M Farasat Shamir^,1,∗, G Mustafa^,2, Saad Waseem^,1, Mushtaq Ahmad^,31National University of Computer and Emerging Sciences, Lahore Campus, Pakistan
²Department of Mathematics, Shanghai University, Shanghai, 200444, China
³National University of Computer and Emerging Sciences, Chiniot-Faisalabad Campus, Pakistan

First author contact: ∗Author to whom any correspondence should be addressed.
Received:2021-03-9Revised:2021-08-11Accepted:2021-08-16Online:2021-09-17


Abstract
In this manuscript, the traversable wormhole solutions have been explored in modified $f(R,G)$ gravity by taking into consideration the model $f(R,G)=R+\gamma {G}^{2}$, where R is the Ricci curvature scalar and G is the Gauss–Bonnet term. An acceptable form of the redshift function has been incorporated which is a non-constant in nature along with the implementation of those already defined in literature, the two shape functions namely $b(r)=\tfrac{r}{\exp (r-{r}_{0})}$ and $b(r)=\tfrac{{r}_{0}\mathrm{log}(r+1)}{\mathrm{log}({r}_{0}+1)}$. It is shown by studying the energy condition and through the graphical analysis that the null energy bounds for the effective energy-momentum tensor are generally violated for the presence of the ordinary matter in modified $f(R,G)$ gravity. Energy conditions associated with the matter content threading the wormhole geometries are evaluated and in general are found to favor the null energy conditions in the vicinity of the wormhole throat, the existence of the non-exotic wormhole geometries threaded by the matter has been confirmed under this gravity.
Keywords： f(R,G) gravity;energy conditions;wormholes;shape function


PDF (1769KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
M Farasat Shamir, G Mustafa, Saad Waseem, Mushtaq Ahmad. Wormhole solutions and energy conditions in f(R,G) gravity. Communications in Theoretical Physics, 2021, 73(11): 115401- doi:10.1088/1572-9494/ac1d9d

1. Introduction

The hypothetical existence of a wormhole has been revealed due to a specific solution of the Einstein field equations which comprises a non-trivial connection of isolated points in space-time. A wormhole could be envisioned as a channel with two ends and each end situating at diverse space-time positions in the Universe. Regardless of the theoretical presence of the wormholes, these contacts endure consistency with the general theory of relativity (GR). Huge distances such as trillions of light-years or more; small distances comprising just a few meters; different universes; or time countless points in our universe could be attached via these wormholes. The unification of the space and time to just a sole spacetime continuum, as projected in the special theory of relativity, could enable someone to go beyond space and time through a wormhole, by incorporating the suitable conditions. Even supposing that wormholes, due to the Schwarzschild solution, are not traversable in both directions, nonetheless, their existence enthused the associated investigations to perceive the idea of the traversable wormhole geometries structured by keeping the wormhole throat open due to the existence of exotic matter, while some others are acknowledged as the Lorentzian wormholes [1], etc.

A number of very accurate independent observational data emerging from reliable resources agree that the Universe is experiencing an epoch of accelerated expansion [2]. Several examples in the literature have previously explained the reasons behind this accelerated expansion extending from the miscellaneous models of dark energy to the substitute theories of gravity. Many valid questions about dark matter and dark energy, initial singularity, cosmological constant, flatness problems, etc have not been truly addressed by the GR. The GR, though, the most beautiful theory of the century, has reported nothing to these glitches which are thought to be the core cause of the accelerating expansion of the Universe. Consequently, as an alternative, modified theories of gravity have fascinated researchers with diverse cosmological tactics and aspects to expose the secrets behind the expanding phenomena. After truly acknowledging the myth of the GR theory, numerous other theories have been premeditated by building some complex Lagrangian. Namely, theories like $f(R)$, $f(R,T)$, $f(R,G)$, and $f(G)$ (here R is the Ricci scalar and G is the Gauss–Bonnet term) have been formulated with the blend of curvature, topological scalars invariants, and their derivatives [3–13]. These theories have spoken the purpose of solving the intricacies owing to the quantum gravity and deliver the researchers with diverse platforms through which the motives for the accelerating expansion of the Universe has been discussed.

Modifications in GR has received tremendous considerations, starting from the expansion of the Universe to the presence of wormholes and other associated concepts [14–16]. Investigating the possible existence of wormhole geometries and their stability in GR requires tremendous attention. So far, a big portion of literature has been published mentioning black holes, however, the emerging center of attention seems to be towards wormholes. Morris and Thorne [17] were the pioneers in introducing the concept of traversable wormholes. Einstein and Rosen [18] suggested that if spacetime could be curved such that it produced a shortcut route then traveling to different places in space could become possible through a bridge, that was hypothesized as a wormhole. Having asymptotically flat both ends, the shape of the wormhole is like a tube, which is further classified as a static and non-static wormhole. Observing wormhole geometries still remains a huge challenge, as Wheeler [1] suggested that the size of wormholes sometimes could be too small to spot them easily. For the presence of a wormhole, a violation of the null energy conditions (NEC) is required [19].

Several efforts have been made to investigate wormhole geometries. Taking into account the energy conditions in discussing wormholes is the most crucial scenario. Energy conditions tell us about the matter that threads the wormhole [20] due to the presence of the term ${R}_{\alpha \beta }{\mu }^{\alpha }{\mu }^{\beta }$ in the Raychaudhuri equations [21]. Wormholes can exist in the presence of a hypothetical matter called an exotic matter which is responsible for the existence of wormhole geometries. In GR, it is believed that for the existence of a traversable wormhole, the violation of NEC is required. However, in modified theories of gravity, there are many studies where NEC need not be violated. Recently, Samanta and Godani [22] worked on $f(R)$ gravity and satisfied all energy conditions at the wormhole throat. Violation of the energy conditions remains a complicated issue as Barcelo and Visser [23] predicted. It is interesting to note that Visser [24, 25] developed a rich class of wormhole solutions confined to the utilization of the ordinary matter with the extraordinary matter used at the wormhole throat.

Banerjee, Jasim and Ghosh [26] worked on traversable wormholes and satisfied NEC with isotropic pressure. In $f(R)$ gravity, Godani and Samanta [27] worked on two shape functions and discussed energy conditions. Previously, Visser and his co-authors [28] discussed traversable wormholes and explained that by appropriately choosing wormhole geometry, the use of violating matter can be limited. General dynamic wormholes and violation of NEC was discussed by Hochberg and Visser [29]. In hybrid metric-Palatini gravity, Capozziello et al [30] explained the probability for the existence of wormholes at the throat with violation of NEC. In $f(R,G)$ gravity, Bhatti et al [31] studied wormhole solutions and energy conditions. In Gauss–Bonnet gravity, static and symmetric wormholes and energy conditions were discussed by Maeda and Nozawa [32]. Lobo worked on traversable wormholes and energy conditions and verified that NEC is violated [33]. In $f(R)$ gravity, Mazharimousavi and Halilsoy [34] constructed wormholes with the introduction of external matter and showed that WEC is satisfied.

Inspired by the above discussions, we have focused on $f(R,G)$ gravity and studied two different forms of shape functions for which we have plotted energy conditions. The paper is structured as follows: In addition to wormhole metrics and its conditions, a short analysis of $f(R,G)$ gravity is carried out in section 2. The derived field equations of $f(R,G)$ gravity along with two shape functions $b(r)=\tfrac{{r}_{0}\mathrm{log}(r+1)}{\mathrm{log}({r}_{0}+1)}$ and $b(r)=\tfrac{r}{\exp (r-{r}_{0})}$ which are already defined in literature by Godani and Samanta [35] are discussed in section 3. In section 4, the energy conditions are also discussed by taking into account these two shape functions and the result of each type of shape function is analyzed and discussed in detail. In the last section, a conclusion of our work is presented.

2. Wormhole geometry in $f(R,G)$ gravity

The coupling of the scalar curvature R with the Gauss–Bonnet term G in the form of $f(R,G)$ gravity [36] has brought a significant change. The action of modified Gauss–Bonnet $f(R,G)$ gravity is defined as [37]:(1)$\begin{eqnarray}S=\displaystyle \frac{1}{2K}\int {{\rm{d}}}^{4}x\sqrt{-g}f(R,G)+{S}_{M}({g}_{\mu \nu },\psi ).\end{eqnarray}$The Gauss–Bonnet term is given by:(2)$\begin{eqnarray}G={R}^{2}-4{R}_{\mu \nu }{R}^{\mu \nu }+4{R}_{\mu \theta \phi \nu }{R}^{\mu \theta \phi \nu }.\end{eqnarray}$Varying equation (1) with respect to the metric tensor, gives us the following field equation:(3)$\begin{eqnarray}\begin{array}{l}{R}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }R={{KT}}_{\mu \nu }+{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }{f}_{R}\\ \ -\ {g}_{\mu \nu }\square {f}_{R}+2R{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }{f}_{G}-2{g}_{\mu \nu }R\square {f}_{G}\\ \ +\ 4{R}_{\mu \theta \phi \nu }{{\rm{\nabla }}}^{\theta }{{\rm{\nabla }}}^{\phi }{f}_{G}-4{R}_{\mu }^{\alpha }{{\rm{\nabla }}}_{\alpha }{{\rm{\nabla }}}_{\mu }{f}_{G}-4{R}_{\nu }^{\alpha }{{\rm{\nabla }}}_{\alpha }{{\rm{\nabla }}}_{\mu }{f}_{G}\\ \ +\ 4{R}_{\mu \nu }\square {f}_{G}+4{g}_{\mu \nu }{R}^{\theta \phi }{{\rm{\nabla }}}_{\theta }{{\rm{\nabla }}}_{\phi }{f}_{G}-\displaystyle \frac{1}{2}{g}_{\mu \nu }V+(1-{f}_{R}){G}_{\mu \nu },\end{array}\end{eqnarray}$where ($\square $) corresponds to the D'Alembert operator.(4)$\begin{eqnarray}V={f}_{R}R+{f}_{G}G-f(R,G),\end{eqnarray}$${f}_{R}=\tfrac{\partial f}{\partial R}$ and ${f}_{G}=\tfrac{\partial f}{\partial G}$, are the partial derivatives with respect to R and G, respectively. The energy-momentum tensor is defined as:(5)$\begin{eqnarray}{T}_{\mu \nu }=(\rho +{p}_{t}){V}_{\mu }{V}_{\nu }-{p}_{t}{g}_{\mu \nu }+({p}_{r}-{p}_{t}){X}_{\mu }{X}_{\nu },\end{eqnarray}$where ${V}_{\mu }={{\rm{e}}}^{\tfrac{a}{2}}{\delta }_{\mu }^{0}$ and ${X}_{\mu }=-{{\rm{e}}}^{\tfrac{b}{2}}{\delta }_{\mu }^{1}$. ${p}_{r}$ are radial pressure whereas ${p}_{t}$ is tangential pressure. We choose spherically symmetric spacetime as:(6)$\begin{eqnarray}\,{{\rm{d}}{s}}^{2}={{\rm{e}}}^{a(r)}{{d}{t}}^{2}-{{\rm{e}}}^{b(r)}{{\rm{d}}{r}}^{2}-{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}),\end{eqnarray}$where $a(r)$ and $b(r)$ are both the functions of radial coordinate r. Here $a(r)$ indicates the redshift function. Now consider ${{\rm{e}}}^{-b(r)}=1-\tfrac{\beta (r)}{r}$, where $\beta (r)$ is known as the shape function of the wormhole geometry. The following conditions should be satisfied for the existence of a traversable wormhole.•At the throat $\beta ({r}_{o})={r}_{o}$ and out of the throat $0\lt 1-\tfrac{\beta (r)}{r}$.
•Radial coordinate r ranges from ${r}_{0}\leqslant r\leqslant \infty $, where r₀ is the throat's radius.
•Redshift function must be finite everywhere to ensure no horizons.
•$\beta ^{\prime} ({r}_{o})\lt 1$ (flaring out condition).
•Asymptotically flatness requires the limit $\tfrac{\beta (r)}{r}\to 0$ as $| r| \to \infty $.
The curvature scalar R turns out to be:(7)$\begin{eqnarray}G=\frac{{{\rm{e}}}^{-2b}}{{r}^{2}}(1-{{\rm{e}}}^{b})(({a}^{{\prime} 2}+2a^{\prime\prime} -a^{\prime} b^{\prime} )-2a^{\prime} b^{\prime} ).\end{eqnarray}$The expression for G turns out to be:(8)$\begin{eqnarray}G=\frac{{{\rm{e}}}^{-2b}}{{r}^{2}}(1-{{\rm{e}}}^{b})(({a}^{{\prime} 2}+2a^{\prime\prime} -a^{\prime} b^{\prime} )-2a^{\prime} b^{\prime} ).\end{eqnarray}$

3.Field equations and energy conditions in $f(R,G)$ gravity

By manipulating equations (2)–(6), the following field equations are obtained:(9)$\begin{eqnarray}\begin{array}{c}\rho =-{{\rm{e}}}^{-b}{f}_{R}^{{\prime} {\prime} }+\frac{1}{{{\rm{e}}}^{2b}}\left(\frac{4{e}^{b}}{{r}^{2}}-\frac{4}{{r}^{2}}\right){f}_{G}^{{\prime} {\prime} }+\frac{1}{{{\rm{e}}}^{2b}}\\ \,\ \times \ \left(\frac{4{b}^{{\prime} 2}}{r}+\frac{2b^{\prime} }{{r}^{2}}+\frac{2b^{\prime} {e}^{b}}{{r}^{2}}\right){f}_{G}^{{\prime} }-{{\rm{e}}}^{-b}\left(\frac{b^{\prime} }{2}+\frac{2}{r}\right){f}_{R}^{{\prime} }\\ \,+\ \frac{{{\rm{e}}}^{-b}}{{r}^{2}}\left(\frac{a^{\prime\prime} {r}^{2}}{2}+\frac{{a}^{{\prime} 2}{r}^{2}}{2}-\frac{a^{\prime} b^{\prime} {r}^{2}}{4}+a^{\prime} r\right){f}_{1R}\\ \,+\ \frac{{{\rm{e}}}^{-2b}}{{r}^{2}}(1-{e}^{b})(({a}^{{\prime} 2}+2a^{\prime\prime} -a^{\prime} b^{\prime} )\\ \ -\ 2a^{\prime} b^{\prime} ){f}_{G}-\frac{f}{2},\end{array}\end{eqnarray}$(10)$\begin{eqnarray}\begin{array}{c}{p}_{r}={{\rm{e}}}^{-b}\left(\frac{a^{\prime} }{2}+b^{\prime} +\frac{2}{r}\right){f}_{R}^{{\prime} }+\frac{1}{{{\rm{e}}}^{2b}}\\ \,\ \times \ \left(\frac{4a^{\prime} b^{\prime} }{r}+\frac{6a^{\prime} }{{r}^{2}}-\frac{2a^{\prime} {{\rm{e}}}^{b}}{{r}^{2}}-\frac{4b^{\prime} {{\rm{e}}}^{b}}{{r}^{2}}+\frac{4b^{\prime} }{{r}^{2}}\right){f}_{G}^{{\prime} }\\ \,-\ \frac{{{\rm{e}}}^{-b}}{{r}^{2}}\left(\frac{a^{\prime\prime} {r}^{2}}{2}+{a}^{{\prime} 2}b^{\prime} -a^{\prime} {b}^{{\prime} 2}-\frac{4{b}^{{\prime} 2}}{r}\right){f}_{1R}\\ -\ \frac{{{\rm{e}}}^{-2b}}{{r}^{2}}(1-{{\rm{e}}}^{b})(({a}^{{\prime} 2}+2a^{\prime\prime} -a^{\prime} b^{\prime} )\\ \ -\ 2a^{\prime} b^{\prime} ){f}_{G}+\frac{f}{2},\end{array}\end{eqnarray}$(11)$\begin{eqnarray}\begin{array}{l}{p}_{t}={{\rm{e}}}^{-b}{f}_{R}^{{\prime\prime} }+\frac{1}{{{\rm{e}}}^{2b}}\left(\frac{2a^{\prime\prime} }{r}+\frac{{a}^{{\prime} 2}}{r}-\frac{a^{\prime} b^{\prime} }{r}+2a^{\prime\prime} b^{\prime} \right.\\ \ \left.-\ a^{\prime} {b}^{{\prime} 2}-\frac{2{b}^{{\prime} 2}}{r}+{a}^{{\prime} 2}b^{\prime} \right){f}_{G}^{{\prime} }+\frac{1}{{{\rm{e}}}^{2b}}\left(\frac{2a^{\prime} }{r}\right){f}_{G}^{{\prime} {\prime} }\\ \ -\frac{{{\rm{e}}}^{-b}}{{r}^{2}}\left(\frac{a^{\prime} r}{2}-\frac{b^{\prime} r}{2}-{{\rm{e}}}^{b}+1\right){f}_{1R}-\frac{{{\rm{e}}}^{-2b}}{{r}^{2}}\\ \ \times \ (1-{{\rm{e}}}^{b})(({a}^{{\prime} 2}+2a^{\prime\prime} -a^{\prime} b^{\prime} )-2a^{\prime} b^{\prime} ){f}_{G}+\frac{f}{2},\end{array}\end{eqnarray}$where the prime (′) here gives the derivative with respect to the radial coordinate r. In the case of wormhole discussions, the redshift function is taken as constant in different modified theories of gravity, which simplifies field equations easily, but in our case, while solving the model, our Gauss–Bonnet term vanishes if we take it as constant. So, we assume a simpler form of the redshift function [38, 39] $a(r)=\tfrac{\eta }{r}$, where η is an arbitrary constant. The difficulty is to simplify the terms ρ, p_r and p_t as these include many unknowns and high-order derivatives. Thus a suitable form of function is chosen to make things easier which is as follows:(12)$\begin{eqnarray}f(R,G)={f}_{1}(R)+{f}_{2}(G),\end{eqnarray}$where ${f}_{1}(R)=R$ is a linear function of Ricci scalar and ${f}_{2}(G)=\gamma {G}^{2}$ is a power law model with $\gamma $ being an arbitrary constant. The exclusive expressions for ρ, p_r, and p_t concerning the matter threading the wormholes comprise the shape function together with the function $f(G)$ and its radial derivatives. These functions are responsible for the matter content required for the existence of the wormhole geometries. The violation of the energy bounds remains the most fundamental and vital reason in approving wormhole geometries in the conventional GR. Nevertheless, the modified gravitational theories have exposed themselves with different character to a great extent, justifiably due to the fact that the Einstein equations are quite different to those of the modified theories of gravity. The concept of these energy conditions is believed to emerge from the existence of a term ${R}_{\varrho \eta }{\kappa }^{\varrho }{\kappa }^{\eta }$ in the Raychaudhuri equations for the time-like congruences well-defined and denoted by ${\mu }^{\varrho }$ and the null geodesics by ${\kappa }^{\varrho }$.

Consequently, we have the following important energy constraints as:$\begin{eqnarray*}\begin{array}{l}{NEC}\iff {T}_{\theta \phi }{k}^{\theta }{k}^{\phi }\geqslant 0,\\ {WEC}\iff {T}_{\theta \phi }{V}^{\theta }{V}^{\phi }\geqslant 0,\\ {SEC}\iff ({T}_{\theta \phi }-\displaystyle \frac{T}{2}{g}_{\theta \phi }){V}^{\theta }{V}^{\phi }\geqslant 0,\\ {DEC}\iff {T}_{\theta \phi }{V}^{\theta }{V}^{\phi }\geqslant 0,\end{array}\end{eqnarray*}$where ${k}^{\theta }$ is null vector and ${V}^{\theta }$ is a time-like vector. For DEC, ${T}_{\theta \phi }{V}^{\theta }$ is not space like. Energy conditions involving the energy density and pressure terms are defined as:$\begin{eqnarray*}\begin{array}{l}{NEC}:\rho +{p}_{r}\geqslant 0,\ \rho +{p}_{t}\geqslant 0,\\ {WEC}:\rho \geqslant 0,\ \ \rho +{p}_{r}\geqslant 0,\ \rho +{p}_{t}\geqslant 0,\\ {SEC}:\rho -{p}_{r}\geqslant 0,\ \rho -{p}_{t}\geqslant 0,\ \rho -{p}_{r}-2{p}_{t}\geqslant 0,\\ {DEC}:\rho \geqslant 0,\ \ \rho -| {p}_{r}| \geqslant 0,\ \rho -| {p}_{t}| \geqslant 0.\end{array}\end{eqnarray*}$It has been stated in the literature that the energy conditions are violated near the wormhole throat [40]. Energy conditions are satisfied while dealing with normal matter because it has positive energy density and positive pressure but exotic matter has negative energy which is the opposite of normal matter and the presence of exotic matter gives rise to the violation of energy conditions. In this work, we choose the particular form of the shape functions, i.e. $b(r)$ from the literature. Two forms of shape function $b(r)$ are chosen, which are defined as:(13)$\begin{eqnarray}b(r)=\displaystyle \frac{{r}_{0}\mathrm{log}(r+1)}{\mathrm{log}({r}_{0}+1)}.\end{eqnarray}$The second specific shape function is defined as:(14)$\begin{eqnarray}b(r)=\displaystyle \frac{r}{\exp (r-{r}_{0})}.\end{eqnarray}$where, r₀ represents the wormhole throat location. For the current study, we choose it as ${r}_{0}=0.2$. Both the considered specific shape functions satisfy the required properties for the existence of a wormhole. In addition, these shape functions satisfy the physical and cosmological implications, such as being theoretically useful as shortcuts in the wormhole spacetime and inducing closed time-like curves, possibly violating causality. In previous literature [41–45], several such specific shape functions are defined and wormholes are studied.

Case 1: $b(r)=\tfrac{r}{\exp (r-{r}_{0})}$$\begin{eqnarray*}\begin{array}{c}\rho =\frac{1}{4{r}^{16}}({r}^{12}\eta (-4+4r+\eta )+8{{\rm{e}}}^{-4r+4{r}_{0}}\\ \ \times \ {r}^{4}\gamma \eta (-3724{r}^{3}-525{r}^{2}\eta +42r{\eta }^{2}+3{\eta }^{3})\\ \ -\ 8{{\rm{e}}}^{-2r+2{r}_{0}}{r}^{4}\gamma \eta (2304{r}^{3}\\ \ +\ 40{r}^{5}(3{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+396{r}^{2}\eta \\ \ -\ 36{r}^{2}\eta -3{\eta }^{3}-8{r}^{3}(-2{{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)(82r+3\eta )+{r}^{2}\left({{\rm{e}}}^{-r+{r}_{0}}\right.\\ \ {\left.-\ {{\rm{e}}}^{-r+{r}_{0}}r\right)}^{2}(1468r+93\eta )+2r({{\rm{e}}}^{-r+{r}_{0}}\\ \ -\ {{\rm{e}}}^{-r+{r}_{0}}r)(2596{r}^{2}-66{r}^{3}\\ \ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+204r\eta -15{\eta }^{2}))\\ \ +\ 16{{\rm{e}}}^{-3r+3{r}_{0}}{r}^{4}\gamma \eta (2988{r}^{3}+12{r}^{5}\\ \ \times \ (3{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+460{r}^{2}\eta -39r{\eta }^{2}\\ \ -\ 3{\eta }^{3}-2{r}^{3}(-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (109r+4\eta )+r({{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (1970{r}^{2}+157r\eta -9{\eta }^{2}))+{{\rm{e}}}^{-r+{r}_{0}}{r}^{5}\\ \ \times \ \eta (-{r}^{7}(-4+4r+\eta )+32\gamma (4{r}^{4}\\ \ \times \ (3{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)-2{r}^{2}\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)(28r+\eta )\\ \ +\ r{\left({{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r\right)}^{3}(79r+4\eta )+2r\\ \ \times \ {\left({{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r\right)}^{2}(161r\\ \ -\ 3{r}^{2}(-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+9\eta )\\ \ +\ ({{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)(368{r}^{2}-34{r}^{3}\\ \ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+26r\eta \\ \ -\ 3{\eta }^{2})))+4{r}^{6}({{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)\end{array}\end{eqnarray*}$(15)$\begin{eqnarray}\begin{array}{l}\ \times \ ({r}^{8}+8\gamma \eta (4{r}^{2}(1+{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}-{{\rm{e}}}^{-r+{r}_{0}}r)+({{\rm{e}}}^{-r+{r}_{0}}\\ \ -\ {{\rm{e}}}^{-r+{r}_{0}}r)(-32r+\eta +-2({{\rm{e}}}^{-r+{r}_{0}}\\ \ -\ {{\rm{e}}}^{-r+{r}_{0}}r)(19r+3r({{\rm{e}}}^{-r+{r}_{0}}\\ \ -\ {{\rm{e}}}^{-r+{r}_{0}}r)+\eta )))))\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{l}{p}_{r}=\displaystyle \frac{1}{4{r}^{16}}(r-{{\rm{e}}}^{-r+{r}_{0}}r)(-128{r}^{8}(1\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)(-2{{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ \gamma {\eta }^{2}+{r}^{13}\eta (-4r+\eta )+8{{\rm{e}}}^{-5r+5{r}_{0}}\\ \ \times \ {r}^{5}\gamma \eta (392{r}^{3}-769{r}^{2}\eta -118r{\eta }^{2}+3{\eta }^{3})\\ \ +\ 8{{\rm{e}}}^{-4r+4{r}_{0}}{r}^{5}\gamma \eta (-680{r}^{3}+12{r}^{3}\\ \ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)(2r-5\eta )\\ \ +\ 2553{r}^{2}\eta +404r{\eta }^{2}-9{\eta }^{3}+2r({{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)(-324{r}^{2}+413r\eta +43{\eta }^{2}))\\ \ +\ {r}^{6}({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (-4{r}^{9}+{r}^{8}\eta -4r({r}^{3}-256\gamma ){\eta }^{2}\\ \ +\ 64\gamma {\eta }^{3}+4({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (4{r}^{6}-{r}^{5}\eta +280r\gamma {\eta }^{2}+48r({{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)\gamma {\eta }^{2}+16\gamma {\eta }^{3}))+8{{\rm{e}}}^{-3r+3{r}_{0}}{r}^{5}\\ \ \times \ \gamma \eta (288{r}^{3}-8{r}^{3}(-2{{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (5r+3({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\ \times \ (r-\eta )-23\eta )-3076{r}^{2}\eta -510r{\eta }^{2}\\ \ +\ 9{\eta }^{3}+r({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (1064{r}^{2}-2466r\eta -252{\eta }^{2}+({{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)(280{r}^{2}-273r\eta -16{\eta }^{2})))\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}{r}^{6}(\eta (7{r}^{8}-2r(-2{r}^{3}+{r}^{6}\\ \ +\ 1152\gamma )\eta -448\gamma {\eta }^{2})+64{r}^{2}\\ \ \times \ (-2{{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\gamma \eta (12\eta \\ \ +\ ({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)(-2r+7\eta ))\\ \ +\ ({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)(4{r}^{6}(-8+{r}^{3})\\ \ -\ {r}^{5}(-8+{r}^{3})\eta +4r({r}^{3}-2096\gamma ){\eta }^{2}\\ \ -\ 768\gamma {\eta }^{3}+32({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ \gamma \eta (32{r}^{2}+6r({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (r-2\eta )-141r\eta -8{\eta }^{2})))\\ \ +\ {{\rm{e}}}^{-2r+2{r}_{0}}{r}^{5}(16{r}^{7}-{r}^{6}(4+3{r}^{3})\eta \end{array}\end{eqnarray*}$(16)$\begin{eqnarray}\begin{array}{l}\ +\ {r}^{2}(-4{r}^{3}+{r}^{6}+12640\gamma ){\eta }^{2}+2240r\gamma {\eta }^{3}\\ \ -\ 24\gamma {\eta }^{4}+8r\gamma \eta (4{r}^{2}(-2{{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)(4r+2({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (5r-8\eta )-51\eta )+({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)\\ \ \times \ (-416{r}^{2}+2560r\eta +254{\eta }^{2}+({{\rm{e}}}^{-r+{r}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{r}_{0}}r)(-408{r}^{2}+697r\eta +40{\eta }^{2}+24r\\ \ \times \ ({{\rm{e}}}^{-r+{r}_{0}}+{{\rm{e}}}^{-r+{r}_{0}}r)(-r+\eta ))))))\end{array}\end{eqnarray}$$\begin{eqnarray*}\,\begin{array}{l}{p}_{t}=\displaystyle \frac{1}{4{r}^{16}}({r}^{12}\eta (2r+\eta )-8{{\rm{e}}}^{-4r+4{{rr}}_{0}}{r}^{4}\\ \ \times \ \gamma \eta (-196{r}^{3}+2663{r}^{2}\eta +686r{\eta }^{2}+35{\eta }^{3})\\ \ +\ 16{{\rm{e}}}^{-3r+3{{rr}}_{0}}{r}^{4}\gamma \eta (-72{r}^{3}\\ \ +\ 3181{r}^{2}\eta +6{r}^{4}(3{{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)\eta \\ \ +\ 832r{\eta }^{2}+41{\eta }^{3}+2{r}^{2}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{{rr}}_{0}}r(3{r}^{2}-65r\eta -5{\eta }^{2})\\ \ +\ 2({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)(-130{r}^{3}\\ \ +\ 667{r}^{2}\eta +131r{\eta }^{2}+6{\eta }^{3}))-8{{\rm{e}}}^{-2r+2{{rr}}_{0}}\\ \ \times \ {r}^{4}\gamma \eta (2{r}^{2}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)\\ \ \times \ (4{r}^{2}-311r\eta -23{\eta }^{2})+\eta (5020{r}^{2}\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\ +\ 32{r}^{4}(3{{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)+1332r\eta \\ \ +\ 61{\eta }^{2})+{\left({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r\right)}^{2}\\ \ \times \ (-464{r}^{3}+785{r}^{2}\eta +128r{\eta }^{2}+4{\eta }^{3})\\ \ +\ 2({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)(-176{r}^{3}\\ \ +\ 2691{r}^{2}\eta +509r{\eta }^{2}+23{\eta }^{3}+3{r}^{2}\\ \ \times \ (-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)(4{r}^{2}-13r\eta \\ \ -\ {\eta }^{2})))-{r}^{4}\left(64{r}^{2}\gamma ({r}^{2}(3{{\rm{e}}}^{-r+{{rr}}_{0}}\right.\\ \ -\ {{\rm{e}}}^{-r+{{rr}}_{0}}r(+(-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)\\ \ \times \ (-15r-\eta )){\eta }^{2}+48{r}^{2}\left({{\rm{e}}}^{-r+{{rr}}_{0}}\right.\\ \ {\left.-\ {{\rm{e}}}^{-r+{{rr}}_{0}}r\right)}^{4}\gamma \eta (-2r+\eta )+16r\left({{\rm{e}}}^{-r+{{rr}}_{0}}\right.\\ \ {\left.-\ {{\rm{e}}}^{-r+r\theta }r\right)}^{3}\gamma \eta (-32{r}^{2}+29r\eta \\ \ +\ 4{\eta }^{2})+({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)(2{r}^{1}0\\ \ -\ {r}^{9}\eta +5248{r}^{2}\gamma {\eta }^{2}+768r\gamma {\eta }^{3}+16\gamma {\eta }^{4}\\ \ -\ 32{r}^{2}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)\end{array}\end{eqnarray*}$(17)$\begin{eqnarray}\,\begin{array}{l}\ \times \ \gamma {\eta }^{2}(14r+\eta ))+16\left({{\rm{e}}}^{-r+{{rr}}_{0}}\right.\\ \ {\left.-\ {{\rm{e}}}^{-r+{{rr}}_{0}}r\right)}^{2}\gamma \eta (-2{r}^{3}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{{rr}}_{0}}r)(-2r+\eta )+\eta (198{r}^{2}+28r\eta \\ \ +\ {\eta }^{2})))+{{\rm{e}}}^{-r+{{rr}}_{0}}{r}^{4}(2{r}^{10}-3{r}^{9}\eta \\ \ -\ {r}^{2}({r}^{6}-10368\gamma ){\eta }^{2}+2816r\gamma {\eta }^{3}\\ \ +\ 112\gamma {\eta }^{4}+16\gamma \eta ({r}^{2}(14{r}^{2}(3{{\rm{e}}}^{-r+{{rr}}_{0}}\\ \ -\ {{\rm{e}}}^{-r+{{rr}}_{0}}r)+(-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)\\ \ \times \ (-242r-17\eta ))\eta +r\left({{\rm{e}}}^{-r+{{rr}}_{0}}\right.\\ \ {\left.-\ {{\rm{e}}}^{-r+{{rr}}_{0}}r\right)}^{3}(-76{r}^{2}+49r\eta +5{\eta }^{2})\\ \ +\ {\left({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r\right)}^{2}(-136{r}^{3}\\ \ +\ 3{r}^{3}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}+{{\rm{e}}}^{-r+{{rr}}_{0}}r)(2r-\eta )\\ \ +\ 547{r}^{2}\eta +89r{\eta }^{2}+3{\eta }^{3})\\ \ +\ ({{\rm{e}}}^{-r+{{rr}}_{0}}-{{\rm{e}}}^{-r+{{rr}}_{0}}r)({r}^{2}(-2{{\rm{e}}}^{-r+{{rr}}_{0}}\\ \ +\ {{\rm{e}}}^{-r+{{rr}}_{0}}r)(8{r}^{2}-65r\eta -5{\eta }^{2})\\ \ +\ 2\eta (862{r}^{2}+149r\eta +6{\eta }^{2}))))),\end{array}\end{eqnarray}$

Case 2: $b(r)=\tfrac{{r}_{0}\mathrm{log}(r+1)}{\mathrm{log}({r}_{0}+1)}$$\begin{eqnarray*}\begin{array}{l}\rho =\displaystyle \frac{1}{4{r}^{16}}({r}^{12}\eta (-4+4r+\eta )\\ \ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{4}}(8{r}_{0}^{4}\gamma \eta (-3724{r}^{3}\\ \ -\ 525{r}^{2}\eta +42r{\eta }^{2}+3{\eta }^{3})\mathrm{log}{\left[1+r\right]}^{4})\\ \ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{3}}(16{{rr}}_{0}^{3}\gamma \eta \mathrm{log}{\left[1+r\right]}^{3}\\ \ \times \ (2988{r}^{3}+460{r}^{2}\eta -39r{\eta }^{2}-3{\eta }^{3}\\ \ +\ \displaystyle \frac{24{r}^{5}{r}_{0}}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}-\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\\ \ \times \ (2{r}^{3}{r}_{0}(109r+4\eta ))+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ ({{rr}}_{0}(1970{r}^{2}+157r\eta -9{\eta }^{2}))))\\ \ -\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}(8{r}^{2}{r}_{0}^{2}\gamma \eta \mathrm{log}{\left[1+r\right]}^{2}\\ \ \times \ (2304{r}^{3}+396{r}^{2}\eta -36r{\eta }^{2}-3{\eta }^{3}\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}{r}^{2}{r}_{0}^{2}(1468r\\ \ +\ \left.93\eta )+\displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}(80{r}^{5}r\right)\\ \ -\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(8{r}^{3}{r}_{0}(82r+3\eta ))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(2{{rr}}_{0}(2596{r}^{2}+204r\eta \\ \ -\ \left.\left.\left.15{\eta }^{2}-\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(66{r}^{3}{r}_{0}\right)\right)\right))\\ \ +\ \displaystyle \frac{1}{\mathrm{log}[1+{r}_{0}]}({r}^{4}{r}_{0}\eta \mathrm{log}[1+r](-{r}^{7}(-4\\ \ +\ 4r+\eta )+32\gamma \left(\displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}{\left[1+{r}_{0}\right]}^{3}}\right.\\ \ \times \ ({{rr}}_{0}^{3}(79r+4\eta ))+\displaystyle \frac{1}{{\left(1+r\right)}^{2}}\\ \ \times \ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}(2{{rr}}_{0}^{2}(161r\end{array}\end{eqnarray*}$(18)$\begin{eqnarray}\begin{array}{l}\ +\ \left.9\eta -\displaystyle \frac{3{r}^{2}{{rr}}_{0}}{{\left(1+r\right)}^{2}\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}\right)+\displaystyle \frac{8{r}^{4}{r}_{0}}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}\\ \ -\ \displaystyle \frac{2{r}^{2}{r}_{0}(28r+\eta )}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}-\displaystyle \frac{2{{rr}}_{0}(28r+\eta )}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\\ \ +\ \left.\left.\left.\displaystyle \frac{{r}_{0}(368{r}^{2}+26r\eta -3{\eta }^{2}-\tfrac{34{r}^{3}{r}_{0}}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]})}{(1+r)\mathrm{log}[1+{r}_{0}]}\right)\right)\right)\\ \ +\ \displaystyle \frac{4{r}^{6}{r}_{0}}{(1+r)\mathrm{log}[1+{r}_{0}]}\left({r}^{8}+8\gamma \eta \right.\\ \ \times \ \left(\displaystyle \frac{4{r}^{2}{r}_{0}\left(1+\tfrac{{r}_{0}}{(1-r)\mathrm{log}[1+{r}_{0}]}\right)}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\right.\\ \ \left.\left.\left.\times \ \left({r}_{0}\left(-32r+\eta -\displaystyle \frac{2{r}_{0}(19r+\eta +\tfrac{3{{rr}}_{0}}{(1+r)\mathrm{log}[1+{r}_{0}]})}{(1+r)\mathrm{log}[1+{r}_{0}]}\right)\right)\right)\right)\right),\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{l}{p}_{r}=\displaystyle \frac{1}{4{r}^{16}(r-\tfrac{{r}_{0}\mathrm{log}[1+r]}{\mathrm{log}[1+{r}_{0}]})}({r}^{13}\eta (-4r+\eta )\\ \ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{5}}\left(8{r}_{0}^{5}\gamma \eta (392{r}^{3}-769{r}^{2}\eta \right.\\ \ \left.-\ 118r{\eta }^{2}+3{\eta }^{3})\mathrm{log}{\left[1+r\right]}^{5}\right)\\ \ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{4}}(8{{rr}}_{0}^{4}\gamma \eta \mathrm{log}{\left[1+r\right]}^{4}\\ \ \times \ (-680{r}^{3}+2553{r}^{2}\eta +404r{\eta }^{2}-9{\eta }^{3}\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(12{r}^{3}{r}_{0}(2r-5\eta ))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(2{{rr}}_{0}(-324{r}^{2}+413r\eta \\ \ +\ 43{\eta }^{2}))))+\displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{3}}(8{r}^{2}{r}_{0}^{3}\gamma \eta \mathrm{log}{\left[1+r\right]}^{3}\\ \ \times \ (288{r}^{3}-3076{r}^{2}\eta -510r{\eta }^{2}+9{\eta }^{3}\\ \ -\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(8{r}^{3}{r}_{0}(5r-23\eta \\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(3{r}_{0}(r-\eta ))))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}({{rr}}_{0}(1064{r}^{2}-2466r\eta \\ \ -\ 252{\eta }^{2}+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ ({r}_{0}(280{r}^{2}-273r\eta -16{\eta }^{2}))))))\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}({r}^{3}{r}_{0}^{2}\mathrm{log}{\left[1+r\right]}^{2})\\ \ \times \ (16{r}^{7}-{r}^{6}(4+3{r}^{3})\eta +{r}^{2}(-4{r}^{3}+{r}^{6}\\ \ +\ 12640\gamma ){\eta }^{2}+2240r\gamma {\eta }^{3}-24\gamma {\eta }^{4}\\ \ +\ 8r\gamma \eta (\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\\ \ \times \ (4{r}^{2}{r}_{0}(4r-51\eta +\displaystyle \frac{2{r}_{0}(5r-8\eta )}{(1+r)\mathrm{log}[1+{r}_{0}]}))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}({r}_{0}(-416{r}^{2}\\ \ +\ 2560r\eta +254{\eta }^{2}+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ ({r}_{0}(-408{r}^{2}+697r\eta +40{\eta }^{2}\\ \ +\ \displaystyle \frac{24{{rr}}_{0}(-r+\eta )}{(1+r)\mathrm{log}[1+{r}_{0}]}))))))\\ \ -\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(128{r}^{8}{r}_{0}\gamma {\eta }^{2}\\ \ \times \ (1+\displaystyle \frac{{r}_{0}}{(1+r)\mathrm{log}[1+{r}_{0}]}))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}({r}^{6}{r}_{0}(-4{r}^{9}\end{array}\end{eqnarray*}$(19)$\begin{eqnarray}\begin{array}{l}\ +\ {r}^{8}\eta -4r({r}^{3}-256\gamma ){\eta }^{2}+64\gamma {\eta }^{3}\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(4{r}_{0}(4{r}^{6}-{r}^{5}\eta \\ \ +\ 280r\gamma {\eta }^{2}+16\gamma {\eta }^{3}+\displaystyle \frac{48{{rr}}_{0}\gamma {\eta }^{2}}{(1+r)\mathrm{log}[1+{r}_{0}]}))))\\ \ +\ \displaystyle \frac{1}{\mathrm{log}[1+{r}_{0}]}({r}^{5}{r}_{0}\mathrm{log}[1+{r}_{0}])(\eta (7{r}^{8}\\ \ -\ 2r(-2{r}^{3}+{r}^{6}+1152\gamma )\eta -4482\gamma {\eta }^{2})\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(64{r}^{2}{r}_{0}\gamma \eta \\ \ \times \ (12\eta +\displaystyle \frac{{r}_{0}(2r+7\eta )}{(1+r)\mathrm{log}[1+{r}_{0}]}))\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}({r}_{0}(4{r}^{6}(-8+{r}^{3})\\ \ -\ {r}^{5}(-8+{r}^{3})\eta +4r({r}^{3}-2096\gamma ){\eta }^{2}\\ \ -\ 768\gamma {\eta }^{3}+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ (32{r}_{0}\gamma \eta (32{r}^{2}-141r\eta -8{\eta }^{2}\\ \ +\ \displaystyle \frac{6{{rr}}_{0}(r-2\eta )}{(1+r)\mathrm{log}[1+{r}_{0}]})))))),\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{l}{p}_{t}=\displaystyle \frac{1}{4{r}^{16}}({r}^{12}\eta (2r+\eta )\\ \ -\ {r}^{4}(64{r}^{2}\gamma {\eta }^{2}\left(\displaystyle \frac{2{r}^{2}{r}_{0}}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}\right.\\ \ +\ \left.\displaystyle \frac{{r}_{0}(-15r-\eta )}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\right)+\displaystyle \frac{48{r}^{2}{r}_{0}^{4}\gamma \eta (-2r+\eta )}{{\left(1+r\right)}^{4}\mathrm{log}{\left[1+{r}_{0}\right]}^{4}}\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}{\left[1+{r}_{0}\right]}^{3}}(16{{rr}}_{0}^{3}\gamma \eta (-32{r}^{2}\\ \ +\ 29r\eta +4{\eta }^{2}))+\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}\\ \ \times \ (16{r}_{0}^{2}\gamma \eta (\eta (198{r}^{2}+28r\eta +{\eta }^{2})\\ \ -\ \displaystyle \frac{2{r}^{3}{r}_{0}(-2r+\eta )}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}))+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ ({r}_{0}(2{r}^{10}-{r}^{9}\eta +5248{r}^{2}\gamma {\eta }^{2}+768r\gamma {\eta }^{3}\\ \ +\ 16\gamma {\eta }^{4}-\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\\ \ \times \ (32{r}^{2}{r}_{0}\gamma {\eta }^{2}(14r+\eta )))))-\displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{4}}\\ \ \times \ (8{r}_{0}^{4}\gamma \eta (-196{r}^{3}+2663{r}^{2}\eta +686r{\eta }^{2}+35{\eta }^{3}))\\ \ +\ \displaystyle \frac{1}{\mathrm{log}{\left[1+{r}_{0}\right]}^{3}}(16{{rr}}_{0}^{3}\gamma \eta \mathrm{log}{\left[1+r\right]}^{3}\\ \ \times \ (-72{r}^{3}+3181{r}^{2}\eta +832r{\eta }^{2}+41{\eta }^{3}\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}(12{r}^{4}{r}_{0}\eta )\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}(2{r}^{2}{r}_{0}(3{r}^{2}-65r\eta \\ \ -\ 5{\eta }^{2}))+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\end{array}\end{eqnarray*}$$\begin{eqnarray*}\,\begin{array}{l}\ \times \ (2{r}_{0}(-130{r}^{3}+667{r}^{2}\eta +131r{\eta }^{2}+6{\eta }^{3}))))\\ \ -\ \displaystyle \frac{8{r}^{2}{r}_{0}^{2}\gamma \eta \mathrm{log}{\left[1+r\right]}^{2}}{\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}(\eta (5020{r}^{2}\\ \ +\ 1332r\eta +61{\eta }^{2}+\displaystyle \frac{64{r}^{4}{r}_{0}}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]})\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}\\ \ \times \ ({r}_{0}^{2}(-464{r}^{3}+785{r}^{2}\eta +128r{\eta }^{2}+4{\eta }^{3}))\\ \ +\ \displaystyle \frac{2{r}^{2}{r}_{0}(4{r}^{2}-311r\eta -23{\eta }^{2})}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}\\ \ +\ \displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}(2{r}_{0}(-176{r}^{3}\\ \ +\ 2691{r}^{2}\eta +509r{\eta }^{2}+23{\eta }^{3}\\ \ +\ \displaystyle \frac{3{r}^{2}{r}_{0}(4{r}^{2}-13r\eta -{\eta }^{2})}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]})))\\ \ +\ \displaystyle \frac{{r}^{3}{r}_{0}\mathrm{log}[1+r]}{\mathrm{log}[1+{r}_{0}]}(2{r}^{10}-3{r}^{9}\eta -{r}^{2}({r}^{6}\\ \ -\ 10368\gamma ){\eta }^{2}+2816r\gamma {\eta }^{3}+112\gamma {\eta }^{4}\end{array}\,\end{eqnarray*}$(20)$\begin{eqnarray}\,\begin{array}{l}\ +\ 116\gamma \eta ({r}^{2}\eta (\displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}(28{r}^{2}{r}_{0})\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}({r}_{0}(-242r-17\eta )))\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{3}\mathrm{log}[1+{r}_{0}]}({{rr}}_{0}^{3}(-76{r}^{2}\\ \ +\ 49r\eta +5{\eta }^{2}))+\displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}{\left[1+{r}_{0}\right]}^{2}}\\ \ \times \ ({r}_{0}^{2}(-136{r}^{3}+547{r}^{2}\eta +89r{\eta }^{2}+3{\eta }^{3}\\ \ +\ \displaystyle \frac{3{r}^{3}{r}_{0}(2r-{eta})}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}))+\displaystyle \frac{1}{(1+r)\mathrm{log}[1+{r}_{0}]}\\ \ \times \ ({r}_{0}(2\eta (862{r}^{2}+149r\eta +6{\eta }^{2})\\ \ +\ \displaystyle \frac{1}{{\left(1+r\right)}^{2}\mathrm{log}[1+{r}_{0}]}({r}^{2}{r}_{0}(8{r}^{2}\\ \ -\ 65{rr}\eta -5{\eta }^{2}))))))).\end{array}\end{eqnarray}$

We have achieved wormhole solutions and evaluated different forms of energy density, radial and tangential pressures in connection to both expressions of the shape function $b(r)$. The graphical analysis for the first specific shape function is provided in figures (1, 2, 3). The behavior of energy conditions for second specific shape function is presented in figures (4, 5, 6). From the figures (1, 4) the presence of exotic matter is confirmed as NEC is violated for both the models. The figures (2, 5) show the graphical behavior of $\rho +{p}_{t}$ and $\rho -{p}_{r}$ for both specific shape functions. The graphical behavior of $\rho -{p}_{t}$ and $\rho -{p}_{r}-2{p}_{t}$ is provided in figures (3, 6).

Figure 1.

New window|Download| PPT slide
Figure 1.The development of ρ and $\rho +{p}_{r}$ is shown. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho \lt 0$ (Yellow), $\rho \gt 0$ (Red), $\rho +{p}_{r}\lt 0$ (Yellow), and $\rho +{p}_{r}\gt 0$ (Red).

Figure 2.

New window|Download| PPT slide
Figure 2.The development of $\rho +{p}_{t}$ and $\rho -{p}_{r}$ is shown. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho +{p}_{t}\lt 0$ (Yellow), $\rho +{p}_{t}\gt 0$ (Red), $\rho -{p}_{r}\lt 0$ (Yellow), and $\rho -{p}_{r}\gt 0$ (Red).

Figure 3.

New window|Download| PPT slide
Figure 3.The development of $\rho -{p}_{t}$ and $\rho -{p}_{r}-2{p}_{t}$ is shown. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho -{p}_{t}\lt 0$ (Yellow), $\rho -{p}_{t}\gt 0$ (Red), $\rho -{p}_{r}-2{p}_{t}\lt 0$ (Yellow), and $\rho -{p}_{r}-2{p}_{t}\gt 0$ (Red).

Figure 4.

New window|Download| PPT slide
Figure 4.The development of ρ and $\rho +{p}_{r}$ is shown. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho \lt 0$ (Yellow), $\rho \gt 0$ (Red), $\rho +{p}_{r}\lt 0$ (Yellow), and $\rho +{p}_{r}\gt 0$ (Red).

Figure 5.

New window|Download| PPT slide
Figure 5.The development of $\rho +{p}_{t}$ and $\rho -{p}_{r}$ is shown. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho +{p}_{t}\lt 0$ (Yellow), $\rho +{p}_{t}\gt 0$ (Red), $\rho -{p}_{r}\lt 0$ (Yellow), and $\rho -{p}_{r}\gt 0$ (Red).

Figure 6.

New window|Download| PPT slide
Figure 6.The development of $\rho -{p}_{t}$ and $\rho -{p}_{r}-2{p}_{t}$ is show,. Here, we assume $\eta =15$ and ${r}_{0}\,=\,0.2$ with $\rho -{p}_{t}\lt 0$ (Yellow), $\rho -{p}_{t}\gt 0$ (Red), $\rho -{p}_{r}-2{p}_{t}\lt 0$ (Yellow), and $\rho -{p}_{r}-2{p}_{t}\gt 0$ (Red).

4. Conclusion

The main focus of this work is to investigate wormhole solutions under the geometry of $f(R,G)$ gravity. In this paper, we have offered an exciting and useful discussion regarding the existence of wormhole geometries through the ordinary matter in modified Gauss–Bonnet $f(R,G)$ gravity, by considering its specific $f(R,G)=R+\gamma {G}^{2}$ model, where G is the Gauss–Bonnet scalar coupled with the Ricci scalar R. We have invested two particular forms of the shape function $b(r)$ together with the non-constant redshift function $a(r)$ into the modified $f(R,G)$ field equations to develop $\rho +{p}_{r}$ and $\rho +{p}_{t}$ of the energy conditions WEC and NEC, which are decisively fundamental constituents, the role of which is very vital and decisive in establishing any possible existence of wormhole geometries in modified $f(R,G)$ gravity. For the ordinary matter responsible for threading these wormhole geometries, we have employed the argument that the validity of the energy conditions is due the curvature incorporating high-order derivatives, which may be termed as the gravitational fluids, authorizing the existence of corresponding wormhole solutions. This is fundamentally unlike their counterparts in the GR.

The reason for taking non-constant redshift is because our Gauss–Bonnet term becomes zero in the case of constant redshift. The existence of wormhole geometries has been investigated by the choice of the two shape functions i.e. $b(r)=\tfrac{r}{\exp (r-{r}_{0})}$ and $b(r)=\tfrac{{r}_{0}\mathrm{log}(r+1)}{\mathrm{log}({r}_{0}+1)}$ under the condition that the redshift function $a(r)$ and the two cases of $b(r)$ obey all the wormhole geometry conditions including the flaring out condition, throat condition and typical asymptotically flatness condition. The energy conditions for both the cases are discussed and plotted. The graphical behavior of ρ, $\rho +{p}_{r}$, $\rho +{p}_{t}$, $\rho -{p}_{r}$, $\rho -{p}_{t}$ and $\rho +{p}_{r}+2{p}_{t}$ are shown in figures (1, 2, 3, 4, 5, and 6). For the exponential shape function, ρ shows positive but decreasing behavior whereas both functions $\rho +{p}_{r}$ and $\rho +{p}_{t}$ attained negative behavior. For the logarithmic shape function behaviour of energy conditions remain the same as for the exponential shape function. Violation of NEC in both cases confirms the presence of exotic matter, hence favoring the existence of wormhole geometries in our $f(R,G)$ under investigation.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1]Wheeler J A 1962 Geometrodynamics New York Academic
[Cited within: 2]

[2]Perlmutter S et al. 1997 Astrophys. J. 483 565
DOI:10.1086/304265 [Cited within: 1]

[3]Harko T et al. 2011 f(R,T) gravity
Phys. Rev. D 84 024020
DOI:10.1103/PhysRevD.84.024020 [Cited within: 1]

[4]Capozziello S De Laurentis M 2011 Extended theories of gravity
Phys. Rep. 509 167
DOI:10.1016/j.physrep.2011.09.003

[5]Capozziello S Francaviglia M 2008 Extended theories of gravity and their cosmological and astrophysical applications
Gen. Rel. Grav. 40 357
DOI:10.1007/s10714-007-0551-y

[6]Capozziello S et al. 2006 Cosmological viability of f(R) gravity as an ideal fluid and its compatibility with a matter dominated phase
Phys. Lett. B 639 135
DOI:10.1016/j.physletb.2006.06.034

[7]Capozziello S Cardone V F Troisi A 2005 Reconciling dark energy models with f(R) theories
Phys. Rev. D 71 043503
DOI:10.1103/PhysRevD.71.043503

[8]Capozziello S Stabile A Troisi A 2007 Spherically symmetric solutions in f(R) gravity via the Noether symmetry approach
Class. Quant. Grav. 24 2153
DOI:10.1088/0264-9381/24/8/013

[9]Capozziello S et al. 2011 Hydrostatic equilibrium and stellar structure in f(R) gravity
Phys. Rev. D 83 064004
DOI:10.1103/PhysRevD.83.064004

[10]Shamir M F Mustafa G Waseem S 2020 Study of non-commutative wormhole solutions in $f(R,G)$ gravity
Int. J. Geom. Methods Mod. Phys. 17 2050129
DOI:10.1142/S0219887820501297

[11]Nojiri S Odintsov S D 2004 Modified gravity with ln R terms and cosmic acceleration
Gen. Rel. Grav. 36 1765
DOI:10.1023/B:GERG.0000035950.40718.48

[12]Lobo F S N Oliveira M A 2009 Wormhole geometries in f(R) modified theories of gravity
Phys. Rev. D 80 104012
DOI:10.1103/PhysRevD.80.104012

[13]Sharif M Ikram A 2016 Energy conditions in f(G,T) gravity
Eur. Phys. J. C 76 640
DOI:10.1140/epjc/s10052-016-4502-1 [Cited within: 1]

[14]Bamba K et al. 2012 Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests
Astrophys.Space Sci. 342 155
DOI:10.1007/s10509-012-1181-8 [Cited within: 1]

[15]Nojiri S Odintsov S D 2007 Introduction to modified gravity and gravitational alternative for dark energy
Int. J. Geom. Methods Mod. Phys. 4 115
DOI:10.1142/S0219887807001928

[16]De Felice A Tsujikawa S 2010 $f(R)$ theories
Living Rev. Rel. 13 3
DOI:10.12942/lrr-2010-3 [Cited within: 1]

[17]Morris M S Thorne K S 1988 Wormholes, time machines and the weak energy conditions
Phys. Rev. Lett. 61 1446
DOI:10.1103/PhysRevLett.61.1446 [Cited within: 1]

[18]Einstein A Rosen N 1935 Can quantum-mechanical description of physical reality be considered complete
Phys. Rev. 47 777
DOI:10.1103/PhysRev.47.777 [Cited within: 1]

[19]Visser M 1995 Lorentzian Wormholes: From Einstein to Hawking New York American Institute of Physics
[Cited within: 1]

[20]Roman T A 2006 Some thoughts on energy conditions and wormholes
Proceedings of the Tenth Marcel Grossmann Meeting on General RelativityWorld Scientific, SingaporeBergliaffa S P Novello M
[Cited within: 1]

[21]Shamir M F Zia S 2018 Existence of static wormhole solutions in f(R,G) gravity
Astrophys. Space Sci. 363 247
DOI:10.1007/s10509-018-3470-3 [Cited within: 1]

[22]Samanta G C Godani N 2019 Validation of energy conditions in wormhole geometry within viable $f(R)$ gravity
Eur. Phys. J. C 79 623
DOI:10.1140/epjc/s10052-019-7116-6 [Cited within: 1]

[23]Barcelo C Visser M 2000 Scalar fields energy conditions and traversable wormholes
Class. Quant. Grav. 17 3843
DOI:10.1088/0264-9381/17/18/318 [Cited within: 1]

[24]Visser M 1989 Traversable wormholes: some simple examples
Phys. Rev. D 39 3182
DOI:10.1103/PhysRevD.39.3182 [Cited within: 1]

[25]Visser M 1989 Traversable wormholes from surgically modified Schwarzschild spacetimes
Nucl. Phys. B 328 203
DOI:10.1016/0550-3213(89)90100-4 [Cited within: 1]

[26]Banerjee A Jasim M K Ghosh S G 2020 Traversable wormholes in $f(R,T)$ gravity satisfying the null energy condition with isotropic pressure
arXiv:2003.01545
[Cited within: 1]

[27]Godani N Samanta G C 2018 Traversable wormholes and energy conditions with two different shape functions in $f(R)$ gravity
Int. J. Mod. Phys. D 28 1950039
DOI:10.1142/S0218271819500391 [Cited within: 1]

[28]Visser M Kar S Dadhich N 2003 Traversable wormholes with arbitrarily small energy condition violations
Phys. Rev. Lett. 90 201102
DOI:10.1103/PhysRevLett.90.201102 [Cited within: 1]

[29]Hochberg D Visser M 1999 General dynamic wormholes and violation of the null energy condition
arXiv preprint gr-qc/9901020
[Cited within: 1]

[30]Capozziello S et al. 2012 Wormholes supported by hybrid metric-Palatini gravity
Phys. Rev. D 86 127504
DOI:10.1103/PhysRevD.86.127504 [Cited within: 1]

[31]Bhatti M Z Yousaf Z Ilyas M 2018 Existence of wormhole solutions and energy conditions in $f(R,T)$ gravity
J. Astrophys. Astron. 39 69
DOI:10.1007/s12036-018-9559-9 [Cited within: 1]

[32]Maeda H Nozawa M 2008 Static and symmetric wormholes respecting energy conditions in Einstein-Gauss-Bonnet gravity
Phys. Rev. D 78 024005
DOI:10.1103/PhysRevD.78.024005 [Cited within: 1]

[33]Lobo F S N 2005 Energy conditions, traversable wormholes and dust shells
Gen. Relativ. Gravit. 37 2023
DOI:10.1007/s10714-005-0177-x [Cited within: 1]

[34]Mazharimousavi S H Halilsoy M 2016 Necessary conditions for having wormholes in f(R) gravity
Mod. Phys. Lett. A 31 1650192
DOI:10.1142/S0217732316501923 [Cited within: 1]

[35]Godani N Samanta G C 2019 Static traversable wormholes in $f(R,T)=R+2\alpha {lnT}$ gravity
Chin. J. Phys. 62 161
DOI:10.1016/j.cjph.2019.09.009 [Cited within: 1]

[36]Cognola G et al. 2006 Dark energy in modified Gauss-Bonnet gravity: Late-time acceleration and the hierarchy problem
Phys. Rev. 73 084007
DOI:10.1103/PhysRevD.73.084007 [Cited within: 1]

[37]Atazadeh K Darabi F 2014 Energy conditions in $f(R,G)$ gravity
Gen. Rel. Grav. 46 1664
DOI:10.1007/s10714-014-1664-8 [Cited within: 1]

[38]Kar S Sahdev D 1995 Restricted class of traversable wormholes with traceless matter
Phys. Rev. D 52 2030
DOI:10.1103/PhysRevD.52.2030 [Cited within: 1]

[39]Sharif M Ikram A 2015 Wormholes supported by f(G) gravity
Int. J. Mod. Phys. D 24 1550003
DOI:10.1142/S0218271815500030 [Cited within: 1]

[40]Hochberg D Visser M 1997 Geometric structure of the generic static traversable wormhole throat
Phys. Rev. D 56 4745
DOI:10.1103/PhysRevD.56.4745 [Cited within: 1]

[41]Samanta G C Godani N 2019 Wormhole modeling supported by non-exotic matter
Mod. Phys. Lett. A 34 1950224
DOI:10.1142/S0217732319502249 [Cited within: 1]

[42]Godani N Samanta G C 2019 Traversable wormholes and energy conditions with two different shape functions in $f(R)$ gravity
Int. J. Mod. Phys. D 28 1950039
DOI:10.1142/S0218271819500391

[43]Böhmer C G et al. 2012 Wormhole geometries in modified teleparallel gravity and the energy conditions
Phys. Rev. D 85 044033
DOI:10.1103/PhysRevD.85.044033

[44]Jamil M et al. 2013 Wormholes in a viable $f(T)$ gravity
Eur. Phys. J. C 73 2267
DOI:10.1140/epjc/s10052-012-2267-8

[45]Sharif M Rani Shamaila 2013 Wormhole solutions in $f(T)$ gravity with noncommutative geometry
Phys. Rev. D 88 123501
DOI:10.1103/PhysRevD.88.123501 [Cited within: 1]