 0\;\text{and}\;\rho+\sum_{j}p_{j}\geq 0,\forall
j,\;\text{where}j=r,t.$ $\displaystyle(\mathcal{DEC})$
$\displaystyle\Leftrightarrow$ $\displaystyle\rho\geq
0,\rho-\left|p_{r}\right|\geq 0\;\text{and}\;\rho-\left|p_{t}\right|\geq 0.$

In our present study, with the help of graphical representation as shown in
Fig. 5-6 we have checked that $\rho+p_{r}<0$, $\rho+p_{t}>0$,
$\rho+p_{r}+2p_{t}<0$, $\rho-p_{t}>0$, and $\rho-p_{t}<0$ for both the
wormhole models, and hence all the energy conditions are violated. The
existence of the exotic matter in the throat of the wormhole is confirmed by
the violation of the null energy criteria.

Figure 5: Energy conditions for Gaussian distribution. The values of the
parameters are the same as Fig. 1.

Figure 6: Energy conditions for Lorentzian distribution. The values of the
parameters are the same as Fig. 2.

###  III.5 Proper radial distance

In this subsection, we are going to discuss the proper radial distance of
$l(r)$ of the wormhole, which can be obtained as,

$\displaystyle l(r)=\pm\int_{r_{0}+}^{r}\frac{1}{\sqrt{1-\frac{b(r)}{r}}}dr,$
(26)

Due to the complex expression of $b(r)$, we performed the above integral
numerically, and its profiles are shown in Fig 7 for different values of
$m^{2}c_{1}$. Fig. 7 shows that $l(r)$ is finite everywhere and decreases from
the upper universe $l(r)~{}\rightarrow~{}+\infty$ to the throat which is
physically reasonable.

Figure 7: The proper radial distance of the wormhole. The values of the
parameters are the same as Fig. 1-2.

###  III.6 Equilibrium condition by analyzing TOV

For our current work, the generalized Tolman-Oppenheimer-Volkov (TOV)
equation, which is given by leon ,

$\displaystyle\frac{dp_{r}}{dr}+\frac{2}{r}(p_{t}-p_{r})+\frac{\phi^{\prime}}{2}(\rho+p_{r})=0,$
(27)

provides the equilibrium condition.
The aforementioned equation is a significant and comprehensive technique that
may be applied to investigate the stability conditions of astrophysical
solutions, such as compact objects and wormholes.
Another way to write the Eq. (27) is as

$\displaystyle F_{g}+F_{a}+F_{h}=0,$ (28)

This describes the equilibrium condition of the wormhole. Here,

$\displaystyle F_{a}$ $\displaystyle=$
$\displaystyle\frac{2}{r}(p_{t}-p_{r}),$ $\displaystyle F_{g}$
$\displaystyle=$ $\displaystyle-\frac{\phi^{\prime}}{2}(\rho+p_{r}),$
$\displaystyle F_{h}$ $\displaystyle=$ $\displaystyle-\frac{dp_{r}}{dr}.$

where $F_{g}$ is the gravitational force, $F_{h}$ is the hydrostatic force,
and $F_{a}$ is the force resulting from the anisotropic matter of the
wormhole. Nonetheless, it is evident from Eq. (28) that the sum of three
distinct forces must equal zero for the system to be in equilibrium. Fig. 8
displays the behavior of $F_{a}$, and $F_{h}$ for a specific selection of
parameter values. However, since $\phi$ is assumed to be constant, the value
of $F_{g}$ is $0$, meaning that the gravitational force does not influence our
model. It is clear from the figure that the other two forces are precisely the
same and opposing one another. Therefore, it is clear that the three forces
work together to establish the balance of the model, which supports the
stability of the system.

Figure 8: Two different forces acting on the wormhole spacetimes are shown in
the figures. For drawing the profiles the values of the parameters used here
are the same as Figs. 1-2.

###  III.7 Measurement of exotic matter

Exotic matter is the term typically used to describe material that violates
the NEC. The following volume integral quantifier (VIQ) can be computed to
examine the amount of exotic matter near the wormhole’s throat Visser:2003yf :

$\displaystyle I_{V}=\oint(\rho+p_{r})dV$ $\displaystyle=$
$\displaystyle-\int_{r_{0}}^{\infty}(1-b^{\prime})\ln\left[e^{\nu}\left(1-\frac{b(r)}{r}\right)^{-1}\right]dr,$
(29)

We can see how VIQ varies in connection with $r$ in Fig. 9. The model
parameters are selected in this case as follows: For the Gaussian distribution
the following values are taken:
$C=0.05,\,m^{2}c_{2}=-0.05,\,M=1.8,\,\theta=0.02,$ and $E_{1}=-1.05$,
$m^{2}c_{1}~{}\in[-0.3,\,0.1]$. For the Lorentzian distribution wormhole
model, we select
$C=0.05,\,m^{2}c_{2}=-0.05,\,M=1.8,\,\phi=0.02,\,E_{2}=-1.05,$ and
$m^{2}c_{1}\in[-0.3,\,0.1]$. In this connection, we want to mention that Lobo
et al. Lobo:2012qq  proposed an asymptotically flat phantom wormhole, and with
the help of the volume integral quantifier, the author showed that as $\alpha$
tends to $-1$, the volume integral quantifier approaches to zero. Bhar
Bhar:2023ruf  also used the volume integral quantifier to measure the amount
of exotic matter to obtain the model of the wormhole in Rastall gravity. We
can see from the figures that in both situations, there exists exotic matter
near the throat of the WH since we have obtained $I_{v}<0$ for when r gets
closer to $r_{0}$. This suggests that a very minimal amount of exotic matter
is needed to open the throat of the wormhole.

Figure 9: The VIQ are shown in the figures. For drawing the profiles, the
values of the parameters used here are the same as Figs. 1-2.

##  IV Photon deflection angle on null geodesics: Analyzing the deviation of
photons’ paths

Let us delve into the investigation of the deviation angle of photons from
null geodesics. To begin, we introduce a comprehensive line element that
encompasses both spherically symmetric and static characteristics misner1973 ;
schutz2014 . This line element can be expressed as follows:

$ds^{2}=-\mathcal{A}(r)dt^{2}+\mathcal{B}(r)dr^{2}+\mathcal{C}(r)d\Omega^{2}.$
(30)

In order to analyze the trajectory of objects in free fall within this
spacetime geometry schutz2014 , we utilize the geodesic equation. This
equation establishes a connection between the momentum one-forms of the
objects and the geometry of spacetime. Mathematically, it can be represented
as:

$\frac{dp_{\beta}}{d\lambda}=\frac{1}{2}g_{\nu\alpha,\beta}p^{\nu}p^{\alpha},$
(31)

where $p^{\alpha}$ is the 4-momentum and $\lambda$ is the affine parameter. To
examine the deviation angle of photons from null geodesics, we introduce the
affine parameter $\lambda$ to characterize the motion. It is important to note
that if the components of $g_{\alpha\nu}$ do not vary with respect to
$x^{\beta}$ for a fixed index $\beta$, then $p_{\beta}$ remains constant
throughout the motion. Focusing on the equatorial slice with $\theta=\pi/2$,
we find that all components $g_{\alpha\beta}$ in Eq. (31) become independent
of $t$, $\theta$, and $\phi$. As a result, we can identify the corresponding
Killing vector fields $\delta^{\mu}\alpha\partial\nu$ with $\alpha$ as a
cyclic coordinate. By denoting the constants of motion as $p_{t}$ and
$p_{\phi}$, we can analyze the system’s dynamics using the following
expressions: $p_{t}=-E$, and $p_{\phi}=L$, where $E$ and $L$ represent the
energy and angular momentum of the photon, respectively. With this in mind, we
can express the geodesic equation as:

$\displaystyle p_{t}=\dot{t}=g^{t\nu}p_{\nu}=E\mathcal{A}^{-1}(r),$ (32)
$\displaystyle p_{\phi}=\dot{\phi}=g^{\phi\nu}p_{\nu}=L\mathcal{C}^{-1}(r).$
(33)

Here, the overdot notation indicates differentiation with respect to the
affine parameter $\lambda$. Continuing from the previous analysis, we can
derive the radial null geodesic as:

$\dot{r}^{2}=E^{2}\mathcal{A}^{-1}(r)\mathcal{B}^{-1}(r)-L^{2}\mathcal{C}^{-1}(r)\mathcal{B}^{-1}(r).$
(34)

To provide a more precise formulation, let’s express the equation for the
photon trajectory in terms of the impact parameter $\eta=L/E$. The equation
becomes:

$\left[\frac{dr}{d\phi}\right]^{2}=\eta^{-2}\mathcal{C}(r)^{2}\mathcal{A}^{-1}(r)\mathcal{B}^{-1}(r)-\mathcal{C}(r)\mathcal{B}^{-1}(r)$
(35)

Now, we consider a photon source with a radius $r_{s}$ that affects the
underlying geometry. To determine the deflection angle of the photons, we need
to find the conditions under which the photons can reach the surface. This
occurs when a solution $r_{0}$ satisfies the conditions $r_{0}>r_{s}$ and
$\dot{r}^{2}=0$, where $r_{0}$ represents the distance of the closest approach
or the turning point. We can express the impact parameter as:

$\eta=\frac{L}{E}=\pm\sqrt{\mathcal{C}(r_{0})\mathcal{A}^{-1}(r_{0})}.$ (36)

In the weak gravity regime, we have $\eta\approx\sqrt{\mathcal{C}(r_{0})}$.
Thus, if a photon originates from the polar coordinate limit, defined as
$\lim\limits_{r\to\infty}\left(r,-\frac{\pi}{2}-\frac{\alpha}{2}\right)$,
passes through the turning point located at $(r_{0},0)$, and approaches
$\lim\limits_{r\to\infty}\left(r,\frac{\pi}{2}+\frac{\alpha}{2}\right)$, we
can define the deflection angle around wormhole throat as $\alpha$. The
deflection angle, denoted by $\alpha$, depends on $r_{0}$ Bhattacharya:2010zzb
. It can be explicitly derived from Eq. (36) as:

$\alpha(r_{0})=-\pi+2\int_{r_{0}}^{\infty}\frac{\mathcal{B}^{1/2}(r)\mathcal{C}^{{}^{-1/2}}(r)}{\left[\left(\mathcal{A}(r_{0})\mathcal{A}^{-1}(r)\right)\left(\mathcal{C}(r)\mathcal{C}^{-1}(r_{0})\right)-1\right]^{1/2}}dr.$
(37)

Figure 10: Deflection angle of photons in both scenarios. For drawing the
profiles the values of the parameters used here are the same as Figs. 1-2.

By employing the specified metric coefficients within the wormhole geometry,
it becomes straightforward to compute the deflection angle around the wormhole
throat in dRGT massive gravity. This can be achieved through numerical
integration of the aforementioned formulas (37) while considering the shape
functions detailed in Eqs. (14) and (19). The plots shown in Fig. 10
illustrate the phenomenon at hand. A negative deflection angle indicates the
presence of repulsive gravity. The concept of the deflection angle around
wormhole throat is introduced to validate the phenomenon of repulsive gravity.
It is noteworthy that a negative deflection angle signifies the presence of
repulsive gravity. Remarkably, the deflection angle consistently maintains
negative values across all values of $r_{0}$ in both solutions. This
consistent negativity serves as a manifestation of the repulsive gravity
effect.

##  V Discussions

In this work, we have provided the wormhole solution in the dRGT massive
gravity, an extension of Einstein gravity. The effects of pressure anisotropy
on the properties of the wormhole in the presence of space non-commutativity
have been investigated. The unique feature of this noncommutative geometry is
that it substitutes the point-like structure of a gravitational source by the
smeared distribution of the energy density. To prevent the appearance of event
horizons, we simplify our calculations by assuming a constant redshift
function, i.e., $\phi^{\prime}=0$. We solved the field equations in this
modified gravity, taking into account two distinct matter densities, and
derived the pressure profiles and shape functions. By carefully selecting the
constant terms, we have also drawn the wormhole shape function, which is
monotonically increasing. Interestingly, as $m^{2}c_{1}$ increases, the throat
radius for both models decreases. Flare-out conditions have also been met for
the various choices of $m^{2}c_{1}$ shown in the figure, which is another
crucial criterion for wormhole geometry. Since the asymptotic flatness is
lost, i.e., $b(r)/r$ does not tend to zero as $r$ tends to $\infty$, the shape
function violates the Morris-Thorne characteristics. With the assistance of
the graphical illustration, the NEC and all other energy criteria have been
checked, and it is observed that for both models, the NEC is violated and it
is necessary to keep the wormhole throat open. This is because of the exotic
anisotropic dark energy nature of coupled curvature and massive gravitons. The
shape function provides the embedding function of the wormhole and, when
rotated along the z-axis, provides the complete visualization of the wormhole
structure, which has been studied in detail. The Tolman-Oppenheimer-Volkov
(TOV) equation is also examined for the study of the equilibrium conditions in
both cases. It is seen that anisotropic and hydrostatic forces interact to
keep the wormhole model stable even though the gravitational force is zero
because of the constant redshift function. Our investigation extended beyond
the TOV equation to explore the repulsive nature of gravity. We discovered
that the presence of repulsive gravity is associated with a negative
deflection angle. To validate this phenomenon, we introduced the concept of
the photon deflection angle on the wormhole. Notably, we observed that the
deflection angle consistently maintains negative values across all values of
$r_{0}$ in the two solutions. This consistent negativity can be interpreted as
a manifestation of the repulsive gravity effect. In this respect, we would
like to draw attention to some previous research on wormhole geometries where
non-commutative geometry is involved. In Rahaman:2014dpa , a novel wormhole
solution influenced by noncommutative geometry was examined. It included the
requirement to permit conformal Killing vectors and to assume a Lorentzian
distribution of particle-like gravitational source. In $f(T,\,T_{G})$ gravity,
the noncommutative wormhole solution was obtained in Mustafa:2019eet .
Chalavadi et al. Chalavadi:2023zcw  obtained the wormhole solution in $f(Q,T)$
gravity by choosing a linear form of $f(Q,T)=\alpha Q+\beta T$, whereas
Debnath et al. Debnath:2023yfm  choose $f(R)$ gravity, Rahaman et al.
Rahman:2022pug  used Einstein-Gauss-Bonnet gravity, and Feng Feng:2022zlc
used exponential gravity to obtain the model of the wormhole in the context of
noncommutative geometry. All of them described the effect of the coupling
parameter of the modified gravity in the spacetime of the wormhole but we are
the first to describe the effect of the graviton mass on the wormhole
spacetime in the presence of noncommutative geometry. Finally, our analysis
shows that there exist traversable wormhole solutions having physically
intriguing features with zero tidal force that have both a Gaussian and
Lorentzian distribution for the matter density in the realm of dRGT massive
gravity.

## acknowledgments

PB is thankful to IUCAA, Government of India for providing visiting
associateship. AE thanks the National Research Foundation of South Africa for
the award of a postdoctoral fellowship.

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