#### Existence of stable wormholes on a non-commutative-geometric background in modified gravity

Eur. Phys. J. C
Existence of stable wormholes on a non-commutative-geometric background in modified gravity
M. Zubair ; 2
G. Mustafa 2
Saira Waheed 1
G. Abbas 0
0 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur , Pakistan
1 Prince Mohammad Bin Fahd University , Al Khobar 31952, Kingdom of Saudi Arabia
2 Department of Mathematics, COMSATS, Institute of Information Technology , Lahore , Pakistan
In this paper, we discuss spherically symmetric wormhole solutions in f (R, T ) modified theory of gravity by introducing well-known non-commutative geometry in terms of Gaussian and Lorentzian distributions of string theory. For analytic discussion, we consider an interesting model of f (R, T ) gravity defined by f (R, T ) = f1(R) + λT . By taking two different choices for the function f1(R), that is, f1(R) = R and f1(R) = R + α R2 + γ Rn, we discuss the possible existence of wormhole solutions. In the presence of non-commutative Gaussian and Lorentzian distributions, we get exact and numerical solutions for both these models. By taking appropriate values of the free parameters, we discuss different properties of these wormhole models analytically and graphically. Further, using an equilibrium condition, it is found that these solutions are stable. Also, we discuss the phenomenon of gravitational lensing for the exact wormhole model and it is found that the deflection angle diverges at the wormhole throat.
1 Introduction
In modern cosmology, the phenomenon of accelerated
cosmic expansion and its possible causes, as confirmed by
numerous astronomical probes, have become a focus of
interest for the researchers [
1–3
]. In this respect, the first
attempt was made by Einstein, by introducing a well-known
C D M model; but in spite of its all beauty and success,
this model cannot be proved to be problem free [
4
]. Later
on, a bulk of different proposals have been presented by
the researchers that can be grouped into two kinds:
modified matter proposals and modified curvature proposals. The
tachyon model, quintessence, the Chaplygin gas and its
different versions, phantom, quintom etc., are all obtained by
introducing some extra terms in the matter section and hence
are members of the modified matter proposal group [
5–15
].
The other idea is to modify curvature sector of Einstein’s
general relativity (GR) by including some extra degrees
of freedom there. One of the primary alterations was the
speculation of the Einstein–Hilbert Lagrangian density with
an arbitrary function f (R) instead of the Ricci scalar R.
This theory has been widely used in the literature [16] to
examine the dark energy (DE) and its resulting speedy
cosmic expansion. Moreover, f (R) theory of gravitation
provides a unified picture of early stages of cosmos
(inflation) as well as the late stages of accelerated cosmos. Some
other well-known examples include Brans–Dicke gravity,
generalized scalar–tensor theory, f (τ ) gravity, where τ is
a torsion, Gauss–Bonnet gravity and its generalized forms
like f (G) gravity, f (R, G) gravity, and f (τ, τG ) theory,
etc. [17–26].
Another significant modification of Einstein gravity
namely f (R, T ) gravity was proposed by Harko et al. [27]
almost five years ago. In this formulation, a generic function
f (R, T ), representing the coupling of the Ricci scalar and
energy-momentum tensor trace, replaces the Ricci scalar R
for the possible modification of curvature sector. Using
metric formalism, they derived the associated field equations for
some specific cases. In [28–32], some interesting
cosmological f (R, T ) models have been developed by employing
various scenarios namely, auxiliary scalar field, dark energy
models and anisotropic universe models. In literature [
33–
47
], different cosmological applications of f (R, T ) gravity
have been discussed like energy conditions,
thermodynamics, exact and numerical solutions of field equations with
different matter content, phase space perturbation, compact
stars and stability of collapsing objects etc.
The existence and construction of the wormhole
solutions is one of the most fascinating topics in modern
cosmology. Wormholes are topological passage-like structures
connecting two distant parts of the same universe or
different universes together through a shortcut called tunnel
or bridge. Generally, in nature, wormholes are categorized
into two sorts namely static wormholes and dynamic
wormholes [
48
]. For the development of wormhole structures, an
exotic fluid (hypothetical form of matter) is required which
violates the null energy condition (NEC) in GR. This
violation of energy condition is regarded as one of the basic
requirements for wormhole construction. The existence of
the wormhole solutions in GR has always been a great
challenge for the researchers. Although GR allows the existence
of wormholes but it is necessary to first modify the matter
sector by including some extra terms (as the ordinary
matter satisfies the energy bounds and hence violates the basic
criteria for existence of the wormhole). These extra terms
are responsible for energy bound violation and hence
permits the existence of wormhole in GR. In 1935, Einstein
and Rosen [
49
] discussed the mathematical criteria of
wormholes in GR and they obtained the wormhole solutions known
as Lorentzian wormholes or Schwarzchild wormholes. In
1988, it was shown [
50
] that wormholes could be large
enough for humanoid travelers and even permit time travel. In
the literature [
51–59
] numerous authors constructed
wormholes by including different types of exotic matter like
quintom, scalar field models, non-commutative geometry and
electromagnetic field etc. and obtained different interesting
and physically viable results. Some important and
interesting results regarding the stable wormhole solutions without
inclusion of any exotic matter are discussed in [
60,61
]. In
Ref. [62], the existence of the wormhole solutions and its
different properties in f (R, T ) theory gravity has been
discussed.
“On a D-brane, the coordinates may be treated as
noncommutative operators”, this is one of the most
interesting aspect of non-commutative geometry of string
theory, providing a mathematical way to explore some
important concepts of quantum gravity [
63,64
]. Basically,
noncommutative geometry is an effort to construct a unified
platform where one can take the spacetime gravitational
forces as a combined form of weak and strong forces
with gravity. Non-commutativity has the important
feature of replacing point-like structures by smeared objects
and hence corresponds to spacetime discretization, which
is due to the commutator defined by [x α, x β ] = i θ αβ ,
where θ αβ is an anti-symmetric second-order matrix. This
smearing effect can be modeled by including a
Gaussian distribution and a Lorentzian distribution of minimal
length √θ instead of the Dirac delta function. The
spherically symmetric, static particle-like gravitational source
representing the Gaussian distribution of non-commutative
geometry with total mass M has an energy density given
by [
65–67
]
Here the total mass M can be considered as wormhole, a
type of diffused centralized object and clearly, θ is the
noncommutative parameter. The Gaussian distribution source
has been utilized by Sushkov to model phantom-energy
upheld wormholes [68]. Also, Nicolini and Spalluci [
69
] used
this distribution to demonstrate the physical impacts of
shortseparation changes of non-commutative coordinates in the
investigation of black holes.
Motivated by this literature, in this manuscript, we will
construct spherically symmetric static wormholes in the
presence of curvature–matter coupling with non-commutative
geometry. In the next section, we will describe the basic
mathematical formulation of f (R, T ) gravity and the
corresponding field equations for static spherically
symmetric spacetime. In Sect. 3, we shall discuss the wormhole
solutions for both Gaussian and Lorentzian distributions of
non-commutative geometry by taking the linear model of
f (R, T ) gravity, i.e., f (R, T ) = R + λT . Section 4
provides wormhole solutions for both these distributions of
non-commutative geometry where the model f (R, T ) =
R + α R2 + γ Rn + λT will be taken into account. In Sect. 5,
the stability of these obtained wormhole solutions will be
discussed through graphs. Section 6 will be devoted to an
investigation of the gravitational lensing phenomenon for the
exact model of Sect. 3 by exploring deflection angle at the
wormhole throat. The last section will summarize the whole
discussion by highlighting the major achievements.
2 Field equations of f ( R, T ) gravity and spherically
symmetric wormhole geometry
In this section, we shall discuss the basic formulation of
f (R, T ) gravity and its corresponding field equations for
spherically symmetric spacetime in the presence of ordinary
matter. For this purpose, we take the following action of this
modified gravity [27]:
S =
Here Lm represents the Lagrangian density of ordinary
matter. By taking the variation of the above action, we have
8π Tμν − fT (R, T )Tμν − fT (R, T ) μν = f R (R, T )Rμν
1
− 2 f (R, T )gμν + (gμν − ∇μ∇ν ) f R (R, T ).
By contracting the above equation, we have a relation
between the Ricci scalar R and the trace T of the
energymomentum tensor as follows:
8π T − fT (R, T )T − fT (R, T )
+ 3 f R (R, T ) − 2 f (R, T ).
= f R (R, T )R
(2)
(3)
These two equations involves covariant derivative and
d’Alembert operator denoted by ∇ and , respectively.
Furthermore, f R (R, T ) and fT (R, T ) correspond to the function
derivatives with respect to R and T , respectively. Also, the
term μν is defined by
μν = gαδβgδμTνμν = −2Tμν + gμν Lm − 2gαβ
∂2 Lm
∂gμν ∂gαβ .
The energy-momentum tensor for an anisotropic fluid is
given by
Tμν = (ρ + pt )VμVν − pt gμν + ( pr − pt )χμχν ,
where Vμ is the 4-velocity vector of the fluid given by V μ =
e−a δ0μ and χ μ = e−bδμ, which satisfy the relations V μVμ =
1
−χ μχμ = 1. Here we choose Lm = ρ, which leads to the
following expression for μν :
μν = −2Tμν − ρgμν .
We relate the trace equation (3) with Eq. (2), then the Einstein
field equations take the form given by
f R (R, T )Gμν = (8π + fT (R, T ))Tμν + [∇μ∇ν f R (R, T )
1
− 4 gμν {(8π + fT (R, T ))T
The spherically symmetric wormhole geometry is defined
by the spacetime:
ds2 = −e2 (r)dt 2 +
dr 2
1 − b(r )/r
where (r ) and b(r ) both are functions of radial coordinate
r and represent redshift and shape functions, respectively
[
50,70
]. In the subsequent discussion, we shall assume the
red shift function to be constant, i.e., (r ) = 0. Here the
radial coordinate r is non-monotonic as it decreases from
infinity to a minimum value r0, representing the location of
wormhole throat, i.e., b(r0) = r0, then it increases back from
r0 to infinity. The most important condition for existence
+ r 2(dθ 2 + sin2θ d 2), (5)
of the wormhole is the flaring out property where the shape
function satisfies the inequality: (b−b r )/b2 > 0, while at the
wormhole throat, it satisfies b(r0) = r0. Further, the property
b (r0) < 1, is also a necessary condition to be satisfied for the
wormhole solutions. Basically these conditions lead to NEC
violation in classical GR. Furthermore, another condition that
needs to be satisfied for wormhole solutions is 1 − b(r )/r >
0. All these conditions collectively provide basic criteria for
the existence of a physically realistic wormhole model.
In order to find the relations for ρ , pr and pt , we substitute
the corresponding quantities for the metric (5) in Eq. (4) and
then by rearranging the resulting equations, we have
(6)
(7)
(8)
(9)
(10)
(11)
b
r 2 =
b
− r 3 =
(8π + fT (R, T ))
f R (R, T )
(8π + fT (R, T ))
f R (R, T )
ρ +
pr +
H
,
f R (R, T )
1
f R (R, T )
×
f R (R, T ) − f R (R, T )
H
− f R (R, T )
,
Since the above system, involving higher-order derivatives
with many unknowns, is very complicated to solve for the
quantities ρ , pr and pt therefore, for the sake of simplicity
in calculations, we assume a particular form of the function
f (R, T ) given by the relation f (R, T ) = f1(R) + f2(T ) with
f2(T ) = λT , where λ is a coupling parameter. After inserting
this form of f (R, T ) and then by simplifying the
corresponding Eqs. (6)–(8), we get
ρ =
b f R
r 2(8π + λ)
f R
2r 2(8π + λ)
(b r − b)
3 Wormhole solutions: Gaussian and Lorentzian
distributions for f1( R) = R model
In this section, we shall consider a specific and interesting
f (R) model [
71,72
] that is given by the linear function of
the Ricci scalar:
Using this relation in Eqs. (12)–(14) and after doing some
simplifications, we get the following set of field equations:
f1(R) = R.
b
ρ = ,
r 2(8π + λ)
b
pr = − r 3(8π + λ) ,
pt =
(b − b r )
2r 3(8π + λ)
.
Here we include the smearing effect mathematically by
substituting Gaussian distribution of insignificant width √θ
in place of the Dirac delta function, where θ is a
noncommutative parameter of Gaussian distribution. Here we
consider the mass density of a static, spherically symmetric,
smeared, particle-like gravitational source given by
The particle mass M , rather than being splendidly restricted
at the point, diffused on a region of the direct estimate √θ .
This is because of the fact that the uncertainty is encoded in
the coordinate commutator.
Comparing Eqs. (16) and (19), and then solving the
resulting differential equation, we get the shape function b(r ) in
terms of error function as follows:
r2 3 1
b(r ) = m0 −2r θ e− 4θ + 2θ 2 π 2 er f
r
√
2 θ
+ C1 , (20)
Here C1 is a constant of integration. Also, λ = −8π which
clearly leads to b(r ) = 0. Using Eq. (20) in (16)–(18), we get
the following relations for the ordinary energy density,
tangential and radial pressures, which will be helpful to discuss
the energy bounds:
2
M e− r4θ
ρ = 8π 3/2θ 3/2 ,
C1 +
pr = −
pt =
2√π
4λπ+8Cπ1 + er f
,
,
In the case of non-commutative geometry with reference
to the Lorentzian distribution, we take the density function
as follows:
where M is a mass which is diffused centralized object such
as a wormhole and θ is a non-commutative parameter.
Comparing (16) and (24) and then solving the resulting differential
equation, we get the following form of the shape function:
√rθ − √θr
2π 2(θ + r 2)
+ C2,
(25)
where C2 is an integration constant. Again using Eq. (25) in
(16) and (18), we get a new set of equations which help us to
discuss the energy conditions for the existence of wormhole
structure. In this case, the expressions for energy density,
radial and tangential pressures are given by
√θ M
ρ = π 2(θ + r 2)2 , (26)
pr = −
pt = −
C2 +
√θ M
2π 2 θ + r 2 2
(λ+8π)M θ+r2 tan−1 √rθ −√θr
2π2(θ+r2)
(λ + 8π )r 3
+
C2 +
(λ+8π)M θ+r2 tan−1 √rθ −√θr
2π2(θ+r2)
2(λ + 8π )r 3
(22)
. (23)
(24)
(27)
(28)
M (8π + λ)
3 3
8π 2 θ 2
where
m0 =
and
2
er f (θ ) = √
Now we will present the graphical illustration of the
obtained shape functions as well as the conditions that need
to be fulfilled for existence of the wormhole. For this purpose,
we take different suitable choices for the involved free
parameters. Firstly, we check the behavior of the shape function b(r )
for the Gaussian distribution where the red shift function has
been taken as a constant. The left graph of Fig. 1 indicates
the positive increasing behavior of the shape function and its
right graph corresponds to the behavior of the shape function
ratio to radial coordinate, i.e., b(rr ) which shows that as the
radial coordinate gets larger values, the ratio b(rr ) approaches
zero, and hence confirms the asymptotic behavior of the
shape function. The left part of Fig. 2 indicates the behavior
of b(r ) − r , which shows that the wormhole throat for this
model is located at r0 = 0.2 where b(r0) = r0. In the right
part of this figure, we check the flaring out condition for this
model by plotting b (r ). It shows that at the wormhole throat
r0 = 0.2, clearly the condition b (r0) < 1 is satisfied. The
graphical behavior of density function as well as the null
energy conditions ρ + pr and ρ + pt are shown in Figs. 3
and 4, respectively. It is clear from these graphs that the
energy density function and the function ρ + pt indicate
the positive but decreasing behavior versus radial coordinate
while ρ + pr shows negative and increasing behavior and
hence violates the NEC. Thus it can be concluded that the
obtained wormhole solutions are acceptable in this modified
gravity.
Fig. 5 This indicates the behavior of b(r ) and b(rr) versus r for the Lorentzian distribution. Here, we choose the free parameters as θ = 0.9, M =
0.0001, C2 = 0.1 and λ = 2
In the case of the wormhole solution with a Lorentzian
distribution, the graphical behavior of the shape function as
well as its corresponding properties is given in Figs. 5, 6
and 7. The left curve of Fig. 5 corresponds to the
behavior of the shape function while the right graph shows the
behavior of b(rr ) . It is clear from the curves that the shape
function is positive and increasing satisfying the asymptotic
flatness condition as r → 0. Figure 6 indicates the location
of wormhole throat and the flaring out condition. It is seen
that the wormhole throat is located at r0 = 0.1 where the
function b(r ) − r crosses the radial coordinate axis. Also, at
this wormhole throat, the flaring out condition b (r0) < 1 is
satisfied for this case as shown in the right part of Fig. 6. The
behavior of the energy density profile and the functions ρ + pr
and ρ + pt is presented in Figs. 7 and 8. These show that the
energy density remains positive and increasing with
increasing values of r . Similarly, the function ρ + pt indicates the
Fig. 8 This shows ρ + pr versus r for the Lorentzian distribution.
Here, we fix θ = 0.9, M = 0.0001, C2 = 0.1 and λ = 2
positive but decreasing behavior whereas the function ρ + pr
shows the negative increasing behavior versus r . This
confirms the violation of NEC in this case and hence allows the
existence of the wormhole. Thus in both cases, all necessary
and important characteristics of the shape function for the
existence of the wormhole are satisfied and thus it can be
concluded that the obtained solutions are physically viable.
4 Wormhole solutions: Gaussian and Lorentzian
distributions for f1( R) = R + α R2 + γ Rn model
In this segment, we will consider another specific f1(R)
model [
73,74
] which is given by the relation
f1(R) = R + α R2 + γ Rn,
(29)
where α and γ are arbitrary constants while n ≥ 3. Using
the model (29) in Eqs. (12)–(14), we get the following set
of equations for energy density, and the radial and tangential
pressures:
ρ =
b (r ) γ 2n−1n br(2r) n−1 + 4αrb2(r) + 1
r 2(8π + λ)
1
pr = 4r 5(8π + λ)(b (r ))3 (−8r αb (r )3(−8r b (r )
+ b (r ) (12 + 2b (r ) − r b (r )) + 2r 2b(3)(r ))
,
(30)
− 2n(−1 + n)nr 5γ
b (r ) n
r 2
2(b (r ))3 + 2
× (−2 + n)r 2(b (r )2 + (b (r ))2(−4 + 8n − r b (r ))
+ 2r b (r )(−4(−1 + n)b (r ) + r b(3)(r )))
+ b(r ) 2nnr 4γ
× b (r )2 + 2(−2 + n)(−1 + n)r 2b (r )2
− (−1 + n)r b (r )((−7 + 8n)b (r )
− 2r b(3)(r ))) + 4(b (r ))3(24αb (r )
+ r (−r − 18αb (r ) + 4r αb(3)(r )))
b (r ) n
r 2
(2n × (−5 + 4n)
1
pt = − 4r 5(8π + λ)(b (r ))2
2nnr 4γ
× (b (r )((−5 + 4n)b(r ) + r (4 − 4n + b (r )))
,
b (r ) n
r 2
(31)
+ 2(−1 + n)r (r − b(r ))b (r )) + 2(b (r ))2(r (b (r )
− (r 2 − 16α + 4αb (r )) + 8r αb (r )) − b(r )
× (−12αb (r ) + r (r + 8αb (r )))) .
(32)
A comparison of Eqs. (19) and (30) (Gaussian distribution)
yields the following non-linear differential equation:
b (r ) γ 2n−1n br(2r) n−1 + 4αrb2(r) + 1
whose analytic solution is also not possible, thus we
evaluate the possible form of the shape function by solving this
equation numerically.
Now we will discuss the behavior of the shape
functions that are obtained by numerical approach as well as
their corresponding important and necessary properties for
the existence of wormhole structure for both Gaussian and
Lorentzian distributions. For this purpose, we utilize a fixed
value n = 3 for the modified model (29) which results in the
cubic form given by f (R) = R + α R2 + γ R3. For the other
higher values, i.e., n > 3, it is observed that the resulting form
of the shape function is not physically viable. For graphical
illustration of the shape functions and their other properties,
we will take different feasible values of the free
parameters. The left part of Fig. 9 indicates that the shape function
remains positive and increasing for the Gaussian distribution
(obtained numerically), while its right part shows behavior
of the function b(rr) versus radial coordinate. Clearly, it
indicates that as the radial coordinate increases, the function
tends to zero and hence leads to the asymptotic behavior of
the shape function. In Fig. 10, the left curve corresponds to
the function b(r ) − r , which provides the location of
wormhole throat at r0 = 0.001 where it cuts the r -axis. Its right
curve provides information as regards the flaring out
condition, i.e., b (r0) < 1 which is clearly compatible at the
obtained wormhole throat. Furthermore, the graphical
illustration of energy density, tangential and radial pressures is
given in Figs. 11 and 12. The left part of Fig. 11 corresponds
to the energy density which shows a positive but decreasing
behavior, while the right curve shows the graph of ρ + pt ,
which is also positive decreasing. Figure 12 indicates the
behavior of ρ + pr , which is clearly negative and increasing
versus r and hence violates the NEC. Thus all the
conditions are satisfied allowing the existence of physically viable
wormhole solution.
Similarly, for the Lorentzian distribution, the graphical
behavior of the shape function and its properties like
asymptotic behavior, wormhole throat and the flaring out condition
are shown in Figs. 13 and 14. It can easily be observed that the
obtained shape function is positive increasing and is
compatible with all conditions. Further, the graphs for the resulting
density profile, ρ + pt and ρ + pr , are given in Figs. 15 and
16, respectively, which confirm the violation of NEC for this
wormhole model. Thus it can be concluded that the obtained
wormhole solutions for this cubic polynomial f1( R) model
are physically interesting for both non-commutative
distributions.
5 Equilibrium condition
In this segment, we explore the stability of obtained solutions
using equilibrium conditions in the presence of Gaussian
and Lorentzian distributions of non-commutative geometry.
For this purpose, we take the Tolman–Oppenheimer–Volkov
equation [
59
], which is given by
d pr + σ
dr 2
2
r
(ρ + pr ) +
( pr − pt ) = 0,
(33)
where σ (r ) = 2 (r ). This equation determines the
equilibrium state of configuration by taking the gravitational,
hydrostatic as well as the anisotropic forces (arising due to
anisotropy of matter) into account. These forces are defined
by the following relations:
Fg f = −
σ (ρ + pr ) ,
2
d pr
Fh f = − dr ,
Fa f = 2
( pt − pr ) ,
r
and thus Eq. (33) takes the form given by
Fa f + Fg f + Fh f = 0.
Since we assumed the red shift function to be a constant so
that (r ) = 0, we have Fg f = 0 and hence the equilibrium
condition reduces to the following form:
Fa f + Fh f = 0.
We shall discuss the stability condition for both exact and
numerical solutions in the presence of both distributions of
non-commutative geometry. Firstly, we calculate Fa f and
Fh f for the Gaussian distribution as follows:
while, for a Lorentzian distribution, these are given by
6π 2C2(θ + r 2)2 − √θ (8π + λ)Mr (3θ + 5r 2) + 3M (8π + λ)(θ + r 2)2 tan−1
−6π 2C2(θ + r 2)2 + √θ (8π + λ)Mr (3θ + 5r 2) − 3M (8π + λ)(θ + r 2)2 tan−1
2π 2(8π + λ)r 4(θ + r 2)2
2π 2(8π + λ)r 4(θ + r 2)2
r
√θ ,
r
√θ .
The graphical behavior of these forces is given in Figs. 17
and 18. The left graph indicates the behavior of these forces
for the Gaussian distribution while the right graph
corresponds to Lorentzian distribution for simple f1( R) model.
It is clear from the graph that both these forces show the
same but opposite behavior and hence cancel each other’s
effect and thus leaving a stable wormhole configuration.
Similarly, we investigate the stability of numerical solutions for
modified cubic f ( R) model using both the Gaussian and the
Lorentzian distributions. The graphical behavior of resulting
forces is given by Fig. 18. Its left part corresponds to behavior
of these forces for the Gaussian distribution whereas the right
graph provides the behavior for the Lorentzian distribution
which clearly indicates that these forces are also balancing
each other’s effect and thus leading to a stable wormhole
structure.
In this section, we will explore the possible detection of
traversable wormhole through gravitational lensing
phenomena. For this purpose, we consider the static spherical
symmetric metric involving x = 2rM representing the radius in
Schwarzschild units and given by
ds2 = − A(x )dt 2 + B(x )dx 2 + C (x )(dθ 2 + sin2 θ dφ2). (34)
rˆ .
Here the closest path taken by the light ray is xˆ = 2M
Here we will consider the obtained exact form of the shape
function in the case of a simple linear f1( R) = R model (Sect.
3). The integration of this shape function from the wormhole
throat r0 to r is given as follows:
Here clearly the coupling constant satisfies λ = −8π .
Basically, we consider the form of static spherically
symmetric wormhole metric (5) where e2 (r) = ( br0 )m where b0
is an integration constant while m = 2(vφ )2, where vφ
indicates the rotational velocity. In [
75–78
], it is pointed out that
m = 0.000001, which is a very small value (nearly zero)
and hence leaving the red shift function as a constant (as we
assumed in previous sections). A comparison of these metrics
leads to the following relations:
A(x ) =
r
b0
m
,
B(x ) = 1 −
b(r ) −1
r
, C (x ) = r 2.
The deflection angle for the light ray is given by
2d
3 − 1
α(xˆ) = − ln
− 0.8056 + I (xˆ).
Here d represents the mouth of wormhole because of exterior
Schwarzschild line element, while the internal metric
contribution is provided by I (xˆ), which implies that the closest
path taken by the ray of light is bigger than the mouth of the
wormhole. This is defined by the relation
2√B(x )dx
C (x )
CC((xxˆ))AA((xxˆ)) − 1
.
I (xˆ) =
G(x ) =
x
ˆ
∞
d
In our case, this integral take the following form:
I (xˆ) =
G(x )dx
x
ˆ
representing the closest approach for the light ray to be inside
the mouth of the wormhole. Here the function G(x ) is given
by
(35)
2
(36)
(37)
(38)
(39)
x y
arctan √ˆ θ
x y√θ
− (xˆ2 y2+θ) − arctan xˆ y0
ˆ √θ
+ (xˆxˆ2yy002√+θθ)
+ xˆ y0
y4−m − y2
In order to investigate the convergence/divergence of this
integral, we can redefine the variable as y = xx for the sake
of simplicity in calculations. Thus the integralˆtakes the
following form:
.
(41)
In the integrand of the above integral, we can assume that
H (y) = f (y)(y4−m − y2), where
1
f (y) = 1 − x y
ˆ
− arctan
λ + 8π
4π 2
xˆ y0
√θ
+
arctan
xˆ y0√θ
(xˆ2 y02 + θ )
x y
ˆ
√θ
√
x y θ
ˆ
− (xˆ2 y2 + θ )
+ xˆ y0.
Taylor’s series can be used to expand the function H (y)
around y = 1 as follows:
H (y) = s(2 − m) f (1)(y − 1) +
(5 − m)(2 − m) f (1)
Here we truncate the Taylor expansion up to second order
where O(y − 1)3 indicates the cubic and higher-order terms
of factor (y − 1). It can easily be observed that the integral
I (xˆ) converges or diverges because of the leading term in
the above expression. Integral can be convergent if the first
(y − 1)1/2 leads the expression where g(1) = 0. If g(1) = 0,
then second term will lead the expression and whose
integration will be ln(y − 1). Since y = 1, it turns out be
undefined there and hence the integral diverges. If we choose
the nearest approach of light ray as the wormhole throat,
i.e., rˆ = r0, then consequently, we have y0 = xx0 and thus
y0 = 1. Using these values in f (y), it can easily ˆbe verified
that f (1) = 0. Hence a photon sphere with radius rˆ (closest
path taken by the light ray) equal to radius of the throat r0, can
be found.
(42)
(43)
2
x 2 1 − x1 λ4+π82π
arctan √xθ
− (xx2√+θθ) − arctan √x0θ
+ (xx002√+θθ)
+ x0
x2−m
xˆ2−m − 1
(40)
7 Conclusions
The existence and construction of the wormhole solutions
in GR with some exotic matter has always been of great
interest for the researchers. The presence of exotic matter is
one of the most important requirement for wormhole
construction as it leads to NEC violation and hence permits the
existence of the wormhole. In the case of modified
theories, construction of wormholes has become more
fascinating topic as these include the effective energy-momentum
tensor that violates NEC without inclusion of any exotic
matter separately. In the present paper, we have constructed
spherically symmetric wormhole solutions in the presence
of two interesting Gaussian and Lorentzian distributions of
non-commutative geometry in f ( R, T ) modified gravity. For
this purpose, in order to make system of equations closed,
we assumed the function f ( R, T ) = f1( R) + λT with two
different forms of f1( R), i.e., the linear form f1( R) = R and
f1( R) = R + α R2 + γ Rn , n ≥ 3.
Firstly, we talked about the possible wormhole
construction for the linear f1( R) model with both Gaussian and
Lorentzian distributions. For Lorentzian and Gaussian
distribution, we found the exact solution. In order to examine
the physical behavior of these obtained solutions, we plotted
b(r ) versus radial coordinate. It is observed that shape
functions show positive increasing behaviors for both these
noncommutative distributions. Further we found the location of
wormhole throats and analyzed some important
characteristics of the shape functions namely asymptotic behavior, the
flaring out condition and the violation of NEC using graphs.
This discussion has been given in Figs. 1, 2, 3, 4, 5, 6, 7
and 8. It is concluded from these graphs that the obtained
shape functions show asymptotic behavior, i.e., b(rr ) → 0 as
r → ∞. Also, for both cases, wormhole throats are located at
r0 = 0.2 and r0 = 0.1. Furthermore, the obtained shape
functions are compatible with the flaring out condition and NEC
as the function ρ + pr indicated negative behavior for both
distributions. Thus the obtained solutions are viable
permitting wormhole to exist in non-commutative f ( R, T ) gravity.
Secondly, we checked the existence of the wormhole for
model f1( R) = R + α R2 + γ Rn , n ≥ 3 by taking both
non-commutative distributions into account. In this case, we
obtained very complicated non-linear differential equations
for b(r ) whose analytic solutions are not possible, therefore
we solved them numerically. It is worthwhile to mention here
that we fixed n = 3 for numerical solutions and their
graphical behaviors as it is found that, for n > 3, the obtained
solutions are not physically interesting (not meeting the
necessary criteria for existence of the wormhole). In the left parts
of Figs. 9 and 13, it is shown that the numerical solutions for
b(r ) indicate increasing positive behavior. Other necessary
conditions like asymptotic behavior of the shape function,
flaring out condition as well as NEC have been given in Figs.
9, 10, 11, 12, 13, 14, 15 and 16. The wormhole throat for
solutions in both distributions are located at r0 = 0.001.
Also, ρ + pr shows negative behavior and hence NEC is
incompatible for this solution. Thus it is concluded that all
the conditions are satisfied for the chosen specific values of
free parameters and hence the obtained wormhole solutions
are viable. It is also interesting to mention here that, for a
different selection of free parameters θ , M, λ etc. (other than
the used values in the present paper), all the functions show
a similar graphical behavior as presented in the figures. Thus
all the necessary conditions for existence of the wormhole
will also be satisfied in these cases and hence the wormhole
solution still exist.
Further, we examined the stability of obtained solutions
using equilibrium condition given by Tolman–Oppenheimer–
Volkov equation. Here we explored the stability for both
models of f1( R) in the presence of Gaussian and Lorentzian
distributions. After evaluating the possible expressions of
anisotropic and hydrostatic forces for these cases, we
examined them graphically as shown in Figs. 17 and 18. It can
easily be observed from the graphs that these forces are almost
equal in magnitude but opposite in behavior, therefore
canceling each other’s effect and hence leaving a balanced final
wormhole configuration. Furthermore, we explored the
possible detection of photon sphere at the wormhole throat. For
this purpose, we followed the procedure given in Refs. [
75–
78
] and explored the convergence of deflection angle. It is
observed that, for the obtained exact solution for f1( R) = R,
the resulting integral diverges at wormhole throat and hence
it is concluded that a photon sphere, with radius r0 (closest
path taken by the light ray) equal to radius of the throat, can
be detected.
Acknowledgements M. Zubair thanks the Higher Education
Commission, Islamabad, Pakistan, for its financial support under the NRPU
project with Grant number 5329/Federal/NRPU/R&D/HEC/2016.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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