1. Introduction
The observational evidence accumulating from recent high-energy astrophysical observational data is arousing the attention of the entire scientific community in suggesting new strategies and in building more and more advanced models for testing gravity in robust field systems which are well-supported by the International Laboratory for Gamma-Ray Astrophysics (INTEGRAL) [1], Swift [2], XMM-Newton [3], Event Horizon Telescope (EHT) [4], Advanced High Energy Astrophysics Telescope (ATHENA) [5], Enhanced x-ray Timing and Polarimetry Mission (eXTP) [1], x-ray Polarimetry Explorer (IXPE) [1], Laser Gravitational Wave Observatory (LIGO) and Virgo [6]. Specific attention is held to wormholes (WHs), which have the exceptional attributes to be horizonless, and equipped with a traversable bridge relating two dissimilar universes or extremely far spacetime regions [7]. In the WH research history, the solution of Morris–Thorne has held a key position as a classic static WH [8], albeit numerous static WH metrics have been gotten prior to this [9]. Ensuing research uncovered that WHs might accept superluminal travel due to the global space-time topology [10] and they might also give rise to time machines [11].
It is generally known that the exotic matter existence, having negative energy, and conflicting with the classical laws of physics are related to the traversable condition in standard general relativity (GR) or to the intriguing prospect of making interstellar travels. A typical way for clarifying such an issue depends on quantum mechanics [12] or on topological contentions [13], if we sketch the WH models in extended/alternative gravity theories.
In other words, cannot be utilized to accomplish stable WH solutions in the GR background. In the literature, several new techniques have been suggested to alleviate this issue. In order to overcome this strange behavior, several researchers focus fundamentally on considering exotic forms of matter. On the contrary, modified/alternative gravity theories are natural suites where this can be tended to holding standard matter [14]. Indeed, in these situations, the above conditions do not apply straightforwardly to matter, and hence, instead of forcing exotic conditions over the thermodynamic observables viz., pressure and energy, we are able to imitate exotic matter by using geometry [15]. This imitates the WH attributes during higher-order curvature terms and/or efficient field theories that may be planned into Lagrangians expanding the Hilbert–Einstein one. The cause is that the gravitational interaction itself can lead to extra terms, which could be described as contributing to an efficient energy–momentum tensor that violates the energy conditions on the right-hand side of Einstein’s field equations. Therefore, the necessary violation of the energy conditions is provided by the modified gravitational interaction itself (see e.g. [13, 16]). In this context, the search for WH solutions in modified gravity theories stunned the scientific community in avoiding the existence of these unusual fluids. More exactly, all known new degrees of freedom in the gravitational sector were introduced by modifications to Einstein’s gravity. In this regard, many WH solutions have been investigated in diverse modified theories of gravity like f(R) gravity WH with and without noncommutative geometry [17, 18], $f({ \mathcal T })$ gravity [19], Einstein–Gauss–Bonnet theory [20], Born–Infeld gravity [21], Lovelock gravity [22], f(R, T) gravity [23] and in others theories of gravity. Capozziello and his collaborators [24–30] have conducted fascinating research on WH geometries in the background of various altered gravity theories.
Motivated by these good discussions, we are considering the teleparallel equivalent of GR (TEGR) as a starting point, we can move toward the modified theory of gravity with matter-coupling. In this respect, Harko and co-workers [31], have formulated such matter-coupled theory namely $f({ \mathcal T },T)$ gravity. In this modified gravitational theory, the gravitational part of Lagrangian is constructed according to an arbitrary function including the torsion scalar ${ \mathcal T }$ and trace T of the stress-energy tensor. The main advantage of $f({ \mathcal T },T)$ modification [32, 33] is the gravitational field equations are explained in an entirely different way whereas compared to other torsion or curvature-based theories of gravity, which is adequately constructed from the tetrad field. For a detailed review of the types of tetrads, one may refer the reader to [34].
The theoretical building of WH geometries depends on how one has an intended metric, which must be fine-tuned by modifying the form of the gravitational potential functions or using an exact equation of state relating pressure to energy density and subsequently solving the Einstein field equations. In our paper, we focus on a new class of phantom traversable WHs in teleparallel gravity with diagonal tetrad under minimal matter coupling by assuming that space-time is spherically symmetric and possesses conformal symmetry. We further provide the discussion of the particles moving in closed orbits around the WHs throat exhibiting quasiperiodic oscillations (QPOs) from the astrophysical point of view. However, exact solutions are found for phantom traversable WHs under the guess of non-static conformal symmetry in an arrangement of a precise methodology that was tested formerly by Boehmer and colleagues [35, 36]. To be more precise, the survey of conformal symmetry provides a characteristic relationship between matter and geometry across the Einstein field equations. For this purpose, the attribute of the conformal Killing operator ${\mathfrak{L}}$ related to the metric g is a linear conformally mapped from the space ${ \mathcal L }(\varepsilon )$ of vector fields on ϵ, which provide2 we present fundamental notations, variables and exact general solutions that are deduced involving non-static conformal symmetries for the WH geometry in $f({ \mathcal T },T)$ gravity. In section 3 we write some essential insight concerning our WH solutions by considering choices for the shape function, a specific linear EoS relating the energy density and the pressure anisotropy, as well as the energy conditions and fundamental frequencies for our phantom traversable WH are discussed. Finally, in section 5 we conclude our results.
$\begin{eqnarray}{{\mathfrak{L}}}_{\varepsilon }{g}_{\epsilon \varepsilon }=h(r){g}_{\epsilon \varepsilon },\,\,\,\,\mathrm{where}\,\,\,\,\varepsilon \in { \mathcal L }(\varepsilon ).\end{eqnarray}$
Here h(r) and g represent the conformal killing vector and the conformally mapped onto itself along ϵ, respectively. It is worth mentioning here that for a static metric, we noticed that neither h(r) nor ϵ should be static. From this mathematical methodology, the equation of state is uniquely established by Einstein’s equations as shown by Herrera and collaborators [37] for a one-parameter group of conformal motions. Subsequently, Maartens and Maharaj [38] extended this specific exact solution for static spherical configurations of charged imperfect fluids by supposing that space-time admits conformal symmetry. Very recently, WH solutions admitting a one-parameter group of conformal motions were studied by Kuhfittig [39]. This present paper is then arranged as follows: In section 2. $f({ \mathcal T },T)$ gravity
It is a fact that all the torsional theories depend on matrices that transform the space-time metric into the Minkowski metric. Here, we consider the following metric5 ). This Weitzenböck notation can be read as:6 ), which turn out to be8 ) can be revised as:8 ) as a function of the contorsion-like tensor as follows:14 ) as:16 )–(18 ), and equations (20 )–(21 ) in equation (15 )
$\begin{eqnarray}{\rm{d}}{s}^{2}={g}_{\alpha \zeta }{\rm{d}}{x}^{\alpha }{\rm{d}}{x}^{\zeta }={\eta }_{{\rm{i}}j}{\theta }^{{\rm{i}}}{\theta }^{j}.\end{eqnarray}$
with $\begin{eqnarray}{\rm{d}}{x}^{\alpha }={{\rm{E}}}_{{\rm{i}}}{}^{\alpha }{\theta }^{{\rm{i}}};\,\,{\theta }^{{\rm{i}}}={{\rm{E}}}^{{\rm{i}}}{}_{\alpha }{\rm{d}}{x}^{\alpha },\end{eqnarray}$
where ηij represents the Mankowski metric with diag( + , − , − , − ) and Eiα represents the components of tetrad and it is satisfying the following relations $\begin{eqnarray}{{\rm{E}}}_{{\rm{i}}}{}^{\alpha }{{\rm{E}}}^{{\rm{i}}}{}_{\zeta }={\delta }_{\zeta }^{\alpha },\,\,{{\rm{E}}}_{\alpha }{}^{\zeta }{{\rm{E}}}^{\alpha }{}_{j}={\delta }_{j}^{\zeta }.\end{eqnarray}$
The Levi–Civita connection is carried out as follows $\begin{eqnarray}{\dot{{\rm{\Gamma }}}}^{\varrho }{}_{\alpha \zeta }=\displaystyle \frac{1}{2}{g}^{\varrho \sigma }({\partial }_{\zeta }{g}_{\sigma \alpha }+{\partial }_{\alpha }{g}_{\sigma \zeta }-{\partial }_{\sigma }{g}_{\alpha \zeta }).\end{eqnarray}$
For all extended teleparallel gravity, the crucial notation, i.e. Weitzenböck relation or connection, can be used instead of the Levi–Civita connection, supplied by equation ( $\begin{eqnarray}{{\rm{\Gamma }}}_{\alpha \zeta }^{\gamma }={{\rm{E}}}_{{\rm{i}}}{}^{\gamma }{\partial }_{\alpha }{{\rm{E}}}^{{\rm{i}}}{}_{\zeta }=-{{\rm{E}}}^{{\rm{i}}}{}_{\alpha }{\partial }_{\zeta }{{\rm{E}}}_{{\rm{i}}}{}^{\gamma }.\end{eqnarray}$
One may deduce the geometrical object i.e. the torsion, from the above relationship equation ( $\begin{eqnarray}{{ \mathcal T }}_{\alpha \zeta }^{\gamma }={{\rm{\Gamma }}}_{\alpha \zeta }^{\gamma }-{{\rm{\Gamma }}}_{\zeta \alpha }^{\gamma }.\end{eqnarray}$
Accordingly, the contortion tensor is determined as follows: $\begin{eqnarray}{K}_{\alpha \zeta }^{\gamma }={{\rm{\Gamma }}}_{\alpha \zeta }^{\gamma }-{\dot{{\rm{\Gamma }}}}_{\alpha \zeta }^{\gamma }=\displaystyle \frac{1}{2}({{ \mathcal T }}_{\alpha }{}^{\gamma }{}_{\zeta }+{{ \mathcal T }}_{\zeta }{}^{\gamma }{}_{\alpha }-{{ \mathcal T }}^{\gamma }{}_{\alpha \zeta }).\end{eqnarray}$
The equation ( $\begin{eqnarray}{K}_{\gamma }^{\alpha \zeta }=-\displaystyle \frac{1}{2}({{ \mathcal T }}^{\alpha \zeta }{}_{\gamma }-{{ \mathcal T }}^{\zeta \alpha }{}_{\gamma }+{{ \mathcal T }}_{\gamma }{}^{\alpha \zeta }).\end{eqnarray}$
The super-potential is determined by combining the torsion and contortion tensors as follows: $\begin{eqnarray}{S}_{\gamma }{}^{\alpha \zeta }=\displaystyle \frac{1}{2}({K}^{\alpha \zeta }{}_{\gamma }+{\delta }_{\gamma }^{\alpha }{{ \mathcal T }}^{\sigma \zeta }{}_{\sigma }-{\delta }_{\gamma }^{\zeta }{{ \mathcal T }}^{\sigma \alpha }{}_{\sigma }).\end{eqnarray}$
Lagrangian density based on torsion ${ \mathcal T }$ is described as: $\begin{eqnarray}{ \mathcal T }={{ \mathcal T }}^{\gamma }{}_{\alpha \zeta }{S}_{\gamma }{}^{\alpha \zeta }\end{eqnarray}$
It is possible to derive the Riemann tensor from equation ( $\begin{eqnarray}{R}_{\gamma }{}^{\alpha \varrho \zeta }={K}_{\alpha \varrho ;\zeta }^{\gamma }-{K}_{\alpha \zeta ;\varrho }^{\gamma }+{K}_{\sigma \zeta }^{\gamma }{K}_{\alpha \varrho }^{\sigma }-{K}_{\sigma \varrho }^{\gamma }{K}_{\alpha \zeta }^{\sigma }.\end{eqnarray}$
We can derive the Ricci scalar as follows $\begin{eqnarray}R=-{ \mathcal T }-2{D}_{\alpha }{{ \mathcal T }}_{\zeta }^{\zeta \alpha },\end{eqnarray}$
where the covariant derivative is indicated by Dα. For an extended teleparallel theory of gravity, here is the modified action with matter coupling: $\begin{eqnarray}s=\int {\rm{d}}{x}^{4}{\rm{E}}\left\{\displaystyle \frac{1}{2{k}^{2}}f({ \mathcal T },T)+{{ \mathcal L }}_{(M)}\right\},\end{eqnarray}$
where T stands for the trace of the energy–momentum tensor and ${{ \mathcal L }}_{(M)}$ represents the Lagrangian density. Further, ${\rm{E}}=\det \left({{\rm{E}}}^{i}{}_{\alpha }\right)=\sqrt{-g}$ and k2 = 8πG. The generalized field equations can be determined from equation ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{8\pi G}{2}{T}_{\varepsilon }^{\epsilon } & = & {S}_{\alpha }^{\zeta \varrho }\left({f}_{{ \mathcal T }{ \mathcal T }}{\partial }_{\varrho }{ \mathcal T }+{f}_{{ \mathcal T }T}{\partial }_{\varrho }T\right)+\left[{{\rm{E}}}^{-1}\right.\\ & & \left.\times {{\rm{E}}}_{\alpha }^{i}{\partial }_{\varrho }({\mathrm{EE}}_{i}^{\gamma }{S}_{\gamma }^{\zeta \varrho }+{{ \mathcal T }}_{\gamma \alpha }^{\gamma }{S}_{\gamma }^{\zeta \gamma }\right]{f}_{{ \mathcal T }}\\ & & +\displaystyle \frac{1}{4}{\delta }_{\alpha }^{\zeta }f+\displaystyle \frac{1}{2}{f}_{T}({T}_{\varepsilon }^{\epsilon }+{p}_{t}{\delta }_{\alpha }^{\zeta }).\end{array}\end{eqnarray}$
It is significant to recognize the existence of an additional force that relies on the substance Lagrangian [40]. In the analysis of the scalar or torsion-matter coupling for perfect fluids, it was discussed in [41] that by recognizing ${{ \mathcal L }}_{(M)}=p$, where p is the pressure, it means that the additional force vanishes. More generally, for the imperfect fluids, this pick has a very fascinating use in the analysis by considering ${{ \mathcal L }}_{(M)}={p}_{t}$, where pt is the pressure component of imperfect fluids. Further, the natural form for Lagrangian matter ${{ \mathcal L }}_{(M)}$, such as ${{ \mathcal L }}_{(M)}=-\rho $, where ρ represents the energy density can be considered more general in the sense that it does not suggest that the additional force has vanished. In this investigation, we have done our analysis with the choice of the matter Lagrangian as ${{ \mathcal L }}_{(M)}={p}_{t}$. For the current study, we are taking k2 = 8πG = 1, to avoid the complication regarding gravitational geometrical units as we are doing work with exact WH solutions. A geometry by static and spherically symmetric space-time metric is expressed as: $\begin{eqnarray}{\rm{d}}{s}^{2}=-{{\rm{e}}}^{\epsilon (r)}{{\rm{d}t}}^{2}+{{\rm{e}}}^{\varepsilon (r)}{{\rm{d}r}}^{2}+{r}^{2}\rm{d}{\theta }^{2}+{r}^{2}{\sin }^{2}\theta \rm{d}{\phi }^{2},\end{eqnarray}$
where ε(r) = 2Φ(r) with Φ(r) is the red-shift function and ${{\rm{e}}}^{\varepsilon (r)}={\left(\tfrac{r+b(r)}{r}\right)}^{-1}$, with b(r) is the shape function. The WH throat joins two asymptotic regions, i.e. b(r0) = r0. The shape function b(r) must satisfy the flaring-out requirement, $\tfrac{b(r)-{rb}^{\prime} (r)}{2{b}^{2}(r)}\gt 0$, which should be valid at or near the throat. near the WH throat, this reduces to $b^{\prime} ({r}_{0})\lt 1$. The shape function should meet the condition $1-\tfrac{b(r)}{r}\gt 0$ for the radial coordinates r > r0 to maintain the proper signature of the metric. To have asymptotically flat geometries, the metric functions need to adhere to the conditions that φ(r) and b(r)/r are tending to zero as r is moving toward ∞. For no-asymptotically flat WHs, this criteria can be relaxed. In the present study, our goal is to find the WH solutions in conformal killing spacetime using the linear equation of state through a particular form of shape function. Then we must have three degrees of freedom in the field equations. Therefore, our matter distribution must be anisotropic in order to have a consistent system. According to the energy–momentum tensor, the anisotropic source of matter is: $\begin{eqnarray}{T}_{\epsilon \varepsilon }=\rho {\upsilon }_{\epsilon }{\upsilon }_{\varepsilon }+{p}_{r}{\chi }_{\epsilon }{\chi }_{\varepsilon }+({\upsilon }_{\epsilon }{\upsilon }_{\varepsilon }-{g}_{\epsilon \varepsilon }-{\chi }_{\epsilon }{\chi }_{\varepsilon }){p}_{t},\end{eqnarray}$
where &ugr;ε denotes the 4-velocity vector of ${\upsilon }^{\epsilon }={{\rm{e}}}^{-\epsilon }{\delta }_{0}^{\epsilon }$ and ${\chi }^{\epsilon }={{\rm{e}}}^{-\varepsilon }{\delta }_{1}^{\epsilon }$, which must satisfy the conditions &ugr;ε&ugr;ε = −χεχε = 1. Now, the diagonal tetrad is calculated as: $\begin{eqnarray}{{\rm{E}}}^{{\rm{i}}}{}_{\alpha }=\left({{\rm{e}}}^{\tfrac{\epsilon (r)}{2}},{{\rm{e}}}^{\tfrac{\varepsilon (r)}{2}},r,r\sin \theta \right).\end{eqnarray}$
The determinant of ${e}_{\gamma }^{\varepsilon }$ is calculated as $\begin{eqnarray}{\rm{E}}={{\rm{e}}}^{\epsilon (r)+\varepsilon (r)}{r}^{2}\sin \theta .\end{eqnarray}$
The appropriate expression for ${ \mathcal T }$ for a WH spacetime is given as: $\begin{eqnarray}{ \mathcal T }(r)=\displaystyle \frac{2{{\rm{e}}}^{-\varepsilon (r)}}{r}\left(\epsilon ^{\prime} (r)+\displaystyle \frac{1}{r}\right).\end{eqnarray}$
In the current analysis, we are working with the zero spin connection. For the zero spin connection choice only the linear model for the $f({ \mathcal T },T)$ gravity is compatible. Here, we employ an appropriate linear model with diagonal tetrad for $f({ \mathcal T },T)$ gravity as follows: $\begin{eqnarray}f({ \mathcal T },T)={\chi }_{1}{ \mathcal T }(r)+\beta T+{\chi }_{2},\end{eqnarray}$
where χ1, β and χ2 are model parameters. The following field equations for $f({ \mathcal T },T)$ gravity are obtained by inserting equations ( $\begin{eqnarray}\begin{array}{l}\rho =-\displaystyle \frac{{{\rm{e}}}^{-\varepsilon (r)}}{4(\beta -1)(\beta +2){r}^{2}}\\ \times \left(2{\chi }_{1}\beta -4{\chi }_{1}+{\chi }_{1}\beta {r}^{2}\varepsilon ^{\prime} (r)\epsilon ^{\prime} (r)-2{\chi }_{1}\beta {r}^{2}\epsilon ^{\prime\prime} (r)\right.\\ -{\chi }_{1}\beta {r}^{2}\epsilon ^{\prime} {\left(r\right)}^{2}+\beta {r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}\\ +2{r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}-{\chi }_{1}\beta r\varepsilon ^{\prime} (r)-2{\chi }_{1}\beta {{\rm{e}}}^{\varepsilon (r)}\\ \left.-3{\chi }_{1}\beta r\epsilon ^{\prime} (r)+4{\chi }_{1}r\varepsilon ^{\prime} (r)+4{\chi }_{1}{{\rm{e}}}^{\varepsilon (r)}\right),\,\,\,\,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{r}=\displaystyle \frac{{{\rm{e}}}^{-\varepsilon (r)}}{4(\beta -1)(\beta +2){r}^{2}}\\ \times \left(2{\chi }_{1}\beta -4{\chi }_{1}+{\chi }_{1}\beta {r}^{2}\varepsilon ^{\prime} (r)\epsilon ^{\prime} (r)-2{\chi }_{1}\beta {r}^{2}\epsilon ^{\prime\prime} (r)\right.\\ -{\chi }_{1}\beta {r}^{2}\epsilon ^{\prime} {\left(r\right)}^{2}+\beta {r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}\\ +2{r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}+3{\chi }_{1}\beta \times r\varepsilon ^{\prime} (r)-2{\chi }_{1}\beta {{\rm{e}}}^{\varepsilon (r)}+{\chi }_{1}\beta r\epsilon ^{\prime} (r)\\ \left.+4{\chi }_{1}{e}^{\varepsilon (r)}-4{\chi }_{1}r\epsilon ^{\prime} (r\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{t}=\displaystyle \frac{{{\rm{e}}}^{-\varepsilon (r)}}{4(\beta -1)(\beta +2){r}^{2}}\\ \times \left({\chi }_{1}{r}^{2}\varepsilon ^{\prime} (r)\epsilon ^{\prime} (r)-2{\chi }_{1}\beta -2{\chi }_{1}{r}^{2}\epsilon ^{\prime\prime} (r)\right.\\ -{\chi }_{1}{r}^{2}\epsilon ^{\prime} {\left(r\right)}^{2}+\beta {r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}+2{r}^{2}{\chi }_{2}{{\rm{e}}}^{\varepsilon (r)}\\ +{\chi }_{1}\beta r\varepsilon ^{\prime} (r)+2{\chi }_{1}\beta \times {{\rm{e}}}^{\varepsilon (r)}-{\chi }_{1}\beta r\epsilon ^{\prime} (r)\\ \left.+2{\chi }_{1}r\varepsilon ^{\prime} (r)-2{\chi }_{1}r\epsilon ^{\prime} (r\right).\end{array}\end{eqnarray}$
An efficient strategy is to employ inheritance symmetry to investigate the natural relationship between matter and geometry using field equations. The symmetry arising from the CKVs is usually measured as inheritance symmetry. Further, we use the concept of conformal symmetry through the usage of the vector field ϖ [42, 43] which is defined as16 ), in equation (25 ):16 )26 ) in equations (22 )–(24 ), we obtain the updated version of field equations as follows
$\begin{eqnarray}{{\mathfrak{L}}}_{\varepsilon }{g}_{\epsilon \varepsilon }={g}_{\varepsilon \varepsilon }{\varepsilon }_{;\lambda }^{\varepsilon }+{g}_{\epsilon \varepsilon }{\varepsilon }_{;\varepsilon }^{\varepsilon }=h(r){g}_{\epsilon \varepsilon }.\end{eqnarray}$
where the Lie derivative is represented by ${\mathfrak{L}}$, with the CKVs ϵϵ, and vector field h(r). The following three expressions are obtained from equation ( $\begin{eqnarray*}{\varepsilon }^{1}{\epsilon }^{{\prime} }(r)=h(r),\,\,{\varepsilon }^{1}=\displaystyle \frac{{rh}(r)}{2},\,\,\,{\varepsilon }^{1}{\varepsilon }^{{\prime} }(r)+2{\varepsilon }_{,1}^{1}=h(r).\end{eqnarray*}$
The following relations are obtained by solving the above system using the space-time equation ( $\begin{eqnarray}{{\rm{e}}}^{\epsilon (r)}={{\rm{e}}}^{2{\rm{\Phi }}(r)}={K}_{1}{r}^{2},\,\,{{\rm{e}}}^{\varepsilon (r)}={\left(\frac{r+b(r)}{r}\right)}^{-1}=\frac{{K}_{2}}{{h}^{2}(r)},\,\,\,\end{eqnarray}$
where, K1 and K2 are considered integration constants. Now, by applying the equation ( $\begin{eqnarray}\rho =\displaystyle \frac{4{\chi }_{1}(\beta +1){h}^{2}(r)+2{\chi }_{1}(\beta +4){rh}(r)h^{\prime} (r)+{K}_{2}\left(2{\chi }_{1}(\beta -2)-(\beta +2){r}^{2}{\chi }_{2}\right)}{4(\beta -1)(\beta +2){K}_{2}{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{p}_{r}=\displaystyle \frac{4{\chi }_{1}(\beta -3){h}^{2}(r)-10{\chi }_{1}\beta {rh}(r)h^{\prime} (r)+{K}_{2}\left((\beta +2){r}^{2}{\chi }_{2}-2{\chi }_{1}(\beta -2)\right)}{4(\beta -1)(\beta +2){K}_{2}{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{p}_{t}=\displaystyle \frac{-4{\chi }_{1}(\beta +1){h}^{2}(r)-2{\chi }_{1}(\beta +4){rh}(r)h^{\prime} (r)+{K}_{2}\left(2{\chi }_{1}\beta +(\beta +2){r}^{2}{\chi }_{2}\right)}{4(\beta -1)(\beta +2){K}_{2}{r}^{2}}.\end{eqnarray}$
3. Phantom traversable WHs
In the current analysis, the EoS is considered the source of fluid in the context of a rapidly accelerated phenomenon of the Universe expansion. The linear EoS is provided as
$\begin{eqnarray}{p}_{r}=\delta \rho ,\end{eqnarray}$
where δ is a constant. As was already noted, the Universe’s accelerated expansion is accessible for fluid having δ ≤ −1/3. The NEC is justified by a fluid that has −1 < δ ≤ −1/3 and ρ > 0, which may be the cause of the Universe’s rapid expansion. The NEC all-time in space is never admitted by phantom fluid. However, linear EoS with δ > −1 and an extra term may defy the NEC in a few areas of the spacetime only.3.1. Shape function
Herein, we shall calculate new shape function by using the equations (27 ) and (28 ) in equation (30 ), and solve it for the shape function, one can get the following shape function:31 ) satisfies the compulsorily required conditions for the existence of a WH. There is no singularity, i.e. the absence of the horizon in the newly calculated shape function within the conformal symmetry under the effect of the linear equation of state confirms their physical viability for the WH existence. The flaring-out condition, i.e. $b^{\prime} (r)\lt 1$, which is the most important condition is satisfied at or near the throat for the newly developed shape function under the current scenario. This reduces to $b^{\prime} ({r}_{0})\lt 1$ near the WH throat. The current shape function via equation (31 ) meets the condition $1-\tfrac{b(r)}{r}\gt 0$ for the radial coordinate r > r0 to maintain the proper signature of the metric. The current shape functionobeys the requirements like the asymptotic flatness of the WH geometry. All the required properties of the obtained WH shape function are presented in figure 1.
$\begin{eqnarray}\begin{array}{l}b(r)=\displaystyle \frac{1}{{K}_{2}}\left[r\left(-{C}_{1}{r}^{-\tfrac{4(\beta (\delta -1)+\delta +3)}{5\beta +(\beta +4)\delta }}+{K}_{2}r\right.\right.\\ \left.\left.-\displaystyle \frac{{K}_{2}\left({r}^{2}{\chi }_{2}(\beta (\delta -1)+\delta +3)-3{\chi }_{1}(\beta -2)(\delta +1)\right)}{6{\chi }_{1}(\beta (\delta -1)+\delta +3)}\right)\right],\end{array}\end{eqnarray}$
where C1 is a constant of integration. According to Morris and Thorne, [8] our obtained shape function given by equation (
Figure 1. Shows the behavior of necessary WH properties with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
3.2. Energy conditions
In the current analysis, we are going to check the effect of phantom matter by using a linear equation of state on the traversability of the WH via energy conditions. The graphical behavior of all the energy conditions against the calculated shape function is given in figures 2–7. We utilize the specific values of the associated variables to investigate the behavior of energy conditions. The density profile stays positive with decreasing development in the present study under the current scenario with these specific values. Figure 3 describes the graphical development of condition ρ + pr for calculated shape function under conformal symmetry. The negative values of ρ + pr confirm the presence of exotic matter in the background of modified teleparallel gravity with matter coupling. Exotic matter is really a crucial prerequisite for the WH’s ability to be traversed. Figure 4 confirms the behavior of ρ − pr with negative nature. In figures 5 and 6, ρ − pt and ρ − pt are provided respectively. It is confirmed that ρ − pt remains positive while ρ − pt remains negative. The strong energy condition ρ + pr + 2pt is given in figure 7, which satisfy the condition. All the required physical behavior of the shape function can be seen from figure 1.
Figure 2. Shows the behavior of ρ with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
Figure 3. Shows the behavior of ρ + pr with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
Figure 4. Shows the behavior of ρ − pr with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
Figure 5. Shows the behavior of ρ + pt with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2=0.35, and δ = −0.3. |
Figure 6. Shows the behavior of ρ − pt with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
Figure 7. Shows the behavior of ρ + pr + 2pt with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, and δ = −0.3. |
4. Fundamental frequencies for our WH solution
Here, we’ll talk about the QPOs that particles on closed orbits around WH throats display from an astrophysical perspective (see for example [45, 46]). The Lagrangian for the motion of a particle around the WH throat for the equation (26 ) under the WH geometry is discussed in order to examine the fundamental frequencies32 ) yields the following relations when $\theta =\tfrac{\pi }{2}$ is assumed to represent the equatorial plane [44]
$\begin{eqnarray}{\mathfrak{L}}=\frac{{r}^{2}}{2}({\dot{\theta }}^{2}+{\sin }^{2}\theta {\dot{\phi }}^{2})-{{\rm{e}}}^{2{\rm{\Phi }}(r)}+\frac{{\dot{t}}^{2}}{2}+{\left(\frac{r+\beta (r)}{r}\right)}^{-1}\frac{{\dot{r}}^{2}}{2}.\end{eqnarray}$
The aforementioned equation ( $\begin{eqnarray}-E=\displaystyle \frac{\partial {\mathfrak{L}}}{\partial \dot{t}}=-{{\rm{e}}}^{2{\rm{\Phi }}(r)}\dot{t},\end{eqnarray}$
and [44] $\begin{eqnarray}L=\displaystyle \frac{\partial {\mathfrak{L}}}{\partial \dot{\phi }}=-{{\rm{e}}}^{2{\rm{\Phi }}(r)}{r}^{2}\dot{\phi },\end{eqnarray}$
where L and E represent candidates for the angular momentum and energy of a particle moving around the predicted WH throat. In addition, we obtain the following formula by utilizing the constraint ${g}_{\mu \nu }{\dot{y}}^{\mu }{\dot{y}}^{\nu }=-1$ as a normalization condition [44] $\begin{eqnarray}{E}^{2}={V}_{\mathrm{eff}}+{{\rm{e}}}^{2{\rm{\Phi }}(r)}{\left(\frac{r+b(r)}{r}\right)}^{-1}{\dot{r}}^{2},\end{eqnarray}$
where Veff mentions the effective potential [44], which is expressed as $\begin{eqnarray}{L}^{2}\times \frac{{{\rm{e}}}^{2{\rm{\Phi }}(r)}}{{r}^{2}}={V}_{\mathrm{eff}}\end{eqnarray}$
The epicyclic frequencies for static and spherically symmetric WH geometries have recently been determined by Falco and his coauthors [45]. The epicyclic oscillations around the Simpson–Visser regular BHs (black holes) like WHs were later addressed by Stuchlík and his collaborator Jaroslav [46]. Moreover, they have investigated the frequencies of the orbital and epicyclic motion in a Keplerian disc with an interior edge at the innermost circular geodesic, which is situated above the BH’s outer horizon or on the opposite side of the WH. When calculating the equatorial circular orbits around the WH throat, the following fundamental frequencies should be taken into account: the Keplerian frequency ${\nu }_{\phi }=\tfrac{{{\rm{\Omega }}}_{\phi }}{2\pi }$, radial and vertical epicyclic frequency ${\nu }_{r}=\tfrac{{{\rm{\Omega }}}_{r}}{2\pi }$,${\nu }_{\theta }=\tfrac{{{\rm{\Omega }}}_{\theta }}{2\pi }$. The formulae below can be used to determine the aforementioned frequencies [47]: $\begin{eqnarray}\begin{array}{l}{{\rm{\Omega }}}_{r}^{2}=-\frac{1}{2{\dot{t}}^{2}{g}_{{rr}}}\frac{{\partial }^{2}{V}_{\mathrm{eff}}}{\partial {r}^{2}},\,\,{{\rm{\Omega }}}_{\theta }^{2}=-\frac{1}{2{\dot{t}}^{2}{g}_{\theta \theta }}\frac{{\partial }^{2}{V}_{\mathrm{eff}}}{\partial {\theta }^{2}},\,\,\,\,\mathrm{and}\\ {{\rm{\Omega }}}_{\phi }=\frac{{\rm{d}}\phi }{{\rm{d}}t}=\frac{-\frac{\partial {g}_{t\phi }}{\partial r}\pm \sqrt{{\left(\frac{\partial {g}_{t\phi }}{\partial r}\right)}^{2}-\frac{\partial {g}_{{tt}}}{\partial r}\frac{\partial {g}_{\phi \phi }}{\partial r}}}{}.\end{array}\end{eqnarray}$
For test particles around a WH throat, the Keplerian frequencies under radial dependency are shown in figures 8 and 9. As a radial coordinate is going away, one can see a slight decrease in the effects of both Keplerian frequencies. The interior edge of the Keplerian disks described by the computed frequencies for this particular examination lies at a marginally stable circular geodesic for the WH geometry. Both kinds of computed frequencies with decreasing behavior are useful for the so-called geodesic replication of high-frequency quasi-periodic oscillations observed in microquasars and dynamic galactic nuclei including a super-massive object. There are some bases in many active galactic nuclei where only WHs with wildly exaggerated parameter values are able to offer a sufficient explanation [46, 48]. An example of one of these bases is the WH throat.
Figure 8. Fundamental radial frequency with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, β = 1.5 and δ = −0.3. |
Figure 9. Fundamental vertical Frequency with χ2 = 2.036 × 10−35, χ1 = 0.9, C1 = 1.2, K2 = 0.35, β = 1.5, and δ = −0.3. |
5. Conclusion
In GR, the WH phenomena via exotic matter have always attracted researchers as the presence of the exotic matter leads to the NEC violation. But regardless of the existence of any such distinct exotic matter, the WH solutions to determining the inclusion of the effective energy–momentum tensor, which is a compulsory part of NEC violation. In the context of teleparallel gravity with matter coupling, the present study investigated spherically symmetric WH geometries using conformal symmetry and a linear equation of state. To complete this project, we have calculated the new WH solution with the phantom regime. The prominent part of the current analysis is exclusively pointed out as follows: It is interesting to mention that the calculated shape function is the most generic and new shape function under conformal symmetry within the phantom regime, which is viable and meets the Morris and Thorne WH conditions. It is necessary to mention that the obtained shape function has a similar behavior and satisfies all the required properties with similar behavior, which has been already reported in the literature [33, 49, 50]. Further, the graphical discussion of all the energy conditions for calculated WH solution via shape function has been provided in figures 3–7.