Traversable wormholes satisfying energy conditions in f(Q) gravity

Abstract

In this article, a new family of asymptotically flat wormhole solutions in the context of symmetric teleparallel gravity, i.e., f(Q) theory of gravity, are presented. Considering a power-law shape function and some different forms of the f(Q) function, we show that a wide variety of wormhole solutions for which the matter fields satisfy some energy conditions, are accessible. We realize that the presence of f(Q) gravity will be enough to sustain a traversable wormhole without exotic matter. The influence of the free parameters of the shape function and the f(Q) models on the energy conditions is investigated. The equation of state and the boundary conditions are analyzed.

1 Introduction

Theoretically, a wormhole provides a short way between different universes or two points in the same universe. Time machine may be a consequence of the existence of wormholes [1]. The first proposal of a wormhole was presented by Flamm [2]. In 1935, Einstein and Rosen constructed the Einstein–Rosen bridge [3] which is basically the maximally extended Schwarzschild solution. It should be noted that the Einstein–Rosen bridge is not a traversable wormhole. Historically, Misner and Wheeler introduced the term “wormhole” in 1957 [4] but the general structure of the traversable wormhole is presented by Morris and Thorne [5]. Traversable wormholes are solutions of the Einstein’s field equations with exotic sources. A traversable wormhole solution should not have any horizon or singularity [6]. It was shown that wormholes need exotic matter to be constructed [6]. In the context of general relativity (GR), violation of null energy condition (NEC) i.e. $T_{\mu \nu }k^{\mu }k^{\nu }\ge 0,$ $k^{\mu }$ is null vector and $T_{\mu \nu }$ is the stress–energy tensor, is the main ingredient of the wormhole theory.

Distance measurements of type Ia supernovae demonstrated accelerated expansion of the Universe which is investigated extensively in the literature [7]. In this view, two main approaches are used to explain the accelerated expansion of the Universe. The first one explains this phenomenon, by introducing a new field in the context of GR, which is famous as dark energy (DE). Fluid with an equation of state (EoS) $p=\omega \rho$ and positive energy density is a good candidate to explain the evolution of the cosmos. The DE admits $-1<\omega \le 0$ while $\omega \le -1$ is defined as the phantom regime. Fluid with $\omega \le -\frac{1}{3},$ may be considered as a source to describe accelerated expansion of the Universe. It should be noted that phantom fluid violates NEC so the wormholes with phantom EoS are studied extensively in the literature [8,9,10,11,12,13,14,15].

One can minimize the amount of exotic material, by constructing a thin-shell wormhole [16,17,18,19,20] or finding the wormhole with variable EoS [21, 22]. In thin-shell wormhole, the exotic matter is confined near the throat of wormhole. On the other side, in [23], it was shown that wormholes with polynomial EoS in contrast to linear EoS violate the energy conditions (ECs) only in a small region of the spacetime.

We do not know much about dark matter and dark energy so in the second approach, alternative theories of gravity have been used to explain the accelerated expansion of the Universe. Since modified gravity theories are able to address several cosmological observations, it would be interesting to study wormholes in these theories and investigate if we can construct them without exotic matter. Several proposals have been presented to explain the exotic matter in the wormhole theory, by using modified theories of gravity. Braneworld [24,25,26,27,28,29,30], Born–Infeld theory [31,32,33], quadratic gravity [34, 35], Einstein–Cartan gravity [36,37,38,39], hybrid metric-Palatini gravity [40, 41], f(R) gravity [42,43,44,45,46] and f(R, T) gravity [47,48,49] are some examples. In most of these theories, Einstein’s field equations is modified. This modification provides the conditions to construct wormhole solutions without exotic matter or at least minimizing the usage of exotic matter.

There are three description of gravity giving the same dynamics. The first one is GR where the gravitational field is solely described by curvature. The other two are constructed from teleparallel geometries (setting curvature to zero) and torsion or non-metricity act as the field strength tensors of the theories [50]. Those theories are denoted as teleparallel equivalent versions of GR. The f(Q) gravity is a class of theories for which both curvature and torsion vanish while gravity is attributed to non-metricity [51]. The f(Q) theory can describe the accelerated expansion of the Universe at least the same level of statistic precision of most renowned modified gravities [52]. Cosmology in f(Q) gravity has been studied in [53,54,55,56,57,58,59,60] (see Ref. [50] for a recent review). Harko et al. have used f(Q) gravity to describe cosmological evolution’s and other aspects [54]. They have shown that an extension of the symmetric teleparallel gravity, where the non-metricity is coupled non-minimally to the matter Lagrangian, in the framework of the metric-affine formalism, can provide alternatives to DE. Some researchers have studied the ECs [61] and Newtonian limit [62] in the background of f(Q) gravity. Anagnostopoulos et al. have used the Big Bang Nucleosynthesis formalism and observations to extract constraints on various classes of f(Q) models [63].

Black hole solutions have been investigated in the context of f(Q) gravity [64,65,66]. Also, the compact star generated by gravitational decoupling in f(Q) gravity theory has been studied in [67]. Sokoliuk et al. have explored the Buchdahl quark stars in the background of f(Q) theory [68]. Recently, wormhole and spherically symmetric configurations have been studied in f(Q) gravity [52, 69,70,71,72,73,74,75,76,77]. In [52], a simple model, $f(Q)=Q+\alpha Q^{2},$ with a polytropic EoS is considered to find the internal spherically symmetric configuration. The static and spherically symmetric solutions with an anisotropic fluid for general f(Q) gravity are presented by Wang et al. [69]. Sharma et al. have studied the wormhole solutions in the context of symmetric teleparallel gravity [70]. They have shown that solutions with specific shape and redshift functions for some of the f(Q) models provide solutions that satisfy ECs in some regions of the spacetime. Mustafa et al. have used the Karmarkar condition in f(Q) gravity formalism to find wormhole solutions which satisfy the ECs [71]. In [72], traversable wormhole geometries in f(Q) by considering two specific EoS are investigated. The solutions in [72] have been obtained for a specific shape function in the fundamental interaction of gravity (i.e., for a linear form of f(Q)). This class of solutions do not respect the ECs. In [73], the authors discuss the existence of wormhole solutions with the help of the Gaussian and Lorentzian distributions of linear and exponential models. They show that the wormhole solutions obtained with these models are physically acceptable and stable but do not respect the ECs. Solutions with constant redshift function and different shape functions are presented in [74] for some known f(Q) functions. This class of solutions violates the ECs. Along this way, Calza and Sebastiani have analyzed a class of topological static spherically symmetric vacuum solutions in f(Q) gravity with constant non-metricity [75]. In [76], we have studied wormholes in the background of f(Q) gravity. We have found solutions which violate the ECs only in some small regions of the spacetime. Falco and Capozziello have studied static and spherically symmetric wormholes in metric-affine theories of gravity [78]. They present general constraints that correspond to the NEC without specifying the f(Q) function or the wormhole functions. In this work, we are particularly interested in presenting some traversable wormhole solutions without assuming any exotic matter.

The organization of the paper is as follows: In Sect. 2, we discuss the conditions and equations governing wormholes then we briefly review f(Q) theory and the ECs generally are given by (18–21). In Sect. 3, by defining a known shape function, we have analyzed some basic f(Q) functions to find solutions which can satisfy the ECs. The physical properties of the solutions that satisfy the ECs are presented in this section. Finally, in Sect. 4, we present our concluding remarks. In this paper, We assume gravitational units, i.e., $c = 8 \pi G = 1.$

2 Basic formulation of wormhole

In this section, we introduce the basic structure of the wormhole theory and a brief review of f(Q) gravity formalism. We use the prescription introduced in Ref. [76] where a detailed discussion about the formulation of the theory can be found. We use the line element of the general static spherically symmetric wormhole as:

$$\begin{aligned} ds^2=-U(r)dt^2+\frac{dr^2}{1-\frac{b(r)}{r}}+r^2(d\theta ^2+\sin ^2\theta , d\phi ^2) \end{aligned}$$

(1)

where $U(r)=\exp (2\phi (r)).$ $\phi (r)$ is called the redshift function and can be used to detect the redshift of the signal by a distance observer. b(r) is called the shape or form function. The shape function should obey the condition

$$\begin{aligned} b(r_0)=r_0, \end{aligned}$$

(2)

where $r_0$ is the wormhole throat. Two other conditions are imposed as follows to have a traversable wormhole,

$$\begin{aligned} b'(r_0)<1 \end{aligned}$$

(3)

and

$$\begin{aligned} b(r)<r,\quad \text {for} \ r>r_0. \end{aligned}$$

(4)

The former is famous as the flaring out condition. In the GR, the flaring-out condition and the NEC are incompatible. In this paper, we will consider the asymptotically flat condition as follows

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{b(r)}{r}=0,\quad \lim _{r\rightarrow \infty }U(r)=1. \end{aligned}$$

(5)

It is worth mentioning that constant redshift function guarantees the absence of horizon around the throat and presents zero tidal force. Physically, wormhole solutions with constant or non-constant redshift function do not have much difference, so for the sake of simplicity, we consider solutions with constant redshift function.

Let us briefly review the f(Q) formalism. The action for symmetric teleparallel gravity is given by

$$\begin{aligned} S=\int \frac{1}{2}f(Q)\sqrt{-g}\;d^{4}x+\int L_{m}\sqrt{-g}\;d^{4}x. \end{aligned}$$

(6)

where f(Q) is a general function of Q,  g is the determinant of the metric, and $L_m$ is the matter Lagrangian density. Now, one can define the non-metricity tensor and its traces by

$$\begin{aligned} Q_{\lambda \mu \nu }= & {} \nabla _{\lambda }g_{\mu \nu }, \end{aligned}$$

(7)

$$\begin{aligned} Q_{\alpha }= & {} Q_{\alpha \;\;\mu }^{\;\;\mu },\quad \widetilde{Q}_{\alpha }=Q^{\mu }_{\;\;\alpha \mu }. \end{aligned}$$

(8)

Further, the non-metricity conjugate is presented by

$$\begin{aligned} P^{\alpha }_{\;\;\mu \nu }= & {} \frac{1}{4}[-Q^{\alpha }_{\;\;\mu \nu } +2Q_{(\mu \;\;\nu )}^{\;\;\alpha }+Q^{\alpha }g_{\mu \nu }\nonumber \\{} & {} -\widetilde{Q}^{\alpha }g_{\mu \nu }-\delta ^{\alpha }_{(\mu }Q_{\nu )}] \end{aligned}$$

(9)

so

$$\begin{aligned} Q=-Q_{\alpha \mu \nu }P^{\alpha \mu \nu }. \end{aligned}$$

(10)

The energy–momentum tensor is shown by

$$\begin{aligned} T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta (\sqrt{-g}L_{m})}{\delta \;g^{\mu \nu }}. \end{aligned}$$

(11)

The field equations are obtained by varying the action (6) with respect to the metric

$$\begin{aligned} T_{\mu \nu }= & {} -\frac{2}{\sqrt{-g}}\nabla _{\gamma }(\sqrt{-g}f_{Q}P^{\gamma }_{\;\;\mu \nu })- \frac{1}{2}g_{\mu \nu }f\nonumber \\{} & {} -f_{Q}(P_{\mu \gamma i}Q_{\nu }^{\;\;\gamma i}-2Q_{\gamma i\mu }P^{\gamma i}_{\;\;\;\;\nu }), \end{aligned}$$

(12)

and

$$\begin{aligned} \nabla _{\mu }\nabla _{\nu }(\sqrt{-g}f_{Q}P^{\gamma }_{\;\;\mu \nu })=0, \end{aligned}$$

(13)

where $f_Q\equiv \frac{df}{dQ}.$ The metric (1) leads to the non-metricity scalar

$$\begin{aligned} Q=-\frac{2}{r}\left( 1-\frac{b(r)}{r}\right) \left( 2\phi ^{\prime }+\frac{1}{r}\right) . \end{aligned}$$

(14)

We shall assume that matter is well described by an anisotropic perfect fluid, i.e., the stress–energy tensor can be written in the form $T^{\mu }_{\nu }=diag[-\rho , p,p_t,p_t],$ where $\rho$ is the energy density, p the radial pressure and $p_t$ the tangential pressure, respectively. Using (1), (14) and (12), one can find the following field equations

$$\begin{aligned} \rho =&\left[ \frac{1}{r}\left( \frac{1}{r}-\frac{rb^{\prime }(r)+b(r)}{r^{2}} {+}2\phi ^{\prime }(r)\left( 1-\frac{b(r)}{r}\right) \!\right. \right] f_{Q} \nonumber \\&+\frac{2}{r}\left( 1-\frac{b(r)}{r}\right) f'_{Q}+\frac{f}{2}. \end{aligned}$$

(15)

$$\begin{aligned} p_{r}=&-\left[ \frac{2}{r}\left( 1-\frac{b(r)}{r}\right) \left( 2\phi ^{\prime }(r)+\frac{1}{r}\right) -\frac{1}{r^{2}}\right] f_{Q}-\frac{f}{2}. \end{aligned}$$

(16)

$$\begin{aligned} p_t(r)=&\left[ \frac{1}{r}\left( \left( 1-\frac{b(r)}{r}\right) \left( \frac{1}{r}+\phi ^{\prime }(r)(3+r\phi ^{\prime }(r)\right) \right. \right. \nonumber \\&\left. \left. +r\phi ^{\prime \prime }\right) -\frac{rb^{\prime }(r) -b(r)}{2r^{2}}(1+r\phi ^{\prime }(r)))\right] f_{Q}\nonumber \\&+\frac{1}{r}\left( \left( 1-\frac{b(r)}{r}\right) \right) (1+r\phi ^{\prime }(r))f_{QQ}+\frac{f}{2}, \end{aligned}$$

(17)

which the prime denotes the derivative $\frac{d}{dr}.$ Now, we have the essential mathematical tools to study the wormhole solutions in the background of f(Q). As was mentioned in the introduction, many algorithms have been used to find wormhole solutions in the f(Q) but we will use a known shape function and some f(Q) models with free parameters to explore new wormhole solutions.

One of the main characteristics of wormhole solutions in GR is the violation of the ECs. The ECs represent paths to accomplish the positiveness of the stress–energy tensor in the presence of matter. The four ECs which are famous as the null energy condition (NEC), dominant energy condition (DEC), weak energy condition (WEC), and strong energy condition (SEC), are defined as:

$$\begin{aligned}{} & {} {\textbf {NEC}}: \rho +p\ge 0,\quad \rho +p_t\ge 0 \end{aligned}$$

(18)

$$\begin{aligned}{} & {} {\textbf {WEC}}: \rho \ge 0, \rho +p\ge 0,\quad \rho +p_t\ge 0, \end{aligned}$$

(19)

$$\begin{aligned}{} & {} {\textbf {DEC}}: \rho \ge 0, \rho -|p|\ge 0,\quad \rho -|p_t|\ge 0, \end{aligned}$$

(20)

$$\begin{aligned}{} & {} {\textbf {SEC}}: \rho +p\ge 0,\, \rho +p_t\ge 0,\rho +p+2p_t \ge 0. \end{aligned}$$

(21)

These conditions are the essential tools to understand the geodesics of the Universe which can be derived from the well-known Raychaudhury equations. The ECs n combination with this equation can explain the attractive nature of gravity [79, 80]. According to [76], by defining the functions,

$$\begin{aligned} H(r)= & {} \rho +p,\, H_1(r)= \rho +p_t,\, H_2(r)= \rho -|p|, \nonumber \\ H_3(r)= & {} \rho -|p_t|,\, H_4(r)= \rho +p+2p_t, \end{aligned}$$

(22)

we can investigate the ECs.

In this article, we present solutions in the background of f(Q) gravity that satisfy the ECs. To have some physically viable and reasonable models of traversable wormholes and to discover the possibility of non-exotic wormholes within the framework of f(Q) gravity, in the next parts of the paper, we will consider a known shape function with some f(Q) models. We will see that the existence of free parameters in the shape function and f(Q) functions provides solutions which respect the ECs.

3 Wormhole solutions

There are several motivations to explore wormhole in theories beyond the standard formulation of gravity. The most important one is solving the problem of exotic matter. As we know, solutions that respect ECs, without any extra condition, in the background of f(Q) gravity introduced by Jimenez et al. [51] are not presented yet. It should be mentioned that in [78] some general constraints that correspond to the NEC without specifying the f(Q) function or the wormhole functions are presented. In this section, we explore solutions which respect the ECs. For the sake of simplicity, we set $r_0 =1$ and $\phi (r)=0.$ By using the Eqs. (15)–(17), and specifying the functions, f(Q),  b(r),  $\phi (r),$ the energy momentum tensor can be explored. Usually, an EoS also assumed, one of the functions f(Q),  b(r),  $\phi (r)$ is considered unknown then one can find it from the field equations. The complexity of equations in the context of f(Q) gravity does not allow us to use this algorithm for finding wormhole solutions that satisfy ECs. We will use known functions with free parameters then by fine-tuning the free parameters, we will construct a wormhole with the most consistency. Along this way, we consider a known shape function

$$\begin{aligned} b(r)=r^{m}. \end{aligned}$$

(23)

This shape function is the most famous one in the wormhole theory. It satisfies all the necessary conditions required to construct a traversable wormhole [12]. It is easy to show that asymptotically flat condition (4) implies that $m<1$ is acceptable. Equation (14) along with the shape function (23) leads to

$$\begin{aligned} Q(r)= -2(r^{-2}-r^{m-3}). \end{aligned}$$

(24)

In the next sections, we use some different models of f(Q) and we seek solutions that satisfy the ECs. We start with some general forms of the f(Q) function to construct the desired wormhole solutions then we will study the physical properties of the solutions.

Fig. 1

The figure represents the $\rho (r,n)$ against radial coordinate and n for $m=-2,$ which is negative in the whole range. See the text for details

Fig. 2

The figure represents the $\rho (r,n)$ against radial coordinate and n for $m=1/2,$ which is negative in the whole range. See the text for details

Fig. 3

The plot depicts the general behavior of $\rho (r,m)$ against r and m for the case $n=2.$ It is clear that $\rho (r,m)$ is negative in the entire range of r and m. It shows that the ECs are violated. See the text for details

3.1 $f(Q)=(-Q)^{n}$

The function

$$\begin{aligned} f(Q)=(-Q)^{n} \end{aligned}$$

(25)

is the first candidate for which we investigate the ECs in the background of f(Q) scenario. It is clear that studying solutions with the form function (23) and a general form of (25) is very complicated. For the sake of simplicity, we use the shape function (23) for some specific values of m i.e., $m=1/2,0, -1/2, -1, -2, -4.$ We have plotted the energy density function for some of these choices in Figs. 1 and 2. These figures imply that the energy density is negative so this class of solutions is not considerable. Although the general form function $b(r)=r^{m}$ with arbitrary m is not examined, the behavior of the energy density for some different value of m corroborate that this class of solutions can not satisfy the ECs.

Alternatively, we consider a f(Q) function with constant n and variable m for the shape function. The energy density is plotted as a function of m and radial coordinate for $n=2$ in Fig. 3. This figure demonstrates that $\rho$ is negative. Thus, this class of solutions is not significant. One can use this method for different values of n but it seems that the result will be the same. As it was seen, the positivity of the energy density is not achievable with this class of shape and f(Q) functions. In the next subsection another f(Q) model is examined.

3.2 $f(Q)=Q^{2}+c$

We continue our study with a f(Q) function in the form

$$\begin{aligned} f(Q)=Q^{2}+c. \end{aligned}$$

(26)

It is a second-order function of Q which c is a constant parameter. One can show that

$$\begin{aligned} \lim _{r\rightarrow \infty }\rho = \frac{c}{2} \end{aligned}$$

(27)

This implies that the constant parameter c can be set as $c=2 \rho _{\infty }$ where $\rho _{\infty }$ is the energy density at large distance. Thus, we are interested in the positive values of c. We have plotted $\rho (r,c)$ as a function of r and c for $m=-2$ in Fig. 4 which shows that the energy density is positive for some range of c. We have also plotted $H(r,c)=\rho +p$ in Fig. 5. It is clear that the ECs are not satisfied in this class of solutions. The same results can be concluded for $m=1/2, -1/2, -1, -2, -3, -4.$

Fig. 4

The figure represents the $\rho (r,c)$ against radial coordinate and c for $m=-2,$ which is positive for some range. See the text for details

Fig. 5

The graphical behavior of H(r, c),  against r and c for the case $m=-2.$ It is clear that this function is negative in the entire range of r and c so the ECs are violated. See the text for details

Fig. 6

The plot depicts the behavior of $\rho (r,n)$ against r and n for the case $c=10$ and $m=-2.$ One can see that $\rho (r,n)$ is a positive function in the entire range. See the text for details

The function (26) is a special case of the function

$$\begin{aligned} f(Q)=(-Q)^{n}+c. \end{aligned}$$

(28)

Studying solutions for this function in the general form is complicated in contrast to the previous functions. As in the previous case, it can be concluded that $c=2 \rho _{\infty }$ so the positive c is acceptable. We investigate solutions for fixed values of c and m and variable n. It should be mentioned that the vanishing c leads to solutions which have been studied in Sect. 3.1. Figure 6 demonstrates that a positive energy density may be reachable for some range of n. To complete our study, we have plotted H(r, n) for $m=-2$ in Fig. 7 which shows that this class of solutions does not satisfy the ECs. The same result is concluded for the values $m=1/2, 0, -1/2, -1, -3, -4,-5, -8$ in the shape function. It seems that the f(Q) function of the form (28) in combination with the shape function (23) can not provide a suitable physical solution.

Fig. 7

The graphical behavior of H(r, n) against r and n for the case $c=10$ and $m=-2.$ It shows that ECs are violated. See the text for details

Fig. 8

The graphical behavior of $\rho (r,n)$ against r and n for the case $f(Q)=(-Q)^{n}+Q,$ and $m=-2.$ It shows that the energy density is negative in some range. See the text for details

3.3 Solutions satisfying energy conditions

Now, let us consider the f(Q) function in the form

$$\begin{aligned} f(Q)= (-Q)^{n}+BQ+c. \end{aligned}$$

(29)

It can be seen that a linear term BQ is added to the previous f(Q) function where B is a constant parameter. In this case, the free parameters are more in contrast to the previous cases so we should fix some initial conditions to study the solutions. It can be deduced that $c=2\rho _{\infty }$ therefore, we consider a vanishing energy density at large scale so the term c is omitted. The first model we study is

$$\begin{aligned} f(Q)= (-Q)^{n}+Q. \end{aligned}$$

(30)

where we have set $B=1$ and $c=0.$

Fig. 9

The figure represents the $\rho (r,n)$ against radial coordinate for $f(Q)=(-Q)^{n}+Q,$ and $m=-1/2$ which shows $\rho (r,n)$ is positive in some regions. See the text for details

Fig. 10

The plot depicts H(r, n) against radial coordinate for $f(Q)=(-Q)^{n}+Q,$ and $m=-1/2$ which shows $H(r,n)>0$ is reachable. See the text for details

Fig. 11

The graphical behavior of $H_1(r,n)$ against r and n for the case $f(Q)=(-Q)^{n}+Q,$ and $m=-1/2.$ It shows that $H_1(r,n)>0$ is not reachable. See the text for details

Fig. 12

The plot depicts $\rho (r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-1/2$ which shows $\rho (r,B)$ is positive in some regions of B. See the text for details

Fig. 13

The plot depicts H(r, B) against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-1/2$ which shows H(r, B) is always positive. See the text for details

Fig. 14

The plot depicts $H_1(r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-1/2$ which shows $H_1(r,B)$ is always negative. See the text for details

Let us investigate this model for some specific values of m of the shape function. We have plotted the energy density as a function of r and n for $m=-2$ in Fig. 8. As one can notice, the energy density is negative in some range of the radial coordinate for the entire range of n. The same result is achieved for the values $m=-3,-4,-5,-8.$ Let us study the specific case $m=-1/2.$ We have plotted $\rho$ as a function r and n in Fig. 9 which shows that energy density is positive in some range of n. Also, the function H is plotted as a function of r and n in Fig. 10. One can deduce from that the condition $H>0$ is reachable. However, Fig. 11 where $H_1$ is plotted demonstrates that this class of solutions can not satisfy the ECs.

Fig. 15

The plot depicts $\rho (r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows $\rho (r,B)$ is positive in some regions of B. See the text for details

Fig. 16

The plot depicts H(r, B) against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows H(r, B) is always positive. See the text for details

Fig. 17

The plot depicts $H_1(r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows $H_1(r,B)$ is positive for some range of B. See the text for details

Fig. 18

The plot depicts $H_2(r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows $H_2(r,B)$ is positive in some regions of B. See the text for details

Due to the high complexity of the f(Q) model in (29), we study the model for $n=2$ and $c=0$

$$\begin{aligned} f(Q)= (-Q)^{2}+BQ. \end{aligned}$$

(31)

Also, it can be considered as a special case of (28) while a linear term BQ is added. It is easy to show that

$$\begin{aligned} B=2 \rho (r_0). \end{aligned}$$

(32)

In [74], it was shown that wormhole solutions could not exist for the specific form function (31). But it should be mentioned that the authors in [74] have used the field equations, presented in [52] instead of field equations in [71]. It should be noted that the choice of gauge may result in different field equations. In [51, 53], authors explained in details about the gauge choice. In this paper, our field equations are the same as [71]. For the case with $m=-1/2,$ Figs. 12 and 13 show that $\rho >0$ and $H>0$ are reachable for some range of B. However, Fig. 14 shows that $H_1$ is not positive. We also checked the cases for $m=1/2,-1/2,-1$ but the results are not interesting either but for $m=-2$ are significant. We would like to clarify the results for the case $b(r)=1/r^{2}$ and $f(Q)=Q^{2}+BQ.$ Figure 15 demonstrates the behavior of energy density as a function of r and B. One can conclude that $\rho >0$ is reachable for some values of B. In Fig. 16, we have plotted H as a function of r and B which implies $H>0$ is reachable. The conditions $H_{1}>0,$ $H_{2}>0,$ $H_{3}>0$ and $H_{4}>0$ are also reachable (see Figs. 17, 18, 19, 20). So, we have found a solution which satisfies all of the ECs for some range of B. It is easy to show that

$$\begin{aligned} \omega _{\infty }=\lim _{r\longrightarrow \infty }\frac{p}{\rho }=\frac{1}{7}, \end{aligned}$$

(33)

which shows the EoS is asymptotically linear [22]

$$\begin{aligned} p(r)=(\omega _{\infty }+g(r))\rho (r). \end{aligned}$$

(34)

It should be noted that

$$\begin{aligned} \lim _{r\longrightarrow \infty }g(r)=0. \end{aligned}$$

(35)

One can see that the this class of solutions are not isotropic.

Let us investigate the effect of the free parameters of (29) on the wormhole solutions. The results for specific values of the parameters B and m (1) and (2). These results have allowed us to give a better understanding of the influence of the free parameters in f(Q) function and shape function on the validation of ECs. It is clear that B has a crucial role in the satisfaction of ECs. Table 1 indicates that for $B=3,$ none of the values we have considered for n leads to a solution that satisfies the ECs. But for $B=5.5,$ some ECs are satisfied for $n=3/2,2.$ It seems that the most of individuals n respect the ECs while B increases. One also may see that the case $n=2/3$ violates all of the ECs independent of the value of B. Generally, due to the non-linearity of the field equations in the f(Q) scenario, we can not introduce any explicit relation that shows how the ECs are affected by the parameters B, n and m.

Fig. 19

The plot depicts $H_3(r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows $H_3(r,B)$ is always positive. See the text for details

Fig. 20

The plot depicts $H_4(r,B)$ against radial coordinate and B for $f(Q)=Q^{2}+BQ,$ and $m=-2$ which shows $H_4(r,B)$ is always positive. See the text for details

Table 1 The results for some individual B and n in $f(Q)=(-Q)^{n}+BQ$ while $b(r)=\frac{1}{r^{2}}$ is considered

Table 2 The results for some individual B and D in $f(Q)=D(-Q)^{2}+BQ$ while $b(r)=\frac{1}{r^{2}}$ is considered

All f(Q) models which have been studied in this paper, are some special cases of the general form

$$\begin{aligned} f(Q)=D(-Q)^{n}+BQ+c. \end{aligned}$$

(36)

It can be verified that the condition $f_{QQ} Q^{\prime }>0$ is not met across the entire span of r for $b(r)=1/r^2.$ It shows that assumption in [78] is not satisfied so we cannot relate our solutions with general constraints that have been presented in [78]. It is easy to check that for $n=0$ and $B=1,$ the model reduces to the standard symmetric teleparallel equivalent of general relativity model with the quantity $(D+c)/2$ playing the role of the cosmological constant [54, 55, 61]. The symmetric teleparallel equivalent of GR can also be achieved for $n=1.$ It was shown that modification from the GR evolution occurs at low curvatures regime for $n < 1$ and occurs at high curvatures regime for $n > 1.$ Hence, it can be deduced that $n > 1$ will be applicable for the early Universe, while $n < 1$ will be applicable to the late-time DE dominated Universe [81].

However, for the sake of completeness, we continue to investigate some other features of the free parameters in this model for wormhole theory. Thus, in summary, we have presented the results of some individual values for B and D while $b(r)=\frac{1}{r^{2}}$ and $n=2$ are considered (Table 2). Remarkably, we have found that the existence of an extra parameter like D can change at least the limits of B for validations of the ECs. For instance, for $B=3,$ one can find that there are not any solutions that respect the ECs in the for $D=1$ but there are for $D=1/2.$ Although the exact relations between the influence of the free parameters in (36) on validation of ECs can not be addressed but one can deduce from the Tables 1 and 2 that the changes in the free parameters affect the ECs.

In [61] a complete test of ECs for some f(Q) gravity models is presented. It was shown that the ECs allowed us to fix our free parameters, restricting the families of f(Q) models compatible with the accelerated expansion of the Universe passes through. Mandal et al. have shown that the ECs directly depend on the free parameters D and n so one cannot take these values arbitrarily, which may violate the ECs as well as the current scenario of the Universe dominated by the dark energy [61].

4 Concluding remarks

GR is the basic theory for the Standard Model of physical cosmology. Despite the success of GR in describing many cosmological phenomena, this theory has some limitations. Expansion of the Universe, cosmological constant problem, coincidence problem, and early universe [82]. The modifications of gravity are proposed as an alternative to investigating these problems.

Traversable wormholes violate the ECs in the context of GR. Although there are not any observational data that verify the existence of wormholes, since they arise as solutions of several gravitational theories exactly as black holes, it is important to study these objects even if we have not detected them yet. Understanding their properties may actually help to detect them. Discovering exact wormhole solutions is an important aspect of wormhole research, and the most crucial challenge is the minimization of the exotic matter involved.

Various techniques have been employed in the existing literature to identify exact wormhole solutions, some proposing approaches to minimize the reliance on exotic matter. Additionally, researchers have uncovered wormholes which respect the ECs in the framework of modified gravity theories. Recently a new proposal teleparallel symmetric equivalent of general relativity has been used to describe wormhole solutions. This new formalism is considered as the third equivalent formulation of GR by means of the Q-scalar motivates novel ways of modifying gravity. In this work, we presented wormhole solutions that satisfy the ECs in the f(Q) theory of gravity. Since the most of the solutions which have been presented before do not respect ECs [69,70,71,72,73,74,75,76,77], our solutions seem to be new and significant.

Finding asymptotically exact wormhole solutions in the context of GR with a known EoS is not a simple task. Due to the higher order terms, this procedure is more complicated in the background of f(Q) gravity. For this reason, we have focused on finding wormhole solutions for some basic models of the f(Q) function and a well-known shape function. The power-law shape function has been used extremely in studying wormhole solutions and is the most famous solution in the different gravity scenarios and for this reason we also adopt it here. We investigated the model $f(Q)=D(-Q)^n+BQ+c$ and we found solutions that satisfy the ECs for specific values of the free parameters.

The non-linearity of the field equations in various modified theories increases the complexity of wormhole solutions. Using the fine-tuning technique for the free parameters can reduce this complexity. By using this technique, we found solutions which satisfy ECs. The results are strongly depend on the numerical values of the model parameters. It was shown that B which has a crucial role in the ECs can be related to the values of the energy momentum tensor at the throat of the wormhole. Also, we have shown that the constant parameter c in f(Q) model must be interpreted as the energy density at the large scale. We have omitted this parameter to have more viable solutions. The boundary conditions have been investigated to improve the viability of the solutions. Furthermore, we have explored the EoS for our solutions which verified that anisotropic matter content is essential to sustain wormholes in this realm. It was shown that the EoS is asymptotically linear.

To summarize, the violation of the ECs is a fundamental inconsistency that needs to be addressed in wormhole theory. By fine-tuning the free parameters of the shape function and f(Q) models, we found solutions which require no-exotic matter. So, this is the significant point of this work where exotic matter is just replaced by a modified form of gravity. Although, we have shown that the presence of f(Q) gravity will be enough to sustain a traversable wormhole without exotic matter. The viability of these models should be tested more precisely in a cosmological background. The combination of these results with astronomical results in the context of f(Q) gravity can provide the best proposal for f(Q) models or at least the ability of each model to describe cosmological phenomena. These solutions may represent an alternative to the standard GR scenario.

Although wormholes have not been detected experimentally yet, in this study, we have explored the possible existence of some wormhole geometries in the context of f(Q) gravity. The theoretical consistency and motivations on these extensions of f(Q) can be established to explore the new avenues in the wormhole theory and cosmological predications of f(Q) theory. Along this way, we have considered a vanishing redshift function, i.e., $\phi (r) = 0,$ but solutions with non-constant redshift function can be explored. Furthermore, our algorithms can be used for some other models of the f(Q) or different forms of shape function.