Possible existence of traversable wormhole in Finsler–Randers geometry

In the present article, we have explored the possible existence of a traversable wormhole in the framework of Finsler–Randers (F–R) geometry. In order to achieve this goal, first, we have constructed gravitational field equations for static, spherically symmetric spacetime with anisotropic fluid distribution in F–R geometry. Next, we have written the deduced form of field equations in the background of Morris–Thorne wormhole geometry. To visualize the shape of the wormhole, we have selected exponential shape function b(r)=rexpηrr0-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(r)=\frac{r}{exp\left( \eta \left( \frac{r}{r_{0}} - 1\right) \right) }$$\end{document} with the constant parameter η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} and the throat radius r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}$$\end{document} and depicted two-dimensional and three-dimensional embedding diagrams corresponding to some considered values of η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} and r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}$$\end{document}. Moreover, all essential requirements to build a wormhole shape have been examined for the reported shape function. Next, We have analyzed wormhole configuration for three cases (I, II, III) corresponding to three selected redshift functions. Furthermore, each case is analyzed by dividing it into two models such as (i) Model-1 (for general anisotropic EoS pt=χpr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{t}=\chi p_{r}$$\end{document}) and (ii) Model-2 (for linear phantom-like EoS pr+ωρ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{r} + \omega \rho =0$$\end{document}). In each model of three cases, we have verified the validity of the wormhole solution in F–R geometry by considering null, weak, strong and dominant energy conditions. Also, the total amount of averaged NEC-violating matter near the wormhole throat has been analyzed by computing volume integral quantifier.


Introduction
In modern cosmology, based on several observational tests, it has been observed that our current universe is in an expansion phase in an accelerating way [1][2][3][4][5][6].As a responsible candidate, the so-called dark energy (DE) plays the main a e-mail: krishnapada0019@gmail.com(corresponding author) b e-mail: ujjaldebnath@gmail.comrole behind this expansion.In GR [7], Einstein connects the Riemannian geometry to gravitation by a system of equations known as Einstein's field equations.However, till today, GR faces some problems regarding DE and direct exploration of the current expanding universe.For this purpose, to perform these problems, several DE models and modified theories of gravities (MTG) have already been established [8][9][10][11].In the recent few years, various results from observable astrophysical studies it has been shown that an anisotropic direction of expansion of the universe may happen if the underlying geometry is anisotropic.In this connection, a new interesting alternative gravitational theory was established on generalized Riemannian geometry known as Finslerian geometry.The most fundamental distinction of Finsler's geometry is that it incorporates the concept of anisotropic intrinsically in the geometry of spacetime.The underlying space of Finsler's geometry is named Finsler space, a metric space.The metric in this space is defined as a function F(x, y) from a tangent bundle of a manifold to R 1 .Here F(x, y) is a non-negative function, mathematically defined as the norm instead of the inner product on a tangent bundle with the position of spacetime coordinate x and a tangent vector y ∈ T x M representing the velocity.Hence the Finslerian geometry is dynamic geometry depending on position and direction, i.e., dynamical coordinates on a tangent bundle of a differentiable manifold, whereas the Riemannian geometry is gravitational geometry.In the framework of this type of geometry, several researchers have studied different cosmological aspects [12][13][14][15][16][17][18][19][20].Now Finsler-Randers spacetime, originated by Randers, is a special type of Finsler spacetime that govern by a generalized Riemannian metric as F(x, y) = g i j y i y j + A λ y λ . ( The physical significance of this metric describes as the additional extra term A λ y λ considered as the minimum cause of relativity violation, or deviation from isotropic to anisotropic nature of spacetime, or violation of Lorentz symmetry.The F-R spacetime gives a special interest to the investigation of the Riemannian model/FRW model in a generalized Riemannian spacetime because the introduction of vector field affects the geometrical terms like curvature, metric and geodesics and also acts as a dark energy fluid.There are a lot of research works for the FRW metric that have already investigated F-R spacetime successfully.Now, we want to focus on a brief review of several research works regarding cosmological and geometrical aspects in the following.In Ref. [15], authors have first constructed gravitational field equations in Randers-type metric corresponding to FRW metric and studied several cosmological aspects, including the Finslerian Raychaudhuri equation.The Lorentz invariance violation and symmetries of Finsler-Randers space have been studied in detail in Ref. [21].A comparative study between the F-R model and the DGP model has been done in detail and concluded these two models are almost cosmological equivalent [22].Finsler-Randers cosmology can imitative a general dark energy scenario and also modified gravity model, shown in the Ref. [23].F-R cosmological model corresponding to modified gravity has been studied in [18], and also authors discussed all energy conditions in this framework in detail.The extended Schwarzschild-like and Schwarzschild De-Sitter solutions in Finsler-Randers geometry and its geometrical applications have been explored by constructing geodesic paths of a particle and as well as a comparison study between GR and S-F-R framework [24,25].Also, from the dynamical system perspective, F-R cosmological model and its outstanding applications are included in Ref. [26].Dynamical analysis of several geometrical aspects like geodesic, angular momentum, the energy of a moving particle, deflection angle etc., is discussed in the Refs.[27][28][29].Based on the F-R metric, authors studied differential geometry with remarkable applications, and also the Raychaudhuri equation, as well as energy conditions, are discussed in detail in the Refs.[30,31]. Due to the strong gravitational field, it was considered that the physical structure of the end status of a static, massive astrophysical object should be spherically symmetric.In this connection, the interior spacetime of a compact object, or compact star, is assumed as spherically symmetric spacetime.Here, for example, a compact object or star may be a black hole, a white dwarf, or a neutron star to a strange star etc.After the discovery of a new and appropriate approach to gravitational theory based on the concept of curvature of spacetime, known as Einstein's General theory of relativity (GR) [7], the investigation of numerous exact solutions of interior and exterior geometry become a new platform of research in the scientific community till today.For this purpose, the spherically symmetric spacetime that is able to explore astrophysical phenomena regarding physical charac-teristics and impact of compact objects has always achieved one of the most interesting problems in particle astrophysics.Now among compact relativistic objects, a more interesting and ideological object (i.e., mathematically possible but without observational evidence) is a wormhole.From the physical point of view, a wormhole is described as a tunnellike structure that connects two different points of spacetime or two points of two different spacetimes.Historically, the notion of the wormhole was actually initiated by Flamm [32] soon after the invention of the Schwarzschild solution [33].The idea of these hypothetical shortcuts was first proposed by Einstein and Rosen, popularly known as the Einstein-Rosen bridge, in order to investigate the co-ordinate or curvature singularity free for the Riessner-Nordstron solution [34,35] and the Schwarzschild solution [36].In a seminal work in 1962, Wheeler [37] first used the term wormhole and describe the geometry of Schwarzschild wormhole and later Hawking [38] transformed this solution into Euclidean wormhole, named as Einstein-Rosen bridge.After some years, it is investigated by Wheeler and Fuller [39] that the Einstein-Rosen bridge type of wormhole may not be stable because when it connects two parts of the same universe, it will pinch off very quickly than light or any particle moving slower than light is not able to fall in from one exterior region to make it to the other exterior region.
The idea of the static and spherically symmetric traversable wormhole was first successfully explored by Morris and Thorne [40] as a possible structure to freely traverse time or interstellar.The intention of time or interstellar travel was able to explain by imposing a special type of matter near the throat that must push the wormhole wall apart and prevents it from squeezing very quickly from gravitational collapse.Hence, the fundamental constituents near the throat of a traversable wormhole must be the exotic matter that violates the null energy condition (NEC), at least in the neighbourhood of the throat.Thus in order to construct a traversable wormhole configuration, the main issue is the violation of energy conditions that claim the presence of exotic matter near the throat of wormhole [41][42][43][44].Moreover, Visser et al. [41,45] revealed an important essential fact that the averaged null energy condition (ANEC) must be violated in the wormhole geometry, and the measurement of this type matter should be infinitesimally small, named as the volume integral quantifier.Till today, there are a lot of research works regarding physical aspects and as well as exact solutions for different types of wormholes such as static wormhole [46][47][48][49][50], dynamical wormhole [51,52] and rotating wormholes [53,54] have already been developed by several authors.Consequently, a lot of research works are done by assuming the wormhole configuration supported by several types of dark energy such as cosmological constant [55][56][57], phantom energy [58][59][60], modified or generalized Chaplygin gas [61][62][63].Besides GR, a large number of wormhole models are developed in the framework of different modified theories of gravities (MTGs).Here an interesting fact is that in the context of MTGs for the construction of wormholes, the exotic matter can be minimized or completely avoided.Again except the Riemannian geometry, wormhole configuration has already been studied by several researchers in the background of different geometrical gravitational theories like Teleparallel geometry, non-commutative geometry and Finslerian geometry etc.Among these geometrical, we want to concentrate on wormhole structure in Finslerian geometry because of some interesting special consequences in this geometry.First of all, Riemannian geometry is a special type of Finslerian geometry, and it is dynamic, already stated in detail in the early stage of this section [64].Now the exact solution of traversable wormhole in the Finslerian framework was first successfully studied by Rahaman et al., [65] by constructing some redshift functions and shape functions with the help of EoS.Recently, a wormhole solution in Finsler geometry with f (R, T ) gravity has been explored in the Refs.[66,67].Also, traversable wormhole supported by Phantom-like dark energy in the framework of Finslerian geometry has been discussed by Singh et al. [68].
In the present study, our main goal is to construct gravitational field equations in the context of Finsler-Randers geometry.For this purpose, we consider a static metric to describe a spherically symmetric spacetime geometry with an anisotropic fluid distribution.Next, our aim is to examine the formation of Morris and Thorne [40] traversable wormhole in F-R geometry.For this context, we are employing two EoS to evaluate Rander terms such as general anisotropic EoS, discussed in the Model-1 and phantom-like EoS, discussed in the Model-2.A brief decoration of the rest portion of the whole study is as follows: In the Sect.2, we have included a details construction of gravitational field equations in the F-R geometry corresponding to a static, spherically symmetric spacetime.Next, the formation of Morris and Thorne wormhole configuration has been discussed in the Sect.3. In this section, we have included a discussion of several aspects like embedding surface, proper radial distance, the validity of energy conditions etc., regarding wormhole configuration.Further, in Sect.4, we have analyzed averaged NEC for our wormhole solutions.At the end section, i.e., Sect.5, we have reported some valuable conclusions from our whole discussions.

Gravitational field equations for F-R metric
A Finsler space is governed by a differentiable function F(x, y), also named as Finsler structure, defined on a tangent bundle T M of a manifold M as The function satisfies some following properties: (i) F is smooth on T M = T M \ {0}, regularity property.(ii) F is a positive one dimensional homogeneous function F(x, λy) = λF(x, y) with λ > 0, homogeneity property, where x ∈ M and y = dx dτ are the respective representation of position and velocity.(iii) The n × n Hessian matrix [64] is positive-definite at every point of T M \ {0}, referred as strong convexity property, where the notation () A special type of Finsler space the so-named the Finsler-Randers space [69], defined by the metric function of the form as where α(x, y) = g i j y i y j , (4) and g i j (x) is the fundamental metric tensor corresponding to Riemannian affine connection.Hence as a consequence, we can say that the geometrical structure of F-R space is obviously a generalization of Riemannian geometry.
Here the spatial coordinates are comoving the time coordinate describes the proper time surveyed by the comoving observer.The vector field y μ = dx μ ds describes the four tangent vectors that represent the velocity of a comoving observer towards a suitable family of world lines (i.e., paths of fluid flow) in a locally anisotropic universe.The arclength s ≡ is the proper time.So we have y μ = (1, 0, 0, 0).u λ represents a weak primordial vector field with |u λ | 1 and u λ = (u 0 , 0, 0, 0) with u 0 (r ).Here u λ is a covector.Now using the Hessian of F, the metric for the Finslerian spacetime can be defined as follows: By simplifying we can derive [15,24] in the following form where With respect to the metric of Finslerian spacetime the usual definition of Christoffel symbols are as follows And the corresponding Cartan tensor defined by E. Cartan as Here it is important to mention that the tensor quantity C i jk plays the role of the torsion tensor in the Finslerian geometrical framework.Thus a Finsler space becomes a Riemannian space when C i jk = 0 and vice-versa.Now, the osculating Riemannian metric can be defined for the case of a connection between a convenient Finsler metric and Riemannian metric as [14,70] The corresponding Christoffel symbols are The full expression of Cartan's torsion tensor can be easily defined as [71] C μνλ = 1 2 All term of the above equation is proportional to the the field u λ thus C μνλ ≈ 0. So the approximation form of Christoffel components are The curvature tensor can be defined as Ricci tensor Scalar curvature Let us considered a static spherically symmetric metric as where ζ and ψ are arbitrary functions of radial coordinate r only.
Now, by imposing the conditions u 0 ≈ 0 and u 2 0 ≈ 0 and putting the metric coefficients g i j , given in the Eq. ( 19), the nonzero components of Ricci tensor (L μν , defined in the Eq. ( 15)) can evaluated as follows: Now the energy-momentum tensor for describing Finslerian anisotropic matter distribution can be given for a comoving obserever as where y μ are fluid 4-velocity vector and z μ are radial 4vector.So the components of T μν can given as follows The trace is Based on the Asanov research [14], the action for the osculating Riemannian spacetime can be defined as where L(x, y(x)) is the osculating Ricci scalar in the context of Finslerian geometry.Now using the variational principle for the above action (28), the field equations of Einstein's GR corresponding to our model can be written as follows So the componentwise field equations can be written as Solving the above field Eqs. ( 30)-( 32), the explicit expressions of ρ, p r and p t can be evaluated as follows:

Wormhole model
In this section, our goal is to examine the formation of a traversable wormhole in the context of F-R geometry.For this purpose, we first recall a brief review of traversable wormholes based on the Morris and Thorne proposition for wormhole description.To represent such type wormhole geometry, first, we consider the following static, spherically symmetric line element [40] where, (r ) and b(r ) are the gravitational redshift function and the shape function, respectively.Now from the physical point of view, to depict the hypothesised picture of a wormhole by the metric (36), the function b(r ) should be in a form such that it connects two asymptotically flat regions of spacetime.Here the minimum value attained by b(r ) is called the throat of the wormhole.As the function b(r ) depicts the shape of the wormhole, it is named by the shape function.According to Morris and Thorne, a traversable wormhole with r 0 as the radius of the throat will be depicted by the metric (36), then the shape function must maintain the following restrictions Motivated from the valuable research works on Cosmological model done by several authors [15,16,18,22,23,26,27,71] in F-R geometry, we have defined the Morris-Thorne wormhole metric in the context of F-R geometry, defined by the metric (3) i.e., may be named by Morris-Thorne-Finsler-Randers metric as F(x, y) = g i j y i y j + u λ y λ , (37) where In the 'Appendix', we have reported a details discussion regarding the computation of field equations corresponding to the metric (37).Now, the field equations ( 30)- (32) for the metric (37) reduce to the following forms In the present work, let us choose the shape function b(r ) in the following form where η ia a constant parameters and r 0 is the radius of wormhole throat.Next, in order to scrutinize the essential requirements of a wormhole shape function, several plots have been drawn for some considered values of η in Fig. 1 corresponding r 0 = 1 and Fig. 2 corresponding to r 0 = 2. From these Figures, it is observed that throat condition, flaring out condition, and asymptotically flatness condition are fulfilled by our considered shape function b(r ) with chosen values of η and r 0 .

Embedding surface
For a better visualization of the wormhole geometry, presented by Eq. ( 36), we have illustrated embedded twodimensional and three-dimensional diagrams for the shape function, stated in Eq. ( 42).Since our wormhole geometry is spherically symmetric, we have used an equatorial slice, represented by θ = π 2 , and fixed a moment of time by t =constant, without loss of generality.We have presented this type of hypersurface by H : θ = π/2, t = constant and hence in this hypersurface, dθ = 0 and dt = 0. So, for this connection, we may write the reduced form of the wormhole metric (36) as In the cylindrical co-ordinates the above metric (43) equivalent to the following form Now, in three-dimensional Euclidean space the embedding surface characterize by Z = Z (r ), so, the reduced form of the metric for this surface may be written as [72] ds Now comparing the Eqs.( 43) and ( 44), we can represent the embedding shape function Z (r ) as follows Utilizing the Eq. ( 46), we have depicted embedded 2D diagram in Fig. 3 and embedded 3D diagram in Fig. 4.

Proper radial distance
In the proper radial co-ordinate system, the wormhole metric (36) can be written as [72,73] Now the proper radial distance of the wormhole is given by where the '±' comes for the both mouths of wormhole.In Fig. 5, we have illustrated the proper radial distance while the radial coordinate r ≥ r 0 .Moreover, in Tables 1  and 2, we have calculated the approximate numerical values of proper radial distance l(r ) and embedding surface function Z (r ) of the wormhole for the first quadrant at several radial coordinate r ≥ r + 0 .And it is observed that the values of l(r ) are slightly more than Z (r ) at any radial coordinate r .

Energy conditions
The energy conditions are defined by a set of inequalities among the thermodynamical parameters like ρ, p r , p t .These conditions originated from Raychaudhuri equations, which represent the action of congruence and the attractive nature of gravity for timelike, spacelike, or lightlike curves [74].The energy conditions are mainly used as the fundamental tools to predict the nature of a strong gravitational field because the geodesic structure of spacetime may be described perfectly by these conditions.Moreover, one can analyze the realisticness of matter distribution and hence the possible existence of wormhole geometry by utilizing these conditions.An interesting fact is that the energy conditions are purely geometrical concepts, and their origin is not dependent on the theory of gravity (for more details, see the Ref. [75]).
Here we will emphasize only four distinct kinds of energy conditions, whereas, in GR, seven different types of energy conditions are analyzed [76].Since the whole of our current discussion is performed with anisotropic fluid, the energy conditions are given as follows: • NEC (Null energy condition): ρ + p r ≥ 0, ρ + p t ≥ 0.
In tensor form NEC is defined as T μν k μ k ν ≥ 0 with null-like vector k μ .• WEC (Weak energy condition): ρ ≥ 0, ρ + p r ≥ 0, ρ + p t ≥ 0. In tensor form WEC defined as T μν U μ U ν ≥ 0 with time-like vector U μ .• SEC (Strong energy condition): ρ + p r ≥ 0, ρ + p t ≥ 0, ρ + p r + 2 p t ≥ 0. In tensor form SEC defined as In tensor form DEC defined as T μν U μ U ν ≥ 0 and T μν U μ is not space-like.Now, the physical meaning of the above energy conditions i.e., NEC, WEC, SEC and DEC may be described as follows: (i) The WEC implies the observer's measurements along any time-like vector about energy density (ρ) must be always positive i.e., ρ ≥ 0. (ii) The SEC implies that the gravitational field should always attractive i.e., may predict a strong gravitational field.(iii) DEC states that the measurements of energy flux by any observer is time-like or null-like and must be always non-negative i.e., ρ ≥ 0. (iv) The basic requirements of SEC and WEC is NEC i.e., WEC ⊂ NEC and SEC ⊂ NEC.
Therefore, if the NEC is hold then the SEC and WEC are automatically hold but the reverse need not always true.Again we have DEC ⊂ WEC ⊂ NEC and hence we can say DEC implies both the NEC and the WEC, while the reverse doesn't need to be true in each case.Moreover, DEC need not imply the SEC.Now, a basic aspect of traversable wormhole i.e., wormhole with the exotic matter violates NEC in the context of GR [41].Also, in order to describe a wormhole geometry the violation of SEC is necessary.Thus again if the NEC is violated then all other standard energy conditions are also violated.By a bouncing Universe, we may treat a Universe that undergoes a collapse, attain a minimum and later on expands [77][78][79][80][81].In classical cosmology, it has been observed that such bounce may not be possible for FRW universe if ρ + 3 p > 0 i.e. active mass is positive with perfect fluid's energy density ρ and pressure p [82].Thus, FRW Universe undegoing a bounce with minimum at which the SEC must be violated.However, the violation of SEC is necessary but not sufficient condition.In the early Universe, some bouncing models may be produced based on the violation of WEC [79].Again, the violation of WEC is also a necessary condition to obtain wormhole solutions.Therefore, there must have an obvious technical connection between wormhole solutions and cosmological bounce.In the Refs.[83,84] authors have discussed the existence of wormhole solutions around the cosmological bounce.So, as a final remark we can say a bounce may be occur in the Universe when WEC, NEC, SEC are not hold at some short distant interval.Hence, we can notice the bouncing conditions for F-R spacetime [31] as ρ + 3 p < 0 and ρ + p < 0 corresponding to isotropic fluid as well as ρ + p r < 0, ρ + p t < 0 and ρ + p r + 2 p t < 0 for anisotropic fluid distributions.
Since the appropriate matter distribution required to create a wormhole is still an open problem in the scientific community, so, we have considered a general anisotropic matter distribution is given by Eq. ( 25).Here the stress of anisotropic force can be represented by the parameter = p t − p r .Physically the nature of this stress is represented as for < 0 (i.e., negative), the resulting anisotropic stress is directed inward, and wormhole geometry becomes an attractive nature, whereas for > 0 (i.e., positive) it is directed outward and geometry becomes in repulsive nature.

Two models
Now, in order to verify the wormhole formation in F-R geometry, we consider two different types of matter content, which are discussed in two models apart in the following: • Model-1: Let us consider the anisotropic matter distribution obeys the following relation [85] where χ , anisotropic parameter may be a constant ( = 1, otherwise it becomes isotropic) or a function of radial co-ordinate r .Putting the analytical expressions of p r and p t , derived in the respective Eqs. ( 40) and ( 41), in the above relation (49), we have derived the following explicit form of u 0 as For simplicity, we have taken the easiest form χ = W 1 r n , where W 1 and n are some constants.Throughout the graphical representation corresponding to this model, Fig. 5 Behavior of l(r ) against radial coordinate r corresponding to the shape function (42) for the throat radius r 0 = 1 (left panel) and r 0 = 2 (right panel) where ω is a positive function of radial co-ordinate r .Now for ω(r ) < −1 then it represents Phantom energy EoS, which have already discussed in the Ref. [68].Utilizing the analytical expressions of ρ r and p r , evaluated in the respective Eqs. ( 39) and ( 40), in the above relation (51), we have derived the explicit form of u 0 as follows In similar way of Model-1 for simplicity, we have also considered ω = W 2 r m and the constants W 2 and m are taken as 3 and 2 respectively throughout the graphical representation corresponding to this model.

Some specific forms of gravitational redshift function
In order to get the energy density and pressure, we assume three cases of gravitational redshift function (r ).

Case-I:
Due to the choice of the redshift function (r ) = constant = K , we get the energy density and pressure in Model-1 and Model-2 in the following: Model-1: Model-2: In the case-I, to verify energy conditions corresponding to our wormhole solutions we have demonstrated sufficient plots thats are included in Figs. 6 and 7 for the respective Model-1 and Model-2.
• For the Model-1, from Fig. 6 and Table 3 • Again for the Model-2, from Fig. 7 and Table 3

Case-II:
(r ) = log ξ r + 1 .Due to the choice of the redshift function (r ) = log ξ r + 1 , (ξ a constant) we get the energy density and pressure in Model-1 and Model-2 in the following: 123 123 In this case, we have scrutinized all required energy conditions by illustrating several plots in Figs. 8 and 9 corresponding to the Model-1 and Model-2 respectively.
• From Fig. 8 and Table 4 corresponding to the Model-1, we have noticed that energy density (ρ) is positive for however, the SEC is violated as ρ + p r +2 p t ≤ 0 for all r ∈ [1, ∞] corresponding to η = 0.3, 0.5, 0.7, 1 except η = 0.1.So, SEC is not satisfied In this case, for the better understanding of energy conditions, we have drawn several plots, which are depicted in Figs. 10 and 11 corresponding to the Model-1 and Model-2 respectively.

Average null energy condition (ANEC) violating matter
In this section, we will discuss the "volume integral quantifier", which basically provides the information about the total amount of exotic matter required to maintains the wormhole structure.According to Visser et al., this quantity only related to ρ and p r but not to tangential pressure p t and is defined in the following integral form [76] where dV = r 2 sin θ drdθ dφ.By imposing a cutoff of the energy-momentum tensor at R ≥ r 0 , we can rewritten the where r 0 is the throat radius.Now, putting the expression of ρ + p r in the above Eq.( 72), one can straightforwardly evaluate the integral I v .Here an important fact that when R → r 0 the integral I v must be → 0. Due to the complicated explicit expression of I v , we examined the nature of the integral against r graphically, included in Figs. 12, 13 and 14.From these figures, we have observed that ANEC is violated near the wormhole throat for Mode-1 and Model-2 in each of cases-I, II, III.So, this nature of I v confirms the existence of exotic matter near the wormhole throat, and we can minimize the total amount of exotic matter by changing the values of our model parameters.Therefore, our proposed spacetime geometries may be demonstrated the existence of traversable wormhole geometry.

Discussions and conclusions
In the present article, we have explored our goal in two parts: firstly, gravitational field equations in Finsler-Randers geometry are constructed for a static, spherically symmetric line element, and secondly, the formation of traversable wormhole has been examined in this geometry.To the best of our knowledge, this is the first attempt at the construction of gravitational field equations in F-R geometry corresponding to spherically symmetric metric and examining traversable wormhole configuration in this geometry.For this purpose, we have deduced the modified field equations corresponding to Morris-Thorne wormhole geometry.Next, we have selected the shape function as in the form b(r ) = r ex p η r r 0 −1 with arbitrary constant parameter η and radius of wormhole throat r 0 .For a better visualization of wormhole shape, we have included two-dimensional and   The significant role of the parameter η for our wormhole configuration has been explored for five considered values like η = 0.1, 0.3, 0.5, 0.7, 1.Furthermore, an interesting fact that our wormhole configuration has been analyzed corresponding to three redshift functions (r ) = constant = K , (r ) = log ξ r + 1 and (r ) = e −γ /r , shortly named as Case-I, Case-II and Case-III respectively.And each of these three cases was analyzed by dividing into two models viz.Model-1 corresponding to the general anisotropic EoS p t = χ p r and Model-2 corresponding to the linear phantomlike EoS p r + ωρ = 0. Importantly, for the judgement of NEC, WEC, SEC and DEC, we have drawn a large but sufficient no. of figures (Figs. 6,7,8,9,10,11) for both models by considering the wormhole throat radius r 0 = 1.Also, for a better understanding of energy conditions, we have reported behaviors of ρ, ρ + p r , ρ + p t , ρ + p r + 2 p t , ρ − |p r | and ρ − |p t | etc. w. r. t. radial coordinate r .near the wormhole throat in tabular forms like Tables 3, 4 and 5.Moreover, we have explored an essential discussion like ANEC regarding the minimum amount of exotic matter near the worm-hole throat.Now some fruitful results of our whole study are reported below as follows: From Figs. 3 and 4, we have confirmed that our proposed shape function (42) satisfied all necessary requirements to form the shape of a wormhole corresponding the respective throat radius r 0 = 1 and r 0 = 2.Because the conditions like Throat Condition, Flaring out condition, and Asymptotically flatness condition hold good for each of selected values η = 0.1, 0.3, 0.5, 0.7, 1.Also, in Fig. 5, we have included the plots of proper radial distance l(r ) from the throat to ∞ and the numerical values of l(r ) and embedding surface function Z (r ) at some radial distance r are reported in the Tables 1  and 2.
For the three cases, i.e., Case-I, II, III, the obtained results show that NEC, WEC, SEC, and DEC are not satisfied appropriately at the neighbourhood of the wormhole throat corresponding to both models.It is important to mention that the fundamental requirement for the formation of a traversable wormhole is fulfilled by the existence of matter that must be completely different from ordinary matter, the so-named as exotic matter.Again the presence of exotic matter may be confirmed by the violation of energy conditions, especially NEC.Therefore the requirements for the existence of a traversable wormhole corresponding to our proposed model are fulfilled as the energy conditions are not satisfied near the wormhole throat.Based on the figure profiles 12, 13 and 14, it is concluded that NEC is violated near the throat, and hence the proposed wormhole is supported by a small amount of exotic matter.
In the present work, we have delivered an accurate construction of gravitational field equations in F-R geometry.Also, we have achieved a detailed set of solutions in the analytic approach and visualized 2D and 3D diagrams of wormhole shape function (42).Finally, it is concluded, based on the overall study, that our proposed wormhole model is viable and realistic.

Fig. 2 2 Fig. 3
Fig. 2 Behavior of b, b r , b , b − rb against the radial coordinate r with r 0 = 2

Fig. 12 Fig. 13 Fig. 14
Fig.12 Behavior of I v against radial coordinate r for the case-I corresponding to the Model-1 (left panel) and Model-2 (right panel)

Table 5
r | and ρ − |p t | against the radial coordinate r