# Energy Constraints for Evolving Spherical and Hyperbolic Wormholes in
$f(R,T)$ Gravity

M. Zubair<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan
Quratulien Muneer<EMAIL_ADDRESS>Department of Mathematics, COMSATS
University Islamabad, Lahore Campus, Lahore, Pakistan  Saira Waheed
<EMAIL_ADDRESS>Prince Mohammad Bin Fahd University, Khobar, Kingdom of
Saudi Arabia

###### Abstract

The primary objective of this article is to study the energy condition bounds
for spherical and hyperbolic wormholes in well-known $f(R,T)$ theory of
gravity. For this purpose, we formulate the field equations for spherically
and pseudospherically geometries using anisotropic matter and linear form of
generic function $f(R,T)$. By imposing different conditions on radial and
tangential pressures or by adopting some known choices for red shift and shape
functions, we present the graphical analysis of energy conditions for both
spherically and pseudospherically symmetric wormholes. It is seen that energy
density for spherically symmetric wormhole is always positive for
$\lambda>-4\pi$ and $\lambda<-8\pi$, while the energy conditions for radial
pressure are negative at throat. Likewise, in case of pseudospherically
symmetric wormhole, it is observed that energy density is always positive for
negative $\lambda$, however conditions based on radial pressure may be
positive or negative for the considered different cases.

Keywords: Modified gravity, Wormholes, Spherically symmetric spacetime,
Pseudospherically symmetric spacetime, Energy conditions.

##  I Introduction

General relativity (GR) embodies the conceptual ingenuity and the very essence
of scientific style in its each advancement. It provides considerable
contributions at observational as well as experimental levels which range from
millimeter to cosmological scales a . However, despite its excellence, the
geometrical singularities challenge its potential and make it impotent in
certain situations. It is argued that simple GR action cannot provide complete
description of accelerated expansion eras of the universe and it is only
possible by summoning new types of matter and energy components of the
universe in its action S . To incorporate these adjustments, modified matter
models or theories have been proposed in literature. In this respect, modified
theories have gained considerable attention of researchers after the
revelation of accelerated expanding nature of universe in its current state.
It is worth mentioning here that although several theories like torsion based
framework and its modified versions, scalar field theories, Gauss-Bonnet
gravity have been presented ND  but the natural extensions of GR namely $f(R)$
theory D  and its extension $f(R,T)$ gravity T  (with $R$ and $T$ as Ricci
scalar and trace of stress energy tensor, respectively) have gained more
attention due to their success and viability. Several works are available in
literature b -23  which present discussions on various cosmic issues and
proved as best candidates for the explanation of cosmic acceleration.

Geometrically, wormholes are tunnels type structures which join two different
asymptotically flat distant universes or two different regions of the same
universe. Such tube-like structures were firstly introduced by Einstein and
Rosen 1* , who investigated exact solutions of GR, and their proposed
structures were named as Einstein-Rosen bridges. Later, Morris and Thorne
(M-T) 1  explored the static spherically symmetric wormhole by adopting GR
gravitational framework and proposed the traversable wormhole by describing
its fundamental aspects, according to which human body can travel via wormhole
tunnels. It is worthy to mention here that wormhole geometry is based on two
functions namely red shift and shape functions. It has been argued 1  that for
the existence of a viable wormhole structure, these functions must satisfy
some axioms, i.e., for $r_{0}\leq r<\infty$, where $r_{0}$ is the throat
radius, the shape function $b(r)$ must satisfy the condition $b(r_{0})=r_{0}$
at $r=r_{0}$ and for $r>r_{0}$, the conditions $1-\frac{b(r)}{r}>0$ and
$b^{{}^{\prime}}(r_{0})<1$ must hold. The wormhole spacetime geometry is
asymptotically flat if it satisfies the condition, i.e., limit
$\frac{b(r)}{r}\rightarrow 0$ as $|r|$ $\rightarrow\infty$. Also, the red
shift $\phi(r)$ must be finite throughout the spacetime. In literature, much
work have already been done on this topic in both GR as well as modified
gravity theories. In 2 , authors have studied spherically symmetric
traversable wormhole spacetime filled with single fluid involving anisotropic
pressure. Cataldo and his collaborators 3  proposed the wormhole shape
function defined by $b(r)=\mu+\nu(r)$ while Samanta et al. 3*  discussed the
exponential form of shape function, i.e., $b(r)=\frac{r}{e^{(r-r_{0})}}$ in GR
framework. Further, in another study 3** , authors have used the power law
type function given by $b(r)=(r_{0})^{\epsilon+1}(r)^{-\epsilon}$ by varying
$\epsilon$. In 3 ; 4 , the authors have studied the existence of spherically
symmetric wormholes as well as pseudospherically symmetric spacetimes
satisfying all necessary conditions for shape and red shift functions. They
concluded that the energy density becomes negative and radial pressure remains
positive for hyperbolic wormhole in GR. In a recent paper 4* , Sharma and
Ghosh proposed a generalized version of Ellis-Bronnikov wormholes which is
embedded in a 4-dimensional wrapped background and discussed the validity of
energy constraints in GR.

Many authors have explored the wormhole solutions in modified theory including
Azizi 4**  who studied wormhole geometries in the framework of $f(R,T)$ theory
by considering a particular EoS for ordinary matter and concluded that
modified stress-energy is responsible for the violation of the NEC. Sahoo with
his collaborators 5  proposed the modeling of traversable wormholes (a
wormhole in which any body can be passed safely) in the frame work of
traceless $f(R,T)$ gravity theory. They discussed different features of energy
condition bounds by considering hybrid form of shape function and a particular
form of EoS parameter. Similarly, Dixit and his collaborators 5*  explored the
traversable wormhole by applying logarithmic shape function in $f(R,T)$
gravity and discussed the behavior of radial and tangential components of
state along with anisotropy parameter in describing the universe geometry.
Mishra et al. 7  presented the traversable wormhole models using different
ansatz of shape functions and EoS parameter in the $f(R,T)$ gravitational
framework. They also examined the suitability of these models by exploring all
energy constraints. Further, Sahoo and his collaborators 8  studied the
wormhole solution for phantom case, i.e., $\omega<-1$ in $f(R,T)$ modified
gravity. They adopted the EoS given by $p_{r}=\omega\rho$ to obtain the shape
function which obeys all its axioms and also examined corresponding energy
condition bounds. Similarly, Moraes and Sahoo 9  proposed the wormhole
solution by applying power law form of shape function and analyzed all energy
constraints in exponential $f(R,T)$ gravity. In literature 10 -20* , wormholes
have been discussed without non-exotic matter in modified theories. In this
respect, Zubair et al. 21  proposed wormhole geometries by adopting simple
linear as well as cubic forms of $f(R,T)$ function. They considered the non-
commutative geometrical aspects of string theory in the context of $f(R,T)$
gravity. In another paper z32 , the researchers have explored the idea of
viable charged wormhole solutions in $f(R,T)$ gravity. They have assumed
simple linear generic model given by $f(R,T)=R+2T$ along with the ordinary
matter as the total pressure of anisotropic fluid. Further, the existence of
static wormhole model have been explored by utilizing different kind of shape
functions in a study z33 . In another paper z34 , authors have investigated
the wormhole modeling by considering a specific general shape function in the
quadratic $f(R,T)$ gravity. In a recent paper z35 , Banerjee and his
collaborator adopted different strategies to construct wormhole geometry using
isotropic pressure in $f(R,T)$ theory and concluded that all energy
constraints are valid for the proposed models.

Above literature motivated us to explore the spherically symmetric and
pseudospherically symmetric wormhole solutions for M-T spacetime in $f(R,T)$
theory. Here, in order to calculate the $\phi(r)$ and $b(r)$ functions, we
consider the matter with EoS parameter for $p_{r}$ or $p_{t}$ to solve
$f(R,T)$ field equations. Also, we shall present the validity of energy
conditions for static spherically and psuedospherically symmetric wormhole
spacetimes defined by the single perfect fluid, i.e., a source of matter with
isotropic pressure in $f(R,T)$ gravity with wide range of $\lambda$. Up to our
knowledge, till now, the only wormhole solution discussed in literature are
non-asymptotically flat wormhole with isotropic pressure as pointed out in
Ref. 28 -30 . In the present work, we shall discuss the asymptotically flat
wormholes which may or may not satisfy all energy constraints with a wide
range of $\lambda$ for both types of wormhole spacetimes.

We have organized this paper in the following pattern. Section II provides the
basic formulation of $f(R,T)$ field equations for static spherically symmetric
spacetime and anisotropic matter. In its subsections, we shall discuss the
energy conditions graphically for different cases: firstly, wormholes with
zero tidal force and filled with perfect fluid, secondly, by taking the linear
equation of state, and lastly, the analytic solution used by Tolman and static
spheres filled with isotropic perfect fluid. Similarly, in sec IV, we express
the $f(R,T)$ field equations for hyperbolic symmetric spacetime. In its
subsection A, we discuss the pseudospherical symmetric wormhole and
corresponding energy conditions using linear equation of state. The zero tidal
hyperbolic wormholes with the corresponding energy conditions will be
explained in subsection B. We choose non-vanishing red shift function and
evaluate the more general form of shape function with isotropic pressure and
explore the energy conditions for the obtained hyperbolic wormhole in
subsection C. Finally, we summarize our discussion.

##  II Field Equations for Spherically Symmetric Spacetime in $f(R,T)$ and
Wormholes Construction

In this section, we shall present a brief review of $f(R,T)$ gravity and some
necessary assumptions used for this work. The $f(R,T)$ theory of gravity is
defined by the total action T  given by

$S=\frac{1}{16\pi}\int f(R,T)\sqrt{-g}d^{4}x+\int L_{m}\sqrt{-g}d^{4}x,$ (1)

where $f(R,T)$ is an arbitrary function of Ricci scalar $R$ and the trace of
energy-momentum tensor $T=g^{\mu\nu}T_{\mu\nu}$ and $L_{m}$ represents the
Lagrangian density of ordinary matter. By varying the above action with
respect to metric tensor $g_{ij}$, we have the following set of equations

$\displaystyle 8\pi T_{ij}-f_{T}(R,T)T_{ij}-f_{T}(R,T)\Theta_{ij}$
$\displaystyle=$ $\displaystyle
f_{R}(R,T)R_{ij}-\frac{1}{2}f(R,T)g_{ij}+(g_{ij}\Box-\nabla_{i}\nabla_{j})f_{R}(R,T).$
(2)

The above equation involves covariant derivative and d’Alembert operator
denoted by $\nabla$ and $\Box$, respectively, where $f_{R}(R,T)$ and
$f_{T}(R,T)$ represent the function derivatives with respect to $R$ and $T$,
respectively. Also, the term $\Theta_{ij}$ is defined by

$\Theta_{ij}=\frac{g^{\alpha\beta}\delta T_{ij}}{\delta
g^{ij}}=-2T_{ij}+g_{ij}L_{m}-2g^{\alpha\beta}\frac{\partial^{2}L_{m}}{\partial
g^{ij}\partial g^{\alpha\beta}},$ (3)

where $T_{ij}=diag(\rho,~{}-p_{r},~{}-p_{t},~{}-p_{t})$, with $\rho$ as the
matter energy density while $p_{r}$ and $p_{t}$ denote the radial and
transverse components of pressure. Also, $L_{m}=-\emph{P}$, where
$\emph{P}=\frac{(p_{r}+2p_{t})}{3}$ is the total pressure. As a result, Eq.(3)
takes the form given by $\Theta_{ij}=-2T_{ij}-\emph{P}g_{ij}$. For the sake of
simplicity in calculations, let us consider the linear form of generic
function $f(R,T)$ given by $f(R,T)=R+\lambda T$ and consequently, the field
equations (2) can be re-written as

$R_{ij}-\frac{1}{2}Rg_{ij}=(8\pi+\lambda)T_{ij}+\frac{\lambda}{2}(\rho-\emph{P})g_{ij},$
(4)

where $\lambda$ is an arbitrary constant.

In 1 , Morris and Thorne introduced the wormhole geometry given by the
following metric:

$ds^{2}=e^{2\Phi(r)}dt^{2}-\frac{dr^{2}}{1-b(r)/r}-r^{2}(d\theta^{2}+sin^{2}\theta
d\Phi^{2}),$ (5)

where the functions $\Phi(r)$ and $b(r)$ refer to red shift and shape
functions, respectively and it is worthy to mention here that the essential
characteristics of a viable wormhole geometry are encoded through these
functions. Thus in order to construct the wormhole geometry, these functions
must satisfy the constraints which are presented by Morris and Thorne in Ref.
1 ; 30 . In the present work, we assume $\phi(r)$ without horizons and hence
it must be finite everywhere. Assuming single imperfect fluid and using
Eqs.(4) and (5), we have

$\displaystyle\frac{b^{\prime}}{r^{2}}=(8\pi+\lambda)\rho+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}),$
(6)
$\displaystyle\frac{1}{r}\bigg{[}\frac{b}{r^{2}}+2\phi^{\prime}(\frac{b}{r}-1)\bigg{]}=(8\pi+\lambda)(-p_{r})+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}),$
(7)
$\displaystyle\frac{1}{2r}\bigg{[}\frac{1}{r}(\phi^{\prime}b+b^{\prime}-\frac{b}{r})+2(\phi^{\prime\prime}+(\phi^{\prime})^{2})b-\phi^{\prime}(2-b^{\prime})\bigg{]}=(8\pi+\lambda)(-p_{t}).+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}).$
(8)

The above equations can be re-arranged to find the explicit expressions of
$\rho,~{}p_{r}$ and $p_{t}$ which can be written as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{2}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}b^{\prime}(48\pi+\lambda(8-r\phi^{\prime})+\lambda(2r(2\phi^{\prime}+r(\phi^{\prime})^{2}+r\phi^{\prime\prime}))-b(3\phi^{\prime}+2r(\phi^{\prime})^{2}+2r\phi^{\prime\prime})\bigg{]},$
(9) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}r(b^{\prime}\lambda(4+r\phi^{{}^{\prime}})-2r(-48\pi\phi^{{}^{\prime}}-10\lambda\phi^{{}^{\prime}}+r\lambda(\phi^{{}^{\prime}})^{2}+r\lambda\phi^{{}^{\prime\prime}}))-b(48\pi(1+2r\phi^{{}^{\prime}})$
(10) $\displaystyle+$
$\displaystyle\lambda(12+21r\phi^{{}^{\prime}}-2r^{2}((\phi^{\prime})^{2}+\phi^{{}^{\prime\prime}})))\bigg{]},$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}-b^{{}^{\prime}}\bigg{[}24\pi(-1+r\phi^{{}^{\prime}}+2r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))+\lambda(-6+3r\phi^{{}^{\prime}}+10r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))\bigg{]}$
(11) $\displaystyle+$ $\displaystyle
r\bigg{[}-b^{{}^{\prime}}(24\pi(1+r\phi^{{}^{\prime}})+\lambda(2+5r\phi^{{}^{\prime}}))+2r(24\pi(\phi^{{}^{\prime}}+r(\phi^{{}^{\prime}})^{2}+r\phi^{{}^{\prime\prime}})+\lambda(4\phi^{{}^{\prime}}+5r((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})))\bigg{]}\bigg{]},$

where prime denotes the derivative with respect to radial coordinate. Also,
from the conservation equation $T^{ij}_{;j}=0$, the hydrostatic equilibrium
equation for the ordinary matter in the interior of wormhole can be defined as

$\displaystyle
p_{r}^{{}^{\prime}}+\phi^{{}^{\prime}}(\rho+p_{r})=\frac{\lambda}{(16\pi+2\lambda)}(\rho^{{}^{\prime}}-\frac{p_{r}^{{}^{\prime}}+2p_{t}^{{}^{\prime}}}{3}).$
(12)

It is obvious that for isotropic fluid, the condition $p_{r}=p_{t}$ must be
imposed. Using this condition, we can obtain a differential equation
connecting both redsihft and shape functions given by

$\displaystyle\phi^{{}^{\prime\prime}}+\phi^{{}^{\prime
2}}-\frac{(b^{{}^{\prime}}r+2r-3b)\phi^{{}^{\prime}}}{2r(r-b)}=\frac{b^{{}^{\prime}}r-3b}{2r^{2}(r-b)}.$
(13)

The above differential equation (DE) can be solved for one of these two
unknowns, for example, one can solve it for $b(r)$ by choosing a specific
known form of function $\phi(r)$. Consequently, it can be written as

$\displaystyle
b(r)=\bigg{(}\int\frac{2r(-\phi^{{}^{\prime}}+r\phi^{{}^{\prime}2}+r\phi^{{}^{\prime\prime}})e^{\int\frac{2r^{2}(\phi^{{}^{\prime}2}+\phi^{{}^{\prime\prime}})-3r\phi^{{}^{\prime}}-3}{r(1+r\phi^{{}^{\prime}})}dr}}{1+r\phi^{{}^{\prime}}}dr+c\bigg{)}\times
e^{-\int\frac{2r^{2}(\phi^{{}^{\prime}2}+\phi^{{}^{\prime\prime}})-3r\phi^{{}^{\prime}}-3}{r(1+r\phi^{{}^{\prime}})}dr},$
(14)

where $c$ is an integration constant. It is interesting to point out that the
$f(R,T)$ field equations has been reduced to two independent differential
equations (DEs) given by Eqs.(9) and (13) which involve four unknowns namely
$\phi(r),~{}\rho(r),~{}b(r)$ and $p(r)$ under the assumption of isotropic
fluid. Therefore, in order to obtain exact solutions, we have to consider some
known forms of two functions and evaluate the others. In the upcoming
subsections, we shall present the wormholes solutions by considering some
interesting cases and also evaluate the energy constraint bounds in each case.

###  II.1 Energy conditions

Energy constraints play an important role in the framework of GR as well as
alternative gravity theories for exploring the physical viability of proposed
models. It is argued that the set of four energy constraints can be formulated
by using the well-known Raychaudhuri’s equations and these bounds were firstly
developed in GR Ray  and then extended to modified gravity theories. These
energy constraints are known as the null energy condition (NEC), the weak
energy condition (WEC), the dominant energy condition (DEC) and the strong
energy condition (SEC). The Raychaudhuri’s equations for congruence of
geodesic with time-like and null-like vectors are given by Ray1

$\displaystyle\frac{d\theta}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{-\theta^{2}}{3}-\sigma_{ij}\sigma^{ij}+\omega_{ij}\omega^{ij}-R_{ij}u^{i}u^{j},$
$\displaystyle\frac{d\theta}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{-\theta^{2}}{3}-\sigma_{ij}\sigma^{ij}+\omega_{ij}\omega^{ij}-R_{ij}k^{i}k^{j},$

where $\sigma_{ij},~{}\omega_{ij}$ and $R_{ij}$ are the shear tensor, rotation
and Ricci tensor, respectively, while $u^{i}$ represents the time-like vector
and $u^{j}$ indicates the null-like geodesics in congruence. It will converge
when $\theta<0$ which further leads to $\frac{d\theta}{d\tau}<0$. By ignoring
second order terms and integrating above equation, we obtain $\theta=-\tau
R_{ij}u^{i}u^{j}$ and $\theta=-\tau R_{ij}k^{i}k^{j}$, where
$\sigma_{ij}\sigma^{ij}\geq 0$ and $\omega=0$. Using these conditions, one can
define the following inequalities

$\displaystyle R_{ij}u^{i}u^{j}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{ij}k^{i}k^{j}\geq
0$

which can be re-written as the linear combination of energy momentum tensor
and its trace by applying field equations as follows

$\displaystyle(T_{ij}-\frac{Tg_{ij}}{2})u^{i}u^{j}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T_{ij}k^{i}k^{j}\geq 0.$

Now, by using above inequalities, we can define the energy constraints as WEC:
$\rho=T_{ij}u^{i}u^{j}\geq 0$, DEC: $T^{00}\geq|T^{ij}|$, i.e., in any
orthonormal basis, the energy dominates as compared to any other components of
$T^{ij}$, while NEC: $T_{ij}k^{i}k^{j}\geq 0$ and SEC:
$(T_{ij}-\frac{Tg_{ij}}{2})u^{i}u^{j}\geq 0$. The NEC and SEC come from
Raychaudhuri equation which depends on the energy momentum tensor and also on
alternative gravity theories, while the WEC and DEC only depend on the energy-
momentum tensor. In upcoming sections, we shall discuss the energy condition
bounds for wormhole models by using these constraints.

###  II.2 Wormhole with Isotropic Pressure using Zero-tidal Force Condition in
$f(R,T)$ Gravity

In this section, we shall discuss the existence of wormhole geometries by
taking two cases into account. In the first case, we consider the zero-tidal-
force wormhole in which $\phi(r)=\phi_{0}$, where $\phi_{0}$ is a constant and
consequently, Eq.(14) leads to the form of shape function given by
$b(r)=cr^{3}$. By applying the initial condition: $b(r_{0})=r_{0}$, one can
determine the value of constant $c$ and finally, the shape function takes the
form as $b(r)=\frac{r^{3}}{r_{0}^{2}}$. Consequently, the wormhole spacetime
turns out to be

$\displaystyle
ds^{2}=dt^{2}-\frac{dr^{2}}{1-(r/r_{0})^{2}}-r^{2}(d\theta^{2}+sin^{2}\theta
d\Phi^{2}).$ (15)

It is worthy to mention here that the above spacetime represents a metric with
constant curvature. It is easy to compute the values of density and pressure
and are given by
$\rho=\frac{2(6\pi+\lambda)}{r^{2}_{0}(4\pi+\lambda)(8\pi+\lambda)}$ and
$p=-\frac{4\pi}{r^{2}_{0}(4\pi+\lambda)(8\pi+\lambda)}$. Consequently, $\rho$
and $p$ are related by the expression
$p=-\rho\frac{2\pi}{(6\pi+\lambda)}$.Also, $b^{\prime}(r_{0})=3\nless 1$ which
implies that flaring out condition is not satisfied, therefore, no zero tidal
force wormhole can be sustained in the present configuration.

In the second case, we consider the constant red shift function and a general
form of shape function given by $b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}$, then
M-T wormhole metric takes the form:

$\displaystyle
ds^{2}=dt^{2}-\frac{dr^{2}}{1-(r/r_{0})^{\alpha-1}}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (16)

where $\alpha<0$. In this case, Eqs.(9)-(11) lead to the following forms of
energy density and pressures

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{2(1+\alpha)(\frac{r}{r_{0}})^{\alpha}(6\pi+\lambda)}{3r^{2}(4\pi+\lambda)(6\pi+\lambda)},$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{(-12\pi+(-2+\alpha)\lambda)(\frac{r}{r_{0}})^{\alpha}}{3r^{2}(4\pi+\lambda)(6\pi+\lambda)}$
$\displaystyle,p_{t}$ $\displaystyle=$
$\displaystyle-\frac{(-2\lambda+\alpha(12\pi+\lambda))(\frac{r}{r_{0}})^{\alpha}}{6r^{2}(4\pi+\lambda)(6\pi+\lambda)}.$

For a viable wormhole geometry, the shape function shows increasing behavior
and satisfies the conditions given by $b(r)-r<0,~{}b^{\prime}(r)<1$ as well as
asymptotically flat condition ($\frac{b(r)}{r}\rightarrow
0,~{}r\rightarrow\infty$). It is easy to check that all of these conditions
are satisfied as shown in the left graph of Figure 1. Next we explore the
validity of energy conditions of the wormhole matter contents versus radial
coordinate. Mathematically, these constraints are defined as WEC ($\rho\geq
0$), NEC ($\rho+p_{i}\geq 0$, $i=r,t$ represents the radial and tangential
pressures), SEC ($\rho+p_{r}+2p_{t}\geq 0$) and DEC ($\rho-p_{r},~{}\rho-
p_{t}$). The right graph of Figure 1 shows the graphical behavior of $\rho$
which represents that wormhole geometry satisfies the WEC for $\lambda<-8\pi$.
From Figure 2, it is observed that the expressions $\rho+p_{i},~{}i=r$ and
$\rho+p_{r}+2p_{t}$ attain positive values for $\lambda<-8\pi$, while
$\rho+p_{t}$ exhibits negative behavior for $\lambda<-8\pi$. Thus, only NEC
for radial pressure and SEC are satisfied for $\lambda<-8\pi$. Next the
graphical behavior of $\rho-p_{r}$ and $\rho-p_{t}$ is provided in Figure 3
which indicates that the DEC with tangential pressure is valid for
$\lambda<-8\pi$, whereas DEC with radial pressure violates for
$\lambda<-8\pi$.

Figure 1: Evolution of shape function $b(r)$ satisfying the conditions and
energy density versus $r$. In the left panel, $b(r)$ (blue), $b^{\prime}(r)$
(red), $b(r)-r$ (gray) and $\frac{b(r)}{r}$ (black). In the right panel, the
plot indicates the behavior of energy density ($\rho$), where the chosen
parameter are $r_{0}=1$ and $\alpha=-.5$.

Figure 2: Evolution of NEC ($\rho+p_{i}$) and SEC ($\rho+p_{r}+2p_{t}$) versus
$r$, $i=r,t$. In the left panel, the yellow plot indicates the behavior of NEC
($\rho+p_{r}$) and gray plot shows behavior of NEC ($\rho+p_{t}$) while in the
right panel, the plot indicates the behavior of SEC where the free parameters
are fixed as $r_{0}=1$ and $\alpha=-.5$.

Figure 3: Evolution of DEC ($\rho-p_{i}$) and properties of shape function
given by Eq.(19) versus $r$. In the left panel, the yellow plot indicates the
behavior of DEC ($\rho-p_{r}$) while gray plot represents the behavior of DEC
($\rho-p_{t}$), where $r_{0}=1$ and $\alpha=-.5$. In the right panel, shape
function (blue), $b^{\prime}(r)$ (red), $b(r)-r$ (gray) and $\frac{b(r)}{r}$
(black), where the chosen parameters are $r_{0}=1$, $\lambda=10$ and
$\omega=-5$.

###  II.3 Spherical Wormholes with Equation of State in $f(R,T)$ Gravity

In this subsection, we will explore the wormhole construction by considering
two interesting cases of EoS parameter. In the first place, we consider the
linear equation of state along with constant red shift function and evaluate
the shape function. The linear equation of state is defined as

$\displaystyle p_{r}=\omega\rho.$ (17)

Using the above EoS along with the Eqs.(9) and (10), we obtain the following
differential equation:

$\displaystyle\frac{1}{r(4\pi+\lambda)(8\pi+\lambda)}(b(48\pi(1+2r\phi^{{}^{\prime}})-\lambda(-12+3r(-7+\omega)\phi^{{}^{\prime}}+2r^{2}(1+\omega)(\phi^{{}^{\prime
2}}+\phi^{{}^{\prime\prime}})))+$ $\displaystyle
r(b^{{}^{\prime}}(48\pi\omega-\lambda(4-8\omega+r\phi^{{}^{\prime}}+r\omega\phi^{{}^{\prime}}))+2r(-48\pi\phi^{{}^{\prime}}+\lambda(2(-5+\omega)\phi^{{}^{\prime}}+r(1+\omega)(\phi^{{}^{\prime}})^{2}+r(1+\omega)\phi^{{}^{\prime\prime}}))))=0.$
(18)

For zero tidal force wormhole, i.e., $\phi(r)=0$, the above differential
equation leads to the following form of shape function:

$\displaystyle
b(r)=Cr^{-\frac{3(4\pi+\lambda)}{12\pi\omega+(-1+2\omega)\lambda}},$ (19)

where C is an integration constant and consequently, metric describing the
wormhole geometry, energy density and tangential pressure take the form given
by

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-\frac{dr^{2}}{(r/r_{0})^{-\frac{(1+\omega)(12\pi+2\lambda)}{12\pi\omega-\lambda+2\omega\lambda}}-1}-r^{2}(d\theta^{2}+sin^{2}\theta
d\Phi^{2}),$ $\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{2(6\pi+\lambda)(\frac{r}{r_{0}})^{-3\frac{4\pi+12\pi\omega+2\omega\lambda}{12\pi\omega-\lambda+2\omega\lambda}}}{r_{0}^{2}(8\pi+\lambda)(12\pi\omega-\lambda+2\omega\lambda)},$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle-\frac{(6\pi(3\omega+2)-(1-3\omega)\lambda)(\frac{r}{r_{0}})^{-3\frac{4\pi+12\pi\omega+2\omega\lambda}{12\pi\omega-\lambda+2\omega\lambda}}}{3r_{0}^{2}(8\pi+\lambda)(12\pi\omega-\lambda+2\omega\lambda)}.$

Now we will discuss the graphical behavior of $b(r)$ defined by Eq.(19) and
check if it satisfies all axioms for the existence of wormhole solution as
discussed in the previous section. From the right plot of Figure 3, it is
clear that the obtained shape function exhibits physically viable behavior.
Further, we plot the energy conditions of wormhole matter content versus
radial coordinate. It is seen from the graphs of Figure 4 that the energy
density remains positive while the expressions $\rho+p_{r}$ and $\rho+p_{t}$
attain negative values for $\lambda>-4\pi$ and thus, the NEC is not satisfied
for this case. Also, from the graphs of Figure 5, it is easy to check that
expressions of DEC ($\rho-p_{i}$, $i=r,t$) indicate positive behavior while
SEC takes negative values for $\lambda>-4\pi$. Therefore, it can be concluded
that DEC is satisfied while SEC violates in this case.

Figure 4: Evolution of energy density $\rho$ and NEC ($\rho+p_{i},~{}i=r,t$)
versus $r$. In the left panel, the plot indicates the behavior of $\rho$ and
in right panel, the gray plot indicates the behavior of $\rho+p_{r}$, while
orange plot indicates the behavior of ($\rho+p_{t}$). Here we fixed $r_{0}=1$
and $\omega=-5$.

Figure 5: Evolution of DEC ($\rho-p_{i},~{}i=r,t$) and SEC
($\rho+p_{r}+2p_{t}$) versus $r$. In the left panel, the orange plot indicates
the behavior of $\rho-p_{r}$, while gray plot indicates the behavior of $\rho-
p_{t}$ and in right panel, the plot indicates the behavior of
$\rho+p_{r}+2p_{t}$.

In the second case, we suppose non-vanishing red shift function, i.e.,
$\phi(r)\neq$ constant. For wormhole construction, we shall construct a master
differential equation for the shape function by using linear equation of state
given by

$\displaystyle p_{r}=p_{t}=\omega\rho,$ (20)

where $\omega$ is a constant. Thus from Eq.(13), it can be written as

$\displaystyle\omega\rho^{\prime}+\phi^{\prime}(1+\omega)\rho=\frac{\lambda(1-\omega)\rho^{\prime}}{16\pi+2\lambda}$
(21)

and consequently, the energy density $\rho$ takes the form as follows

$\displaystyle\rho(r)=ce^{-\frac{(1+\omega)(16\pi+2\lambda)\phi(r)}{16\pi\omega+3\lambda\omega-\lambda}},$
(22)

where $c$ is an integration constant. It is interesting to mention here that
if we consider vanishing red shift function, then we shall obtain $\rho$ as a
constant and this case we have already discussed in the previous subsection.
From Eqs.(9), (13) and (20), we obtain a differential equation for $b(r)$ as

$\displaystyle(6-3r-r^{2})(-16\pi+\lambda(-3+\omega))^{2}b^{2}+2r^{2}\bigg{(}-(16\pi-\lambda(-3+\omega))(16\pi(-1+r+2\omega)-\lambda(1+r(-3+\omega)$
$\displaystyle+5\omega))b^{\prime}+(-128\pi^{2}\omega(\omega-1)+\lambda^{2}(1-2\omega-3\omega^{2})+8\pi\lambda(1-2\omega+5\omega^{2}))b^{\prime
2}+r(256\pi^{2}\omega-16\lambda\pi(-1+\omega^{2})$
$\displaystyle+\lambda^{2}(3-10\omega+3\omega^{2}))b^{\prime\prime}\bigg{)}+r(16\pi-\lambda(-3+\omega))b((16\pi(-2+r+3\omega+2r\omega)-\lambda(3-5r+7\omega+7r\omega))b^{\prime}$
$\displaystyle-2(-(-3+2r)(16\pi+3\lambda-\lambda\omega)-r(\lambda+16\omega\pi-3\omega\lambda)b^{\prime\prime})))=0.$
(23)

It is very difficult to find the analytic solution of above differential
equation without assuming any condition. However, one may find the solution of
this equation by taking some assumptions into account. Since first term of
Eq.(23) vanishes when $\omega=\frac{16\pi+3\lambda}{\lambda}$, $\forall$
$\lambda$, so by taking this condition, it is easy to obtain the solution:
$b(r)=A,~{}e^{2\phi(r)}=1-\frac{A}{r}$ and $\rho=p=0$ which represents the
Schwarzschild solution. It is worthy to mention here that Schwarzschild
solution describes the non-traversable wormhole 30 . Next we consider the
power law form of shape function given by $b(r)=\frac{A}{r^{n}}$ which is a
viable choice as it ensures the validity of M-T constraint given by
$\frac{b(r)}{r}\leq 1$ for $n>-1$ and $r\rightarrow\infty$. On substitution of
this choice $b(r)=\frac{A}{r^{n}}$ into Eq.(23), it is easy to check that this
equation will be satisfied when the constraints $n=-3$ or -$\frac{3}{2}$ hold
(for $\omega=\frac{16\pi+3\lambda}{\lambda}$ and $\lambda=\pm 4\sqrt{2}\pi$).

###  II.4 Spherical Wormhole with $e^{\phi(r)}$ = $(\frac{r}{r_{0}})^{\beta}$
in $f(R,T)$ Gravity

In this subsection, we shall study the spherical wormhole solutions by
imposing the most general form of red shift function given by
$e^{\phi(r)}=(\frac{r}{r_{0}})^{\beta}$, where $\beta$ is any arbitrary
constant and try to explore the corresponding form of shape function. By
inserting $e^{\phi(r)}=(\frac{r}{r_{0}})^{\beta}$ in Eq.(13), we obtain the
shape function given by

$\displaystyle
b(r)=\frac{\beta^{2}-2\beta}{\beta(\beta-2)-1}r-cr^{-\frac{2\beta^{2}-5\beta-3}{1+\beta}},$
(24)

where $c$ is an integration constant.

Figure 6: Evolution of energy density versus $r$ and NEC. The left plot
indicates the behavior of energy density. In the right panel, the gray plot
indicates the behavior of $\rho+p_{r}$ and orange plot shows the behavior of
$\rho+p_{t}$ for $r_{0}=1$ and $\beta=.6$.

Figure 7: Evolution of SEC ($\rho+p_{r}+2p_{t}$) and DEC ($\rho-p_{i}$) versus
$r$. In the left panel, the plot indicates the behavior of SEC and in the
right panel, the plot indicates the behavior of DEC, where $\rho-p_{r}$ (gray)
and $\rho-p_{t}$ (orange) for $r_{0}=1$ and $\beta=.6$.

By considering the wormhole condition $b(r_{0})=r_{0}$ in Eq.(24), we obtain
the following expression for wormhole metric, $\rho$ and $p$:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(\frac{r}{r_{0}})^{2\beta}dt^{2}-\frac{dr^{2}}{1-\frac{\beta^{2}-2\beta-2}{\beta^{2}-2\beta-1}-c(\frac{r}{r_{0}})^{-\frac{2(\beta^{2}-2\beta-1)}{1+\beta}}}-r^{2}d\Omega^{2},$
(25)
$\displaystyle\rho=\frac{1}{12r^{2}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}\frac{6\beta(8\pi(-2+\beta)+(-3+\beta)\lambda)}{\beta(\beta-2)-1}+\frac{12(\frac{r}{r_{0}})^{-2\frac{\beta(\beta-2)-1}{1+\beta}}(1+2\beta)(4\pi(-3+\beta)+(-2+\beta)\lambda)}{(1+\beta)(\beta(\beta-2)-1)}\bigg{]},$
$\displaystyle
p=\frac{1}{12r^{2}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}-\frac{6\beta(8\pi\beta+\lambda+\lambda\beta)}{\beta(\beta-2)-1}+\frac{12(\frac{r}{r_{0}})^{-2\frac{\beta(\beta-2)-1}{1+\beta}}(1+2\beta)(4\pi(1+\beta)+\beta\lambda)}{(1+\beta)(\beta(\beta-2)-1)}\bigg{]}.$

After the re-scaling $\zeta^{2}=(\beta(\beta-2)-1)r^{2}$, the metric (25) can
be written as

$\displaystyle
ds^{2}=(\frac{\zeta}{\zeta_{0}})^{2\beta}dt^{2}-\frac{d\zeta^{2}}{(\frac{\zeta}{\zeta_{0}})^{-2\frac{\beta(\beta-2)-1}{1+\beta}}-1}-\frac{\zeta^{2}}{(\beta(\beta-2)-1)}(d\theta^{2}+\sin^{2}\theta
d\Phi^{2}).$ (26)

As we are interested in exploring the impact of matter terms (trace of energy-
momentum tensor) involved in this theory, therefore we will fix the value of
$\beta$ and plot $\rho$ in a wider range of $\lambda$. Moreover, we choose
$r_{0}=1$ and $\beta=0.6$ and analyze all energy conditions. It is easy to see
that these constraints are valid for $\lambda>-4\pi$ as shown in the graphs of
Figures 6 and 7.

##  III Field Equations for hyperbolic wormhole space time in $f(R,T)$ Gravity

In this section, we shall describe different cases of wormhole construction by
taking hyperbolic spacetime into account. The spacetime representing wormhole
geometry in hyperbolic coordinates is defined as

$ds^{2}=e^{2\Phi(r)}dt^{2}-\frac{dr^{2}}{1-b(r)/r}-r^{2}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}),$ (27)

where $\Phi$ and $b(r)$ have the usual meaning, i.e., wormhole red shift and
shape functions, respectively. From Eqs.(4) and (27), we obtain the following
set of field equations:

$\displaystyle\frac{b^{\prime}-2}{r^{2}}$ $\displaystyle=$
$\displaystyle(8\pi+\lambda)\rho+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}),$
(28)
$\displaystyle\frac{2\phi^{{}^{\prime}}}{r}(1-\frac{b}{r})-\frac{b}{r^{3}}+\frac{2}{r^{2}}$
$\displaystyle=$
$\displaystyle(8\pi+\lambda)(-p_{r})+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}),$
(29)
$\displaystyle\frac{1}{2r}\bigg{[}\frac{1}{r}(\phi^{\prime}b+b^{\prime}-\frac{b}{r})+2(\phi^{\prime\prime}+(\phi^{\prime})^{2})b-\phi^{\prime}(2-b^{\prime})\bigg{]}$
$\displaystyle=$
$\displaystyle(8\pi+\lambda)(-p_{t})+\frac{\lambda}{2}(\rho-\frac{p_{r}+2p_{t}}{3}).$
(30)

From the above equations, the explicit form of energy density, radial and
tangential pressures denoted by symbols $\rho,~{}p_{r}$ and $p_{t}$,
respectively, can be obtained as follows

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}(48(-2+b^{\prime})\pi
r+\lambda(b(2-r\phi^{\prime}-2r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))+r(b^{{}^{\prime}}(8-r\phi^{{}^{\prime}})+2(-10+r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))))\bigg{]},$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}b(48\pi(1+2r\phi^{{}^{\prime}})+\lambda(10+23r\phi^{{}^{\prime}}+2r^{2}((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})))-r(96\pi(1+r\phi^{{}^{\prime}})$
(32)
$\displaystyle+\lambda(-b^{{}^{\prime}}(4+r\phi^{{}^{\prime}})+2(14+12r\phi^{{}^{\prime}}+r^{2}((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))))\bigg{]},$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}-b\bigg{[}24\pi(-1+r\phi^{{}^{\prime}}+2r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}}))+\lambda(-4+7r\phi^{{}^{\prime}}+10r^{2}((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})))\bigg{]}+$
(33) $\displaystyle
r\bigg{[}-b^{{}^{\prime}}(24\pi(1+r\phi^{{}^{\prime}})+\lambda(2+5r\phi^{{}^{\prime}}))+2(24r\pi(\phi^{{}^{\prime}}+r(\phi^{{}^{\prime}})^{2}+r\phi^{{}^{\prime\prime}})+\lambda(-2+6r\phi^{{}^{\prime}}+5r^{2}((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})))\bigg{]}\bigg{]},$

where the prime denotes the derivatives with respect to radial coordinate. It
is worthy to mention here that the conservation equation for hyperbolic
spherically symmetric spacetime is same as for symmetric metric and hence
given by Eq.(12). It is easy to check that we have system of three equations
with five unknown functions. To develop solutions representing hyperbolic
wormholes, we have to fix two of these unknowns. In this work, we will solve
these equations by considering the following three possibilities:

  * •

EoS for radial or tangential pressure;

  * •

Condition for isotropic pressure, i.e., $p_{r}=p_{t}$;

  * •

Some known and interesting choices for $\phi(r)$ and $b(r)$.

In the following subsections, we shall discuss these cases separately and
present the graphical analysis of obtained solutions.

###  III.1 Hyperbolic Wormhole with Equation of State in $f(R,T)$ Gravity

In this subsection, we shall try to explore the existence of pseudospherical
wormhole solutions filled by the matter with anisotropic pressure by taking
the following EoS for the radial pressure into account:

$\displaystyle p_{r}=\omega\rho.$ (34)

From Eq.(LABEL:28) and the above EoS parameter, it can be written as

$\displaystyle\frac{1}{r(4\pi+\lambda)(8\pi+\lambda)}\bigg{[}-b(48\pi(1+2r\phi^{{}^{\prime}})+\lambda(-r\phi^{{}^{\prime}}(-23+\omega)-2(-5+\omega)+2r^{2}((\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})(1+\omega)))$
$\displaystyle+r(48\pi(2+2r\phi^{{}^{\prime}}+(-2+b^{{}^{\prime}})\omega)+\lambda(-b^{{}^{\prime}}(4-8\omega+r\phi^{{}^{\prime}}(1+\omega))+2(14+12r\phi^{{}^{\prime}}-10\omega+r^{2}(\phi^{{}^{\prime}})^{2}+\phi^{{}^{\prime\prime}})(1+\omega)))))\bigg{]}=0.$

For zero tidal force wormhole, one needs to impose the condition $\phi(r)=0$
and consequently, we obtain

$\displaystyle
b(r)=2r+A(2r(\lambda-12\pi\omega-2\lambda\omega))^{\frac{24\pi-\lambda(-5+\omega)}{24\pi\omega+\lambda(-2+4\omega)}},$
(35)

where $A$ is an integration constant. For this shape function, the graphical
illustration of different axioms which are necessary for a viable wormhole
geometry is provided in Figure 8.

Figure 8: Evolution of shape function given by Eq.(ref31) to check its
consistency with the axioms. Here, shape function (blue dashes),
$b^{\prime}(r)$ (red dashes), $b(r)-r$ (gray dashes) and $\frac{b(r)}{r}$
(black), where the chosen parameters are $\omega=-16,~{}\lambda=-15$ and
$r_{0}=1$.

Figure 9: Evolution of energy density and NEC ($\rho+p_{i}$) versus $r$. In
the left panel, the plot indicates the behavior of energy density. In the
right panel, the gray plot shows the behavior of NEC ($\rho+p_{r}$) and orange
plot shows the behavior of NEC ($\rho+p_{t}$), where $\omega=-16$ and
$r_{0}=1$.

Figure 10: Evolution of SEC ($\rho+p_{r}+2p_{t}$) and DEC ($\rho+p_{i}$)
versus $r$. In the left panel, the plot indicates the behavior of SEC. In the
right panel, the gray plot shows the behavior of DEC ($\rho-p_{r}$) and orange
plot shows the behavior of DEC ($\rho-p_{t}$) for $\omega=-16$ and $r_{0}=1$.

Consequently, one may write wormhole metric, energy density and tangential
pressure as follows

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-\frac{dr^{2}}{(\frac{r}{r_{o}})^{-\frac{24\pi(\omega-1)+\lambda(7-5\omega)}{24\pi\omega-2\lambda+4\lambda\omega}}-1}-r^{2}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}),$ (36) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{3r_{0}(\frac{r}{r_{o}})^{\frac{24\pi-\lambda(-5+\omega)}{24\pi\omega-2\lambda+4\lambda\omega}}}{r^{3}(24\pi\omega-2\lambda+4\lambda\omega)},$
(37) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle-\frac{3r_{0}(-1+\omega)(\frac{r}{r_{o}})^{\frac{24\pi-\lambda(-5+\omega)}{24\pi\omega-2\lambda+4\lambda\omega}}}{2r^{3}(24\pi\omega-2\lambda+4\lambda\omega)}.$
(38)

Next we shall describe the energy conditions of wormhole matter content
graphically with respect to radial coordinate. These graphs are provided in
Figures 9 and 10. The left panel of Figure 9 provides the validity of WEC,
i.e., $\rho\geq 0$ which indicates that energy density remains positive for
$\lambda<0$. From the right panel of same graph, it is easy to observe that
NEC for both radial and tangential pressures remain invalid. Likewise, the
left and right graphs of Figure 10 shows that SEC violates while DEC remains
valid for both radial and tangential pressures. Thus, it can be concluded that
the WEC and DEC are valid in the present setup whereas the SEC and NEC are
incompatible for the choice $\lambda<0$.

Figure 11: Evolution of energy density ($\rho$) and NEC ($\rho+p_{i}$) versus
$r$. In the left panel, the right plot represents the behavior of energy
density ($\rho$). In the right panel, the orange plot shows the behavior of
$\rho+p_{r}$ and gray plot represents the behavior of $\rho+p_{t}$ for
$\alpha=0.5$ and $r_{0}=1$.

###  III.2 Zero tidal force Hyperbolic Wormholes along with Isotropic fluid or
a Power Law Shape Function in $f(R,T)$ Gravity

In this section, we shall construct wormhole geometry by considering two
cases. In the first case, we shall construct M-T hyperbolic wormhole by taking
an interesting choice of shape function given by
$b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}$ along with zero red shift function,
i.e., $\phi(r)=0$. The graphical illustration of this shape function has
already been discussed in the previous section which indicated that it is
viable choice for $\alpha<0$. It is worthy to mention here that in case of
hyperbolic geometry, the behavior of this shape function is accurate and
consistent with all axioms for $\alpha>0$. Inserting the value of red shift as
well as shape function in Eq.(34), we obtain the metric, $\rho$ and $p$ in the
following form:

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-\frac{dr^{2}}{1-(\frac{r}{r_{o}})^{\alpha-1}}-r^{2}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}),$ (39) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\pi(-48r+24\alpha
r_{0}(\frac{r}{r_{o}})^{\alpha})+(-10r+(1+4\alpha)r_{0}(\frac{r}{r_{0}})^{\alpha})\lambda}{6r^{3}(4\pi+\lambda)(8\pi+\lambda)},$
(40) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{\pi(-24\pi-4\lambda)r_{0}((\frac{r}{r_{o}})^{\alpha})+r(-4\lambda+(24\pi+2\lambda)\alpha(\frac{r}{r_{0}})^{\alpha})}{12r^{3}(4\pi+\lambda)(8\pi+\lambda)}.$
(41)

For the obtained solutions, we plot the models of all energy conditions versus
radial coordinates as shown in Figures 11 and 12. From these graphs, it can be
seen that all energy conditions are valid for $\lambda<-30$ except the DEC
($\rho-p_{r}$) for radial pressure.

Figure 12: Evolution of SEC ($\rho+p_{r}+2p_{t}$) and DEC ($\rho-p_{i}$)
versus $r$. In the left panel, the plot shows the behavior of SEC. In the
right panel, gray plot represents the behavior of DEC ($\rho-p_{r}$) and
yellow plot represents the behavior of DEC ($\rho-p_{t}$) with $\alpha=0.5$
and $r_{0}=1$.

In the second case, we will construct new pseudospherical wormholes with
isotropic pressure by imposing the condition given by $p_{r}=p_{t}$. For
hyperbolic spacetime, using Eq.(LABEL:28), we obtain the following
differential equation:

$\displaystyle\phi^{{}^{\prime\prime}}+\phi^{{}^{\prime
2}}+\frac{(b^{{}^{\prime}}r-3r+5b)\phi^{{}^{\prime}}}{2r(b-r)}=\frac{-b^{{}^{\prime}}r-b+4r}{2r^{2}(b-r)}$
(42)

which must be satisfied by the shape and red shift functions. This
differential equation can be re-arranged and the shape function can be
determined as

$\displaystyle
b(r)=\bigg{(}\int\frac{2(2+3r\phi^{{}^{\prime}}+r^{2}(\phi^{{}^{\prime}2}+\phi^{{}^{\prime\prime}}))e^{-\int\frac{1+2r^{2}(\phi^{{}^{\prime}2}+\phi^{{}^{\prime\prime}})+5r\phi^{{}^{\prime}}}{r(1+r\phi^{{}^{\prime}})}dr}}{1+r\phi^{{}^{\prime}}}dr+B\bigg{)}\times
e^{-\int\frac{1+2r^{2}(\phi^{{}^{\prime}2}+\phi^{{}^{\prime\prime}})+5r\phi^{{}^{\prime}}}{r(1+r\phi^{{}^{\prime}})}dr},$
(43)

where $B$ is an integration constant. Now we have two set of differential
equations with four unknowns namely shape function, red shift function,
density and pressure. For solving these equations, one should use some valid
assumptions. In the present work, we pick an interesting choice of red shift
function given by $\phi(r)=\phi_{0}$=constant (zero tidal force). By inserting
this condition in Eq.(43), we obtain a spacetime with constant curvature
defined by $b(r)=2r+\frac{B}{r}$. In this case, the wormhole spacetime, energy
density and isotropic pressure can be written as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-\frac{dr^{2}}{(\frac{r}{r_{0}})^{-2}-1}-r^{2}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}),$ $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{(r_{0})^{2}}{2r^{4}(4\pi+\lambda)},$ $\displaystyle p_{r}$
$\displaystyle=$ $\displaystyle
p_{t}=-\frac{(r_{0})^{2}}{2r^{4}(4\pi+\lambda)}.$

Also, by evaluating the flaring out condition at throat, we obtain
$b^{\prime}(r_{0})=2\nless 1$ which shows that no zero tidal force hyperbolic
wormhole exists in $f(R,T)$ gravity.

###  III.3 Hyperbolic Wormhole with Non-vanishing RedShift Function and
Isotropic Pressure in $f(R,T)$

Here we shall consider an interesting power law form of red shift function
given by $e^{\phi(r)}=(\frac{r}{r_{0}})^{\beta}$ with $\beta$ as some
arbitrary constant. By inserting the value of $e^{\phi(r)}$ in Eq.(42), we
obtain the following form of shape function given by

$\displaystyle
b(r)=\frac{\beta^{2}-2\beta-2}{\beta(\beta-2)-1}r-cr^{-\frac{2\beta^{2}-5\beta-3}{1+\beta}},$
(44)

where $c$ is a constant of integration. The graphical behavior of this shape
function is shown in Figure 13 which indicates that all axioms, essential for
a valid shape function, are satisfied.

Figure 13: Evolution of shape function satisfying all axioms and energy
density versus $r$. In the left panel, shape function (blue plot),
$b^{\prime}(r)$ (red plot), $b(r)-r$ (gray plot) and $\frac{b(r)}{r}$ (black
plot). In the right panel, the plot represents the behavior of energy density
where the chosen free parameters are $r_{0}=.1$ and $\beta=3.5$.

By imposing, the wormhole condition $b(r_{0})=r_{0}$, Eq.(44) leads to the
following expression for wormhole metric, $\rho$ and $p$:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(\frac{r}{r_{0}})^{2\beta}dt^{2}-\frac{(\beta(\beta-2)-1)}{(\frac{r}{r_{0}})^{-2\frac{\beta(\beta-2)-1}{1+\beta}}-1}dr^{2}-r^{2}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}),$ $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{6r^{3}(1+\beta)(-1-2\beta+\beta^{2})(8\pi+\lambda)(4\pi+\lambda)}\bigg{[}(24\pi(r\beta(2+\beta+\beta^{2})+(\frac{r}{ro})^{(3+5\beta-2\beta^{2})/(1+\beta)}r_{0}(-3-5\beta+2\beta^{2}))$
$\displaystyle+$
$\displaystyle(r\beta(9+5\beta-4\beta^{2})+(\frac{r}{ro})^{(3+5\beta-2\beta^{2})/(1+\beta)}r_{0}(-13-21\beta+10\beta^{2})\lambda)\bigg{]},$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{6r^{3}(1+\beta)(-1-2\beta+\beta^{2})(8\pi+\lambda)(4\pi+\lambda)}\bigg{[}(24\pi(1+\beta)(r\beta^{2}+(\frac{r}{ro})^{(3+5\beta-2\beta^{2})/(1+\beta)}r_{0}(1+2\beta))$
$\displaystyle+$
$\displaystyle(r\beta(3+7\beta+4\beta^{2})+(\frac{r}{ro})^{(3+5\beta-2\beta^{2})/(1+\beta)}r_{0}(1+9\beta+14\beta^{2})\lambda)\bigg{]}.$

By the re-scaling $\zeta^{2}=(\beta(\beta-2)-1)r^{2}$ of the above wormhole
metric, it can be written as

$\displaystyle
ds^{2}=(\frac{\zeta}{\zeta_{0}})^{2\beta}dt^{2}-\frac{d\zeta^{2}}{(\frac{\zeta}{\zeta_{0}})^{-2\frac{\beta(\beta-2)-1}{1+\beta}}-1}-\frac{\zeta^{2}}{(\beta(\beta-2)-1)}(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}).$ (45)

Since our main objective is to explore the impact of matter terms in the
considered gravitational theory, therefore we fix the value of $\beta$ and
plot energy density by varying the value of $\lambda$. Moreover, we choose
$r_{0}=1$ and $\beta=3.5$. We analyze the energy conditions graphically for
the obtained model. From the right graph of Figure 13, it is seen that the
energy density remain positive versus radial coordinate. From the graphs of
Figure 14, it can be observed that the NEC ($\rho+p_{r}\geq 0$) is valid for
$\lambda<-35$ while for tangential pressure, it is invalid. From the same
Figure, it is seen that the validity of DEC ($\rho-p_{i}\geq 0$) is guaranteed
for tangential pressure, whereas SEC violates for $\lambda<-35$ as shown in
Figure 15).

Figure 14: Evolution of NEC ($\rho+p_{i}$) and DEC ($\rho-p_{i}$) versus $r$.
In the left panel, $\rho+p_{r}$ (yellow plot) and $\rho+p_{t}$ (gray plot)
while, in the right panel, $\rho-p_{r}$ (yellow plot) and $\rho-p_{t}$ (gray
plot) for $r_{0}=.1$ and $\beta=3.5$. Figure 15: Evolution of SEC
($\rho+p_{r}+2p_{t}$) versus radial coordinate.

##  IV Conclusion and Final Remarks

The construction of viable wormhole geometries in modified gravity theories
has always been a fascinating topic for researchers. In the present work, we
propose interesting exact solutions for both classes of static spherically
symmetric and pseudospherically symmetric spacetimes representing wormholes in
the well-famed $f(R,T)$ gravitational framework. The considered gravitational
theory is an interesting choice in this respect as it allows the interaction
of curvature and matter. For the sake of simplicity, we have considered linear
model of $f(R,T)$ function representing minimal interaction of matter and
curvature. Our primary objective is to check what ranges of coupling parameter
$\lambda$, present in $f(R,T)$ function, can allow the existence of wormholes
as well as the validity of energy condition bounds. In this work, we have
obtained some specific solutions either by considering EoS or by assuming some
interesting choices of red shift or shape functions. The obtained solutions
are then examined graphically and the validity of energy condition bounds have
been explored in each case. It is seen that for all obtained spherically
symmetric wormholes, we have positive energy density at the throat and radial
pressure is negative for $\lambda>-4\pi$ and $\lambda<-8\pi$ satisfying all
axioms for shape function. For pseudospherically symmetric wormhole, energy
density is positive for all conditions and radial pressure may be positive or
negative for different ranges of $\lambda$.

For zero tidal force wormhole one needs to assume $\phi(r)=$constant, and we
have discussed two possibilities. Firstly, we obtain shape function using the
main condition for perfect fluid, i.e., $p_{r}=p_{t}$. Secondly, we have
imposed a restricted choice of shape function for both spherically and
hyperbolic symmetric metrics. In both cases, energy density remains positive
and radial pressure is negative for $\lambda>-4\pi$, but for pseudospherically
symmetric wormhole, the energy density is also positive and radial pressure is
negative for $\lambda>-4\pi$ in first possibility. But for the restricted
choice of shape function, the energy density and pressure are positive for
$\lambda\leq-10\pi$.

We have also discussed a possibility with isotropic fluid using the linear
EoS. We have obtained the shape function by imposing zero-tidal force
condition for red shift function in both symmetric and pseudospherically
symmetric wormhole cases by taking linear EoS into account. We have observed
that energy density is positive and radial pressure is negative for
$\lambda>-4\pi$ for spherically symmetric wormhole. Whereas, for hyperbolic
wormhole energy density remains positive and pressure turns out as negative
for $\lambda<0$. Lastly, we have imposed a restricted choice of red shift
function to evaluate the most general form of $b(r)$ using isotropic pressure
condition. For this condition, in case of symmetric wormhole, same results
have been obtained for energy density and radial pressure for $\lambda>-4\pi$.
In case of hyperbolic wormhole, both energy density and radial pressure are
positive for $\lambda\leq-12\pi$.

At the end, we have applied the energy conditions to material content of
symmetric and pseudospherically symmetric wormhole solutions for all
restricted choices of red shift and shape functions as well as linear equation
of state with isotropic pressure. One can easily observe from all figures that
either NEC, DEC and SEC are violated or valid for different ranges of
$\lambda$ while WEC is always valid for both symmetric and pseudospherically
symmetric wormholes. A brief comparison of both classes of wormholes is listed
in Table 1. In this table, we indicate that which energy condition is
satisfied or remained invalid corresponding to specified wormhole geometry,
which can be explained as follows:

Metric | Constraints | WEC ($\rho\geq 0$) | NEC ($\rho+p_{r}\geq 0$) | DEC ($\rho-p_{r}\geq 0$) | SEC ($\rho+p_{r}$
---|---|---|---|---|---
|  |  | ($\rho+p_{t}\geq 0$) | ($\rho-p_{t}\geq 0$) |  $+2p_{t}\geq 0$)
$ds^{2}=dt^{2}-\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\alpha-1}}-r^{2}d\Omega^{2}$ |  $\phi(r)$=constant, | Yes | No | Yes | Yes
|  $b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}$, |  | Yes | No |
| $\lambda<-8\pi$ |  |  |  |
$ds^{2}=dt^{2}-\frac{dr^{2}}{(\frac{r}{r_{0}})^{\frac{-(1+\omega)(\lambda_{1})}{\lambda_{2}}}-1}$ |  $\phi(r)=0$, | Yes | No | Yes | No
$-r^{2}d\Omega^{2}$, $\lambda_{1}=12\pi+2\lambda$ |  $b(r)=r_{0}(\frac{r}{r_{0}})^{\frac{-3\lambda_{3}}{\lambda_{2}}}$, |  | No | Yes |
$\lambda_{2}=12\pi\omega-\lambda+2\omega\lambda$ |  $\lambda_{3}=4\pi+\lambda$, |  |  |  |
|  $p_{r}=\omega\rho$, |  |  |  |
| $\lambda>-4\pi$ |  |  |  |
$ds^{2}=(\frac{r}{r_{0}})^{2\beta}dt^{2}-\frac{\beta_{1}dr^{2}}{(\frac{r}{r_{0}})^{\frac{-2\beta_{1}}{1+\beta}}-1}$ |  $\phi(r)=\ln(\frac{r}{r_{0}})^{2\beta}$, | Yes | Yes | Yes | No
$-r^{2}d\Omega^{2}$, $\beta_{1}=\beta^{2}-2\beta-1$ | $b(r)=\frac{(\beta^{2}-2\beta)r}{\beta_{1}}$ |  | Yes | Yes |
|  $-\frac{r_{0}}{\beta_{1}}(\frac{r}{r_{0}})^{\frac{-(2\beta+1)*(\beta-3)}{1+\beta}}$, |  |  |  |
| $\lambda>-4\pi$ |  |  |  |
$ds^{2}=dt^{2}-\frac{dr^{2}}{(\frac{r}{r_{0}})^{-\frac{\varsigma_{2}}{\varsigma_{1}}}-1}-$ |  $\phi(r)=0$, | Yes | No | Yes | No
$r^{2}d\Omega_{ps}^{2}$, $\varsigma_{1}=24\pi\omega-2\lambda+4\lambda\omega$ |  $b(r)=2r-r_{0}(\frac{r}{r_{0}})^{\frac{\varsigma_{3}}{\varsigma_{1}}}$, |  | No | Yes |
$\varsigma_{2}=-24\pi(\omega-1)+\lambda(7-5\omega)$ |  $\varsigma_{3}=24\pi+\lambda(-5+\omega)$, |  |  |  |
|  $p_{r}=\omega\rho$, |  |  |  |
| $\lambda<0$ |  |  |  |
$ds^{2}=dt^{2}-\frac{dr^{2}}{1-(\frac{r}{r_{0}})^{\alpha-1}}-$ |  $\phi(r)=0$, | Yes | Yes | No | Yes
$r^{2}d\Omega_{ps}^{2}$ | $b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}$ |  |  |  |
| $\lambda\leq-10\pi$ |  |  |  |
$ds^{2}=(\frac{r}{r_{0}})^{2\beta}dt^{2}-\frac{\beta_{1}dr^{2}}{(\frac{r}{r_{0}})^{\frac{-2\beta_{1}}{1+\beta}}-1}$ |  $\phi(r)=\ln(\frac{r}{r_{0}})^{2\beta}$, | Yes | Yes | No | No
$-r^{2}d\Omega_{ps}^{2}$, $\beta_{1}=\beta^{2}-2\beta-1$ | $b(r)=\frac{(\beta^{2}-2\beta-2)r}{\beta_{1}}$ |  | No | Yes |
|  $-\frac{r_{0}}{\beta_{1}}(\frac{r}{r_{0}})^{\frac{-(2\beta+1)*(\beta-3)}{1+\beta}}$, |  |  |  |
| $\lambda\leq-12\pi$ |  |  |  |
Table 1: In this table, we have listed all obtained wormhole solutions. It
indicates that which of the energy conditions, i.e., WEC, NEC, DEC and SEC are
valid. It is worthy to mention here that for wormhole with isotropic pressure,
the SEC becomes $\rho+3p$. Also, here
$d\Omega^{2}=d\theta^{2}+r^{2}\sin^{2}\theta d\Phi^{2}$ and
$d\Omega_{ps}^{2}=d\theta^{2}+r^{2}\sinh^{2}\theta d\Phi^{2}$.

  1. 1.

In case of spherical wormhole,

     * •

when $\phi(r)$ is constant and $b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}$, the
WEC, SEC, NEC with transverse pressure and DEC with radial pressure are valid
for $\lambda<-8\pi$, while the NEC with radial pressure and DEC with
transverse pressure are not satisfied for $\lambda<-8\pi$.

     * •

For $\phi(r)=0$ and
$b(r)=r_{0}(\frac{r}{r_{0}})^{\frac{-3(4\pi+\lambda)}{12\pi\omega\lambda+2\omega\lambda}}$,
the WEC and DEC are satisfied for $\lambda>-4\pi$, while the NEC and SEC are
not satisfied for $\lambda>-4\pi$.

     * •

For $\phi(r)=\ln(\frac{r}{r_{0}})^{2\beta}$ and
$b(r)=\frac{\beta^{2}-2\beta}{\beta^{2}-2\beta-1}r-\frac{r_{0}}{\beta_{1}}(\frac{r}{r_{0}})^{\frac{-(2\beta+1)(\beta-3)}{1+\beta}}$,
all energy constraints are valid except SEC for $\lambda>-4\pi$.

  2. 2.

In case of hyperbolic wormhole,

     * •

when $\phi(r)=0$ and
$b(r)=2r-r_{0}(\frac{r}{r_{0}})^{\frac{24\pi-5\lambda+\omega\lambda}{24\pi\omega-2\lambda+4\lambda\omega}}$,
the WEC and DEC are valid for $\lambda<0$, while the NEC and SEC are not valid
for $\lambda<0$.

     * •

For $\phi(r)=0$ and
$b(r)=r_{0}(\frac{r}{r_{0}})^{\alpha}{12\pi\omega\lambda+2\omega\lambda}$, the
WEC, NEC and SEC are valid, while the DEC is violated for $\lambda\leq-10\pi$.

     * •

For $\phi(r)=\ln(\frac{r}{r_{0}})^{2\beta}$ and
$b(r)=\frac{\beta^{2}-2\beta-2}{\beta^{2}-2\beta-1}r-\frac{r_{0}}{\beta_{1}}(\frac{r}{r_{0}})^{\frac{-(2\beta+1)(\beta-3)}{1+\beta}}$,
the WEC, the NEC with radial pressure and the DEC with transverse pressure are
satisfied for $\lambda\leq-12\pi$, while the SEC and NEC, and DEC with other
components of pressure are not satisfied for $\lambda\leq-12\pi$.

Many authors have studied the topic of wormholes existence in the framework of
$f(R,T)$ theory. In this context, Sahoo and his collaborators 8  studied the
spherically symmetric wormhole solution for phantom case, i.e., $\omega<-1$ in
$f(R,T)$ modified gravity. They adopted the EoS given by $p_{r}=\omega\rho$ to
obtain the form of shape function which can satisfy all the necessary axioms
for the existence of wormhole geometry and also examined corresponding energy
condition bounds. Similarly, Moraes and Sahoo 9  proposed the wormhole
solution by applying the power law form of shape function and analyzed all
energy constraints in exponential $f(R,T)$ gravity. In another study, Zubair
et al. 21  proposed wormhole geometries by adopting simple linear as well as
cubic forms of $f(R,T)$ function and further they considered the non-
commutative geometrical aspects of string theory in the context of $f(R,T)$
gravity. In a paper z32 , the researchers have explored the idea of viable
charged wormhole solutions in $f(R,T)$ gravity. They have assumed simple
linear generic model given by $f(R,T)=R+2T$ along with the ordinary matter as
the total pressure of anisotropic fluid. Further, the existence of static
wormhole model have been explored by by some researchers where they utilized
different kind of shape functions z33 . In another paper z34 , authors have
investigated the wormhole modeling by considering a specific general shape
function in the quadratic $f(R,T)$ gravity. In a recent papers z35 , Banerjee
and his collaborator adopted different strategies to construct wormhole
geometry using isotropic pressure in $f(R,T)$ theory and concluded that all
energy constraints are valid for the proposed models, while Bhar and his
collaborators Bhar , proposed the M-T wormhole solution which admits conformal
motion in $f(R,T)$ gravity. They employed the phantom energy EoS, i.e.,
$p_{r}=\omega\rho$, $\omega<-1$ to constraint their model and showed that
wormhole solution exist for both positive and negative values of coupling
constant $\lambda$.

In majority of the previously listed works, authors have considered a specific
form of shape function and analyze the possibility of static and spherically
symmetric wormhole existence satisfying energy condition bounds. In the
present work, we have explored both the spherically symmetric and
pseudospherically symmetric wormhole solutions for M-T spacetime in $f(R,T)$
theory. The present work can be considered as an extension of the papers 3
and 4 , where authors have discussed similar cases of wormhole existence by
using spherically and pseudospherically symmetric spacetimes in the framework
of GR. In this work, we have not only checked the validity of energy bounds by
taking some type of shape functions into account but also explored the
possible forms of $\phi(r)$ and $b(r)$ functions by taking the matter with EoS
parameter for $p_{r}$ or $p_{t}$ through $f(R,T)$ field equations. Further, we
have presented the validity of energy conditions bounds for both spherical and
hyperbolic wormholes in the presence of single perfect fluid, i.e., a source
of matter with isotropic pressure with a wider range of parameter $\lambda$.
Up to the best of our knowledge, the only wormhole solution discussed in
literature are non-asymptotically flat wormhole with isotropic pressure as
pointed out in the References 28 -30 . In the present work, we have discussed
the asymptotically flat wormholes which may or may not satisfy all energy
constraints with a wider range of $\lambda$ for both types of wormhole
spacetimes.

## Acknowledgments

M. Zubair thank the Higher Education Commission, Islamabad, Pakistan for its
financial support under the NRPU project with grant number
5329/Federal/NRPU/R&D/HEC/2016.

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