.

# Traversable wormhole on the brane with non-exotic matter: a broader view

Rikpratik Sengupta1, Shounak Ghosh2, Mehedi Kalam1 and Saibal Ray3 1
Department of Physics, Aliah University, Kolkata 700160, West Bengal, India  2
Department of Physics, Indian Institute of Engineering Science and Technology,
Shibpur, Howrah 711103, West Bengal, India  3 Department of Physics,
Government College of Engineering and Ceramic Technology, Kolkata 700010, West
Bengal, India.  $^1$<EMAIL_ADDRESS>$^2$
<EMAIL_ADDRESS>$^1$<EMAIL_ADDRESS>$^3$
<EMAIL_ADDRESS>
###### Abstract

In this article, the possibility of construction of a traversable wormhole on
the Randall-Sundrum braneworld with non-exotic matter employing the Kuchowicz
potential has been studied. We have obtained the solution for the shape
function of the wormhole and studied its properties along with validity of
Null Energy Condition (NEC). The junction conditions at the surface of the
wormhole are used to evaluate the model parameters. We also evaluate the
surface density and surface pressure for the wormhole. We study the
geometrical nature of the wormhole and consider the radial and tangential
tidal constraints on a traveller trying to traverse the wormhole. Besides, a
linearized stability analysis is performed to obtain the region of stability
for the wormhole. Our analysis, besides giving an estimate for the bulk
equation of state (EoS) parameter, imposes restrictions on the brane tension,
which is a very essential parameter in braneworld physics, and very
interestingly the restrictions imposed by our physically plausible and
traversable wormhole model are in conformity with those imposed by other
braneworld geometries which are not associated with a wormhole solution.
Besides, it is important to study such constraints imposed by geometrical
objects such as wormholes on any gravity theory operating at high-energy
scales like braneworld, as wormholes are believed to have been formed from
massive compact objects of high energy densities. Also, we go on to justify
that the possible detection of a wormhole may well indicate that we live on a
three-brane universe.

##  1 INTRODUCTION

One of the most fascinating and interesting theoretical solutions for the
spacetime that can be found from Einstein’s General Relativity (GR) is a
wormhole, representing a bridge, i.e., ’throat’ like connection between two
spacetimes of the same universe or two different universes. The geometrical
structure of wormhole is tubular which spreads and turns out as asymptotically
flat at the infinities and having no singularity or event horizon. The concept
of wormhole was first theorized in 1916 [1], though the name was not wormhole
at the time. In 1935, Einstein and Rosen [2] used the theory of GR to
elaborate the idea, suggesting the existence of "bridges" through spacetime,
that connects two isolated areas of spacetime, theoretically creating a
shortcut that could reduce travel time and distance, these shortcuts were
termed as Einstein-Rosen bridges, or wormholes. The mathematical structure of
wormhole was first obtained by Fuller and Wheeler [3] by solving the Einstein
Field Equation (EFE) for spherically symmetric spacetime. They have shown that
such solutions were unstable as the throat of the wormhole would have pinched
off, due to this any traversing signal can be trapped in an infinite curvature
region, that makes the wormhole non-traversable. The idea of traversable
wormholes was introduced first by Morris and Thorne [4], where they studied
the wormhole solutions as a tool for understanding of GR in a simpler way.
According to their prescription for a stable traversable wormhole, the
corresponding energy-momentum tensor must violate the null energy condition
(NEC) of GR in such a way that pinching-off of the throat can be prevented,
which is known as the flaring out state of the wormhole. Further, a
sustainable wormhole may allow the possibility of time-travel, allowing closed
time-like curves violating causality [5]. The infringement of NEC indicates
the existence of exotic matter in the throat of the wormhole. Though such
‘exotic’ matter has not been observed till date, its presence has been
hypothesized in theories for attempting to explain the present cosmic
acceleration [6, 7, 8, 9]. Lot of researchers have studied the traversable
wormholes admitting exotic matter under the framework of GR which can be found
in [10, 11, 12, 13, 14, 15, 16]. Presence of stable wormhole demands the
violation of NEC in the spacetime which immediately suggests the presence of
exotic matter in it in the context of GR. So, in GR it is not possible to
construct stable wormhole with matter satisfying the NEC. All these factors
motivate us to study the formation of wormhole with ordinary matter under the
framework of braneworld theory of gravity.

Though Einstein’s GR is one of the most successful theory to unreveal various
mysteries of the universe, but it still requires some modification to overcome
the shortcomings in the theoretical as well as observational aspects. The
confirmation of accelerating expansion of the universe based on some
observational evidences as well as the existence of dark matter led to a
theoretical challenge to GR [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In
order to address these questions various modified theories of gravity have
been developed successively with a modification either in the matter-energy
part or in the geometrical part of Einstein field equation. In the context of
modified theories of gravity, one can claim that Einstein’s GR fails to
provide adequate descriptions at large energy scales and a more generalized
action is required for the gravitational field. It was Hilbert who obtained
the field equations using the action principle admitting linear function of
the scalar curvature (or the Ricci Scalar) $R$ in the gravitational
Lagrangian. Most of the modified theories have been developed by incorporating
different functional form of $R$ in the gravitational Lagrangian of the
corresponding Einstein-Hilbert action. There have been number of attempts made
to modify GR at higher energy scales [28, 29] in order to obtain a set of non-
singular solutions [30], and also at energy scales corresponding to the
present epoch of cosmic evolution to accommodate naturally the current cosmic
acceleration without incorporating exotic matter [31, 32].

On the other hand, a class of modified theories of gravity have been developed
considering higher dimensions to overcome the problems in GR. Kaluza and Klein
[33, 34] first attempted such modification by adopting one compact extra
dimension. Through this modification, they were able to unify the interactions
of gravity and electromagnetism. Recently, Randall and Sundrum gave a concrete
form to the idea of braneworld [35, 36] as a higher dimensional approach to
modify GR from a purely geometrical approach [37]. Braneworlds have been used
very effectively as a platform to investigate cosmological [38, 39, 40, 41,
42, 43, 44, 45], astrophysical [46, 47, 48, 49, 50, 51] and collapse [52, 53,
54, 55] problems. Due to the non-locality and non-closure properties of the
modified EFE on the brane, the Minimum Geometric Deformation (MGD) approach
has been introduced [56, 57, 58, 59], which are the key tools for studying
collapse problems on the brane. While the particles and forces, which are
operating within the realm of standard model of particle physics, are believed
to be confined to the brane, the gravitational force is let free to traverse
the higher dimensional embedding spacetime known as the bulk. This propagation
of gravity has many interesting effects on the brane or our observed universe.
To mention a few, the weakness of the gravitational force compared to the
other forces at the fundamental level can be explained by its simultaneous
presence in the higher dimensional embedding space, leaving behind an
effective reduced manifestation on the 3-brane. Also, a possible explanation
of the dark matter can be provided by the gravitational effect of the second
brane in case of the two brane RS- model or simply the gravitational effect of
the bulk in case of single brane model.

Wormhole solutions in the context of modified gravity without accounting for
extra dimensions have been investigated in [60, 61, 62, 63, 64, 65, 66]. In
higher dimensional modified gravity approach, wormhole solutions can be found
in [67, 68, 69, 70, 71, 72, 73]. The possibility of existence of wormholes in
the central and outer region of galactic halos has been investigated in [74]
and [75, 76, 77], respectively. Here, we attempt to investigate the
possibility of construction of a traversable wormhole on the braneworld, such
that the metric potential in the temporal direction is of the Kuchowicz type
[78]. The motivation behind the choice of this particular function due to it,s
well-behaved, analytic and non-singular nature over a finite range of the
radial parameter. The Kuchowicz potential has been widely applied for the
modelling compact objects [79, 80, 81] to study their properties which provide
satisfactory results for those objects. This motivates us to extend the idea
of the implementation of this potential function to construction of braneworld
wormholes. Moreover, it can be found in literature [82, 83, 84, 85] that the
traversable wormholes can be constructed from black holes. As black holes form
an undisputed class of compact objects, so it would be particularly
interesting to study the possibility of construction of a traversable wormhole
by using the Kuchowicz potential as the redshift function. The field equations
in RS brane gravity include modifications due to higher dimensional
contribution from the bulk. So it is very difficult to obtain an exact
solution for the wormhole with both the metric potentials unknown. In fact,
this is also the case in many instances when wormhole solutions are obtained
in the relativistic context using exotic matter. Again using the Kuchowicz
metric potential Ghosh et al. [79] has obtain a set of acceptable regular
solution for gravastars without having the central singularity at the
interior. In a similar way with the choice of Kuchowicz potential as redshift
function for the construction of wormhole may be done as the throat of the
wormhole must not get pinched-off in order to ensure traversability. Also, due
to lack of experimental evidence in favour of exotic matter so far, we choose
to consider a linear Equation of State (EoS) corresponding to perfect fluid
matter on the brane, describing ordinary non-exotic matter. Such ordinary
matter does not violate the NEC but due to the modified gravity framework we
are using, the effective modification is induced in the matter source which
may lead to the violation of the NEC even in the absence of the exotic matter.

Here, we have tried to construct solution for a static Lorentzian wormhole on
the brane which does not evolve with time and hence there is no possibility of
expansion of the throat like an Euclidean wormhole. Since the Lorentzian
wormhole is static and local object, we have considered the regularity of the
redshift function at finite $r$ only. Such similar redshift functions like the
Krori-Barua potential have been used to construct static Lorentzian wormhole
in literature[86]. Moreover, we have not assumed any form for the shape
function like many other works and using the Kuchowicz potential as a redshift
function we obtain a solution for the shape function on the brane naturally,
which satisfies all the criteria laid down by Morris and Thorne to represent a
physically traversable wormhole.

It would be extremely satisfactory if the redshift function would have
remained finite at infinite distances from the throat and that is why in many
works in the relativistic context, the redshift function is taken simply to be
a constant or an inversely dependent function of r from the mathematical point
of view, but there is no physical justification for choosing such an ad hoc
redshift function. Most of the well-known metric potentials for spherically
symmetric solutions have this problem of diverging at infinity just like the
Kuchowicz function. We have chosen the Kuchowicz potential as the redshift
function, from the physical motivation of it’s property of providing regular
solution at the interior of gravastar and as it is well-behaved at all points
other than $r$ approaching infinity.

Plenty of wormhole solutions have been studied in the last three decades in
the General Relativistic context, as they emerge naturally as solutions to the
Einstein Field Equations just like black holes or gravitational waves. It is
now largely accepted that in order to construct wormholes in the framework of
General Relativity, exotic matter is required as it violates the Null Energy
Condition (NEC) required to make the wormhole throat stable and prevent it
from collapsing. However, such exotic matter has never been observed and in
the context of particle physics, the requirement for such “beyond the standard
model” exotic matter can be met by the introduction of extra dimensional idea
like the braneworld. For eg., the braneworld models can explain the
observational effect of dark matter without involving any exotic matter, just
by considering the gravitational effect of matter present in the higher
dimensional bulk spacetime[87]. So, we choose to explore the possibility of
construction of a traversable wormhole in a modified gravity context involving
the physical idea of higher dimensional Randall- Sundrum braneworld in the
absence of exotic matter.

There may be a possibility of indirect evidence for wormholes like the one
being considered by us, from experiments in particle physics looking out for
extra dimensions. As we shall see from our analysis, our model can
theoretically constrain some of the fundamental parameters involving higher
dimensions, which may also be constrained from an experimental point of view
in terms of fundamental particle characteristics, if extra spacetime
dimensions are indeed detected.

Recently there have been investigations exploring the possibility of
asymptotically flat, traversable wormholes without using exotic phantom matter
within the context of GR[88, 89]. The model obtained by [88] describes such
wormholes which are $Z_{2}$-symmetric about the throat, supported by a Maxwell
and two Dirac fields without any exotic coupling. However, the model is
plagued with certain unrealistic physical features like the simultaneous
existence of particles and antiparticles at the throat without getting
annihilated, besides nonsmooth metric and matter fields. This physical
inconsistencies were removed by [89] in their latest model by not considering
the $Z_{2}$-symmetry relative to the throat and they found that wormholes
asymmetric relative to the throat also do not require any exotic phantom
matter at the throat to ensure asymptotic flatness or traversability. Moreover
in absence of the $Z_{2}$-symmetry there is no coexistence of particles and
antiparticles at the throat and also the supporting metric and matter fields
exhibit smoothness.

In the present article we have studied the mathematical model of the wormhole
on the brane including the modified EFE on the braneworld, solution for the
wormhole shape function, validity of the NEC on the brane for ordinary perfect
fluid matter and obtaining the possible numerical values of the constants from
the junction conditions for plotting the physical parameters of interest
pertaining to the wormhole. We shall then discuss the physical relevance of
the obtained solutions and plots for the physical parameters and draw
conclusion based on it in the following section.

##  2 Mathematical Model of the Brane Wormhole

In this Section we will try to construct a detailed mathematical model of the
brane wormhole for linear EoS, where one of the metric potentials is of the
Kuchowicz type. We shall start with the modified EFE on the Randall-Sundrum
(RS) II braneworld. We briefly discuss the setup for the higher dimensional
gravity framework that we have used. The modifications to the standard
relativistic field equations for a static wormhole have been presented and
then solved analytically to obtain the shape function.

###  2.1 Modified EFE on the Brane

We begin with the general form of the modified EFE on the 3-brane [37] and
move on to obtain the set of EFEs for the case of a spherically symmetric,
static metric. The RS II braneworld model considers a single 3-brane embedded
in higher dimensional bulk having one extra dimension which is the compact
space $S_{1}/\bf{Z}_{2}$, where $\bf{Z}_{2}$ refers to the orbifold symmetry
or the reflection symmetry.

The general form of the modified EFE on the 3-brane is given as

$G_{\mu\nu}=-\Lambda g_{\mu\nu}+\kappa^{2}T_{\mu\nu}^{mod},$ (1)

which may be expanded as

$G_{\mu\nu}=-\Lambda
g_{\mu\nu}+\kappa^{2}T_{\mu\nu}+\kappa_{5}^{4}S_{\mu\nu}-E_{\mu\nu}.$ (2)

The modified stress-energy tensor $T_{\mu\nu}^{mod}$ contains the corrections
to the standard EFE, besides the matter stress-energy tensor $T_{\mu\nu}$ on
the brane. Here, the 4-dimensional constant have been represented by
$\kappa^{2}=\frac{\sigma\kappa_{5}^{4}}{6}$, where $\sigma$ denotes the brane
tension. In the high energy regime the brane tension is small and tends to
zero for extremely high energies. We take $\kappa^{2}=8\pi G=1$ on the brane.

The effective cosmological constant on the brane is given by

$\Lambda=\frac{\kappa_{5}^{2}}{2}\left(\Lambda_{5}+\frac{\kappa_{5}^{2}\sigma^{2}}{6}\right).$
(3)

In the RS II set-up the bulk is Anti-de Sitter ($AdS_{5}$), making the bulk
cosmological constant $\Lambda_{5}$ negative, and it is fine-tuned with the
brane tension in such a way as to give the value of $\Lambda$ to be zero.

The corrections to the EFE are of two types-(i) Local correction
($S_{\mu\nu}$) which comprises of terms quadratic in the stress-energy tensor
on the brane, and (ii) Non-local correction $(E_{\mu\nu})$ which comprises of
the projection of the bulk Weyl tensor on the brane, transferring the
gravitational effect from the bulk.

The local correction is given by

$S_{\mu\nu}=\frac{TT_{\mu\nu}}{12}-\frac{T_{\mu\alpha}T^{\alpha}_{\nu}}{4}+\frac{g_{\mu\nu}}{24}\left[3T_{\alpha\beta}T_{\alpha\beta}-(T^{\alpha}_{\alpha})^{2}\right],$
(4)

where $T=T^{\alpha}_{\alpha}$ and we consider matter on the brane with the
perfect fluid stress-energy tensor

$T_{\mu\nu}=\rho u_{\mu}u_{\nu}+ph_{\mu\nu}.$ (5)

Here $u_{\mu}$ is the 4-velocity, $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ is
the induced metric on the brane, $\rho$ and $p$ are the energy density and
pressure,respectively.

The non-local correction has the form

${E}_{\mu\nu}=C^{(5)}_{ACBD}n^{C}n^{D}g_{\mu}^{A}g_{\nu}^{B}=-\frac{6}{\sigma}\left[Uu_{\mu}u_{\nu}+Pr_{\mu}r_{\nu}+h_{\mu\nu}\left(\frac{U-P}{3}\right)\right],$
(6)

where $C^{(5)}_{ACBD}$ is the bulk Weyl tensor, $n^{C}$ is unit normal
spacelike vector, $r_{\mu}$ denotes projected radial vector, $U$ and $P$
denote the bulk energy density and bulk pressure respectively. ${E}_{\mu\nu}$
is symmetric and traceless. It has no components orthogonal to the brane.

For a static, spherically symmetric matter distribution on the 3-brane, the
line element is given by

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (7)

The modified EFE on the brane, given by Eq. (2) can be computed as

$\displaystyle
e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=\left(\rho\left(1+\frac{\rho}{2\sigma}\right)+{\frac{6U}{\sigma}}\right),$
(8) $\displaystyle
e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=\left(p+{\frac{\rho\left(p+\frac{\rho}{2}\right)}{\sigma}}+{\frac{2U}{\sigma}}+{\frac{4P}{\sigma}}\right),$
(9) $\displaystyle
e^{-\lambda}\left(\frac{\nu^{\prime\prime}}{2}-\frac{\lambda^{\prime}\nu^{\prime}}{4}+\frac{{\nu^{\prime}}^{2}}{4}+\frac{\nu^{\prime}-\lambda^{\prime}}{2r}\right)=\Bigg{(}p+{\frac{\rho\left(p+\frac{\rho}{2}\right)}{\sigma}}+{\frac{2U}{\sigma}}-{\frac{{2P}}{\sigma}}\Bigg{)}.$
(10)

The brane energy-momentum tensor and the overall effective energy momentum
tensor are both conserved separately. So, the conservation equation on the
brane reads the same as in GR

$\frac{dp}{dr}=-\frac{1}{2}\frac{d\nu}{dr}(p+\rho).$ (11)

###  2.2 Solution to the Wormhole Shape Function

The line element for a static, spherically symmetric wormhole can be written
as

$ds^{2}=-e^{\nu(r)}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (12)

where $b(r)$ denotes the shape function of the wormhole.

Rewriting the above field equations (8)-(10) for a wormhole with static and
spherically symmetric matter distribution, in terms of the shape function
$b(r)$ we get eventually

$\displaystyle\frac{b^{\prime}}{r^{2}}=\rho\left(1+\frac{\rho}{2\sigma}\right)+{\frac{6(X\rho+Y)}{\sigma}},$
(13)
$\displaystyle\left(1-\frac{b}{r}\right)\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=p+{\frac{\rho\left(2p+\rho\right)}{2\sigma}}+{\frac{2(X\rho+Y)}{\sigma}}+{\frac{4\omega(X\rho+Y)}{\sigma}},$
(14)
$\displaystyle\left(1-\frac{b}{r}\right)\left(\nu^{\prime\prime}+{{\nu^{\prime}}^{2}}+\frac{\nu^{\prime}}{r}\right)-\frac{b^{\prime}-b}{2r}\left({\nu^{\prime}}+\frac{1}{r}\right)=p+{\frac{\rho\left(2p+\rho\right)}{2\sigma}}+{\frac{2(X\rho+Y)}{\sigma}}-{\frac{2\omega(X\rho+Y)}{\sigma}}.$
(15)

It can be observed from the Eqs. (8)-(11) that there are six unknowns
parameter and we have only four equations on the brane. In order to solve
these equations we have followed the previous prescriptions in the brane
gravity framework and considered the bulk EoS as $P=\omega U$ [90, 91], where
$\omega$ is (EoS) parameter, and the energy density of the brane and bulk are
related by $U=X\rho+Y$ [90], where $X$ and $Y$ are the constants and
determined from the boundary conditions.

Now, we consider the metric potential $e^{\nu(r)}$ to be of the Kuchowicz type
[78] which is given as

$e^{\nu(r)}=e^{Br^{2}+2\ln C},$ (16)

where $B$ and $C$ are arbitrary constants. The dimension of $B$ is $[L^{-2}]$
and $C$ is a dimensionless constant. The redshift function diverges as $r$
tends to infinity. We are considering a model here for a static wormhole and
as there is no time evolution or expansion of the wormhole throat, so the
finiteness of the redshift function is physically significant in the vicinity
of the wormhole. It can be seen that for all finite values of $r$, the
redshift function exhibits regularity and is capable of producing a well
behaved and stable, traversable wormhole with the obtained shape function
satisfying all the criteria as per the Morris-Thorne prescription, as
discussed in details in the concluding section.

As argued earlier, due to lack of observational confirmation of exotic matter
which is usually preferred for constructing traversable wormholes, we chose to
propose an ansatz for the EoS describing ordinary non-exotic matter on the
brane as follows

$p(r)=\mu\rho(r),$ (17)

where $\mu>0$.

Using Eqs. (11), (16) and (17) we get

$\rho(r)=C_{1}e^{-\frac{(\mu+1)Br^{2}}{2\mu}},$ (18)

where $C_{1}$ is the integration constant which can be determined from the
matching condition. The variation of the energy density $\rho(r)$ on the brane
along the radial distance $r$ is plotted in Fig. 1. The figure clearly
indicates that the matter density at the throat is much higher and gradually
decreasing with respect to the radial parameter attains a minimum value at the
boundary.

Figure 1: Variation of the density with respect to $r$.

The obtained $\rho$ can be plugged in along with Eq. (16) in the field Eq.
(15) to obtain the shape function for the wormhole given by

$\displaystyle
b(r)=\frac{2re^{-2Br^{2}}}{Bk^{4}\sigma}\left(-\frac{\mu{C_{1}}(\sigma\mu
k^{4}-2X(\omega-1)){e^{\frac{Br^{2}(3\mu-1)}{2\mu}}}}{(3\mu-1)(2Br^{2}+1)}+\frac{k^{4}}{(\mu-1)(2Br^{2}+1)}\left(-\frac{{C_{1}}^{2}(2\mu+1)\mu{e^{{\frac{Br^{2}(\mu-1)}{\mu}}}}}{4}\right.\right.$
$\displaystyle\left.\left.+\frac{\sigma(\mu-1)}{2}\left(B(2Br^{2}+1)+Y(\omega-1)\right)e^{2Br^{2}}+C_{2}B\right)\right),$
(19)

where $C_{2}$ is the integration constant.

Figure 2: (Left Panel)Variation of the shape function with respect to $r$;
(Right Panel)Variation of the shape function with respect to $r$ in vicinity
of $r=2.5$

The shape function is found to be dependent on all the brane and bulk
parameters including the brane tension. The variation of the shape function
$b(r)$ along the radial distance $r$ has been plotted in Fig. 2. Apparently,
in the vicinity or $r=2.5$ $b(r)=r$,but on close inspection by enlarging the
plot in the above mentioned range it can be seen that $b(r)<r$ (right panel of
Fig. 2).

The obtained shape function is found to satisfy all the criteria to represent
a traversable wormhole, without the requirement for exotic matter violating
the null energy condition (NEC). The physical justification for this comes
from the fact that, as a result of the higher dimensional corrections to 4-D
gravity, there is an “effective matter” description on the brane, which
results in the violation of the NEC, as we shall find in the following
susection.

The properties of the shape function that can be inferred from the plot and
its feasibility in construction of wormhole has been discussed later in the
concluding section.

###  2.3 Validity of NEC

One of the most important energy conditions appearing in GR is the NEC given
by $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$, where $k^{\mu}$ denotes a null vector.
This reduces to $\rho+p\geq 0$ for a perfect fluid. So, for the ordinary
matter we have considered on the brane having EoS of form given by Eq. (17),
the NEC holds good. However, as we are concerned with the effective stress-
energy tensor on the brane due to the local and non-local correction terms, so
it is the effective energy density and pressure that we must consider for
constructing the NEC on the brane. Hence, the NEC becomes
$\rho^{eff}(r)+p_{r}^{eff}(r)\geq 0$, where it is to be noted from Eqs. (9)
and (10) that the effective pressures in the radial and transverse directions
differ due to difference in contribution from the bulk pressure term.

Here

$\rho^{eff}=\left[\rho\left(1+\frac{\rho}{2\sigma}\right)+{\frac{6U}{\sigma}}\right],$
(20)

and

$p_{r}^{eff}=\left(p+{\frac{\rho\left(p+\frac{\rho}{2}\right)}{\sigma}}+{\frac{2U}{\sigma}}+{\frac{4P}{\sigma}}\right).$
(21)

The sum of the effective energy density and effective radial pressure can be
computed to be

$\displaystyle\rho_{eff}+p_{reff}$ (22) $\displaystyle=$
$\displaystyle\frac{1}{k^{4}\sigma}\left(k^{4}{C_{1}}^{2}(\mu+1)e^{-\frac{(\mu+1)Br^{2}}{\mu}}+4C_{1}\left(\frac{\sigma(\mu+1)k^{4}}{4}+X(\omega+2)\right)e^{-\frac{(\mu+1)Br^{2}}{2\mu}}+4Y(\omega+2)\right).$

Figure 3: Variation of the NEC with respect to $r$.

The variation of $\rho_{eff}+p_{r}^{eff}$ along $r$ has been plotted in Fig.
3. We find that the NEC is violated for effective matter distribution on the
brane.

We have not used exotic matter to violate the Null Energy Condition (NEC). The
higher dimensional contributions in Equation 2 appear as an effective
modification to the stress energy tensor on the brane, but we have not
introduced any such matter physically. This is the benefit of using the RS
brane gravity to construct a traversable wormhole as compared to General
Relativity, as we do not have to introduce any additional exotic matter which
has not been detected, but rather the NEC violation is enforced from the
higher dimensional contribution itself.

###  2.4 The Junction Conditions

The exterior spacetime is comprises of vacuum, the line element for which may
be described by the Schwarzschild solution on the brane[92]. As shown by [92]
in their seminal work, the static, spherically symmetric vacuum solution on
the brane is modified from GR by the effect of a "tidal charge" which is
produced a result of the projection of the higher dimensional bulk gravity
effects onto the 3-brane. This leads to the effective brane Schwarzschild
solution behaving as Reissner-Nordstr$\ddot{o}$m (RN) solution of GR even in
the absence of any electric charge. This is a very interesting consequence of
single brane RS gravity. So the line element for the spacetime exterior metric
to the wormhole is written as

$ds^{2}=\left(1-\frac{2M}{r}+\frac{Q}{2r^{2}}\right)dt^{2}-\left(1-\frac{2M}{r}+\frac{Q}{2r^{2}}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (23)

where $M$ is the total mass of the wormhole and $Q$ is the tidal charge. The
tidal charge parameter can take both positive and negative values. The value
may be quite large in very strong gravity regimes but even a small value will
ensure significant higher dimensional correction to GR.

Presence of matter on the surface of wormhole must produce an extrinsic
discontinuity which generates an intrinsic surface energy density and surface
pressure. The surface behaves as the junction between the spacetime inside the
wormhole and its exterior, which makes the wormhole a geodesically complete
manifold with ordinary matter characterizing the configuration of the
wormhole. Thus, according to the fundamentals of junction conditions there
should be a smooth matching between the two spacetimes at the junction. Again,
the continuity of the metric coefficients at the junction surface do not
guarantee the continuity of its derivatives. So, in order to determine the
surface stress-energy $S_{ij}$ one can use formalism suggested by Darmois-
Israel [93, 94].

Now intrinsic surface stress energy tensor $S_{i}^{j}$ can be obtained from
Lanczos equation [95, 96, 97, 98] in the following form

$S_{j}^{i}=-\frac{1}{8\pi}(\kappa_{j}^{i}-\delta_{j}^{i}\kappa_{k}^{k}),$ (24)

the discontinuity in the second fundamental form can be written as

$\kappa_{ij}=\kappa_{ij}^{+}-\kappa_{ij}^{-},$ (25)

whereas the second fundamental form is given by

$\kappa_{ij}^{\pm}=-n_{\nu}^{\pm}\left[\frac{\partial^{2}X_{\nu}}{\partial\xi^{i}\partial\xi^{j}}+\Gamma_{\alpha\beta}^{\nu}\frac{\partial
X^{\alpha}}{\partial\xi^{i}}\frac{\partial
X^{\beta}}{\partial\xi^{j}}\right]|_{S},$ (26)

where the unit normal vector $n_{\nu}^{\pm}$ are defined as

$n_{\nu}^{\pm}=\pm\left|g^{\alpha\beta}\frac{\partial f}{\partial
X^{\alpha}}\frac{\partial f}{\partial
X^{\beta}}\right|^{-\frac{1}{2}}\frac{\partial f}{\partial X^{\nu}},$ (27)

with $n^{\nu}n_{\nu}=1$. Here $\xi^{i}$ is the intrinsic coordinate of the
surface of the wormhole and $f(x^{\alpha}(\xi^{i}))=0$ is the parametric
equation of it. Here $+$ and $-$ corresponds to exterior and interior
spacetime respectively of the wormhole.

Now the surface stress energy tensor, for the spherically symmetric spacetime,
can be written as $S_{i}^{j}=diag(-\Sigma,\mathcal{P})$. Where $\Sigma$ and
$\mathcal{P}$ are the surface energy density and surface pressure
respectively, which can be obtained at the surface of the wormhole (i.e at
$r=R$) by the following equations

$\displaystyle\Sigma=-\frac{1}{4\pi
R}\bigg{[}\sqrt{e^{\lambda}}\bigg{]}_{-}^{+}$ $\displaystyle=\frac{1}{4\pi
R}\left[\sqrt{\left(1-\frac{2M}{R}+\frac{Q}{2r^{2}}\right)}-\sqrt{1-\frac{2e^{-2BR^{2}}}{Bk^{4}\sigma}\left(H_{1}-\frac{\mu
C_{1}(\sigma\mu
k^{4}-2X(\omega-1)){e^{\frac{BR^{2}(3\mu-1)}{2\mu}}}}{{(3\mu-1)(2BR^{2}+1)}}\right)}\right],$
(28) $\displaystyle\mathcal{P}=\frac{1}{16\pi
R}\bigg{[}\bigg{(}\frac{2f+f^{\prime}R}{\sqrt{f}}\bigg{)}\bigg{]}_{-}^{+}=\frac{1}{16\pi
R}\left[\frac{2(1-\frac{M}{R})}{\sqrt{1-\frac{2M}{R}+\frac{Q}{2r^{2}}}}-\frac{F1}{F2}\right],$
(29)

where

$\displaystyle
H_{1}=-\frac{{C_{1}}^{2}(2\mu+1)k^{4}\mu{e^{{\frac{BR^{2}(\mu-1)}{\mu}}}}}{4(\mu-1)(g_{3}-1)}+\frac{(4B^{2}k^{4}R^{2}\sigma+Y(\omega-1)){e^{2BR^{2}}}+C_{2}Bk^{4}\sigma}{4(2g_{3}-1)},$
(30) $\displaystyle F1=\frac{2C_{1}g_{1}(2X(\omega-1)-\sigma\mu
k^{4})(\mu-1)\left(g_{2}(2g_{3}-1)-2\mu\right)}{k^{4}\sigma(3\mu-1)(\mu-1)(2g_{3}-1)^{2}B}+\left(-3k^{4}{C_{1}}^{2}(2\mu+1)\right.$
$\displaystyle\left.\left.\left(\left(2g_{2}g_{3}-\mu\right)\right)g_{1}+2(\mu-1)\left(C_{2}Bk^{4}\sigma(2BR^{2}g_{3}-1){e^{-2BR^{2}}}-Y(\omega-1)\right)\right)(3\mu-1)\right),$
(31) $\displaystyle
F2=\sqrt{1-\frac{2e^{-2BR^{2}}}{Bk^{4}\sigma(2BR^{2}+1)}\left(-\frac{\mu
C_{1}(\sigma\mu
k^{4}-2X(\omega-1)){e^{\frac{BR^{2}(3\mu-1)}{2\mu}}}}{{(3\mu-1)}}+H_{1}\right)}.$

Here, $g_{1}={e^{-{\frac{BR^{2}(\mu+1)}{\mu}}}}$, $g_{2}=BR^{2}(\mu+1)$ and
$g_{3}=(BR^{2}+1)$.

Now for the static wormhole the surface pressure as well as the surface energy
must vanish at the boundary surface. So we have $\sigma=\mathcal{P}=0$ at the
boundary, which eventually provides the condition

$b(r)|_{r=R}=2M+\frac{Q}{2R^{2}}.$ (33)

In order to calculate the different unknown constants we have used the
following boundary conditions:

(i) From the junction condition at the boundary (from Eq.(33))
$b(r)|_{r=R}=2M+\frac{Q}{2R^{2}}$.

(ii) Continuity of the metric potential $g_{tt}$ and its derivative
$\frac{\delta g_{tt}}{\delta r}$ at the surface boundary, i.e., at $r=R$.

Now we choose appropriate realistic values for the physical parameters namely
$\ \sigma=10^{3}\ Mev/Fermi^{3},Q=0.00018,M=1.7\ M_{\odot},\ r_{0}=0.75,\ R=4\
km$, where $r_{0}$ is the throat radius (in $km$). Making use of the above
conditions, we obtain the values of different parameters associated with the
wormhole as:
$X=-1943.448951,Y=1684.121721,C_{1}=64.85766573,C_{2}=-61.40896649,\
\omega=0.9157485630$. These values of the parameters have been used to plot
Figs. (1)-(5). All the plots have been started from the throat (i.e.
$r=r_{0}$) upto the surface boundary (i.e at $r=R$).

We have not adopted any ad-hoc values for the free model parameters. We
explicitly obtain their values from the Israel-Darmois junction conditions.
Only the essential physical parameters like realistic physical value of brane
tension for the energies corresponding to the present universe and tidal
charge have been taken from other works[90]. There is no constraint on upper
limit of wormhole mass for our model, as we have not used any unobserved
exotic matter for wormhole construction. So, we have chosen a realistic value.

###  2.5 Tidal acceleration

In order to ensure the traversability of our wormhole, it has to be accounted
that tidal forces do not become so large that the observer travelling through
the wormhole gets ripped apart. This implies that the tidal acceleration
experienced by the traveller should not exceed a realistic value, like the
acceleration due to gravity on the earth [4]. The tidal acceleration has both
radial and tangential components which must be individually constrained to be
less than the acceleration due to gravity on the earth. These two radial and
tangential tidal constraints may be expressed as inequalities in terms of the
Riemann curvature tensor of the form

$|R_{rtrt}|=|(1-\frac{b}{r})\big{[}\frac{\nu"}{2}+\frac{\nu^{\prime
2}}{4}-\frac{b^{\prime}r-b}{2r(r-b)}.\frac{\nu^{\prime}}{2}\big{]}|\leq
g_{earth}.$ (34)

$\gamma^{2}|R_{\theta t\theta t}|+\gamma^{2}v^{2}|R_{\theta r\theta
r}|=|\frac{\gamma^{2}}{2r^{2}}\big{[}v^{2}(b^{\prime}-\frac{b}{r})+(r-b)\nu^{\prime}\big{]}|\leq
g_{earth}.$ (35)

where $\gamma=\frac{1}{\sqrt{1-v^{2}}}$ and $v$ is the traveller’s velocity.
If we consider realistically, velocity of the traveller $v\ll 1$ which implies
$\gamma\approx 1$ and substituting the form of $b(r)$ given in Eq. (19), one
can get at the throat of the wormhole

$v\leq 0.01207442556\sqrt{g_{earth}}.$ (36)

The velocity of the traveller is restricted by the above inequality if the
tangential tidal acceleration is to be kept sufficiently small such that the
wormhole remains traversable. We also find that the radial tidal acceleration
is sufficiently small enough at the throat to allow traveller to pass through
without getting ripped apart. Thus, the tidal acceleration as obtained above,
also ensures the traversability of our proposed wormhole model.

###  2.6 Linearized stability analysis

For performing qualitative stability analysis, we consider the radius of the
wormhole throat to be a function of proper time. Let us denote the throat
radius as $r_{0}=x(\tau)$. The energy density turns out to be

$\sigma=-\frac{1}{2\pi x}\sqrt{f(x)+\dot{x}^{2}},$ (37)

and the pressure is given by

$p=\frac{1}{8\pi}\frac{f^{\prime}(x)}{\sqrt{f(x)}}-\frac{\sigma}{2}.$ (38)

Here $f(x)=1-\frac{2M}{x}+\frac{Q}{2x^{2}}$, where $M$ is the mass of the
wormhole and $Q$ is the tidal charge which arises due to higher dimensional
effects.

From the conservation equation we get an equation of motion of the form

$\dot{x}^{2}+V(x)=0,$ (39)

where the potential $V(x)$ is given by

$V(x)=f(x)-[2\pi x\sigma(x)]^{2}.$ (40)

It is now possible to perform stability analysis of the wormhole by
considering a linearization around $x_{0}$, which we assume to be a static
solution to the equation of motion Eq.(41).

Figure 4: Plot of $\beta$ vs $x_{0}$.

We consider Taylor series expansion of the potential around the assumed static
solution upto second order, which gives

$V(x)=V(x_{0})-V^{\prime}(x_{0})(x-x_{0})+\frac{1}{2}V"(x_{0})(x-x_{0})^{2}+O[(x-x_{0})^{3}],$
(41)

where prime denotes derivative with respect to x.

As we have considered a static wormhole spacetime in Eq.(12),$V(x_{0})=0$ and
$V^{\prime}(x_{0})=0$. The wormhole will be stable if and only if
$V"(x_{0})>0$. We define the parameter $\beta$

$\beta(\sigma)=\frac{\delta p}{\delta\sigma}.$ (42)

It turns out that

$V^{\prime\prime}(x)=f^{\prime\prime}(x)-8\pi^{2}[(\sigma+2p)^{2}+\sigma(\sigma+p)(1+2\beta)$
(43)

Therefore, for the condition we obtain for the wormhole to be stable is

$\beta<\frac{\frac{f^{\prime\prime}(x_{0})}{8\pi^{2}}-(\sigma+2p)^{2}-2\sigma(\sigma+p)}{4\sigma(\sigma+p)},$
(44)

which on substituting the values of $\sigma$ and $p$ can be written as

$\beta<\frac{x_{0}^{2}(f_{0}^{\prime})^{2}-2x_{0}^{2}f_{0}^{\prime\prime}f_{0}}{4f_{0}(a_{0}f_{0}^{\prime}-2f_{0})}-\frac{1}{2}$
(45)

We have obtained the stable regions denoted by regions 1,2 and 3 in Fig. (4).
So, our model is found to be stable by performing a linearized stability
analysis.

###  2.7 Acceleration and nature of the wormhole

An interesting feature of the wormhole can be understood from the redshift
function. Its derivative with respect to the radial coordinate determines
whether the wormhole has an "attractive" or "repulsive" geometrical nature.
The 4-velocity of an observer at rest is given by

$U^{\mu}=\frac{dx^{\mu}}{d\tau}=(e^{-\frac{\nu(r)}{2}},0,0,0),$ (46)

where $\tau$ denotes proper time. The 4-acceleration of the static observer is
given by $a^{\mu}=U^{\mu}_{;\nu}U^{\nu}$, which has the radial component given
by

$a^{r}=\frac{\nu^{\prime}}{2}\bigg{(}1-\frac{b(r)}{r}\bigg{)}.$ (47)

The geodesic equation gives the equation of motion for an initially static
test particle, moving in the radial direction as

$\frac{d^{2}r}{dt^{\tau}}=-\Gamma^{r}_{tt}\bigg{(}\frac{dt}{d\tau}\bigg{)}^{2}=-a^{r}$
(48)

The wormhole geometry is “attractive” if $a^{r}>0$, which implies that an
observer should have an outward-directed radial acceleration to prevent being
pulled into the wormhole. Alternatively, if it turns out that $a^{r}<0$, then
it implies that the wormhole geometry is “repulsive” meaning that an observer
must have an inward-directed radial acceleration in odrer to prevent oneself
from being pushed away from the wormhole.

$\displaystyle
a^{r}=\frac{1}{B\left(\mu+1\right)^{2}{k}^{4}\sigma}\left(8Br\pi\mu{C_{1}}\left(\mu+1\right)\left({k}^{4}\sigma+6X\right){{\rm
e}^{-\frac{1}{2}{\frac{\left(\mu+1\right)B{r}^{2}}{\mu}}}}-4\sqrt{2}{\pi}^{\frac{3}{2}}\sqrt{{\frac{\left(\mu+1\right)B}{\mu}}}{\mu}^{2}{C_{1}}\left({k}^{4}\sigma+6X\right)\right.$
$\displaystyle\left.{\rm
erf}\left(\frac{1}{2}\sqrt{2}\sqrt{{\frac{\left(\mu+1\right)B}{\mu}}}r\right)+2\left(\mu+1\right)B\mu\pi{{\rm
e}^{-{\frac{\left(\mu+1\right)B{r}^{2}}{\mu}}}}{{C_{1}}}^{2}{k}^{4}r-\sqrt{{\frac{\left(\mu+1\right)B}{\mu}}}{\mu}^{2}{\pi}^{\frac{3}{2}}{\rm
erf}\left(\sqrt{{\frac{\left(\mu+1\right)B}{\mu}}}r\right){{C_{1}}}^{2}{k}^{4}\right..$
$\displaystyle\left.+{B}^{2}\left(\mu+1\right)^{2}\left(\sigma\left(r-{\\_C2}\right){k}^{4}-16\pi
Y{r}^{3}\right)\right)$ (49)

Figure 5: Plot of radial component of acceleration with respect to $r$.

As we can see from Fig. 5 for our wormhole model on the RS braneworld, it
turns out that the radial component of 4-acceleration is positive, which
indicates an attractive wormhole geometry.

As, we shall see in the following section, our model also interestingly
reproduces some theoretical constraints on a few essential braneworld
parameters, not previously discussed in context of wormholes. In a nutshell,
we have constructed a physically viable traversable wormhole without
introducing any exotic matter at the throat, as necessary in the relativistic
context. This is possible due to the higher dimensional contribution to
gravity.

##  3 DISCUSSIONS AND CONCLUSION

In this work, we have successfully obtained a physically acceptable solution
describing a traversable wormhole under the framework of braneworld gravity
excluding exotic matter. Here we have calculated the shape function by
implementing Kuchowicz type metric function as $g_{tt}$ in the field
equations. The form of the adopted Kuchowicz potential to study wormhole is a
well-behaved, analytic and non-singular function within the finite range of
the radial coordinate. Though there is an inconsistency at
$r\rightarrow\infty$ with the Kuchowicz metric function, but it provides the
physically acceptable solutions for the wormhole providing a solution for the
shape function which is well-behaved and satisfies all the essential criteria
for being a shape function as prescribed by Morris and Thorne. Besides, it
brings out the shape of the wormhole as well. Such a similar potential with
identical features at r tending to infinity (Krori-Barua potential) has been
used to construct wormhole solution[86]. Various properties have been studied
in details to check the viability of our model. We have also estimated values
of the model parameters from junction conditions. We have actually used the
Israel-Darmois junction conditions to obtain the values of the free constant
parameters in our model. The junction conditions have been employed to obtain
the values for the three free constant parameters in our model namely $X$, $Y$
and $\omega$ along with the two integration constants $C_{1}$ and $C_{2}$. For
obtaining these values from the junction conditions using MAPLE, we have taken
certain realistic values of the physical parameters involved in our model.
There are two physical parameters for the brane, namely the brane tension and
tidal charge. Our choice of the brane tension is for intermediate energy
scales comparable to the present epoch of cosmic evolution [90] and the tidal
charge which arises as a consequence of higher dimensional effect on any
spherical symmetric matter distribution in its exterior spacetime, causes
considerable modification to GR for very small values comparable to the order
we have chosen. For the wormhole there is one physical parameter that we have
chosen namely the throat radius, which is arbitrarily chosen and is not
unique. It is usually chosen to be less than unity in most models. Varying it
slightly does not impact our analysis significantly. One more physical
parameter for the wormhole that we have assumed is the wormhole mass.
Generally in the relativistic context it is taken to be very small as it is
composed of unobserved exotic matter. So in order to minimize the quantity of
the exotic matter used, it is chosen to be very small. But as in the RS brane
scenario, we do not require any exotic matter to construct the wormhole, there
is no need for choosing the mass to be extremely small to minimize the amount
of composite matter, and we have taken a probable realistic value of 1.75
solar mass for our analysis. As we shall note, we can find a satisfactory
estimate of the bulk EoS parameter besides drawing constraints on the brane
tension. In this Section we are going to discuss some of the important results
that we have obtained from our present study.

1\. Matter density: By solving the conservation equation along with the EoS we
have obtained the matter density of the wormhole and its variation has been
shown in Fig. 1. From the plot of the density with respect to the radial
distance, it can be confirmed that the matter near the throat is much denser
than the boundary of the WH. This is in well agreement with the tubular shape
of wormhole as, on going radially outward from the throat the density
decreases, i.e., matter get distributed over the asymptotically flat
spacetime.

2\. Properties of Shape function: For the construction of a wormhole the shape
function plays an important role. In order to obtain a physically acceptable
traversable wormhole, the shape function must satisfy a number of conditions
[4]. Here we discuss those conditions and corresponding behaviour for our
model:

(i) First of all at the throat radius $r=r_{0}$, the shape function $b(r)$
must be equal to the throat radius $r_{0}$ itself. From Fig. 2 one can observe
that our model satisfies this condition.

(ii) The metric coefficient $(g_{rr})$ must be regular and well-behaved, i.e.,
$b(r)<r$ for $r>r_{0}$. Again from Fig. 2 we have observed this condition is
also satisfied for our present study of wormhole.

(iii) In addition to the above, the nature of the plot for the shape function
must be such that if the curve is rotated with respect to the $b(r)$ axis, the
$b(r)>0$ half for both positive and negative $r$, must approximately resembles
with the upper half of a wormhole. Fig. 2 tells us that it does so.

3\. Flaring-out condition: To have a stable wormhole, it must obey the
flaring-out condition at the throat in order to ensure that there is no pinch-
off making the wormhole non-traversable. The flaring-out condition states that
the first order derivative of the shape function with respect to $r$ at the
throat must be less than $1$. Now, this implies that the NEC must be violated.
We see from Eq. (17) that it is not the case for our matter distribution, but
on the brane we are interested only about the effective matter distribution
which as seen from Eq. (22) and Fig. 3 violates NEC. This is a huge advantage
of considering wormhole on the brane as contrast to GR, because we find that
there is no need to consider exotic matter but the effective matter
description due to local and non-local higher dimensional corrections enforce
violation of NEC even with ordinary matter on the brane.

4\. Geometrical nature and traversability condition: By evaluating the radial
component of the 4-acceleration to be positive, we infer that the wormhole has
an "attractive" geometry which implies that the radial acceleration of the
observer must be directed outward in order to avoid being pulled into the
wormhole. Also, the radial tidal acceleration is computed at the throat and
found to satisfy the tidal constraint which ensures travesarbility of the
traveller without being pulled apart by the tidal forces. An additional
constraint is obtained on the velocity of the traveller from the tangential
tidal constraint.

5\. Linear stability: A linearized stability analysis is performed involving
the sound speed and a constraint is obtained on the sound speed in terms of
the parameters of the solution for the wormhole spacetime. The constraint is
used to obtain the stable regions for the wormhole in Fig. (4).

6\. Constraints on brane tension and estimate for $\omega$: We have found two
very interesting features in connection with the brane tension, which is
perhaps one of the most important regulating parameter in braneworld physics
from our study:

(i) If the brane tension $\sigma$ is made negative, it turns out that the
values of the parameters to be obtained from the matching conditions become
imaginary, which strongly rules out the possibility of a negative tension
brane. Such a brane is well-known to exhibit instabilities and have also been
ruled out by previous investigations involving braneworld black holes [99] and
fundamental braneworld physics [100].

(ii) As the value of the brane tension $\sigma$ is lowered from the value
$10^{3}\ TeV^{4}$ as considered for our analysis, we can see that on lowering
$\sigma$ upto $1TeV^{4}$, the solution is found valid for wormhole
construction, but on lowering $\sigma$ further in $0.01-0.99\ TeV^{4}$ range,
the essential conditions for wormhole formation are violated. Also, the
condition for violation of the NEC for effective matter description on the
brane no longer holds true. Thus, we can claim that from our present study
that we have imposed a theoretical constraint on the brane tension such that
always $\sigma>1$. This is exactly in confirmation with similar constraint
drawn on the brane tension from the perspective of previous theoretical
investigations [28]. A possible explanation of this may arise from the fact
that as we approach extremely high energy scales then, $\sigma\rightarrow 0$,
as discussed before. So, for energy scales corresponding to $\sigma<1$,
quantum gravity effects must dominate and the effective description on the
brane is no longer valid.

Further the value of the bulk EoS parameter $\omega$ we obtain from the
matching condition is $0.9157485630$ which is close to $+1$ and justifies the
bulk space to be as $AdS_{5}$ (negative cosmological constant).

The RS braneworld gravity is inspired from the higher dimensional
Superstring/M-theories and the extra terms which appear in the field equation
(2), may be physically interpreted as follows. The third term representing the
local correction basically encodes the quadratic correction terms to the
stress energy tensor. This gives us a modified effective fluid description on
the brane, using ordinary matter. The final term represents the correction
induced on the brane due to the non-local bulk effect, and we find that the
matter on the bulk makes the higher dimensional bulk spacetime to be of Anti
de Sitter (AdS) nature for successfully obtaining traversable wormhole
solution. This is again in good agreement with RS braneworld gravity.

We can talk about direct correspondence between the model we have used and the
F(R) models described in [32] on at least two instances:. Firstly, the effect
of the string inspired Gauss Bonnet (GB) term in modifying the field equations
have been described in [32]. Such similar involvements in the braneworld
context is important in establishing the instability of a brane having a
negative brane tension[100]. Our theoretical model of the wormhole is also
found to be inconsistent for a negative tension brane. So, it can be said that
the modification to the EFE has similar physical impact as it rules out
negative tension branes.

Secondly, in [32], for a particular form of F(R) gravity, the hierarchy
problem is found to be solved. It resolves the issue as to why the
gravitational force is weaker than the other fundamental interactions. It has
been shown in [35] that on introducing a second brane the hierarchy problem
can be solved too. The physical explanation is that the full strength of
gravity can be realized only if we are in the higher dimensional bulk
spacetime, but not on the (3+1)-brane as gravity is free to propagate in the
bulk, unlike the other standard model forces. Similarly, the explanation for
dark matter and galaxy rotation curves as provided by modified F(R) gravity
discussed in ref. 32 can also be provided in a higher dimensional RS
braneworld setup[87]. The effect of higher dimensional bulk gravity or the
gravitational effect of the matter on the second brane is found to explain the
galaxy rotation curves. Also, the introduction of a viscosity term in the
matter sector may provide a modified description equivalent to the effective
matter description on the RS brane. We will consider such a modification in
the matter sector identical to RS gravity in constructing traversable wormhole
model in our future work.

###  3.1 Observational status of wormholes

Recently large number of attempts have been made through observational
evidences in favor of the idea of braneworld [101, 102, 103, 104, 105, 106] as
well as on the possible ways for detection of wormholes [107, 108, 109, 110,
111, 112]. Dai and Stojkovic [113] have suggested a possibility of observing a
wormhole from the idea of non-conservation of individual fluxes in the two
different spacetimes connected by a wormhole, which may result in mutual
effect on objects in close approximate of either mouths of the wormhole due to
one another. They have suggested the application of this idea to study the
possible effect on the orbit of stars near the black hole at our galactic
centre, which may possibly harbor a traversable wormhole. The possible
production of micro-lensing effects by wormholes, resembling gamma ray bursts,
have been studied by Torres et al. [114] and successfully imposed a constraint
on the upper limit of the mass density of wormholes using the BATSE data.
Doroshkevich et al. [115] have suggested a possible technique of detection of
wormholes by detecting the possible radiation pulse emitted from the other
side of a wormhole. The quasinormal ringing of black holes have been studied
by Konoplya and Zhidenko [116], who have investigated the scattering
properties in the vicinity of a rotating traversable wormhole. A class of non-
symmetric wormholes is found to exhibit super radiance and symmetric wormholes
may be easily distinguished from black holes as their simultaneous ringing at
various dominant multipoles are not identical. As per the findings of
Churilova et al. [117], for wormholes with a variable redshift function and
tidal force in the radial direction, there exists long-lived quasinormal modes
or quasi-resonances in the wormhole background. Wielgus et al. [118] have
constructed a reflection-asymmetric wormhole from two distinct RN spacetimes
with different mass and charge. Such a wormhole may be detected by the
presence of unique, single or double photon rings in its shadow due to
reflection of a considerable amount of infalling radiation back to the
spacetime containing the source.

In many works on wormhole, the redshift function is simply assumed ad hoc to
be a constant or an inversely proportional first order function of $r$ from a
view point of mathematical simplicity or a well-known shape function known to
satisfy all the properties of a traversable wormhole is assumed from the
beginning. As a contrast to this, we have chosen a physically satisfactory
well-behaved metric potential for spherically symmetric spacetime in the form
of the Kuchowicz function, as the redshift function.

Two of the conditions for having a traversable wormhole are that the metric
must be asymptotically flat and regular everywhere, and the Kuchowicz metric
does not satisfy these conditions. The redshift function diverges as r tends
to infinity. We are considering a model here for a static Lorentzian wormhole
and as there is no time evolution or expansion of the wormhole throat, so the
finiteness of the redshift function is physically significant only in the
vicinity of the wormhole. It can be seen that for all finite values of r, the
redshift function exhibits regularity and is capable of producing a well
behaved and stable, traversable wormhole with the obtained shape function
satisfying all the criteria as per the Morris-Thorne prescription, as has been
discussed in the second point.

Using this setup, we have obtained naturally a solution for the shape function
which agrees with all the essential criteria required to form a traversable
wormhole (as prescribed by Morris and Thorne), just by naturally solving the
field equations on the brane and also reproduces roughly the shape of the
outer surface of a wormhole. We also find that there is no need for any exotic
matter and the Null Energy Condition is violated automatically which we
interpret to be the contribution of the gravitational effect of higher
dimensions, just like it provided the dark matter effect. The novelty of our
work also lies in the fact that we have imposed constraints on the brane
tension to be positive with a lower limit for successful description of a
traversable wormhole which turns out to be in exact accordance with constraint
obtained from other physical aspects.

The observational status of wormhole physics in general has been discussed
briefly in subsection 3.1. From our model we can say that, if there is any
particular signature in favour of higher dimensions from particle physics
experiments in the near future, then the study of certain features of a higher
dimensional brane universe like the brane tension or the nature of the higher
dimensional bulk matter may be used to account for the possibility of physical
existence of wormholes. It is evident from our model that in order to satisfy
the physical characteristics of a wormhole-like object, some constraints are
imposed upon the essential features of a higher dimensional universe. These
features like the tension or the nature of bulk matter are in turn linked to
properties of fundamental particles in particle physics. So, the validity of
the constraints imposed by our wormhole model on these features may be used to
rule out or favour our wormhole model.

In a nutshell, we have chosen such a gravitational framework for constructing
wormhole, where the need for beyond the standard model exotic matter from a
particle physics point of view is met up by the behaviour of gravity in the
extra-dimensional scenario. So, obtaining the violation of the NEC without any
exotic matter can be justified clearly. Also, the fact that the obtained shape
function of the wormhole satisfies all the Morris-Thorne criteria and that the
other physical features of the wormhole like the traversability and stability
turn out to be satisfactory recommends that the idea of wormhole is more
naturally compatible in the higher dimensional braneworld framework than in
standard GR, as we do not have to put any exotic form of matter to stabilize
the wormhole spacetime.

As a final comment we conclude that, our present analysis satisfies all the
required conditions to ensure traversability in a static wormhole spacetime on
the brane, without any exotic matter. Thus, we can justify the applicability
of the Kuchowicz metric function to wormholes as well, in the presence of non-
exotic matter on the braneworld. The result of our investigation is further
strengthened by a probable estimation for $\omega$ and quantitative
restriction imposed on $\sigma$ to be non-negative and greater than unity,
which are in excellent agreement with the previously worked out models for
braneworld black holes and fundamental braneworld physics. Though the
existence of wormholes is yet to be directly confirmed, we may go on to state
that if a wormhole is detected in the future, then our universe has a very
high likelihood of being a 3-brane embedded in higher dimensions, due to our
successful construction of a wormhole on the brane in the absence of any kind
of exotic matter, exactly satisfying the properties of the braneworld
involving essential parameters like the brane tension. It is difficult to test
the results experimentally at this moment. However, if a wormhole is detected
in the near future and some constraint is obtained on its mass and throat
radius, then there is a possibility that such a constraint restricts the brane
tension to be positive as in our model and other brane gravity models[99, 100]
besides putting a lower constraint on the positive brane tension as obtained
by us.

There has been an effective gravitational effect on the non-exotic matter
present on the brane that has led to the violation of the NEC even in the
absence of any exotic matter. The effect of viscous matter or effect of
torsion and shear on such ordinary matter can be explored in future work, both
in a relativistic as well as higher dimensional braneworld context. It would
be interesting to study whether such a modification in the matter sector can
allow the construction of traversable wormholes without exotic matter.

Moreover, there are other higher dimensional braneworld models like the DGP,
or generalizations of the Randall-Sundrum scenario like the Sahni-Shtanov
brane models. It would be interesting to explore the development of a
traversable wormhole, particularly in the Sahni-Shtanov context, as such
higher dimensional gravitational effects are well-known to admit non-singular
bouncing solutions violating the Null Energy Condition in the cosmological
context as well. Also, it would be of interest to explore the possibilities of
construction of traversable wormholes in other physically attractive modified
gravity scenarios which do not admit higher spacetime dimensions. It could
then be realized whether the traversability of wormhole in the absence of
exotic matter is a feature peculiar to extra spacetime dimensions or any
gravitational framework admitting a class of modification to the relativistic
Einstein-Hilbert action.

## Acknowledgments

The authors are thankful to the anonymous referees for their help in improving
the manuscript substantially. MK and SR are thankful to the Inter-University
Centre for Astronomy and Astrophysics (IUCAA), Pune, India for providing the
Visiting Associateship under which a part of this work was carried out. RS is
thankful to the Govt. of West Bengal for financial support through SVMCM
scheme. SG is thankful to the Directorate of Legal Metrology under the
Department of Consumer Affairs, West Bengal for their support.

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