# Gravitational lensing by traversable wormholes supported by three-form
fields

Daris Samart<EMAIL_ADDRESS>Khon Kaen Particle Physics and Cosmology Theory
Group (KKPaCT), Department of Physics, Faculty of Science, Khon Kaen
University, Khon Kaen, 40002, Thailand  Natthason Autthisin
<EMAIL_ADDRESS>Khon Kaen Particle Physics and Cosmology Theory
Group (KKPaCT), Department of Physics, Faculty of Science, Khon Kaen
University, Khon Kaen, 40002, Thailand  Phongpichit Channuie
<EMAIL_ADDRESS>College of Graduate Studies, Walailak University, Thasala,
Nakhon Si Thammarat, 80160, Thailand  School of Science, Walailak University,
Thasala, Nakhon Si Thammarat, 80160, Thailand

###### Abstract

In this paper, we study the deflection angle of light by traversable
wormholes, which are supported by the three-form fields. The specific forms of
the redshift and shape functions that produce results compatible with the
energy conditions at throat of the wormholes are used. Having used the well-
defined parameter sets of the three-form wormholes, we investigate the photon
geodesic motion under the effective potential of the wormhole background. As a
result, we discover that the radius of the photon sphere is a very useful
observable that is used to analyze the geometrical structures of a physical
wormhole.

##  I Introduction

Over the past few decades, wormholes have become one of the most intensively
studied topics in the literature. The wormhole solutions represent a conduit
between the points of two parallel universes or even two different points of
the same universe. The existence of traversable wormholes is known as
Einstein-Rosen bridges proposed by Einstein and Rosen in 1935 Einstein and
Rosen (1935). However, Wheeler and his colleague later showed that wormholes
would not be stable and can not be traversable Wheeler (1955); Fuller and
Wheeler (1962). In 1988, Morris, Thorne, and Yurtsever have demonstrated that
the traversable wormholes were explicitly constructed Morris and Thorne
(1988). Later, other types of traversable wormholes were studied as plausible
solutions to the equations of general relativity, including analyses initiated
by Matt Visser M (1995)

Searching for a proper form of the exotic matter inside the wormholes is a
central topic in the wormhole research. In this work, we will use the the
three-form fields as the exotic matter that can support the throat of the
traversable wormholes. The three-form fields naturally exist in the string
theory Ovrut and Waldram (1997); Klebanov and Strassler (2000); Bousso and
Polchinski (2000); Frey and Mazumdar (2003); Groh et al. (2013); Bielleman et
al. (2015); Bandos and Meliveo (2012); Farakos et al. (2017); Bandos et al.
(2018); Farakos et al. (2018). Moreover, it has been used to explain the
several problems in cosmology such as inflation in very early universe and
structure formation Koivisto and Nunes (2009); Germani and Kehagias (2009); De
Felice et al. (2012a, b); Kumar et al. (2014), and even the dark energy
problem Koivisto and Nunes (2009, 2010); Ngampitipan and Wongjun (2011);
Koivisto and Nunes (2013). The traversable wormholes with the three-form
fields were constructed by authors of Ref.Barros and Lobo (2018). It was shown
that the three-form fields are an interesting exotic matter possibly used to
support the wormholes throat. Consequently, the three-form fields are also
used to study relativistic stars and black holes Barros et al. (2020);
Bouhmadi-López et al. (2021a); Barros et al. (2021). Recently, the three-form
wormholes are studied in detail by using the Higgs type three-form potential
Bouhmadi-López et al. (2021b).

To prove the existence of the wormholes, there are several methods that can be
used to figure the wormholes. Gravitational lensing is one of the interesting
phenomena which can be used to probe the astrophysical objects such as black
holes. Gravitational lensing in wormholes has been explored in many aspects
and compared with the study of black holes Nandi et al. (2006); Rahaman et al.
(2007); Dey and Sen (2008); Bhattacharya and Potapov (2010); Abe (2010);
Nakajima and Asada (2012); Sharif and Iftikhar (2015); Tsukamoto (2016);
Tsukamoto and Harada (2017); Nandi et al. (2017, 2018). Using the Gauss-Bonnet
theorem, gravitational lensing by many types of black holes and wormholes have
also been extensively studied, e.g., Jusufi and Övgün (2018); Jusufi et al.
(2019); Övgün et al. (2019). The gravitational lensing tells us about the
geometry of space-time around massive object and wormhole is one of them.
Therefore, this is a promising idea to search for the existence of the
wormholes. In addition, gravitational lensing occurs when the light travel
passing through the space-time around the massive objects Congdon and Keeton
(2018). Consequently, this leads to the bending of light between source and
observers according to the curve space-time around the massive object. One of
the striking predictions of the gravitational lensing is the Einstein’s rings
Shaikh et al. (2019a), commonly observed phenomena of the light bending in
astrophysics.

In this work, we use the gravitational lensing as the main tool to search for
the existence of the wormholes. Gravitational lensing is one of the classical
predictions by GR. This phenomenon tells us about distribution of matter and
manifests the curved space-time. This makes the curve geodesic line and leads
to the bending of light between the light source and observers due to the
massive object in the space-time Carroll (2019); Hartle (2003). The trajectory
of the light passing through the wormholes will be curved by the space-time
geometry of the wormholes. We will calculate and discuss a strong
gravitational lensing and an Einstein’s rings which are very useful in the
search of the traversable wormholes.

The present work is organized as follows: In Sec.II, all relevant equations of
the three-form wormholes are briefly constructed and some implications are
revisited. Next, in Sec.III, we set up the analytical expression of thee
effective potential and deflection angle of the three-form wormholes. The
numerical results are presented and its physical interpretations are given in
section IV. Finally, we conclude our findings in the last section V.

##  II Background equations with three-form fields

###  II.1 Gravitational three-form fields action and spherical symmetric
solution ansatz

We start with the gravitational action of the Einstein gravity including the
three-form fields. In this case, the action is of the form:

$\displaystyle
S=\int\,d^{4}x\left[\frac{1}{2\,\kappa^{2}}\,R-\frac{1}{48}\,F_{\mu\nu\rho\sigma}\,F^{\mu\nu\rho\sigma}-V(A^{2})\right],$
(1)

where $\kappa^{2}\equiv 8\pi\,G$, $A^{2}=A_{\mu\nu\rho}\,A^{\mu\nu\rho}$ and
$R$ is a Ricci scalar. Additionally, $F_{\mu\nu\rho\sigma}$ is a field
strength tensor of the three-form fields, $A_{\mu\nu\rho}$. It is defined as

$\displaystyle
F_{\mu\nu\rho\sigma}=\nabla_{\mu}\,A_{\nu\rho\sigma}-\nabla_{\sigma}\,A_{\mu\nu\rho}+\nabla_{\rho}\,A_{\sigma\mu\nu\rho}-\nabla_{\nu}\,A_{\rho\sigma\mu}\,.$
(2)

Varying the action in (1), the Einstein field equation is given by

$\displaystyle G_{\mu\nu}=\kappa^{2}\,T_{\mu\nu}\,,$ (3)

where $G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,R$ . The energy-
momentum tensor of the three-form fields reads

$\displaystyle
T_{\mu\nu}\equiv\frac{1}{6}\,F_{\mu}^{~{}\,\rho\sigma\tau}\,F_{\nu\rho\sigma\tau}+6\frac{\partial\,V(A^{2})}{\partial(A^{2})}\,A_{\mu}^{~{}\,\rho\sigma}\,A_{\nu\rho\sigma}-\left[\frac{1}{48}\,F_{\mu\nu\rho\sigma}\,F^{\mu\nu\rho\sigma}-V(A^{2})\right]\,g_{\mu\nu}\,.$
(4)

The equation of motion of the three-form is obtained by varying with respect
to the $A_{\mu\nu\rho}$ and we find

$\displaystyle\nabla_{\sigma}F^{\sigma\mu\nu\rho}-12\frac{\partial\,V(A^{2})}{\partial(A^{2})}\,A^{\mu\nu\rho}=0\,.$
(5)

The three-form gauge field, $A_{\mu\nu\rho}$ is represented by the following
relation

$A_{\mu\nu\rho}=\sqrt{-g}\epsilon_{\mu\nu\rho\sigma}B^{\sigma},$ (6)

where the $B^{\sigma}$ is the one-form dual vector of the three-form fields,
$A_{\mu\nu\rho}$. The solution of the $B^{\sigma}$ can be expressed by using
ansatz as a function of $\zeta(r)$ via Barros and Lobo (2018)

$B^{\sigma}=\left(0,\sqrt{1-\frac{b(r)}{r}}\zeta(r),0,0\right)^{\intercal}.$
(7)

where $\zeta(r)$ is a generic parametrization of the three-form fields and we
will solve the EFE in Eq.(3) in order to obtain the solution of $\zeta(r)$.

Before imposing the ansatz of the three-from field solution, it is worth
introducing the spherical symmetry line element of the traversable wormholes
used in the present work. We consider a general spherically symmetric space-
time geometry and static traversable wormholes proposed by Morris-Thorne
Morris and Thorne (1988). It reads

$ds^{2}=e^{2\Phi(r)}dt^{2}+\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (8)

where $r,\,\theta$ and $\phi$ are the spherical coordinates, $\Phi(r)$ and
$b(r)$ are two arbitrary functions of radius, $\Phi(r)$ is called the redshift
function since it determines the gravitational redshift, and $b(r)$ determines
the spatial shape of the wormhole we called the shape function. The
traversable wormholes with three-form fields solutions for $\Phi(r)$ and
$b(r)$ and its relevant quantities will be solved in the next section.
Specifically, we will first use the wormhole line element to specify the
ansatz of the three-form field at this moment.

According to Eqs.(6) and (7), the nonzero components of three-form fields are
given by

$\begin{split}A_{t\theta\phi}=A_{\theta\phi t}=A_{\phi
t\theta}=-A_{t\phi\theta}=-A_{\theta t\phi}=A_{\theta t\phi}=-A_{\phi\theta
t}=\sqrt{g\left(\frac{b(r)}{r}-1\right)}\zeta(r)\end{split}$ (9)

and $A^{2}$ is given by

$A^{2}\equiv A^{\alpha\beta\gamma}A_{\alpha\beta\gamma}=-6\zeta(r)^{2}.$ (10)

Having uses Eqs.(2) and (6), we can re-write the kinetic term of the three-
form fields as

$-\frac{1}{48}F^{2}=-\frac{1}{48}F^{\alpha\beta\gamma\delta}F_{\alpha\beta\gamma\delta}=\frac{1}{2}(\nabla_{\mu}B^{\mu})^{2}=-6\Upsilon(r),$
(11)

where the function $\Upsilon(r)$ with the traversable wormhole line element in
Eq.(8) reads

$\Upsilon=4\left(1-\frac{b}{r}\right)\left[\zeta\left(\Phi^{\prime}+\frac{2}{r}+\zeta^{\prime}\right)\right]^{2}.$
(12)

We have completed a general form of the three-form field solutions in the
spherical symmetric background. This will be used to construct the traversable
wormholes in the next section.

###  II.2 Traversable wormhole with three-form fields and and its energy
conditions

In the present section, we continue to derive the EFE with the three-form
field in order to construct the traversable wormholes with the three-form
field solution. More importantly, we also include an ordinary matter term,
$\mathcal{L}_{m}$, into the gravitational action in Eq.(1) and the ordinary
matter term is considered as an anisotropic fluid in this work. Taking a
standard variation of the action with respect to the metric tensor,
$g_{\mu\nu}$ and setting $\kappa^{2}\equiv 8\pi G=1$, one gets

$G_{\mu\nu}={T}{{}^{\textrm{eff}}_{\mu\nu}},$ (13)

where the effective energy-momentum tensor defined by

$T^{\textrm{eff}}_{\mu\nu}=T^{(A)}_{\mu\nu}+T^{(m)}_{\mu\nu},$ (14)

where an $(A)$ superscript refers to the three-form $A_{\alpha\beta\gamma}$
and an $(m)$ denotes an ordinary matter of the energy-momentum tensor,
respectively, while the energy-momentum tensor of the anisotropic fluid is
given by
$T_{\nu}^{(m)\,\mu}=\big{(}-\rho_{m},\,-\tau_{m},\,p_{m},\,p_{m}\big{)}$.

Using the metric tensor in Eq.(8), the energy-momentum tensor of the three-
form, $T^{(A)}_{\mu\nu}$ in Eq.(4) can be calculated for diagonal the
components as

$\displaystyle{T}^{(A)}{{}^{t}_{t}}$ $\displaystyle=$
$\displaystyle-\rho_{A}=-\frac{1}{8}\Upsilon-V+\zeta\frac{\partial
V}{\partial\zeta},$ (15) $\displaystyle{T}^{(A)}{{}^{r}_{r}}$ $\displaystyle=$
$\displaystyle-\tau_{A}=-\frac{1}{8}\Upsilon-V,$ (16)
$\displaystyle{T}^{(A)}{{}^{\theta}_{\theta}}$ $\displaystyle=$
$\displaystyle{T}^{(A)}{{}^{\phi}_{\phi}}=p_{A}=-\frac{1}{8}\Upsilon-V+\zeta\frac{\partial
V}{\partial\zeta}.$ (17)

From a definition of the effective energy-momentum tensor in Eq.(14), we can
calculate the EFE in Eq.(13) for each diagonal component. One finds

$\displaystyle\frac{b^{\prime}}{r^{2}}=\rho_{\textrm{eff}}=\rho_{m}+\rho_{A},$
(18) $\displaystyle
2\left(1-\frac{b}{r}\right)\frac{\Phi^{\prime}}{r}-\frac{b}{r^{3}}=\tau_{\textrm{eff}}=\tau_{m}+\tau_{A},$
(19)
$\displaystyle\left(1-\frac{b}{r}\right)\left(\Phi^{\prime\prime}-\frac{b^{\prime}r-b}{2r(r-b)}\Phi^{\prime}+(\Phi^{\prime})^{2}+\frac{\Phi^{\prime}}{r}-\frac{b^{\prime}r-b}{2r^{2}(r-b)}\right)=p_{\textrm{eff}}=p_{m}+p_{A}.$
(20)

According to the conservation of the effective energy-momentum tensor,
$\nabla_{\mu}{T}^{(\rm eff)\,\mu}_{\nu}=0$, we obtain the equation of motion
of the effective energy-momentum tensor as

$\tau^{\prime}_{\textrm{eff}}+\frac{2}{r}(\tau_{\textrm{eff}}+p_{\textrm{eff}})+\Phi^{\prime}(\tau_{\textrm{eff}}-\rho_{\textrm{eff}})=0.$
(21)

Substituting all components of the effective energy-momentum tensor from
Eqs.(18), (19) and (20) into EFE in Eq.(13), we find

$\displaystyle\zeta\left[r\Phi^{\prime}\left(b^{\prime}-\frac{b}{r}\right)+4+2b^{\prime}+2r^{2}\Phi^{\prime\prime}\left(\frac{b}{r}-1\right)-6\frac{b}{r}\right]$
$\displaystyle\qquad\,+\,2r^{2}\frac{\partial
V}{\partial\zeta}+\zeta^{\prime}r\left[3\frac{b}{r}-4+b^{\prime}+2\Phi^{\prime}r\left(\frac{b}{r}-1\right)\right]+2\zeta^{\prime\prime}r^{2}\left(\frac{b}{r}-1\right)=0.$
(22)

In practice, it is very difficult to solve the exact solutions of the
wormholes because of the unknown functions such as $\Phi$, $b$, $\rho_{m}$,
$\tau_{m}$, $p_{m}$, $\zeta$ and $V$. Therefore, we need to assume a specific
form of the redshift function, $\Phi(r)$ and shape function, $b(r)$ which are
compatible with all requirements of the traversable wormhole solutions. More
importantly, it has been demonstrated in the Ref.Barros and Lobo (2018) that
the following specific forms of the functions $\Phi(r)$, $b(r)$ and $\zeta(r)$
are consistent to the EFE in Eq.(13) with the line element in Eq.(8) as well.
The specific forms of the solutions in this work are taken from Refs.Lobo
(2005, 2008); Capozziello et al. (2012). The functions $\Phi(r)$ and $b(r)$
can be parameterized as

$\Phi(r)=\Phi_{0}\left(\frac{r_{0}}{r}\right)^{\alpha}\quad\textrm{and}\quad
b(r)=r_{0}\left(\frac{r_{0}}{r}\right)^{\beta}$ (23)

where $\alpha,\beta,$ and $\Phi_{0}$ are dimensionless and the ranges of the
model parameters are assumed to be $\beta>-1,\alpha>0$ and $\beta>0$.
$\Phi(r)$ and $\zeta(r)$ are also dimensionless function, whereas the $b(r)$
and $r_{0}$ are quantities with dimension of length.

In this work, we use a specific form of the three-form field potential as
Barros and Lobo (2018),

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle\zeta^{2}+C\,.$ (24)

where $C$ in an integration constant. The author of Ref.Barros and Lobo (2018)
has shown that the relevant energy conditions for the three-form wormhole are
satisfied by using the the form of $\Phi(r)$, $b(r)$ and $V(r)$ in Eqs.(23)
and (24), respectively, with the following sets of the parameters:
$\bullet$ Case 1

$\displaystyle\Phi_{0}=-1,\quad\alpha=1,\quad\beta=1,\quad C=-0.1\,.$ (25)

$\bullet$ Case 2

$\displaystyle\Phi_{0}=-2,\quad\alpha=1,\quad\beta=-1/2,\quad C=0\,,$ (26)

The authors of Ref.Barros and Lobo (2018) showed that all two cases are the
best parameter sets that produce results compatible with the relevant energy
conditions. Therefore, they are suitable to be used to construct wormhole
solutions. Then, we employ above two cases of the parameters to study
gravitational lensing of the three-form wormholes in the next section.

##  III Effective potential and Deflection angle of the three-form wormhole

In order to study the trajectory of the photon in the three-form wormholes
background, the effective potential from the geodesic equation plays the major
role leading to analysis of the deflection angle of the photon trajectory.
Detailed derivations of the relevant quantities for deflection angle are given
in the standard framework and they can be found in general textbooks in GR
Carroll (2019); Hartle (2003). Then we will not repeat them here. We aim to
derive in this section the analytical form of the effective potential and
deflection angle of light of wormholes with the three-from fields. We study
the numerical calculations in the next chapter.

In the previous section, we have reviewed how to construct wormholes solutions
supported by three-form fields. In this section, we use the redshift function
$\Phi(r)=\Phi_{0}(\frac{r_{0}}{r})^{\alpha}$ and the shape function
$b(r)=r_{0}(\frac{r_{0}}{r})^{\beta}$ that can be used to construct wormhole
solutions as mentioned previously. We divide the analysis of the effective
potential from the geodesic equation and the deflect angle in this section
into two cases with two sets of parameters in Eqs.(25,26) as shown in the
previous section. In addition, we consider different values of $\alpha$,
$\beta$ and $\Phi_{0}$ relevant to the calculations to determine the effective
potential as well as the deflection angle.

###  III.1 Case I : $\alpha=1$, $\beta=1$ and $\Phi_{0}=-1$

We set $\Phi_{0}=-1$, $\alpha=1$ and $\beta=1$ leading to $\Phi(r)=-r_{0}/r$
and $b(r)=r_{0}^{2}/r$, respectively. From the standard approach in GR as done
in Refs.Shaikh et al. (2019b, a); Godani and Samanta (2021), the effective
potential, $V$, is defined by

$\displaystyle V=\frac{e^{-\frac{2r_{0}}{r}}}{r^{2}}\,,\qquad\text{with}\quad
r^{2}=l^{2}+r_{0}^{2}\,,$ (27)

where $l$ is the proper radial coordinate of the travesable wormholes. With
the parameters given in case 1, we find

$V=\frac{1}{l^{2}+r_{0}^{2}}\,e^{-\frac{2r_{0}^{2}}{\sqrt{l^{2}+r_{0}^{2}}}}.$
(28)

We will plot potential $V$ with respect to proper radial coordinate, $l$, in
the section IV.1. The numerical results will be displayed in the next section.

According to the standard calculations (see Refs.Shaikh et al. (2019b, a);
Godani and Samanta (2021) for detail derivations), the deflection angle with
the parameter set given in case 1 can be obtained as

$\alpha(u)=2\int_{r_{\rm
tp}}^{\infty}dr\,\frac{e^{-\frac{r_{0}}{r}}}{r^{2}\sqrt{\left(1-\frac{r_{0}^{2}}{r^{2}}\right)\left(\frac{1}{u^{2}}-\frac{e^{-}\frac{2r_{0}}{r}}{r^{2}}\right)}}-\pi.$
(29)

In addition, the impact parameter, $u$, of the case 1 is given by

$u=r_{\rm tp}\,e^{\frac{r_{0}}{r_{\rm tp}}}\,.$ (30)

Moreover, the turning point of the effective potential, $r_{\rm tp}$ is
defined by

$r_{\rm tp}=-\frac{r_{0}}{W\big{(}\frac{r_{0}}{u}\big{)}}$ (31)

where $W$ function is the Product-Log function. The numerical results of the
deflection angle will be depicted and discussed in the next section.

###  III.2 Case II : $\alpha=1$, $\beta=-\frac{1}{2}$ and $\Phi_{0}=-2$

For the case 2, we fix $\Phi_{0}=-2$, $\alpha=1$ and $\beta=1$ to obtain
$\Phi(r)=-2r_{0}/r$ and $b(r)=\sqrt{r_{0}r}$, respectively. Having used the
definition of the effective potential as shown previously, one can write the
expression of the effective potential, $V$, as

$V=\frac{1}{l^{2}+r_{0}^{2}}e^{-\frac{4r_{0}}{\sqrt{l^{2}+r_{0}^{2}}}}.$ (32)

The numerical results of the potential $V$ with respect to proper radial
coordinate $l$ will be plotted and the graph is depicted in Fig.(2) in the
next section. With the same procedure previously, the deflection angle of case
2 parameters is given by

$\alpha(u)=2\int_{r_{\rm
tp}}^{\infty}dr\,\frac{e^{-\frac{2r_{0}}{r}}}{r^{2}\sqrt{\left(1-\frac{r_{0}}{r\sqrt{\frac{r_{0}}{r}}}\right)\left(\frac{1}{u^{2}}-\frac{e^{-}\frac{4r_{0}}{r}}{r^{2}}\right)}}-\pi.$
(33)

The impact parameter, $u$, of case 2 reads

$u=r_{\rm tp}e^{\frac{2r_{0}}{r_{\rm tp}}}\,.$ (34)

In addition, the turning point, $r_{\rm tp}$ for case 2 reads

$r_{\rm tp}=-\frac{2r_{0}}{W\big{(}-\frac{2r_{0}}{u}\big{)}},$ (35)

The numerical plot of the deflection angle with respect to impact parameter
will be depicted in the next section. In this section, we have computed the
effective potential, $V$, and the deflection angle, $\alpha$, for the the
photon trajectories in the spacetime back ground of the three-form wormholes.
The numerical results and some physical implication will be presented in the
next section.

##  IV Numerical results

In this section, we will present the numerical results for the effective
potential and the deflection angle by using the parameters given in case 1 and
case 2 where they are satisfied the NEC and WEC for the three-form wormholes
as demonstrated in Ref.Barros and Lobo (2018).

###  IV.1 Effective potential

We start with a numerical study of the effective potential in Eqs.(28) and
(32) for parameters of case 1 and case 2, respectively.

Figure 1: Plot for effective potential $V$ as a function of radial coordinate
$l$ with different values of $r_{0}$. In this figure, we have used the
parameters of case 1, $\Phi_{0}=-1,\alpha=1$ and $\beta=1$. with the ansatz of
$\Phi(r)$ and $b(r)$ in Eq.(23).  Figure 2: Plot for effective potential $V$
as a function of radial coordinate $l$ with different value of $r_{0}$. In
this case 2, we have considered $\Phi_{0}=-2,\alpha=1$ and $\beta=-1/2$ with
the ansatz of $\Phi(r)$ and $b(r)$ in Eq.(23).

Before discussing the numerical of the effective potential of the photon
trajectory in the three-form wormholes background, it is worth considering the
photon moving in the potential until reaching the “turning point” where
$d\widetilde{V}/dr=0$. Here $\widetilde{V}\equiv L^{2}\,V$ with $L$ is the
angular momentum of the photon circular motion. The constant radius $r_{\rm
ph}$ is the constant radius of the photon motion in the circular orbit. The
circular orbit of the photon occurs when

$\frac{2e^{2\Phi(r_{\rm ph})}\Phi^{\prime}(r_{\rm ph})}{r_{\rm
ph}^{2}}-\frac{2e^{2\Phi(r_{\rm ph})}}{r_{\rm ph}^{3}}=0.$ (36)

The radius of the photon circular orbits is given by

$r_{\rm ph}=\frac{1}{\Phi^{\prime}(r_{\rm ph})}.$ (37)

The radius $r_{\rm ph}$ is known as the photon sphere radius occurring at
strong gravity region. In general, the conditions for examining the photon
sphere radius are written as

$\tilde{V}(r_{\rm ph})=E^{2}\,,\quad\frac{d\tilde{V}}{dr}\bigg{|}_{r=r_{\rm
ph}}=0\,,\quad\frac{d^{2}\tilde{V}}{dr^{2}}\bigg{|}_{r=r_{\rm ph}}<0\,,$ (38)

where $r_{\rm ph}$ and $E$ are the photon sphere radius corresponding to the
maximum of the effective potential and the energy of the photon, respectively.

We have plotted effective potential for different values of $r_{0}$. As a
result, we found that the numerical results in Fig.(1) are compatible with the
conditions (38) for all values of $r_{0}$ in the plots.

The light deflection occurs when the turning point of the photon approaches
the throat $r_{\rm tp}=r_{0}$ of the wormholes. This implies the occurrence of
the bending of light due to the photon sphere at the throat of wormholes
Godani and Samanta (2021). In case 2, as a result, Fig.(2) shows the maximum
value of $\tilde{V}$ corresponding to the photon sphere radius, $r_{\rm ph}$.
The light with turning point $r_{\rm tp}>r_{0}$ always take a turn outside the
photon sphere. This implies that light diverges at the photon sphere outside
the throat as the interpretation from Ref.Shaikh et al. (2019a).

###  IV.2 Deflection angle

Figure 3: Plot deflection angle $\alpha$ with respect to impact parameter $u$
for three different values of $r_{0}$ by using the ansatz of $\Phi(r)$ and
$b(r)$ and the parameter set in case 1 in Eqs.(23) and (25), respectively.
Figure 4: Plot deflection angle $\alpha$ with respect to impact parameter $u$
for three different values of $r_{0}$ by using the ansatz of $\Phi(r)$ and
$b(r)$ and the parameter set in case 2 in Eqs.(23) and (26), respectively.

We have plotted numerical integration of the deflection angle from Eqs.(29)
and (33) with respect to the impact parameter. Fig.(3) and Fig.(4) show the
divergence of $\alpha$ as the $u$ increase. the deflection of light occurs
when the impact parameter approches the critical value $u\rightarrow u_{0}$.
It is diverging to infinity corresponding to the throat radius $r_{0}$. When
the throat of wormhole $r_{0}$ is large $\alpha$ is diverging slowly. We close
this section by discussing the implications of the gravitational lensing
created by the three-form wormholes. In case 1, on the one hand, the numerical
results from Fig.(1) and (3) show that the potential, $tilde{V}$, has the
maximum value at the throat and deflection angle of the photon getting diverge
at the throat. This implies that the photon sphere can be used to figure out
the throat of the three-form wormholes. On the other hand, the numerical
results of case 2 shown in Fig.(2) and (4) show that the potential $\tilde{V}$
has the maximum value at the photon sphere radius. This means that the light
bending occurs at the photon sphere outside the throat. In addition, the
results of case 2 tell us that the observers will see two sets of Einstein’s
relativistic rings at and outside the throat. Comparing the numerical results
of the deflection angle from two case studies, the solution from case 1 with
$\Phi_{0}=-1$ has the deflection angle less than that of case 2 with
$\Phi_{0}=-2$, meaning that the coefficient of the redshift function,
$\Phi_{0}$, plays an important role in of the deflection angle analysis.
Moreover, the deflection angles of both sets of the parameters from Fig.(3)
and (4) imply that the larger wormholes produce the bigger deflection angles.

##  V Conclusions

In this work, we focus on the study of gravitational lensing from the
traversable wormholes with the three-form fields. We used specific solutions
to construct the wormhole solutions. We employ two sets of parameters and
specific form of the three-form potential that are given by Barros and Lobo
(2018) which are most suitable to construct the energy densities and NEC
profiles of the three-form fields and the matter source which does not violate
the energy conditions. The null geodesics condition is used to find the
geodesics of the wormhole metric to obtain effective potential. We have used
the ansatz of the redshift and the shape functions in Eq.(23) with the
parameters of case 1 and case 2 to explain the existence of the photon sphere
occurring in the wormhole background. As the results present in the present
work, the geometrical shapes of the wormholes can be represented by using the
observed photon sphere. Moreover, the Einstein’s ring which is an useful
physical observable in astronomy and cosmology can be used to probe the
existence of the wormholes, see for example Refs.Tsukamoto et al. (2012);
Shaikh et al. (2019b). In this work, we have provided an useful information
that can be developed to predict the Einstein ring in the three-form
wormholes. In addition, the gravitational signal of the wormholes has been
drawn a lot of attention due to the active research fields in the
gravitational wave observations. We will leave this interesting topic for the
future work.

## Acknowledgement

D. Samart is financially supported by the Mid-Career Research Grant 2021 from
National Research Council of Thailand under a contract No. N41A640145. P.
Channuie acknowledged the Mid-Career Research Grant 2020 from National
Research Council of Thailand (NRCT5-RSA63019-03) and the National Science,
Research and Innovation Fund (NSRF) with grant No.R2565B030.

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