Weak Deflection Angle by Kalb–Ramond Traversable Wormhole in Plasma and Dark Matter Mediums

Abstract

This paper is devoted to computing the weak deflection angle for the Kalb–Ramond traversable wormhole solution in plasma and dark matter mediums by using the method of Gibbons and Werner. To acquire our results, we evaluate Gaussian optical curvature by utilizing the Gauss–Bonnet theorem in the weak field limits. We also investigate the graphical influence of the deflection angle $\tilde{\alpha }$ with respect to the impact parameter $\sigma$ and the minimal radius $r_0$ in the plasma medium. Moreover, we derive the deflection angle by using a different method known as the Keeton and Petters method. We also examine that if we remove the effects of plasma and dark matter, the results become identical to that of the non-plasma case.

1. Introduction

A visual illustration of a black hole (BH) shows that it dissipates energy through radiation, then compresses, and sooner or later dissolves [1]. Albert Einstein first estimated the presence of BHs by his theory of general relativity [2]. A BH is considered as an area of space having too strong a gravitational pull such that the fastest-moving objects and even light cannot escape it. Four types of BHs that exist are intermediate BHs, stellar BHs, miniature BHs, and supermassive BHs. Black holes are certainly very simple as they have two primary parts, the event horizon and the singularity. The event horizon is the boundary that indicates the limit of a BH. At the event horizon, the escape velocity becomes equal to the velocity of light. The singularity is a point in space where the existing mass has infinite density. According to GR, spacetime singularities lead to various issues, both scientifically and physically [3].

Just like BHs, wormholes (WHs) appear as effective solutions to the Einstein field equations. The simplest solution to the Einstein field equations is the Schwarzschild solution. The idea of a WH was first given by Flamm in 1916, soon after the invention of Schwarzschild’s BH solution. In general, a WH is the link between two separate regions of space that are far away from each other via a tunnel [4]. In 1935, Einstein and Rosen [2] additionally investigated the theory of inter-universe connections. These spacetime connections were known to be “Einstein–Rosen Bridges”. However, the idea of “WHs” was formulated by Wheeler in 1957 [5,6]. After that, he illustrated that WHs would be unsteady and non-traversable for even a photon.

Later on, the term traversable WH was developed by Morris and Thorne in 1988 [7]. They established flat traversable WHs with exotic matter that do not satisfy the null energy conditions [8]. Exotic matter creates problems for making stable WHs. Morris, Thorne, and Yurtsever also showed that traversable WHs can be made stable and flat by applying the Casimir effect. Another type of traversable WH, a thin-shell WH, was introduced by Matt Visser in 1989 [9], in which a path through the WH may be made in such a way that the traversing path does not cross the region of exotic matter. Furthermore, the metric of the Ellis WH was considered firstly in [10], and the analysis of the Ellis WH returned in the standard work of Morris and Thorne, where they introduced traversable WHs. The deflection of light was initially suggested by Chetouani and Clement in the Ellis WH [11]. Nakajima and Asada also studied gravitational lensing by the Ellis WH [12].

It is a known fact that weak and strong gravitational lensing (GL) is a very productive area to find not only dark and heavy objects, but also BHs and WHs. To identify a WH, a possible way is the implementation of GL. The GL by WHs has been reviewed on a large scale in the literature of theoretical physics, as well as astrophysics [3,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57].

Gravitational lensing was suggested by Soldner for the first time in the background of Newtonian theory. The basic theory of GL was formed by Liebes, Refsdal, and many other scientists [58,59,60]. When light released by distant galaxies passes through heavy objects in the universe, the gravitational attraction by those heavy objects can cause the light to deviate from its pathway. This process is known as GL. Gravitational lensing is a very applicable method to understand dark matter, galaxies, and the universe. Three types of GL listed in the literature are: (i) strong GL, (ii) weak GL, and (iii) micro-GL. Strong GL indicates the approximate magnification, location, and time delay of the images with the aid of BHs. Strong GL is also helpful to see various objects such as boson stars, fermion stars, and monopoles [61]. Weak GL is used to discover the mass of astronomical objects without demanding their changing nature. Weak lensing also differentiates dark energy from modified gravity and investigates how the universe is expanding rapidly. Gravitational lensing has been calculated for many spacetimes by applying various methods [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84]. In the past few years, we have examined many research works that relate GL to the Gauss–Bonnet theorem (GBT).

Using the GBT, Gibbons and Werner [84] revealed that it is possible to compute the deflection angle in weak field limits using the Gauss–Bonnet theorem. After that, Werner expanded this technique to Kerr BHs [85] by using Nazim’s method for the Randers–Finsler metric. Then, Ono et al. enlarged the study of weak GL in stationary axisymmetric spacetimes using the finite distance method [86,87,88,89]. The theory of GL comprises three physical processes named (i) geometric optics, (ii) the thin lens approximation, and (iii) the perturbation theory of GL [90,91,92].

For a distant observer from a source, the bending angle of light can be determined by using the GBT in the weak field limits [84]. By considering an oriented surface, let us describe a domain $D_R$ surrounded by the beam of light with a circular boundary $C_R$ having the Euler characteristic element $\mathcal{X}$ and metric g at the focus area where the light rays coincide with the source and the viewer. Therefore, when the GBT is applied within the optical metric, it provides us the bending angle of light stated as [84]:

$$\int {\int }_{{\mathcal{D}}_R}\mathcal{K}dS+{\oint }_{\partial {\mathcal{D}}_R}kdt+\sum _n{\theta }_n=2\pi \mathcal{X}\left({\mathcal{D}}_R\right),$$

where $D_R$ is the region that comprises the source of the light waves, the observer, and the focal point of the lens, $\mathcal{K}$ shows the Gaussian optical curvature, $dS$ is the surface element, k is known as geodesic curvature, and $\partial {\mathcal{D}}_R$ shows that this portion is surrounded by the outermost light rays. The asymptotic bending angle of light can be computed as [84]:

$$\tilde{\alpha }=-\int {\int }_{D_{\infty }}\mathcal{K}dS,$$

where $\tilde{\alpha }$ represents the bending angle and $D_{\infty }$ indicates the infinite domain bounded by the rays of photons, apart from the lens. Moreover, by utilizing the GBT, the bending angle for static and axisymmetric rotating Teo WHs was investigated by Jusufi and Övgün [93]. Recently, Övgün studied the deflection angle by Damour–Solodukhin WHs [94].

An approximate form of GL by the help of spherically symmetric lenses up to the post-post-Newtonian (PPN) was newly originated by Keeton and Petters [90,91]. Sereno and de Luca [92] expanded the PPN approximation to Kerr BHs. The Keeton and Petters method allows the calculations of those observable quantities that can be basically independent of coordinates and, therefore, physically applicable.

Dark matter comprises those particles that do not absorb, reflect, or emit light, so that they cannot be detected by observing electromagnetic radiation. Dark matter is a substance that cannot be directly visualized. Dark matter produces up to $27\%$ of the overall mass–energy of the universe, and the remaining part consists of dark energy. Dark matter can only be noticed by its gravitational interaction. However, it has a weak non-gravitational interaction and is a non-relativistic nature [95]. Some of the dark matter exists as: weakly interacting massive particles (WIMPs), axions, sterile neutrinos, super WIMPs, etc. Dark matter particles are formed by non-baryonic particles. The neutrino is the only familiar non-baryonic particle, and it is taken as the first dark matter candidate. The refractive index used in dark matter maintains the propagation speed. According to the literature of the 20th Century, we have seen many times that matter mostly consists of protons and neutrons. However, the matter that we see is not a suitable form of matter. In this universe, there are some other types of matter whose masses are five-times greater than regular matter. Such unknown matter is called “dark matter”. We have not been able to detect its existence in the laboratory until now. For the proper understanding of dark matter, we have to understand and then use various branches of astronomy and physics such as particle physics to describe the relation between dark matter and standard matter. Cosmology, general relativity, and astrophysics are applicable for the wide-ranging study of dark matter in the literature [96,97]. For the inspection of the bending angle through dark matter, we take the refractive index as [96,97]:

$$n\left(\omega \right)=1+\beta A_0+A_2{\omega }^2,$$

where $\omega$ is the frequency of a photon. It is noticed here that $\beta =\frac{{\rho }_0}{4m^2{\omega }^2}$ represents the mass density of the scattered dark matter particles, $A_0=-2e^2{ϵ}^2$ and $A_{2}j\ge 0$. The higher-order terms such as $O\left({\omega }^2\right)$ and onward are linked with the polarizability of the dark matter candidate. The term $O\left({\omega }^2\right)$ stands for the neutral dark matter candidates, while $O\left({\omega }^{-2}\right)$ denotes the charged dark matter candidates.

The main goal of this paper is to examine the deflection of light for the Kalb–Ramond traversable WH solution by using various methods.

This paper is organized as follows. In Section 2, we analyze the Kalb–Ramond traversable WH solution in detail. In Section 3, we compute the deflection angle for the Kalb–Ramond traversable WH solution in the plasma medium. Section 4 is related to the graphical inspection of the deflection angle of the Kalb–Ramond traversable WH solution in the framework of the plasma medium. In Section 5, we investigate the bending angle of the Kalb–Ramond traversable WH solution by using the Keeton and Petters method. In Section 6, we enlarge our observations and derive the value of the deflection angle in the dark matter medium. In Section 7, we conclude all the results.

2. Kalb–Ramond Traversable Wormhole Solution

This section is devoted to describing some properties of a traversable WH solution. The metric is spherically symmetric and independent of time for the sake of simplification. A wormhole is like a tunnel connecting two regions of space, having no horizon because the presence of a horizon does not allow two-sided traveling. The time taken by the traveler to pass through the WH should be limited and fairly short. The gravitational force that the traveler inside the WH experiences should be slightly small.

Local Lorentz violation effects also result in modified gravitational dynamics. However, the Lorentz symmetry may be violated close to the Planck scale. When one or more of the tensor fields gain nonzero vacuum expectation values (VEVs), Lorentz symmetry breaks spontaneously. The Kalb–Ramond (KR) field, an antisymmetric tensor field $B\mu \nu$ that appears in string theories, is another cause for spontaneous Lorentz symmetry breaking (LSB). A non-vanishing VEV breaks the gauge and the Lorentz symmetry by allowing a self-interaction potential. The tensor antisymmetric VEV can be split into two vectors, called pseudo-electric and pseudo-magnetic vectors, in the same way that the electromagnetic field strength can. A background KR field recently altered about a static and spherically symmetric WH. The parameter-dependent power-law correction to the Schwarzschild solution results from a non-minimal interaction between the KR VEV and the Ricci tensor. The effects of this LSB solution on the WH temperature and the shadows have also been investigated [98]. The static spherically symmetric spacetime of a Morris–Thorne traversable WH is expressed by the metric [7]:

$$d{s}^2=-e^{2\varphi \left(r\right)}d{t}^2+\frac{d{r}^2}{1-\frac{\Omega \left(r\right)}{r}}+r^{2}d{\theta }^2+r^2{sin}^2\theta d{\varphi }^2,$$

(1)

where the value of $\Omega \left(r\right)$ is defined as [98]:

$$\Omega \left(r\right)=r{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}.$$

Here, $\lambda$ is the Lorentz violating parameter $\lambda ={\left|b\right|}^2{\xi }_2$, b and ${\xi }_2$ are constants [98], $\varphi \left(r\right)$ is the redshift function, and $\Omega \left(r\right)$ is the shape function of the WH. Both of these functions are adjustable. The redshift function is finite everywhere. The shape function $\Omega \left(r\right)$ calculates the shape of the WH or describes the WH physically. The radial variable r has a minimum value of $r_0$ and a maximum value of infinity, or $r_0\le r<\infty$. $r_0$ is a positive constant as it decreases from positive infinity to a least value and, after that, starts moving towards positive infinity.

After putting the value of the shape function, the spacetime metric becomes:

$$d{s}^2=-d{t}^2+\frac{d{r}^2}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2).$$

(2)

Now, for the Kalb–Ramond traversable WH solution, we can write our metric in general form as:

$$d{s}^2=-g_{tt}d{t}^2+g_{rr}d{r}^2+r^{2}d{\theta }^2+r^2{sin}^2\theta d{\varphi }^2,$$

(3)

where, $g_{tt}=1$ and $g_{rr}=\frac{1}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}$. For the calculation of the deflection angle by the Kalb–Ramond traversable WH solution, the optical path can be obtained by the null geodesic condition $d{s}^2=0$. Furthermore, we assume that the photon is moveable in the equatorial plane ($\theta =\frac{\pi }{2}$). In order to calculate the deflection angle, one can obtain the optical path metric in explicit form given as follows [98]:

$$d{t}^2=\frac{d{r}^2}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}+r^{2}d{\varphi }^2.$$

(4)

The above metric will be used to calculate the deflection angle of light by the Kalb–Ramond traversable WH solution, which is described.

3. Deflection Angle in the Plasma Medium

In this section, we evaluate the impact of the plasma medium on weak GL for the Kalb–Ramond traversable WH solution (4). The value of the refractive index $n\left(r\right)$ for the given WH solution is defined as [99]:

$$n\left(r\right)=\sqrt{1-\frac{{\omega }_e^2\left(r\right)}{{\omega }_{\infty }^2\left(r\right)}\Omega \left(r\right)}.$$

(5)

In the refractive index $n\left(r\right)$, ${\omega }_e$ indicates the plasma frequency of an electron, while ${\omega }_{\infty }$ denotes the frequency of a photon, which is noticed by the observer at infinity. Therefore, the optical metric can be written as:

$$d{t}^2=g_{lm}^{opt}d{x}^{l}d{x}^m=n^2\left(r\right)\left[\frac{d{r}^2}{f\left(r\right)}+r^{2}d{\varphi }^2\right],$$

(6)

where $f\left(r\right)$ is defined as

$$f\left(r\right)=\frac{1}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}.$$

(7)

Now, the relative Gaussian optical curvature can be determined by the expression present in [100,101]:

$$\mathcal{K}=\frac{R}{2},$$

(8)

where R is the Ricci scalar. We calculate the Gaussian optical curvature for the Kalb–Ramond traversable WH solution as:

$$\mathcal{K}\simeq -\frac{r_0}{2r^3}+\frac{5}{4}\frac{r_0^2}{r^4}\frac{w_e^2}{w_{\infty }^2}-\frac{r_0}{r^3}\frac{w_e^2}{w_{\infty }^2}+\mathcal{O}\left(r_0^3\right).$$

(9)

The Gaussian curvature depends on minimal radius $r_0$ and radial parameter r. For the sake of simplicity, we consider the optical curvature up to order two, as well as to match our results given in the non-plasma medium case.

The infinitesimal surface element for the Kalb–Ramond traversable WH solution can be computed as:

$$dS=\sqrt{g}drd\varphi =\left(r-\frac{r{w}_e^2}{w_{\infty }^2}\right)drd\varphi +\mathcal{O}\left(r_0^3\right).$$

To find the deflection angle in the plasma medium, we utilize the GBT. As the beam of light approaches from infinity up to a large distance and remembering that we are present in weak field limitations, here, the beam of light becomes almost straight. Hence, we utilize the straight line approximation $r=\frac{\sigma }{sin\varphi }$, where $\sigma$ expresses the impact parameter, and the GBT is stated as [84]:

$$\tilde{\alpha }=-{\int }_0^{\pi }{\int }_{\frac{\sigma }{sin\varphi }}^{\infty }\mathcal{K}dS.$$

(10)

The resulting expression can be expressed for $\lambda =\frac{3}{2}$. The Lorentz violating parameter must be restricted to the range $\lambda >1/2$ to achieve an asymptotically flat spacetime. If we take $\lambda \simeq 0$, the geometry of the metric will not be asymptotically flat. For this value of $\lambda$, deflection angle $\tilde{\alpha }$ for the Kalb–Ramond traversable WH can be obtained as:

$$\tilde{\alpha }\simeq \frac{r_0}{\sigma }+\pi {\left(\frac{r_0}{4\sigma }\right)}^2+\frac{w_e^2}{w_{\infty }^2}\frac{r_0}{\sigma }+\frac{r_0^2\pi }{8{\sigma }^2}\frac{w_e^2}{w_{\infty }^2}+\mathcal{O}\left(r_0^3\right).$$

(11)

The above result shows that the light rays are moving in the plasma medium. If we remove the plasma effects, this angle will convert into the non-plasma medium. The deflection angle $\tilde{\alpha }$ depends on the minimal radius $r_0$ and the impact parameter $\sigma$, where the bending angle is directly proportional to $r_0$ and inversely proportional to $\sigma$.

4. Graphical Behavior of the Deflection Angle

This section is based on the explanation of the graphical analysis of the deflection angle for the Kalb–Ramond traversable WH solution in the plasma medium. For this purpose, we analyze the deflection angle with respect to the impact parameter $\sigma$ and the minimal radius $r_0$.

4.1. $\tilde{\alpha }$ versus $\sigma$

Figure 1 shows the behavior of $\tilde{\alpha }$ with respect to the impact parameter $\sigma$ for plasma impact $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$. For $r_0\le 3$, we observe that, in the first graph, as $\sigma$ increases, $\tilde{\alpha }$ decreases exponentially and shows a convergent behavior, which converges to zero. On the other hand, we evaluated that, with the increase of $r_0$, the deflection angle also increases. Similarly, for $r_0\ge 3$ in the second plot, the angle shows a decreasing behavior when the value of the impact parameter increases.

Figure 2 shows that, for $r_0\le 3$, $r_0>3$ by assuming plasma impact $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-2}$, one can analyze a similar behavior to that of $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$. We also investigated that, for small values of $r_0$, as $\sigma$ increases, $\tilde{\alpha }$ exponentially decreases. Moreover, it is to be observed that, as the value of $\sigma$ increases, $\tilde{\alpha }$ decreases. This shows that the deflection angle has an inverse relation with $\sigma$ and a direct relation with $r_0$.

4.2. $\tilde{\alpha }$ versus $r_0$

Figure 3 shows the behavior of the deflection angle $\tilde{\alpha }$ of the light with respect to $r_0$ for the impact parameter $0<\sigma <50$ and plasma impact $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$. We examine that for $0<\sigma <50$, the left graph indicates that as the linear behavior as $r_0$ increases, the bending angle $\tilde{\alpha }$ also increases, and as $\sigma$ increases, the deflection angle decreases. For larger $\sigma$, the angle approaches zero. A similar case is given in the right plot for $\sigma >50$: angle increases with increasing values of $r_0$, and vice versa.

Figure 4 shows that, for $\sigma >50$ and $0<\sigma \le 50$ having plasma impact value $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-2}$, we can analyze similar behavior as for the case $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$. We evaluated that as the values of $r_0$ increases, the deflection angle also increases, and as $\sigma$ increases, the bending angle decreases, which shows the divergent behavior of the graphs. Furthermore, we investigated that bending angle $\tilde{\alpha }$ has a direct relation with impact parameter $r_0$ and an inverse relation with $\sigma$.

5. Deflection Angle Using the Keeton and Petters Method

Keeton and Petters established a completely beneficial framework for computing corrections in a standard asymptotically flat metric theory of gravity [90,91]. The central focus is to illustrate a way to manage lensing in computing gravity theories using PPN corrections up to third-order.

The non-linear spherically symmetric and asymptotically Minkowski spacetime is defined by:

$$d{s}^2=-A\left(r\right)d{t}^2+B\left(r\right)d{r}^2+C\left(r\right)d{\Omega }^2.$$

(12)

Due to the spherical symmetry, the geodesics of Equation (12) lie in the equatorial plane, while $d{\Omega }^2=d{\theta }^2+{sin}^2\theta d{\varphi }^2$ represents the standard unit metric. The above equation becomes:

$$d{s}^2=-A\left(r\right)d{t}^2+B\left(r\right)d{r}^2+C\left(r\right)d{\theta }^2.$$

(13)

For $A\left(r\right)\to 1$, $B\left(r\right)\to 1$, and $C\left(r\right)\to r^2$, the spacetime metric is flat in the absence of the lens, and we suppose that A, B, and C are all +ve in the outside region of the lens from where the required light rays pass. The Kalb–Ramond traversable WH solution can be presented as:

$$d{s}^2=-d{t}^2+\frac{d{r}^2}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}+r^{2}d{\theta }^2,$$

(14)

where the metric functions are:

$$A\left(r\right)=1,$$

(15)

and

$$B\left(r\right)=\frac{1}{1-{(\frac{r}{r_0})}^{\frac{2}{1-2\lambda }}}.$$

(16)

To find the PPN coefficients, we compare the coefficients of the extended form of metric function with the coefficients of the standard form of the general metric in a PPN series to third-order. The general form of the PPN series for Eq.(5.1) given as [90]:

$$A\left(r\right)=1+2a_1\left(\frac{\varphi }{c^2}\right)+2a_2{\left(\frac{\varphi }{c^2}\right)}^2+2a_3{\left(\frac{\varphi }{c^2}\right)}^3...$$

(17)

$$B\left(r\right)=1-2b_1\left(\frac{\varphi }{c^2}\right)+4b_2{\left(\frac{\varphi }{c^2}\right)}^2-8b_3{\left(\frac{\varphi }{c^2}\right)}^3...$$

(18)

Here, $\varphi$ denotes the three-dimensional Newtonian prospective. In our paper, we consider

$$\left(\frac{\varphi }{c^2}\right)=\frac{r_0}{r}.$$

(19)

We obtain the values of the coefficients of PPN metric as

$$a_1=0,a_2=0,a_3=0,b_1=\frac{1}{2},b_2=\frac{1}{4},b_3=\frac{1}{8}.$$

After finding the PPN coefficients, we then determine the coefficients in the extended form of the bending angle. The extended form of the light deflection angle can be written as follows

$$\alpha \left(b\right)=A_1\left(\frac{m}{b}\right)+A_2{\left(\frac{m}{b}\right)}^2+A_3{\left(\frac{m}{b}\right)}^3+O{\left(\frac{m}{b}\right)}^4.$$

(20)

We take $r_0=m$ and $\sigma =b$. To compute the coefficients of the bending angle, we utilize the following equation:

$$\begin{array}{ccc}\hfill A_1& =& 2\left(a_1+b_1\right),\hfill \\ \hfill A_2& =& \left(2a_1^2-a_2+a_1{b}_1+b_2-\frac{b_1^2}{4}\right)\pi ,\hfill \\ \hfill A_3& =& \frac{2}{3}[35a_1^3+15a_1^2{b}_1-3a_1\left(10a_2+b_1^2-4b_2\right)\hfill \\ & +& 6a_3+b_1^3-6a_2{b}_1-4b_1{b}_2+8b_3].\hfill \end{array}$$

(21)

Using the Equation (21), we can determine the values of the coefficients presented as follows

$$A_1=1,A_2=\frac{3\pi }{16},A_3=\frac{5}{12}.$$

After substituting the values of the coefficients in Equation (20), we obtain the resulting bending angle of the Kalb–Ramond traversable WH as:

$$\tilde{\alpha }\left(\sigma \right)=\left(\frac{r_0}{\sigma }\right)+\frac{3\pi }{16}{\left(\frac{r_0}{\sigma }\right)}^2+\frac{5}{12}{\left(\frac{r_0}{\sigma }\right)}^3+O{\left(\frac{r_0}{\sigma }\right)}^4.$$

(22)

This expression shows the deflection angle by using the Keeton and Petters method, which is dependent on $r_0$ and $\sigma$. The bending angle is directly proportional to $r_0$ and inversely proportional to $\sigma$.

6. Deflection Angle of a Photon in the Dark Matter Medium

Here, in this section, we examine how dark matter influences the weak deflection angle. For this purpose, we take the refractive index for the dark matter medium [42,96,97]:

$$n\left(\omega \right)=1+\beta A_0+A_2{\omega }^2.$$

(23)

The two-dimensional optical geometry for the Kalb–Ramond traversable WH solution is:

$$d{t}^2=n^2\left(\frac{d{r}^2}{1-{\left(\frac{r}{r_0}\right)}^{\frac{2}{1-2\lambda }}}+r^{2}d{\varphi }^2\right),$$

(24)

with the condition:

$$\frac{dt}{d\varphi }{|}_{C_R}=n{\left(r^2\right)}^{\frac{1}{2}}.$$

(25)

Applying this condition to a non-singular domain $C_R$ outside of the light ray will yield the deflection angle. Hence, with the help of the GBT, we can calculate the weak deflection angle for the Kalb–Ramond traversable WH solution in the dark matter medium.

$$\underset{R\to \infty }{lim}{\int }_0^{\pi +\alpha }\left[{\mathcal{K}}_g\frac{dt}{d\varphi }\right]{|}_{C_R}d\varphi =\pi -\underset{R\to \infty }{lim}\int {\int }_{D_R}\mathcal{K}dS.$$

(26)

We compute the Gaussian optical curvature as follows:

$$\mathcal{K}=\frac{r_0}{2r_3\left(1+\beta A_0+A_2{\omega }^2\right)}.$$

(27)

After this, we determine:

$$\underset{R\to \infty }{lim}{\mathcal{K}}_g\frac{dt}{d\varphi }{|}_{C_R}=1.$$

(28)

Now, when we apply the limit $R\to \infty$, then we can evaluate the deflection angle for the Kalb–Ramond traversable WH solution by using the GBT as below:

$$\tilde{\alpha }=-{\int }_0^{\pi }{\int }_{\frac{\sigma }{sin\varphi }}^{\infty }\mathcal{K}dS.$$

(29)

We obtain the weak deflection angle of the Kalb–Ramond traversable WH solution in the dark matter medium after putting the value of $\mathcal{K}$ and $dS$:

$$\tilde{\alpha }\simeq \frac{r_0}{\sigma {\left(1+\beta A_0+A_2{\omega }^2\right)}^6}+\frac{r_0^2{\beta }^6{A}_0^{12}\pi }{16{\sigma }^2{\left(1+\beta A_0+A_2{\omega }^2\right)}^6}|+\mathcal{O}\left(r_0^3\right).$$

(30)

This result shows that, when the effect of dark matter vanishes, the bending angle reduces to the deflection angle in the non-plasma medium. The effect of dark matter shows a larger deflection than in the general case. The deflection angle $\tilde{\alpha }$ depends on $\sigma$ and $r_0$. We can see that dark matter gives a larger deflection angle than in the general case.

7. Conclusions

This paper was concerned with calculating the bending angle $\tilde{\alpha }$ for the Kalb–Ramond traversable WH solution in plasma and dark matter mediums. We evaluated the metric and explained its mathematical interpretation. For the calculation of the bending angle, we obtained the Gaussian optical curvature. After that, we applied the GBT and derived the value of the deflection angle for the Kalb–Ramond traversable WH solution in the plasma medium, Equation (11) for $\lambda =\frac{3}{2}$.

If we neglect or have the value of the plasma effect $\frac{{\omega }_e}{{\omega }_{\infty }}$ approach zero, then the effect of the plasma vanishes, and the resulting expression of the bending angle for the plasma medium reduces to the non-plasma case.

The graphical behavior of the deflection angle in the plasma medium was also calculated.

$\tilde{\alpha }$ with respect to impact parameter $\sigma$:

For $r_0\le 3$ and $r_0>3$ having the effect of the plasma $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$, the deflection angle $\tilde{\alpha }$ gradually approaches zero as $\sigma$ increases, in both graphs. We have also observed that $\tilde{\alpha }$ exponentially decreases and indicates a convergent behavior for large values of $\sigma$. We examined that for $r_0\le 3$ and $r_0>3$ having the plasma effect $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-2}$ as $r_0$ approaches infinity, $\tilde{\alpha }\to \infty$ at smaller $\sigma$. We also investigated that $\tilde{\alpha }$ has an inverse relation with $\sigma$ and a direct relation with $r_0$.

$\tilde{\alpha }$ with respect to the minimal radius $r_0$:

We= analyzed that, for $0<\sigma \le 50$ and $\sigma >50$, supposing $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$, the deflection angle $\tilde{\alpha }$ increases as the value of $r_0$ increases. The linear behavior of both graphs was examined. We also investigated that, as $\sigma$ increases, the angle decreases, and vice versa. We reviewed that, for $0<\sigma \le 50$ and $\sigma >50$ at $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-2}$, deflection angle $\tilde{\alpha }$ exhibited similar behavior as that for $\frac{{\omega }_e}{{\omega }_{\infty }}={10}^{-1}$. We also examined that $\tilde{\alpha }$ increases for larger $r_0$ and $\tilde{\alpha }$ decreases as $\sigma$ increases. We investigated a divergent behavior in both cases. We evaluated that $\tilde{\alpha }$ has a direct relationship with $r_0$ and an inverse relation with $\sigma$.

Furthermore, we also derived the deflection angle by using the Keeton and Petters, method which is the approximate form of GL, and used a spherically symmetric spacetime metric. For this purpose, we found the coefficients of the PPN metric by comparing the expanded metric function with the standard PPN metric. Later, we determined the coefficients of the bending angle and, again, by comparing them with the general form of Schwarzschild metric to obtain the final results given in Equation (22).

Lastly, we calculated the value of deflection angle $\tilde{\alpha }$ for the Kalb–Ramond traversable WH solution in the dark matter medium. Dark matter contains particles that cannot absorb, emit, or reflect light rays. For this purpose, we calculated Gaussian optical curvature Equation (27) and determined the deflection angle (30).

In the plasma, the Keeton and Petters method, and the dark matter medium case, the results reduced to the non-plasma case. The resulting equations depend on the minimal radius $r_0$ and impact parameter $\sigma$, showing that the deflection angle is directly proportional to $r_0$ and inversely to $\sigma$. The plots between the angle and minimal radius or impact parameter demonstrates that, when $r_0$ increases, the bending angle also move towards infinity. Furthermore, we observed the convergent and divergent behavior with respect to $\sigma$ and $r_0$, respectively. When we removed the effect of plasma and dark matter, the resultant expression became similar to that of the Bumblebee traversable wormhole solution. In the case of the Keeton and Petters method, only the first term became similar to the general case.