# Wormholes in $\kappa(R,T)$ gravity

Susmita Sarkar<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, India  Nayan Sarkar<EMAIL_ADDRESS>Department of Mathematics, Jadavpur University, Kolkata-700032, India  Farook
Rahaman<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata 700 032, West Bengal, India.  Y Aditya<EMAIL_ADDRESS>Department of Mathematics, ANITS(A), Visakhapatnam 531 162, A.P., India .

###### Abstract

The wormhole solution could be found by solving the Einstein field equations
with violating the null energy condition (NEC). We represent wormhole
solutions in $\kappa(R,T)$ gravity in two different ways. At first, we find
the shape function by considering a redshift function and linear equation of
state (EoS). The solution represents a wormhole for the real feasible matter.
In the second part, we consider four pairs of two redshift functions and two
shape functions and analyze the obtained solutions. Some of the models suggest
that for particular values of the parameters, the existence of wormholes are
supported by an arbitrarily small quantity of exotic matter.

##  I  Introduction

In general relativity, one of the noteworthy features is the possible
existence of hypothetical geometry consisting nontrivial topological
structure. This topological feature of space-time as the solution of
Einstein’s general relativity field equations called as wormholes. Wormhole
(WH) is a hypothetical path which connects two different space-times. In
recent times it becomes a very interesting research object in theoretical
Astrophysics as if traversable WH exits then time machine can be constructed
mv95 . Flamm fl16  has first introduced the concept of WH and discussed 2D
embedding diagram of Schwarzschild WH. Later a similar attempt has been made
by Einstein and Rosen en35  and proposed the non-traversable Schwarzschild WH
solutions. Also, this doesn’t allow a two-way transmission between two
different regions of the space-time that leads to the contraction of WH
throat.

The concept of Lorentzian traversable WH proposed by Morris and Thorne ms88
has attracted many researchers. In this case, the throat of WH is threaded by
a variety of exotic matter which causes repulsion against the collapse of WH
throat. Due to this condition of the traversable WH, it must satisfy the
flare-out condition to preserve the geometry. Also, because of the absence of
the event horizon in this traversable WH, the observers can traverse freely
from one place of the universe to another place. Morris and Thorne ms88  have
also found that a kind of exotic matter is responsible for traversable WH
which violates the null energy condition(NEC) and weak energy condition (WEC).
NEC is the weakest one whose violation gives rise to the violation of other
energy conditions. Thus, it is a fascinating challenge to search for a
realistic model that minimize the usage of exotic matter or satisfies the
energy conditions.

One of the most striking research areas over the last two decades has been the
scenario of the accelerated expansion of our universe. It is thought that the
reason behind this cosmic accelerated expansion of the universe is some exotic
fluid or unidentified energy with huge negative pressure dubbed as dark
energy(DE). Observational data from various experiments such as high redshift
supernovae, galaxy clustering and cosmic microwave background anisotropy
support this accelerated expansion of our universe ag98 ; sp99 ; ds03 ; mt04 .
According to the latest observations, our universe constitutes about 71.4% of
DE, 24% dark matter and 4.6% ordinary matter. After many efforts by
astronomers to find the nature and composition of DE but till we have just
known that it has some repulsive force or negative pressure. To describe the
DE, one can use the equation of state(EoS) parameter $\omega=p/\rho$, where
$p$ and $\rho$ denote the pressure and energy density of the DE. Vacuum energy
($\omega=-1$) is the most convenient explanation for the DE that can be able
to push matter apart pj03 , but it faces some theoretical problems like the
fine tuning and cosmic coincidence.

Modified theories of gravitation are the another approaches to address the DE
problem. These modified theories can be obtained by generalizing or modifying
the Einstein-Hilbert action in standard GR. This idea gives rise to various
alternative theories of gravitation like $f(\tau)$ gravity rf07 ; gr09 ; ev10
, Gauss-Bonnet gravity sm05 ; gc06 , $f(R)$ gravity sc03 ; sn03  and $f(R,T)$
gravity th11  etc (where $R$ is the curvature scalar, $\tau$ denotes the
torsion scalar, $T$ is the trace of the energy momentum tensor). For an
exhaustive study of modified gravities, the reader may see the reviews sc11 ;
sn11 ; sn17 . On the other hand, the models in these modified theories may
overcome the violation of energy conditions and lead to the WHs threaded by
ordinary matter. Basically, in modified theories, the effective stress-energy
tensor (extra terms or modified terms) is responsible for the violation of
energy conditions while ordinary matter satisfies these conditions. Harko et
al. th13  have presented theoretical WH geometries that can be obtained
without the presence of exotic matter but are validated in the context of
$f(R)$ modified gravity with higher order curvature terms. Rahaman et al. fr14
have derived some new exact solutions of static noncommutative WHs in $f(R)$
theory of gravity supported by the Lorentzian density distribution. Lobo and
Oliveira fs09  have constructed traversable WH geometries within the framework
of $f(R)$ theories of gravity. Mehdizadeh et al. mr15  have explored
traversable WHs in Einstein-Gauss-Bonnet which satisfies the WEC. Zubair et
al. mz16  have discussed static spherically symmetric WHs in $f(R,T)$ modified
theory of gravity with matter contents as isotropic, anisotropic and
barotropic fluids, and shown that the WH solutions can be obtained without
exotic matter in few regions of space-time. Moraes et al. ph17  have studied
general solutions for WHs with constant throat radius in $f(R,T)$ gravity.
Sharif and Ikram ms15  have studied the traversable WH solutions in the
framework of $f(G)$ gravity (where $G$ is the Gauss-Bonnet curvature
invariant) and found that the effective stress-energy tensor violates the NEC
throughout the WH throat. Sarkar et al. sa18  have discussed WH in a modified
theory of gravity in the background five dimensional Kaluza–Klein cosmology.
Samanta et al. sam18  have presented a comparative study of traversable WHs
with exponential shape function in $f(R)$ and $f(R,T)$ modified gravities and
in general relativity. Rosa et al. ro18  have explored WH solutions in a
generalized hybrid metric-Palatini matter theory. Elizalde and Khurshudyan
el18  have investigated WH formation in $f(R,T)$ gravity with radial pressure
which admits an equation of state parameter of varying Chaplygin gas. Zubair
et al. zu18  have studied static spherically symmetric WHs in generalized
$f(R,\phi)$ gravity. Jusuf et al. ju18  have discussed the deflection of light
by black holes and WHs in the context of massive theory of gravity. Godani and
Samanta go19  have obtained traversable WHs with two different shape functions
in $f(R)$ gravity.

The field equations obtained in the above modified theories are much more
complicated than those of GR and in the process of modification some of the
beauty of the original theory is also lost. In view of this, very recently,
Gines R. Perez Teruel gr18  has introduced a new modified theory named as
$\kappa(R,T)$ gravity. In the formulation of this theory he doesn’t begin with
the standard modified theory of gravity approach. Rather, he added the
possible source terms directly to GR field equations by following Maxwell’s
and Einstein’s original approaches. In particular, $\kappa(R,T)$ gravity is
based on a natural extension of GR, where the modified field equations are
obtained by adding the terms that only include the scalar curvature $R$ and
the trace $T$ of the stress-energy tensor. So far, no one has investigated WH
solutions in the context of $\kappa(R,T)$ theory of gravity, hence it would be
interesting to study WH solutions in the framework of $\kappa(R,T)$ modified
theory of gravity.

The innovation points of this work are: (i) For the choice of the redshift
function and linear equation of state(EoS), the obtained solution represents
the wormhole structure with a peculiar property that the original matter
distribution is of real feasible matter and it provides the fuel to sustain
the wormhole in $\kappa(R,T)$ gravity. (ii) The solutions corresponding to
four pairs of two shape functions and two redshift functions also represent
the wormholes and here some of the models suggest that for particular values
of the parameters, the existence of wormholes are supported by an arbitrarily
small quantity of exotic matter.

The context has been designed as follows: In Sec. II we have described
$\kappa(R,T)$ gravity and written down the Einstein field equations in this
gravity also mentioned the required conditions on the redshift and shape
function to present the wormhole structure. The shape function and the
corresponding solution are analyzed by redshift function and EoS taking into
the account in Sec.III. In Sec.IV we have shown the wormhole solutions by
considering different redshift and shape functions. Finally, the discussion
has been made in Sec.V.

Figure 1:  The shape function(Left) and $\frac{b(r)}{r}$(Right) are plotted
against the radial coordinate $r(>$ the throat radius) corresponding to
$m=0.3,C=1$.

##  II Einstein’s field equations in $\kappa(R,T)$ gravity

Here, we will present a brief introduction to $\kappa(R,T)$ gravity, the
corresponding field equations and required conditions on redshift and shape
functions.

Various modified theories of gravitations are proposed in view of the
important degree of arbitrariness inherent in the choice of the gravity
Lagrangian. Many of these theories are so similar that it is difficult to
differentiate one from the other. The Lagrangian formalism has undoubted
advantages at the level of symmetries implementation and conservation-laws
derivation, but possible theoretical alternatives to standard Lagrangian
theories also deserve consideration. In this point of view, the importance of
Non-Lagrangian theories in other branches of theoretical physics such as
quantum field theory is being acknowledged in the last years. Also, these
theories offer new opportunities in the search of new types of invariants.

Very recently, Teruel gr18  has formulated an example of a Non-Lagrangian
modified theory of gravitation, namely $\kappa(R,T)$ gravity, inspired by
Maxwell’s approach to Electrodynamics, adding new possible source terms
directly in the field equations. The field equations in $\kappa(R,T)$ modified
gravity are obtained by adding new possible source terms directly to GR field
equations asgr18

$R_{ij}-\frac{1}{2}Rg_{ij}-\Lambda g_{ij}=\kappa(R,T)T_{ij}$ (1)

where $g_{ij}$ is the metric potential, $R_{ij}$ is the Ricci tensor,
$\Lambda$ is a cosmological constant, $T_{ij}$ is the energy-momentum tensor
of the matter source, and $\kappa(R,T)$ corresponds to the Einstein
gravitational constant and it is proposed as a function of the traces
$T=g_{ij}T^{ij}$, and Ricci scalar $R=g_{ij}R^{ij}$. Clearly, the
gravitational constant $\kappa$ depends on the scalars, so we can explore the
possibility of a varying gravitational constant, i.e. generalization of the
original Einstein’s gravitational constant (not at the level of an action
functional). A varying gravitational constant in the action leads to a Brans-
Dicke scalar-tensor theory type cb61 ; cb05  with entirely different field
equations from Eq.(1). Since the left hand side of the field Eq.(1) is
divergence free, we have

$\displaystyle\nabla^{j}\left(\kappa(R,T)T_{ij}\right)$ $\displaystyle=$
$\displaystyle 0$ (2)

Then, these field equations imply the non-covariant conservation of $T_{ij}$
that can be expressed as

$\displaystyle\nabla^{j}T_{ij}$ $\displaystyle=$
$\displaystyle-\frac{\nabla^{j}\kappa(R,T)}{\kappa(R,T)}T_{ij}.$ (3)

Teruelgr18  has proposed and analyzed some cosmological implications of two
particular models. The first one is proposed by setting, $\kappa(T)=8\pi
G-\lambda T$, and corresponds to a matter-matter coupling. The second model is
characterized by a gravitational constant that varies as
$\kappa^{{}^{\prime}}(R)=8\pi G+\alpha R$, which will provide a coupling
between matter and curvature terms.

Let us consider a Morris-Thorne wormhole, which is represented by a static
spherically symmetric metric in $Schwarzschild$ co-ordinate ($t$, $r$,
$\theta$, $\phi$) as

$ds^{2}=e^{2f(r)}dt^{2}-\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (4)

Figure 2:  $\frac{db(r)}{dr}$(Left) and the original matter density(Right) are
are plotted against the radial coordinate $r(>$ the throat radius)
corresponding to $m=0.3,C=1$ and $m=0.3,C=1,a=0.02$, respectively.

where $f(r)$ and $b(r)$ are the functions of the radial coordinate $r$, called
redshift function and shape function, respectively. The minimum radius
$r=r_{0}$ in the metric coefficient $g_{rr}$ is termed as the throat radius of
the wormhole, where $b(r_{0})=r_{0}$ . Also, the absence of the event horizon
is required for a traversable wormhole.
Therefore, to represent a wormhole structure the redshift and shape functions
must satisfy the following conditions fr06  :

  * (i)

For avoiding an event horizon the redshift function $f(r)$ should be finite.

  * (ii)

The shape function $b(r)$ should satisfy the flare-out condition at the throat
$r=r_{0}$ i.e. $b^{\prime}(r_{0})<1$.

  * (iii)

$b(r)$ should be less than r for $r>r_{0}$.

  * (iv)

The shape function should be asymptotically flat i.e.
$\frac{b(r)}{r}\rightarrow$ 0 as r$\rightarrow\infty$

To know the nature of matter that may support the wormhole we need to solve
the Einstein field equations. The Einstein field equations in $\kappa(R,T)$
gravity for the line element(4) along with $\Lambda=0$ and
$\kappa(R,T)=8\pi-\lambda T$ $(G=c=1)$ are:

$\displaystyle 8\pi\rho(r)\left[1-a(\rho(r)-3p(r))\right]$ $\displaystyle=$
$\displaystyle\frac{b^{\prime}(r)}{r^{2}}$ (5) $\displaystyle 8\pi
p(r)\left[1-a(\rho(r)-3p(r))\right]$ $\displaystyle=$
$\displaystyle\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}$

where $"\prime"$ denotes the derivative with respect to the radial coordinate
$r$. $\rho(r)$, $p(r)$ are the matter density and pressure of the original
matter configuration, respectively and $a=\frac{\lambda}{8\pi}$ is non zero
constant.
The effective density and pressure of the effective matter configuration are
given by

$\displaystyle\rho_{e}(r)$ $\displaystyle=$
$\displaystyle\rho_{effective}(r)=8\pi\rho(r)\left[1-a(\rho(r)-3p(r))\right]$
(7) $\displaystyle p_{e}(r)$ $\displaystyle=$ $\displaystyle
p_{effective}(r)=8\pi p(r)\left[1-a(\rho(r)-3p(r))\right]$ (8)

One can easily note that there is two independent field
equations(5)-(LABEL:pe) of four unknowns $\rho(r)$, $p(r)$, $f(r)$ and $b(r)$
and hence it is difficult to find the exact expressions of $\rho(r)$, $p(r)$
in terms of the radial coordinate $r$ only. Therefore, in the next section, we
will proceed to solve this system of equations accurately by assuming a linear
equation of state(EoS) and a redshift function.

Figure 3: The sum of original matter density and pressure(Left) and The sum of
effective matter density and pressure(Right) are plotted against the radial
coordinate $r(>$ the throat radius) corresponding to $m=0.3,C=1,a=0.02$.

##  III Wormhole Solution

To obtain the exact solutions of Einstein’s field equation in the framework of
$\kappa(R,T)$ gravity we can proceed as follows :
From Eqs. (5)- (LABEL:pe) we get

$\frac{\rho(r)}{p(r)}=\frac{\frac{b^{\prime}(r)}{r^{2}}}{\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}}$
(9)

Now, we consider a redshift function $f(r)=A$(a constant), where $A$ being a
constant avoids the event horizon and linear equation of state(EoS)
$p(r)=m\rho(r)$, where m is a constant satisfying $0<m<1$ for the real matter
configuration. Later we will provide a restriction on m as $m\neq\frac{1}{3}$
for our solution. For these EoS and redshift function, Eq.(9) yields

$\displaystyle b^{\prime}(r)+\frac{1}{mr}b(r)$ $\displaystyle=$ $\displaystyle
0$ (10)

The differential Eq.(10) gives the following shape function

$b(r)=Cr^{-\frac{1}{m}}$ (11)

where C is an integration constant.

From the equation $b(r_{0})=r_{0}$ we get the throat radius of the wormhole as
$r_{0}=C^{\frac{m}{m+1}}$. For the choice of $C=1,$ the throat radius of the
wormhole is $r_{0}=1$. Also, we provide some graphical representations to get
an overview of the obtained shape function as well as the conditions that need
to satisfy for the existence of wormhole structure. Fig.1 shows the positive
and decreasing behavior of the shape function with respect to the radial
coordinate $r$(Left) and the right plot of Fig.1 presents the behavior of the
ratio function $\frac{b(r)}{r}$, which shows that $\frac{b(r)}{r}$ tends zero
for large values of the radial coordinate $r$ and hence the shape function
satisfies the asymptotic behavior. Also, the shape function satisfies the
flare out condition, (see Fig.2 (Left)). Consequently, our obtained shape
function is well-behaved to present a wormhole structure.

Therefor, we obtain a wormhole representing line element as:

$ds^{2}=e^{2A}dt^{2}-\left(1-Cr^{-\frac{m+1}{m}}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (12)

Now, we obtain the expression of original matter density from Eq. (5) using
Eq.(11) and the EoS as:

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2a(1-3m)}\left[1\pm\sqrt{1+\frac{aC(1-3m)}{2\pi
m}r^{-\left(\frac{3m+1}{m}\right)}}\right]$ (13)

with $m\neq\frac{1}{3}$.

Figure 4:  The shape function(Left) and $\frac{b(r)}{r}$(Right) are plotted
against the radial coordinate $r(\geq$ the throat radius) corresponding to
$\beta=r_{0}=1.5$ and $n=0.1,0.3,0.6,0.9$.

For further study, we consider the following expression of $\rho(r)$:

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2a(1-3m)}\left[1+\sqrt{1+\frac{aC(1-3m)}{2\pi
m}r^{-\left(\frac{3m+1}{m}\right)}}\right]$ (14)

Actually, we are interested to find the original matter as the real feasible
matter configuration. For this purpose, we have taken one expression of
$\rho(r)$ given in above Eq.(14). The original matter configuration becomes
the exotic matter for another expression of $\rho(r)$.
Therefore, we obtain the following expressions

$\rho(r)+p(r)=\frac{(1+m)}{2a(1-3m)}\left[1+\sqrt{1+\frac{aC(1-3m)}{2\pi
m}r^{-\left(\frac{3m+1}{m}\right)}}\right]$ (15)

and

$\displaystyle\rho_{e}(r)+p_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{8\pi(1+m)}{2a(1-3m)}\left[1+\sqrt{1+\frac{aC(1-3m)}{2\pi
m}r^{-\left(\frac{3m+1}{m}\right)}}\right]\left[1-\frac{1}{2}\left\\{1+\sqrt{1+\frac{aC(1-3m)}{2\pi
m}r^{-\left(\frac{3m+1}{m}\right)}}\right\\}\right]$

It is well-known that the exotic matter is necessary to hold a wormhole
structure and the exotic matter is characterized by the violation of null
energy condition(NEC). The energy conditions are defined as:

  * (i)

The weak energy condition(WEC): $T_{\alpha\beta}U^{\alpha}U^{\beta}\geq 0$
i.e. $\rho(r)\geq 0$ and $\rho(r)+p(r)\geq 0$, where $T_{\alpha\beta}$ is the
energy momentum tensor with $U^{\alpha}$ being a timelike vector.

  * (ii)

The null energy condition(NEC):
$T_{\alpha\beta}\upsilon^{\alpha}\upsilon^{\beta}\geq 0$ i.e.
$\rho(r)+p(r)\geq 0$ and for the modified matter configuration
$\rho_{e}(r)+p_{e}(r)\geq 0$, with $\upsilon^{\alpha}$ being a null vector.

The density $\rho(r)$ of the original matter configuration is positive against
the radial coordinate $r$ (see Fig.2 (Right)). The original matter
distribution satisfies the week energy condition(WEC) ($\rho(r)\geq
0,~{}\rho(r)+p(r)\geq 0$) as well as the null energy condition(NEC)
($\rho(r)+p(r)\geq 0$) (see Fig.2(Right) and Fig.3(Left)) and hence the
original matter distribution is of real feasible matter while the right plot
of Fig.3 shows that $\rho_{e}(r)+p_{e}(r)$ is negative i.e. the modified
matter violates the null energy condition(NEC). Thus, we obtain an interesting
result that a real feasible matter distribution will provide the fuel to
construct and sustain a wormhole in $\kappa(R,T)$ gravity.

Figure 5:  $\frac{db(r)}{dr}$(Left) and the original matter density(Right) are
plotted against the radial coordinate $r(\geq$ the throat radius)
corresponding to $\beta=r_{0}=1.5$, $n=0.1,0.3,0.6,0.9$ and $\beta=r_{0}=1.5$,
a = 0.02, $n=0.1,0.3,0.6,0.9$, respectively.

##  IV Wormhole solution with specific redshift and shape functions

In this section, we will examine the existence of the wormhole structures by
considering four pairs of two redshift functions and two shape functions into
the account. One of the most important factor to measure the amount of exotic
matter need to form a wormhole is ”volume integral quantifier”, which is
defined as:

$I_{v}=\int\\{\rho(r)+p(r)\\}dV$ (17)

where $dV$ is the volume element, given by $dV=r^{2}\sin\theta drd\theta$. One
can note that the volume integral quantifier $I_{v}~{}\rightarrow$ 0 if
$\rho(r)+p(r)\rightarrow 0$ and this reveals that an arbitrarily small amount
of exotic matter is responsible to construct a wormhole fr06 . Using this
property later we will show that an infinitesimal exotic matter is sufficient
to construct the wormholes.
Now, Eqs. (5) and (LABEL:pe) are the quadratic equations in $\rho(r)$ and
$p(r)$, respectively. So, these equations give two different expressions for
$\rho(r)$ and $p(r)$. After solving Eqs.(5)-(LABEL:pe) we obtain the general
expressions of $\rho(r)$ and $p(r)$ in term of the redshift and shape
functions as:

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{b^{\prime}(r)\left[2\pi r^{2}b(r)(1+2rf^{\prime}(r))-4\pi
r^{4}f^{\prime}(r)\pm\sqrt{2\pi
r(b(r)-2r^{2}f^{\prime}(r)+2b(r)rf^{\prime}(r))^{2}f_{1}}\right]}{4\pi
ar[r(b^{\prime}(r)-6rf^{\prime}(r))+b(r)(3+6rf^{\prime}(r))][b(r)-2r^{2}f^{\prime}(r)+2rbf^{\prime}(r)]}$
$\displaystyle p(r)$ $\displaystyle=$ $\displaystyle\frac{4\pi
r^{4}f^{\prime}(r)-2\pi r^{2}b(r)(1+2rf^{\prime}(r))\mp\sqrt{2\pi
r(b(r)-2r^{2}f^{\prime}(r)+2b(r)rf^{\prime}(r))^{2}f_{1}}}{4\pi
ar^{2}[r(b^{\prime}(r)-6rf^{\prime}(r))+b(r)(3+6rf^{\prime}(r))]}$ (19)

whereas

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle r(2\pi
r^{2}-ab^{\prime}(r)+6arf^{\prime}(r))-3b(a+2arf^{\prime}(r))$

Therefore, anyone can easily discuss the nature of the matter configuration by
finding the density $\rho(r)$ and pressure $p(r)$ from Eqs.(LABEL:rho1)-(21)
for the arbitrary choice of redshift and shape functions in $\kappa(R,T)$
gravity.
For further study we consider

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{b^{\prime}(r)\left[2\pi r^{2}b(r)(1+2rf^{\prime}(r))-4\pi
r^{4}f^{\prime}(r)-\sqrt{2\pi
r(b(r)-2r^{2}f^{\prime}(r)+2b(r)rf^{\prime}(r))^{2}f_{1}}\right]}{4\pi
ar[r(b^{\prime}(r)-6rf^{\prime}(r))+b(r)(3+6rf^{\prime}(r))][b(r)-2r^{2}f^{\prime}(r)+2rbf^{\prime}(r)]}$
$\displaystyle p(r)$ $\displaystyle=$ $\displaystyle\frac{4\pi
r^{4}f^{\prime}(r)-2\pi r^{2}b(r)(1+2rf^{\prime}(r))+\sqrt{2\pi
r(b(r)-2r^{2}f^{\prime}(r)+2b(r)rf^{\prime}(r))^{2}f_{1}}}{4\pi
ar^{2}[r(b^{\prime}(r)-6rf^{\prime}(r))+b(r)(3+6rf^{\prime}(r))]}$ (21)

because the other expressions are not well-behaved for our models.

Figure 6:  The sum of original matter density and pressure(Left) and the sum
of effective matter density and pressure(Right) are plotted against the radial
coordinate $r(\geq$ the throat radius) corresponding to $\beta=r_{0}=1.5$,
$a=0.02$, $n=0.1,0.3,0.6,0.9$ and $\beta=r_{0}=1.5$, $n=0.1,0.3,0.6,0.9$,
respectively.

###  IV.1  The shape function $b(r)=\beta\left(\frac{r}{\beta}\right)^{n}$

Here, we consider the shape function
$b(r)=\beta\left(\frac{r}{\beta}\right)^{n}$fr19  to analyze the solutions
with respect to two different redshift functions, separately. The throat
radius of the wormhole is $r_{0}=\beta$ where
$\beta\left(\frac{r_{0}}{\beta}\right)^{n}=r_{0}$. The shape function is
positive and increasing in nature, shown in Fig.4(Left). The Fig.4(Right)
ensures that $\frac{b(r)}{r}$ tends to zero for the large values of the radial
coordinate $r$ i.e. the shape function has asymptotic behavior. It is obvious
from Fig.5(Left) that $b^{\prime}(r\geq r_{0})<1$ for $n<1$ i.e. the shape
function satisfies the flare-out condition for $n<1$.
Next, we will discuss the solutions for the above shape function along with
two different redshift functions, separately.

Figure 7: The original matter density(Left) and The sum of original matter
density and pressure(Right) are plotted against the radial coordinate $r(\geq$
the throat radius) corresponding to $\beta=r_{0}=1.5$, $a=0.02$, $\alpha=0.01$
and $n=0.1,0.3,0.6,0.9$.

####  IV.1.1  The redshift function $f(r)=0$

The constant redshift function $f(r)$, known as the tidal force solution is
always admissible as the constant redshift function is finite and avoids the
event horizon.

For the tidal force solution and above shape function, Eqs.(LABEL:rho1)-(21)
yields the expressions of original matter density $\rho(r)$ and pressure
$p(r)$ as:

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle\frac{n}{4\pi
ar^{2}(3+n)}\left[2\pi r^{2}-\sqrt{2\pi r\left\\{2\pi
r^{3}-a(3+n)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}}\right]$ (22)
$\displaystyle p(r)$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
ar^{2}(3+n)}\left[2\pi r^{2}-\sqrt{2\pi r\left\\{2\pi
r^{3}-a(3+n)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}}\right]$ (23)

Therefore,

$\displaystyle\rho(r)+p(r)$ $\displaystyle=$ $\displaystyle\frac{n-1}{4\pi
ar^{2}(3+n)}\left[2\pi r^{2}-\sqrt{2\pi r\left\\{2\pi
r^{3}-a(3+n)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}}\right]$ (24)

Also, from Eqs.(7) and (8) we obtain

$\displaystyle\rho_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{n\beta\left(\frac{r}{\beta}\right)^{n}}{r^{3}}$ (25)
$\displaystyle p_{e}(r)$ $\displaystyle=$
$\displaystyle-\frac{\beta\left(\frac{r}{\beta}\right)^{n}}{r^{3}}$ (26)

and hence

$\displaystyle\rho_{e}(r)+p_{e}(r)$ $\displaystyle=$
$\displaystyle(n-1)\frac{\beta\left(\frac{r}{\beta}\right)^{n}}{r^{3}}$ (27)

To check the behavior of the obtained original matter density $\rho(r)$ and
the null energy condition(NEC) we draw the graphs for $\rho(r)$ and
$\rho(r)+p(r)$ and $\rho_{e}(r)+p_{e}(r)$ against the radial coordinate $r$ in
Figs.5(Right)-6. The right sketch of Fig.5 shows the positive behavior of the
original matter density with respect to the radial coordinate $r$ and Fig.6
indicates that $\rho(r)+p(r)<0$ and $\rho_{e}(r)+p_{e}(r)<0$ i.e. the original
matter distribution as well as the modified matter distribution in
$\kappa(R,T)$ gravity both violate the null energy condition(NEC) and hence
there exists a wormhole structure supported by the matter configuration in
$\kappa(R,T)$ gravity. Moreover, from Eq.(24) it is clear that $\rho(r)+p(r)$
tends to zero when $n$ tends to unity i.e. the volume integral quantifier
$I_{v}$ tends to zero when $n$ tends to unity. So, an infinitesimal amount of
exotic matter is sufficient to hold a wormhole.

Figure 8: The sum of effective matter density and pressure(Left) and the shape
function(Right) are plotted against the radial coordinate $r(\geq$ the throat
radius) corresponding to $\beta=r_{0}=1.5$, $\alpha=0.01$, $n=0.1,0.3,0.6,0.9$
and $\gamma=r_{0}=1.5$ and $\delta^{2}=0.1,0.3,0.6,0.9$, respectively.

####  IV.1.2  The redshift function $f(r)=\frac{\alpha}{r}$.

The shape function $f(r)=\frac{\alpha}{r}$ is always finite for $r\geq$
$r_{0}$, where $\alpha$ is constant and hence this redshift function avoids
the event horizon. Therefore, from Eqs.(LABEL:rho1)$-$(21) we get the
expressions for $\rho(r)$ and $p(r)$ as:

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle
n\beta\left(\frac{r}{\beta}\right)^{n}\left[2\pi
r^{3}\left\\{2\alpha+\beta\left(\frac{r}{\beta}\right)^{n}\right\\}-4\pi\alpha
r^{2}\beta\left(\frac{r}{\beta}\right)^{n}-f_{4}\right]\left[{4\pi
ar\left\\{2\alpha
r+(r-2\alpha)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}f_{5}}\right]^{-1}$
(28) $\displaystyle p(r)$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
ar^{2}f_{5}}\left[2\pi
r^{3}\left\\{2\alpha+\beta\left(\frac{r}{\beta}\right)^{n}\right\\}-4\pi\alpha
r^{2}\beta\left(\frac{r}{\beta}\right)^{n}-f_{4}\right]$ (29)

Therefore,

$\displaystyle\rho(r)+p(r)$ $\displaystyle=$ $\displaystyle\left[2\pi
r^{3}\left\\{2\alpha+\beta\left(\frac{r}{\beta}\right)^{n}\right\\}-4\pi\alpha
r^{2}\beta\left(\frac{r}{\beta}\right)^{n}-f_{4}\right]\left[rn\beta\left(\frac{r}{\beta}\right)^{n}-\left\\{2\alpha
r+(r-2\alpha)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}\right]$
$\displaystyle\left[4\pi ar^{2}\left\\{2\alpha
r+(r-2\alpha)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}f_{5}\right]^{-1}$
(30)

whereas

$\displaystyle f_{4}$ $\displaystyle=$ $\displaystyle\left\\{2\alpha
r+(r-2\alpha)\beta\left(\frac{r}{\beta}\right)^{n}\right\\}\sqrt{2\pi(2\pi
r^{4}-af_{5})}$ $\displaystyle f_{5}$ $\displaystyle=$ $\displaystyle
r\left[6\alpha+(3+n)\beta\left(\frac{r}{\beta}\right)^{n}\right]-6\alpha\beta\left(\frac{r}{\beta}\right)^{n}$
(32)

Also, from Eqs.(7)$-$(8) we get the following result

$\displaystyle\rho_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{n\beta\left(\frac{r}{\beta}\right)^{n}}{r^{3}}$ (33)
$\displaystyle p_{e}(r)$ $\displaystyle=$ $\displaystyle-\frac{2\alpha
r+(r-2\alpha)\beta\left(\frac{r}{\beta}\right)^{n}}{r^{4}}$ (34)

and hence

$\displaystyle\rho_{e}(r)+p_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{[2\alpha+r(n-1)]\beta\left(\frac{r}{\beta}\right)^{n}-2\alpha
r}{r^{4}}$

Fig.7(Left) shows the positive and decreasing behavior of original matter
density $\rho(r)$. The matter configuration in $\kappa(R,T)$ gravity violates
the null energy condition(NEC), clear from Figs.7(Right)-8(Left) and these
confirm that the solution supports a wormhole structure. Moreover, from
Eq.(LABEL:rhop32) we can see that $\rho(r)+p(r)$ tends to zero whenever $n$
tends to unity. Therefore, in this case, also an arbitrarily small amount of
exotic matter is responsible to construct a wormhole design.

Figure 9:  $\frac{b(r)}{r}$(Left) and $\frac{db(r)}{dr}$(Right) are plotted
against the radial coordinate $r(\geq$ the throat radius) corresponding to
$\gamma=r_{0}=1.5$ and $\delta^{2}=0.1,0.3,0.6,0.9$.

###  IV.2 The shape function
$b(r)=\gamma\left\\{1+\delta^{2}\left(1-\frac{\gamma}{r}\right)\right\\}$ .

we consider another form of the shape function
$b(r)=\gamma\left\\{1+\delta^{2}\left(1-\frac{\gamma}{r}\right)\right\\}$fr19
where $\gamma,\delta$ are constants. The throat radius of the wormhole is
obtained as $r_{0}=\gamma$ by using
$\gamma\left\\{1+\delta^{2}\left(1-\frac{\gamma}{r_{0}}\right)\right\\}=r_{0}$.
The shape function is positive and increasing in nature (see Fig.8(Right)) and
it has asymptotic behavior (see Fig.9(Left)). The shape function also
satisfies the flare-out condition, shown in Fig.(9)(Right) provided
$\delta^{2}<1$.
Here, we take the same two redshift functions $f(r)=0$ and
$f(r)=\frac{\alpha}{r}$ to analyze the corresponding solutions, separately.
Therefore, in similar way, we obtain the following expressions for these sets
of redshift and shape functions:

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{2\pi\gamma\delta^{2}r^{2}+\gamma\delta^{2}\sqrt{2\pi\left[2\pi
r^{4}-a\gamma\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}\right]}}{4\pi
ar^{2}\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}}$ (36)
$\displaystyle p(r)$ $\displaystyle=$
$\displaystyle-\frac{[2\pi\gamma\delta^{2}r^{2}+\gamma\delta^{2}\sqrt{2\pi\left[2\pi
r^{4}-a\gamma\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}\right]}]\\{r+(r-\gamma)\delta^{2}\\}}{4\pi
ar^{2}\gamma\delta^{2}\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}}$
(37) $\displaystyle\rho(r)+p(r)$ $\displaystyle=$
$\displaystyle\left\\{\frac{2\pi\gamma\delta^{2}r^{2}+\gamma\delta^{2}\sqrt{2\pi\left[2\pi
r^{4}-a\gamma\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}\right]}}{4\pi
ar^{2}\left\\{3r(1+\delta^{2})-2\gamma\delta^{2}\right\\}}\right\\}\left\\{1-\frac{\left[r+(r-\gamma)\delta^{2}\right]}{\gamma\delta^{2}}\right\\}$
(38)

$\displaystyle\rho_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{(\delta\gamma)^{2}}{r^{4}}$ (39) $\displaystyle p_{e}(r)$
$\displaystyle=$
$\displaystyle-\frac{\gamma\left\\{(1+\delta^{2})r-\gamma\delta^{2}\right\\}}{r^{4}}$
(40) $\displaystyle\rho_{e}(r)+p_{e}(r)$ $\displaystyle=$
$\displaystyle-\frac{\gamma\left\\{(1+\delta^{2})r-2\gamma\delta^{2}\right\\}}{r^{4}}$
(41)

Figure 10: The original matter matter density(Left) and the sum of original
matter density and pressure(Right) are plotted against the radial coordinate
$r(\geq$ the throat radius) corresponding to $\gamma=r_{0}=1.5$, $a=0.02$ and
$\delta^{2}=0.1,0.3,0.6,0.9$.

And

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{\gamma^{2}\delta^{2}\left[f_{6}-\left\\{2\alpha\gamma^{2}\delta^{2}+r^{2}(2\alpha+\gamma+\gamma\delta^{2})-r\gamma(\gamma\delta^{2}+2\alpha+2\alpha\delta^{2})\right\\}\sqrt{2\pi
r(2\pi r^{5}-af_{7})}\right]}{4a\pi
r^{2}\left\\{2\alpha\gamma^{2}\delta^{2}+r^{2}(2\alpha+\gamma+\gamma\delta^{2})-r\gamma(\gamma\delta^{2}+2\alpha+2\alpha\delta^{2})\right\\}f_{7}(r)}$
$\displaystyle p(r)$ $\displaystyle=$
$\displaystyle-\frac{f_{6}-\left\\{2\alpha\gamma^{2}\delta^{2}+r^{2}(2\alpha+\gamma+\gamma\delta^{2})-r\gamma(\gamma\delta^{2}+2\alpha+2\alpha\delta^{2})\right\\}\sqrt{2\pi
r(2\pi r^{5}-af_{7})}}{4a\pi r^{3}f_{7}}$ (42) $\displaystyle\rho(r)+p(r)$
$\displaystyle=$
$\displaystyle\left\\{\frac{f_{6}-\left\\{2\alpha\gamma^{2}\delta^{2}+r^{2}(2\alpha+\gamma+\gamma\delta^{2})-r\gamma(\gamma\delta^{2}+2\alpha+2\alpha\delta^{2})\right\\}\sqrt{2\pi
r(2\pi r^{5}-af_{7})}}{4a\pi r^{2}f_{7}}\right\\}$ (43)
$\displaystyle\times\left\\{\frac{\gamma^{2}\delta^{2}}{2\alpha\gamma^{2}\delta^{2}+r^{2}(2\alpha+\gamma+\gamma\delta^{2})-r\gamma(\gamma\delta^{2}+2\alpha+2\alpha\delta^{2})}-\frac{1}{r}\right\\}$

$\displaystyle\rho_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{(\gamma\delta)^{2}}{r^{4}}$ (44) $\displaystyle p_{e}(r)$
$\displaystyle=$
$\displaystyle\frac{r\gamma^{2}\delta^{2}-2\alpha\left\\{r^{2}-\gamma(r+\delta^{2}-\gamma\delta^{2})\right\\}-r^{2}\gamma-r^{2}\gamma\delta^{2}}{r^{5}}$
(45) $\displaystyle\rho_{e}(r)+p_{e}(r)$ $\displaystyle=$
$\displaystyle\frac{2r\gamma^{2}\delta^{2}-2\alpha\left\\{r^{2}-\gamma(r+\delta^{2}-\gamma\delta^{2})\right\\}-r^{2}\gamma-r^{2}\gamma\delta^{2}}{r^{5}}$

whereas

$\displaystyle f_{6}$ $\displaystyle=$ $\displaystyle 2\pi
r^{3}(2r^{2}\alpha+r^{2}\gamma-2r\alpha\gamma+r^{2}\gamma\delta^{2})-4\pi
r^{4}\alpha\gamma\delta^{2}-2\pi
r^{4}\gamma^{2}\delta^{2}+4\pi\alpha\gamma^{2}\delta^{2}r^{3}$ $\displaystyle
f_{7}$ $\displaystyle=$ $\displaystyle
6\alpha\gamma^{2}\delta^{2}+3r^{2}(2\alpha+\gamma+\gamma\delta^{2})-2r\gamma(\gamma\delta^{2}+3\alpha+3\alpha\delta^{2})$

Figure 11: The original matter matter density(Left) and the sum of original
matter density and pressure(Right) are plotted against the radial coordinate
$r(\geq$ the throat radius) corresponding to $\gamma=r_{0}=1.5$, $a=0.02$,
$\alpha=0.01$ and $\delta^{2}=0.1,0.3,0.6,0.9$.

Figure 12: The sum of effective matter density and pressure(Left) and the sum
of effective matter density and pressure(Right) are plotted against the radial
coordinate $r(\geq$ the throat radius) corresponding to $\gamma=r_{0}=1.5$,
$\delta^{2}=0.1,0.3,0.6,0.9$ and $\gamma=r_{0}=1.5$, $\alpha=0.01$,
$\delta^{2}=0.1,0.3,0.6,0.9$, respectively.

In similar argument, anyone can notice that the obtained solutions for the
shape function
$b(r)=\gamma\left\\{1+\delta^{2}\left(1-\frac{\gamma}{r}\right)\right\\}$
along with the redshift functions $f(r)=0$ and $f(r)=\frac{\alpha}{r}$
represent the wormhole structures by violating the null energy condition(NEC),
separately (see Figs.10(Right),11(Right) and 12). Moreover, the original
matter densities are positive and decreasing against the radial coordinate
$r$, clear from Figs.10(Left)-11(Left). Finally, we can easily see from
Eqs.(38) and (43) that $\rho(r)+p(r)$ does not tends to zero for any numerical
values of $\delta$ and hence there are no possibility to exist the
infinitesimal exotic matters that may support the wormholes for these
solutions.

##  V Results and Discussions

In this article, we have found wormholes supporting solutions of the Einstein
field equations in the framework of $\kappa(R,T)$ gravity. We have obtained
the solutions into two different ways: (i) The solution is obtained by
considering a redshift function and a linear equation of state(EoS) and (ii)
The solutions are obtained by considering four pairs of two redshift functions
and two shape functions.
In the first part, we have taken a redshift function $f(r)=A$, $A$ is constant
and a linear equation of state (Eos) $p(r)=m\rho(r)$, where m is a constant
along with the restriction $0<m<1$ and $m\neq\frac{1}{3}$. These redshift
function and EoS taking into account we have determined the shape function and
the obtained shape function is positive, decreasing in nature and satisfied
the flare-out condition along with the asymptotic behavior (see
Figs.1-2(Left)). Therefore, the shape function satisfied all the necessary
conditions to construct a wormhole design and consequently this shape function
constructs a wormhole like geometry. The obtained original matter density is
positive and decreasing in nature, also the original matter configuration
satisfied the weak energy condition(WEC) i.e. $\rho(r)\geq 0$,
$\rho(r)+p(r)\geq 0$ as well as the null energy condition(NEC) i.e.
$\rho(r)+p(r)\geq 0$ whereas the modified matter violates the null energy
condition(NEC) i.e. $\rho_{e}(r)+p_{e}(r)<0$), shown in Figs.2(Right)-3. Thus,
we have obtained a peculiar result that the original matter distribution is
the real feasible matter, which provides the fuel to construct and sustain a
wormhole in $\kappa(R,T)$ gravity.
Secondly, we have considered four pairs of two different redshift functions
and two different shape functions. Both the shape functions are positive,
increasing and satisfied all the conditions to present the wormhole
structures, shown in Figs.4, 5(Left), 8(Right) and 9. The solutions
corresponding to the shape function
$b(r)=\beta\left(\frac{r}{\beta}\right)^{n}$ and redshift functions
$f(r)=0,~{}f(r)=\frac{\alpha}{r}$ violate the null energy condition,
separately, clear from Figs.6, 7(Right) and 8(Left) i.e. the solutions
representing matter distributions hold the wormhole structures in
$\kappa(R,T)$ gravity. Moreover, an arbitrarily small amount of exotic matter
is responsible to construct a wormhole design for these solutions. Also, the
matter distributions represented by the solutions corresponding to the shape
function
$b(r)=\gamma\left\\{1+\delta^{2}\left(1-\frac{\gamma}{r}\right)\right\\}$ and
redshift functions $f(r)=0,~{}f(r)=\frac{\alpha}{r}$ retain the wormhole
structures by violating the null energy condition(NEC) (see Figs.10(Right),
11(Right) and 12), separately but there are no possibility to exist the
infinitesimal exotic matters that may support the wormholes.

Now it will be very interesting to present a comparative analysis with the
other WH works in modified theories of gravitation. Godani and Samanta go19
have investigated the traversable WHs and energy conditions within the
framework of $f(R)$ modified theory gravity. They have used the shape function
of the form $b(r)=\beta\left(\frac{r}{\beta}\right)^{n}$, and they have found
that the energy density is positive and all energy conditions are violated
which supports the existence of WHs. On the other hand, Samanta et al. sam18
have presented the WHs with exponential shape function within the framework of
$f(R)$ gravity. They have concluded that the WH may not contain same type of
matter for all values of $r$ and it is filled some abnormal type of matter,
which is may be called as dark energy. But, in our case, the WHs in
$\kappa(R,T)$ gravity violates the energy conditions for all values of $r$ and
WH contains same type of matter. Hence, our results are coincide with the
results obtained by Godani and Samanta go19 .

### Acknowledgments

Farook Rahaman would like to thank the authorities of the Inter-University
Centre for Astronomy and Astrophysics, Pune, India for providing the research
facilities. Susmita Sarkar and Nayan Sarkar are thankful to UGC (Grant No.:
1162/(sc)(CSIR-UGC NET , DEC 2016)) and CSIR (Grand
No.-09/096(0863)/2016-EMR-I.), Govt. of India for financial support
respectively.

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