1 Introduction

General theory of relativity provides an elegant mathematical description of the spacetime geometry. One of the viable solutions to Einstein’s field equations yields a static and spherically symmetric black hole [1]. The recent detection of gravitational waves (GWs) [2] confirmed that stellar-mass black holes really exist in Nature. Interestingly, Ludwing Flamm [3] realized in 1916 that Einstein’s equations exhibit another solution currently known as a white hole. However, unlike black holes, it was believed that white holes eject matter and light from their event horizon. These two solutions could represent two different regions in spacetime connected by a conduit. This conduit was named later a “bridge” and in 1935, Einstein and Rosen used the theory of general relativity to propose the existence of “bridges” through spacetime [4]. Historically some decades later, Misner and Wheeler first introduced the term “wormhole” in Ref. [5]. Since then, wormholes become a focus on studying new realms of research. However, an original version of wormholes was later ruled out because they are not traversable. This means that its throat opens and closes so quickly.

However, in order to prevent the wormhole’s throat from closing, one can add a scalar field coupled to gravity. This new ingredient provides a more general class of wormholes firstly proposed by Ellis [6] and independently by Bronnikov [7]. However, the main problem of wormholes is that they are supported by exotic matter; a kind of matter which violates the known energy conditions. Some conditions introduced by Morris and Throne in 1988 for the wormholes to be traversable can be found in Ref. [8]. These solutions are obtained by considering an unusual type of matter which can maintain the structure of the wormhole. In addition, this exotic matter with negative energy density satisfies the flare-out condition and violates weak energy condition [8, 9]. Thin-shell wormholes have been studied and analyzed in several contexts involving particular models of exotic matters and theories of gravity, e.g. see Refs. [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] and the references therein.

In the present work, we study the thin-shell wormholes in dRGT massive gravity. The study of massive gravity has begun prior to the discovery of the accelerated expansion of the universe. In 1939, Fierz and Pauli used a linear theory of massive gravity as a mass term of the graviton [37]. Unfortunately, there are some flaws of the proposed model [38, 39] where the asymptotic-massless limits of the linear theory do not satisfy the GR prediction. Later on, the author of Ref. [40] suggested that the nonlinear approach in the massive gravity theory might be able solve the problem. However, this inevitably leads to a new problem called the Boulware–Deser (BD) ghost [41]. In 2010, the BD ghost was completely eliminated by a new nonlinear version of massive gravity proposed by de Rham, Gabadadze, and Tolley (dRGT) [42]. Since then the dRGT was targeted as one of the compelling scenarios when studying the universe on a cosmic scale. Additionally, there have been many interesting articles regarding the applications of the dRGT massive gravity to the exotic objects, e.g., black holes [43,44,45].

In this work, we study the thin-shell wormholes in dRGT massive gravity. The structure of the present work is as follows: we first present a short review of the dRGT model of the nonlinear massive gravity in Sect. 2. In Sect. 3, we consider the mathematical setup in order to study the thin-shell wormhole. We study junction conditions allowing one to glue two identical dRGT spacetimes. In addition, we make stability analyses of the dRGT thin-shell wormhole by considering four existing exotic fluid models in Sect. 4. We also check the null, weak, and strong conditions at the wormhole throat for all models present in Sect. 5. Moreover, we quantify the wormhole geometry by using the embedding diagrams to represent a thin-shell wormhole in the dRGT massive gravity in Sect. 6. Finally, we discuss our main results and conclude our findings in the last section. In this work, we use the geometrical unit such that \(G=1\).

2 A short recap of massive gravity

In this section, we assume that the universe is undergoing an accelerating phase [46, 47]. To begin the formalism, let us follow the work proposed in Ref. [42] and first define the tensor \(H_{\mu \nu }\) as the covariantization of the metric perturbation, \(g_{\mu \nu }=\eta _{\mu \nu } + h_{\mu \nu } = H_{\mu \nu } + \eta _{ab}\partial _{\mu }\phi ^{a}\partial _{\nu }\phi ^{b}\). Note that the four Stückelberg fields \(\phi ^{a}\) transform as scalars, and \(\eta _{ab}= (-1,1,1,1)\). The helicity-0 mode \(\pi \) of the graviton can be obtained by expressing \(\phi ^{a} = (x^{a}-\eta ^{a\mu }\partial _{\mu }\pi )\) such that

$$\begin{aligned} H_{\mu \nu } = h_{\mu \nu } + 2\Pi _{\mu \nu } - \eta ^{\alpha \beta }\Pi _{\mu \alpha }\Pi _{\beta \nu }, \end{aligned}$$
(1)

with \(\Pi _{\mu \nu }=\partial _{\mu }\partial _{\nu }\pi \). The authors of Ref. [42] define the tensor quantity \({{\mathcal {K}}}^{\mu }_{\nu }(g,H)\) as follows:

$$\begin{aligned} {{\mathcal {K}}}^{\mu }_{\nu }(g,H) {=} -\sum ^{\infty }_{n=1}d_{n}(H^{n})^{\mu }_{\nu },\,\,\mathrm{with}\,\,d_{n} {=} \frac{(2n)!}{(1-2n)(n!)^{2}4^{n}},\nonumber \\ \end{aligned}$$
(2)

where \(H^{\mu }_{\nu }=g^{\mu \alpha }H_{\alpha \nu }\) and \((H^{n})^{\mu }_{\nu }=H^{\mu }_{\alpha _{1}}H^{\alpha _{1}}_{\alpha _{2}}H^{\alpha _{2}}_{\alpha _{3}}...H^{\alpha _{n-1}}_{\nu }\) denotes the product of n tensors \(H^{\alpha }_{\beta }\). Therefore, the action representing the dRGT model on the manifold \({{\mathcal {M}}}\) is given byFootnote 1

$$\begin{aligned} S = \int _M \mathrm{d}^4 x \sqrt{-g} \frac{1}{16\pi G}\big ( R + m^2_g \mathcal {U}(g,\phi ^a) \big ), \end{aligned}$$
(3)

where \(\sqrt{-g}\) is the volume element in 4-dimensional manifold \(\mathcal {M}\) and the potential \(\mathcal {U}\) is defined by

$$\begin{aligned} \mathcal {U} = \mathcal {U}_2 + \alpha _3 \mathcal {U}_3 + \alpha _4 \mathcal {U}_4, \end{aligned}$$
(4)

where \(\mathcal {U}_{2},\,\mathcal {U}_{3}\) and \(\mathcal {U}_{4}\) are given by

$$\begin{aligned} \mathcal {U}_2= & {} [\mathcal {K}]^2 - [\mathcal {K}^2], \nonumber \\ \mathcal {U}_3= & {} [\mathcal {K}]^3 - 3[\mathcal {K}][\mathcal {K}^2] + 2 [\mathcal {K}^3], \nonumber \\ \mathcal {U}_4= & {} [\mathcal {K}]^4 - 6[\mathcal {K}]^2[\mathcal {K}^2] + 8[\mathcal {K}][\mathcal {K}^3] + 3[\mathcal {K}^2]^2 - 6[\mathcal {K}^4].\nonumber \\ \end{aligned}$$
(5)

Here a bracket \([\quad ]\) represents the trace of the tensor, \([{{\mathcal {K}}}] \equiv {{\mathcal {K}}}^{\mu }_{\mu },\,[{{\mathcal {K}}}^{2}] \equiv {{\mathcal {K}}}^{\mu }_{\alpha _1}{{\mathcal {K}}}^{\alpha _1}_{\mu },\,[{{\mathcal {K}}}^{3}] \equiv {{\mathcal {K}}}^{\mu }_{\alpha _1}{{\mathcal {K}}}^{\alpha _1}_{\beta _1}{{\mathcal {K}}}^{\beta _1}_{\mu }\) and \([{{\mathcal {K}}}^{4}] \equiv {{\mathcal {K}}}^{\mu }_{\alpha _1}{{\mathcal {K}}}^{\alpha _1}_{\beta _1}{{\mathcal {K}}}^{\beta _1}_{\alpha _2}{{\mathcal {K}}}^{\alpha _2}_{\mu }\). Note here that parameters \(\alpha _{3}\) and \(\alpha _{4}\) of the dRGT theory can be related to the graviton mass. Performing variation of the gravitational action in Eq. (3) with respect to the metric, \(g_{\mu \nu }\), yields the Einstein equation of the dRGT massive gravity,

$$\begin{aligned} G_{\mu \nu } + m^2_g X_{\mu \nu } = 0, \end{aligned}$$
(6)

where \(X_{\mu \nu }\) is defined by

$$\begin{aligned} X_{\mu \nu }= & {} \frac{1}{\sqrt{-g}}\frac{\delta \sqrt{-g} \mathcal {U}}{ \delta g^{\mu \nu }} \nonumber \\= & {} \mathcal {K}_{\mu \nu } -\alpha \left[ \big (\mathcal {K}^2 \big )_{\mu \nu } - [\mathcal {K}]\mathcal {K}_{\mu \nu } + \frac{1}{2} g_{\mu \nu }\big ( [\mathcal {K}]^2 - [\mathcal {K}^2]\big )\right] \nonumber \\&+\, 3\beta \left[ \big (\mathcal {K}^3\big )_{\mu \nu } - [\mathcal {K}]\big ( \mathcal {K}^2\big )_{\mu \nu } + \frac{1}{2} \mathcal {K}_{\mu \nu }\big ( [\mathcal {K}]^2 - [\mathcal {K}^2]\big )\right. \nonumber \\&\left. -\frac{1}{6} g_{\mu \nu }\big ( [\mathcal {K}]^3 - 3[\mathcal {K}][\mathcal {K}^2] + 2[\mathcal {K}^3]\big )\right] , \end{aligned}$$
(7)

where the parameters \(\alpha \) and \(\beta \) are related to \(\alpha _{3,4}\) from the action in Eq. (3) via

$$\begin{aligned} \alpha = 1 + 3\alpha _3,\qquad \beta = \alpha _3 + 4\alpha _4. \end{aligned}$$
(8)

In addition, one finds that the Einstein equation of the dRGT massive gravity is reduced to the standard GR by setting \(m_g \rightarrow 0\). Having used the static and spherically symmetric spacetime, we obtain an explicit form of the line element of the dRGT massive gravity which reads [42, 48]

$$\begin{aligned} \mathrm{d}s^2 = g_{\mu \nu } \mathrm{d}x^{\mu }\mathrm{d}x^{\nu } = - f(r) \mathrm{d}t^2 + \frac{\mathrm{d}r^2}{f(r)} + r^2 \mathrm{d}\Omega ^2, \end{aligned}$$
(9)

where \(\mathrm{d}\Omega ^2 = \mathrm{d}\theta ^2 + \text {sin}^2\theta \mathrm{d}\phi ^2\) and the function f(r) is the vacuum solution without any kind of matter. The latter can be written as

$$\begin{aligned} f(r) = 1 - \frac{2M}{r} - \frac{\Lambda r^2}{3} + \gamma r + \zeta , \end{aligned}$$
(10)

where M is the mass parameter, \(\Lambda \) is the effective cosmological constant, and \(\gamma \) and \(\zeta \) are new parameters and they are linear combinations of the parameters in the dRGT massive gravity via the following relations:

$$\begin{aligned}&\Lambda \equiv -3m_g^2(1+ \alpha + \beta ), \quad \gamma \equiv -m_g^2k(1 + 2\alpha + 3\beta ),\nonumber \\&\zeta \equiv m_g^2k^2(\alpha + 3\beta ). \end{aligned}$$
(11)

The behaviors of how a function f(r) depends on r is displayed in Fig. 1. According to the Einstein equation of the dRGT massive gravity in Eq. (6), one might identify the \(m_g^2X_{\mu \nu }\) term as the effective energy-momentum tensor. Using an explicit form of the metric tensors in Eq. (10), allows us to directly compute components of \(m_g^2X_{\mu \nu }\) in Eq. (7). The components of \(m_g^2 X_{\nu }^\mu \) read [49,50,51]

$$\begin{aligned} \rho _g(r)\equiv & {} -\frac{m_g^2}{8\pi G}X_t^t\nonumber \\= & {} \frac{m_g^2}{8\pi G}\left( \frac{3r -2k}{r} + \frac{\alpha (3r-k)(r-k)}{r^2} + \frac{3\beta (r-k)^2}{r^2}\right) ,\nonumber \\ \end{aligned}$$
(12)
$$\begin{aligned} p_g^{(r)}(r)\equiv & {} -\frac{m_g^2}{8\pi G}X_r^r \nonumber \\= & {} \frac{m_g^2}{8\pi G}\left( \frac{3r -2k}{r} + \frac{\alpha (3r-k)(r-k)}{r^2} + \frac{3\beta (r-k)^2}{r^2}\right) ,\nonumber \\ \end{aligned}$$
(13)
$$\begin{aligned} p_g^{(\theta ,\phi )}(r)\equiv & {} -\frac{m_g^2}{8\pi G}X_{\theta ,\phi }^{\theta ,\phi }\nonumber \\= & {} \frac{m_g^2}{8\pi G}\left( \frac{3r -k}{r} + \frac{\alpha (3r-2k)}{r} + \frac{3\beta (r-k)}{r}\right) .\nonumber \\ \end{aligned}$$
(14)

With the help of Eq. (11), the parameters k, \(\alpha \) and \(\beta \) are re-written in terms of \(\Lambda \), \(\gamma \) and \(\zeta \) by

$$\begin{aligned} k= & {} \frac{\gamma +\sqrt{\gamma ^2 + (m_g^2+\Lambda )\zeta }}{m_g^2+\Lambda }, \nonumber \\ \alpha= & {} -\frac{\gamma ^2 + (2 m_g^2 +\Lambda )\zeta -\gamma \sqrt{\gamma ^2 + (m_g^2 + \Lambda )\zeta }}{m_g^2\zeta }, \nonumber \\ \beta= & {} \frac{2 \Lambda }{3 m_g^2} + \frac{\gamma ^2 + m_g^2\zeta -\gamma \sqrt{\gamma ^2 + (m_g^2 + \Lambda )\zeta }}{m_g^2\zeta }. \end{aligned}$$
(15)

In addition, we found that the parameters k, \(\alpha \) and \(\beta \) are finite in the \(\zeta \rightarrow 0\) limit, i.e.,

$$\begin{aligned} k = \frac{2\gamma }{m_g^2+\Lambda },\qquad \alpha = -3 \beta = -\frac{3}{2}-\frac{\Lambda }{2 m_g^2}. \end{aligned}$$
(16)

In this limit, the above relations are well defined for tuning the positive value and the smallness of the effective cosmological constant, \(\Lambda \), constrained by the observed values from the cosmological data. More importantly, it has been shown that the effective energy-momentum tensor, \(X_{\mu \nu }\), exhibits a behavior like anisotropic dark energy fluid i.e., \(p_g^{(r)} = -\rho _g\); see more detailed discussions and applications in Refs. [49,50,51].

Fig. 1
figure 1

Behaviors of f(r) against r. For simplicity, we have used \(M=1.0,\,G=1.0,\,\zeta =0.0\). We labeled lines with orange and purple colors for \(\Lambda =0.0001,\,\gamma =0.001\) and \(\Lambda =0.0001,\,\gamma =0.00002\), respectively

Full size image

We close this section by adding matter fluid to the Einstein equation. By using the standard method in GR, the Einstein equation with the (fluid) matter field of the dRGT massive gravity is given by

$$\begin{aligned} G_{\mu \nu }= & {} 8\pi G T_{\mu \nu },\qquad T_{\mu \nu } = T_{\mu \nu }^{(f)} + T_{\mu \nu }^{(g)}, \\ T_{\mu \nu }^{(f)}= & {} (\sigma + p) u_\mu u_\nu + p g_{\mu \nu }, \nonumber \end{aligned}$$
(17)
$$\begin{aligned} T_{\mu \nu }^{(g)}= & {} -\frac{m_g^2}{8\pi G}X_{\mu \nu } \nonumber \\= & {} \Big ( \rho _g + p_g^{(\perp )}\Big )u_\mu u_\nu + p_g^{(\perp )} g_{\mu \nu } + \Big ( p_g^{(r)} - p_g^{(\perp )}\Big )\chi _\mu \chi _\nu , \end{aligned}$$
(18)

where \(\sigma \) and p are, respectively, the energy density and the isotropic pressure of the fluid matter, while the components of the effective energy-momentum tensor of the dRGT massive gravity \(\big (T_{\mu \nu }^{(g)}\big )\) are already given in Eqs. (12–14). The \(u_\mu \) is a time-like unit vector, the \(\chi _\mu \) is the space-like unit vector orthogonal to the \(u_\mu \) and the angular plane and \(p_g^{(\perp )} = p_g^{(\theta )} = p_g^{(\phi )}\). The total energy-momentum tensor can be recast in the following form:

$$\begin{aligned} T_\nu ^\mu= & {} T_{\nu }^{(f),\,\mu } + T_{\nu }^{(g),\,\mu }\nonumber \\= & {} \left( \begin{array}{cccc} -\sigma -\rho _g&{} \quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad p + p_g^{(r)}&{}\quad 0&{}\quad 0\\ 0&{} \quad 0&{}\quad p + p_g^{(\perp )}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad p + p_g^{(\perp )} \end{array} \right) .\nonumber \\ \end{aligned}$$
(19)

Moreover, the energy-momentum tensor given in Eq. (17) satisfy the conservation law:

$$\begin{aligned} \nabla ^\mu T_{\mu \nu } = 0, \qquad \nabla ^\mu T_{\mu \nu }^{(f)} = 0, \qquad \nabla ^\mu T_{\mu \nu }^{(g)} = 0. \end{aligned}$$
(20)

3 The thin-shell wormholes in dRGT spacetimes

3.1 Junction conditions in dRGT theory

In this work, the wormhole helps joining two different dRGT spacetimes. It behaves like a surface between two bulks and is called thin shell [52, 53]. In order to join two different manifolds, we will construct a thin-shell wormhole of the dRGT massive gravity following the standard approach proposed in Refs. [54,55,56]. Consider two distinct spacetime manifolds, \({{\mathcal {M}}}_{+}\) and \({{\mathcal {M}}}_{-}\), with metrics given by \(g^{+}_{\mu \nu }(x^{\mu }_{+})\) and \(g^{-}_{\mu \nu }(x^{\mu }_{-})\), in terms of independently defined coordinate systems \(x^{\mu }_{+}\) and \(x^{\mu }_{-}\). The manifolds are bounded by hypersurfaces \(\Sigma _{+}\) and \(\Sigma _{-}\) with induced metrics \(h^{+}_{ab}\) and \(h^{-}_{ab}\), respectively. Regarding the Darmois–Israel formalism, the coordinates on \({{\mathcal {M}}}\) can be chosen as \(x^{\mu }=(t, r, \theta , \phi )\), while for the coordinates on the induced metric \(\Sigma \), we write \(y^{a} = (\tau , \theta , \phi )\), these being the intrinsic coordinates. Note that the hypersurfaces are isometric, i.e., \(h^{+}_{ab}(y^{a})=h^{-}_{ab}(y^{a})=h_{ab}(y^{a})\). A single manifold \({{\mathcal {M}}}\) is obtained by gluing together \({{\mathcal {M}}}_{+}\) and \({{\mathcal {M}}}_{-}\) at their boundaries, i.e., \({{\mathcal {M}}}={{\mathcal {M}}}_{+}\cup {{\mathcal {M}}}_{-}\), with the natural identification of the boundaries \(\Sigma _{+}=\Sigma _{-}=\Sigma \).

We describe two different manifolds as follows [55, 56]:

$$\begin{aligned} \mathcal {M}_{\pm } = \{ x^{\mu }_{\pm }\,\,|\, t_\pm \ge T_\pm (\tau ) \text { and } r \ge a(\tau ) \}, \end{aligned}$$
(21)

where the plus (minus) sign means the upper (the lower) spacetime manifold. Before considering the junction condition for matching the boundaries between two different manifolds, it is worth mentioning that at the end of Sect. 2, we have written the Einstein equations including matter terms for the whole theory, in the bulk spacetime. However, in the latter, the matter is assumed to exist only on the thin shell comprising the throat. The line elements of the manifolds are given by

$$\begin{aligned} \mathrm{d}s_\pm ^2 = g_{\mu \nu }^\pm \mathrm{d}x^{\mu }\mathrm{d}x^{\nu } = - f_\pm (r) \mathrm{d}t^2 + \frac{\mathrm{d}r^2}{f_\pm (r)} + r^2 \mathrm{d}\Omega ^2. \end{aligned}$$
(22)

Both different manifolds are linked by the (co-moving) thin shell and the hypersurface \(\Sigma \) is parametrized by [55,56,57,58]

$$\begin{aligned} \Sigma = \{ y^{a}\,\,|\, t_\pm = T_\pm (\tau ) \text { and } r = a(\tau ) \}. \end{aligned}$$
(23)

Thus the line element of the thin shell reads

$$\begin{aligned} \mathrm{d}s_{\Sigma }^2= & {} g_{\alpha \beta }\mathrm{d}x^{\alpha }\mathrm{d}x^{\beta } = g_{\alpha \beta } \bigg ( \frac{\partial x ^ {\alpha }}{\partial y^a} \mathrm{d}y^a \bigg ) \bigg ( \frac{\partial x ^ {\beta }}{\partial y^b} \mathrm{d}y^b \bigg ) \nonumber \\= & {} h_{ab}\mathrm{d}y^a \mathrm{d}y^b = -\mathrm{d}\tau ^2 + a^2(\tau ) \mathrm{d}\Omega ^2, \end{aligned}$$
(24)

where \(y^a = y^a(x^{\mu })\) is a coordinate on the hypersurface \(\Sigma \) and \(h_{ab}\) is called the induced metric or the first fundamental form on the hypersurface \(\Sigma \), [55,56,57]

$$\begin{aligned} h_{ab} \equiv g_{\alpha \beta } e^{\alpha }_{a} e^{\beta }_{b}. \end{aligned}$$
(25)

It is worth noting here that the throat must satisfy by the Israel junction conditions which provide the following coordinate choice:

$$\begin{aligned} -f_\pm (a)\dot{T}_\pm ^2 + \frac{\dot{a}^2}{f_\pm (a)} = 1,\quad \forall \tau , \end{aligned}$$
(26)

where dots denote derivative with respect to \(\tau \) (\(\mathrm{d}/\mathrm{d}\tau \)) and \(\tau \) simply is a local time on the thin shell. The induced metric is a tangent component of \(g_{\alpha \beta }\) on the hypersurface \(\Sigma \). Then the normal vector component \(n_{\alpha }\) of the metric tensor on the hypersurface is defined as follows [55,56,57]:

$$\begin{aligned} n_{\alpha } \equiv \frac{F(r, a(\tau ))_{, \alpha }}{|F(r, a(\tau ))_{, \beta } F(r, a(\tau ))^{, \beta }|^{1/2}}, \end{aligned}$$
(27)

where \(F(r, a(\tau )) \equiv r - a(\tau ) = 0\) is the hypersurface function and \(a(\tau )\) is the throat radius of the thin-shell wormhole. Note that the Greek indices (\(x^{\alpha }, x^{\beta }, \dots \)) are the coordinates on the manifold \(M_{\pm }\), while the Latin indices (\(x^{a}, x^{b}, \dots \)) are the coordinates on the hypersurface \(\Sigma \). The metric tensor on the hypersurface can be split into two parts [55,56,57] as

$$\begin{aligned} g_{ab} = h_{ab} + \epsilon n_a n_b, \end{aligned}$$
(28)

where \(\epsilon \) represents the types of thin shell with \(\epsilon = -1,0,+1\) being the space-like, null-like and time-like, respectively.

Next we will briefly derive the junction conditions on the hypersurface between the boundaries of two different manifolds (\(\partial \mathcal {M}\)). It is well known that in order to connect two manifolds one needs to add the action of the boundary terms or the Gibbons–Hawkings terms into the total action. Then the total gravitational action of the dRGT massive gravity with the matter fields and the boundary terms is given by [54, 59]

$$\begin{aligned} S_{\mathrm{total}}= & {} \int _{\mathcal {M}_+} \mathrm{d}^4 x \sqrt{-g^+}\nonumber \\&\left( \frac{1}{16\pi G} \big ( R^+ + m^2_g \mathcal {U}(g^+,\phi ^a) \big ) + \mathcal {L}_{\mathrm{matter}}^+\right) \nonumber \\&+ \frac{1}{8\pi G} \int _{\partial \mathcal {M}_+} \mathrm{d}^3 y \sqrt{-h^+} K^+ \nonumber \\&+ \int _{\mathcal {M}_-} \mathrm{d}^4 x \sqrt{-g^-}\nonumber \\&\left( \frac{1}{16\pi G} \big ( R^- + m^2_g \mathcal {U}(g^-,\phi ^a) \big ) + \mathcal {L}_{\mathrm{matter}}^-\right) \nonumber \\&+ \frac{1}{8\pi G} \int _{\partial \mathcal {M}_-} \mathrm{d}^3 y \sqrt{-h^-} K^- \nonumber \\&+ \int _{\Sigma } \mathrm{d}^3y \sqrt{-h} \,\mathcal {L}_{\mathrm{matter}}^{\Sigma }, \end{aligned}$$
(29)

where \(\sqrt{-h}\) is the volume element on the 3-dimensional hypersurface and K is the trace of the extrinsic curvature \(K_{ab}\) on the thin shell with \(K \equiv K^a_a = h_{ab} K^b_a\). Here \(\mathcal {L}_{\mathrm{matter}}^{\Sigma } = \mathcal {L}_{f}^{\Sigma } + \mathcal {L}_{g}^{\Sigma }\) contains two types of fluids (perfect fluid, \(\mathcal {L}_{f}^{\Sigma }\) and massive gravity fluid, \(\mathcal {L}_{f}^{\Sigma }\)) which are localized on the hypersurface. The extrinsic curvature can be calculated via the following equation:

$$\begin{aligned} K_{ab}= & {} -n_{\alpha } \bigg [ \frac{\mathrm{d}^2 x^{\alpha } }{\mathrm{d}y^a \mathrm{d}y^b} + \Gamma ^{\alpha }_{\beta \gamma } \frac{\mathrm{d}x^{\beta }}{\mathrm{d}y^{a}} \frac{\mathrm{d}x^{\gamma }}{\mathrm{d}y^{b}} \bigg ]. \end{aligned}$$
(30)

Let us first discuss the matter Lagrangian on the hypersurface, \(\mathcal {L}_{\mathrm{matter}}^{\Sigma }\). It has been demonstrated in Refs. [54, 59] for the scalar field matter case that the Lagrangian of the matter field on the hypersurface has the same form of that in the bulk, but the metric tensor \(g_{\mu \nu }\) in the bulk is replaced by the induced metric on the hypersurface, \(h_{ab}\).

Varying the total gravitational action in Eq. (29) with respect to the induced metric tensor, \(h^{ab}\) and using the standard technique of the hypersurface in GR, the junction condition is given by

$$\begin{aligned} \delta _{b}^a \Delta K - \Delta K_{b}^a = 8 \pi G S_b^a , \end{aligned}$$
(31)

where we have the notation \(\Delta A \equiv A_+ - A_-\). The new effective energy-momentum tensor \(S_{ab}\) on the thin shell is defined by

$$\begin{aligned} S_{b}^a\equiv & {} t_{b}^a + Y_{b}^a. \end{aligned}$$
(32)

In addition, the definition of the energy-momentum tensor of the fluid on the thin shell, \(t_{b}^a\) takes the form

$$\begin{aligned} t_{b}^a= & {} - \frac{1}{\sqrt{-h}} \frac{\delta }{\delta h_a^{b}} \Big ( \sqrt{-h} \,\mathcal {L}_{f}^{\Sigma } \Big ) = (\sigma + p)u^a u_b + p h_b^a. \end{aligned}$$
(33)

For the massive gravity fluid, the \(Y_{b}^a\) tensor is given by

$$\begin{aligned} Y^a_b= & {} - \frac{1}{\sqrt{-h}} \frac{\delta }{\delta h_a^{b}} \Big ( \sqrt{-h} \,\mathcal {L}_{g}^{\Sigma } \Big ) \nonumber \\= & {} (\rho _g + p_g^{(\perp )})u^a u_b + p_g^{(\perp )} h_b^a. \end{aligned}$$
(34)

Furthermore, it is very convenient to represent the \(S_b^a\) tensor in the matrix form,

$$\begin{aligned} S_{b}^a\equiv & {} \left( \begin{array}{ccc} -\rho _{\mathrm{eff.}}&{}\quad 0&{}\quad 0\\ 0&{}\quad P_{\mathrm{eff.}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad P_{\mathrm{eff.}} \end{array} \right) \nonumber \\= & {} \left( \begin{array}{ccc} -\sigma -\rho _g&{}\quad 0&{}\quad 0\\ 0&{}\quad p + p_g^{(\perp )} &{}\quad 0\\ 0&{}\quad 0&{}\quad p + p_g^{(\perp )} \end{array} \right) , \end{aligned}$$
(35)

where the explicit forms of the \(\rho _g\) and \(p_g^{(\perp )}= p_g^{(\theta ,\phi )}\) are given in Eqs. (12) and (14). We will see in the latter case that the equation of motion of the dRGT massive gravity wormholes takes a very simple form, like the standard GR case with two types of fluids.

3.2 The thin-shell wormhole dynamics in dRGT spacetimes

We next consider the relation between the thin-shell wormhole on the hypersurface \(\Sigma \) and the spacetime on manifold \(\mathcal {M}_{\pm }\). Considering the parameterization of the coordinate on the hypersurface in Eq. (23), we find

$$\begin{aligned} -\mathrm{d}\tau ^2 = -f_\pm (a) dT_\pm ^2 + \frac{\mathrm{d}a^2}{f_\pm (a)} \end{aligned}$$
(36)

and

$$\begin{aligned} a^2 \mathrm{d}\Omega ^2 = r^2 \mathrm{d}\Omega ^2. \end{aligned}$$
(37)

The relation between T and \(\tau \) is given by

$$\begin{aligned} {\dot{T}}_\pm = \frac{1}{f_\pm (a)}\sqrt{f_\pm (a) + {\dot{a}}^2} \end{aligned}$$
(38)

and

$$\begin{aligned} {\ddot{T}}_\pm = -\frac{{\dot{f}}_\pm }{f_\pm ^2}\sqrt{f_\pm +{\dot{a}}^2} + \frac{2 {\dot{a}} \ddot{a} + {\dot{f}}_\pm }{2 f_\pm \sqrt{f_\pm + {\dot{a}}^2 }}, \end{aligned}$$
(39)

where \({\dot{f}} = \frac{\mathrm{d}f}{\mathrm{d}\tau } = \frac{\mathrm{d}f(a)}{\mathrm{d}a} \frac{\mathrm{d}a}{\mathrm{d}\tau } = f'(a) \,{\dot{a}}\) and a prime denotes the first derivative with respect to a. Now we are ready to compute the non-vanishing components of the extrinsic curvature of the wormhole in dRGT massive gravity. Having used the line element in Eq. (24), one obtains the non-vanishing components of \(K_b^a\) as

$$\begin{aligned} K^{\tau \pm }_{\tau }= & {} \pm \frac{1}{\sqrt{f_{\pm } + {\dot{a}}^2 }} \bigg ( \ddot{a} + \frac{f'_{\pm }}{2} \bigg ), \end{aligned}$$
(40)
$$\begin{aligned} K^{\theta \pm }_{\theta }= & {} K^{\phi \pm }_{\phi } = \pm \frac{1}{a} \big (\sqrt{ f_{\pm } + {\dot{a}}^2} \big ). \end{aligned}$$
(41)

We note that the extrinsic curvature is a diagonal matrix. It is worth mentioning the parameters in the dRGT massive gravity. By using the observational constraints on the dRGT theory, we find

$$\begin{aligned} \zeta = 0,\quad \Rightarrow \quad \alpha = -3\beta = -\frac{3}{2} - \frac{\Lambda }{2 m_g^2},\quad k = \frac{2\gamma }{m_g^2+\Lambda }.\nonumber \\ \end{aligned}$$
(42)

Using Eqs. (12) and (14) with \(r=a\) at the boundaries \(\partial \mathcal {M}_\pm \), the components of the effective energy-momentum tensor, \(S_b^a\), in Eq. (35) are given by the following explicit expressions:

$$\begin{aligned} S_\tau ^\tau= & {} -\rho _{\mathrm{eff.}} = -\sigma - \rho _g(a)\nonumber \\= & {} -\sigma + \frac{1}{8\pi G}\left( \frac{2\gamma }{a} - \Lambda \right) ,\nonumber \\ S_\theta ^\theta= & {} S_\phi ^\phi = P_{\mathrm{eff.}} = p + p_g^{(\perp )}(a)\nonumber \\= & {} p + \frac{1}{8\pi G}\left( \frac{\gamma }{a} - \Lambda \right) . \end{aligned}$$
(43)

Setting \(\zeta = 0\), we find that there is only one free parameter of the dRGT massive gravity since the graviton mass, \(m_g\) and the cosmological constant, \(\Lambda \) can be fixed by using the observed values of those two quantities.

In this work, we assume the \(Z_2\) symmetry between two bulk spacetime manifolds. This means that \(f_+(a) = f_-(a) = f(a)\), \(\Lambda _+ = \Lambda _- = \Lambda \), and \(\gamma _+ = \gamma _- = \gamma \). The \((\tau \tau )\) component of the junction condition of the thin-shell wormhole in Eq. (31) reads

$$\begin{aligned} \frac{2}{a} \big (\sqrt{ f + {\dot{a}}^2} \big ) = -8\pi G \sigma + \left( \frac{2\gamma }{a} - \Lambda \right) . \end{aligned}$$
(44)

On the other hand, the angular component of Eq. (31) is given by

$$\begin{aligned} \frac{1}{\sqrt{f + {\dot{a}}^2 }} \big ( 2\ddot{a} + f' \big ) = 8\pi G p + \left( \frac{\gamma }{a} - \Lambda \right) . \end{aligned}$$
(45)

In addition, the continuity of the perfect fluid matter gives a relation between the energy density and pressure on the thin shell as

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} \tau }\big ( a \sigma \big ) + p \frac{\mathrm{d}a}{\mathrm{d} \tau } = 0. \end{aligned}$$
(46)

It is also written in terms of the first order derivative of \(\sigma \) with respect to a as

$$\begin{aligned} \frac{\mathrm{d} \sigma }{\mathrm{d} a} = -\bigg ( \frac{\sigma + p}{a} \bigg ). \end{aligned}$$
(47)

The second order derivative of \(\sigma \) with respect to a yields

$$\begin{aligned} \frac{\mathrm{d}^2 \sigma }{\mathrm{d}a^2} = \frac{\sigma + p}{a^2} \bigg ( 2 + \frac{\mathrm{d}p}{\mathrm{d} \rho } \bigg ), \end{aligned}$$
(48)

where \(p = p(\sigma )\). The above equations are useful for analyzing the stability of the wormhole with several types of the perfect fluid matters. We study their effects on particular models in the next section.

4 Stability analysis of the dRGT thin-shell wormhole

The stability of the wormhole can be quantified via the study of the effective potential of the wormhole dynamics. The equation of motion for determining the stability of the throat \(a(\tau )\) is directly derived from Eq. (44) to obtain

$$\begin{aligned} \frac{1}{2} {\dot{a}}^2 + V(a) = 0, \end{aligned}$$
(49)

where the effective potential V(a) is written by

$$\begin{aligned} V(a) = \frac{1}{2}f(a) - \frac{a^2}{8} \left[ 8\pi G \sigma - \left( \frac{2\gamma }{a} - \Lambda \right) \right] ^2. \end{aligned}$$
(50)

This single dynamical equation (49) completely determines the motion of the wormhole throat. Notice that if we consider only the massive gravity correction term, \(m^{2}_{g}{{\mathcal {U}}}\), without invoking exotic fluids, i.e., \(\sigma =0=p\), the potential (50) becomes

$$\begin{aligned} V(a) = \frac{1}{2}f(a) - \frac{a^2}{8} \left[ \frac{2\gamma }{a} - \Lambda \right] ^2. \end{aligned}$$
(51)

In this situation, we find that \(V''(a_0)<0\) meaning that it is not possible to obtain stable wormholes if \(\Lambda >0\). We assume that the throat of the thin-shell wormhole is static at \(a = a_0\) and satisfies the relation

$$\begin{aligned} f(a_0) > 0, \end{aligned}$$
(52)

in order to avoid the event horizon \(r_{\text {EH}}\) from the wormhole, \(f(r_{\text {EH}}) = 0\). In order to analyze the stability of the throat, we consider a small perturbation to the potential and are able to determine whether the throat is stable or not. Here the usual Taylor series expansion is applied to the potential V(a) around the static radius \(a_0\) as follows:

$$\begin{aligned} V(a)= & {} V(a_0) + V'(a_0) (a - a_0)+ \frac{1}{2} V''(a_0)(a - a_0)^2 \nonumber \\&+ \mathcal {O}((a - a_0)^3). \end{aligned}$$
(53)

When evaluating at the static solution \(a=a_{0}\), we obtain the expected result \(V(a_0) = 0\) and \( V'(a_0) = 0\) if \(a_0\) is the static radius. Then Eq. (53) reduces to

$$\begin{aligned} V(a) = \frac{1}{2} V''(a_0)(a - a_0)^2 + \mathcal {O}((a - a_0)^3). \end{aligned}$$
(54)

Therefore, the equation of motion for the wormhole throat approximately takes the form

$$\begin{aligned} {\dot{a}}^2 + \frac{1}{2} V''(a_0)(a - a_0)^2 = 0. \end{aligned}$$
(55)

Using Eq. (50) with the help of Eq. (47) and Eq. (48), we find

$$\begin{aligned} V''(a_0)= & {} \frac{1}{2} f''(a_0) + \frac{\mathrm{d} p}{\mathrm{d} \sigma } \bigg ( -2 G (p + \sigma ) \pi \Lambda \nonumber \\&- 16 G^2 \pi ^2 \sigma (p + \sigma ) + \frac{4 G \pi \gamma (p + \sigma )}{a} \bigg ) \nonumber \\&- 16 G^2 p^2 \pi ^2 + 4 G p \pi \Lambda - \frac{1}{4} \Lambda ^2. \end{aligned}$$
(56)

Thus, the wormhole is stable if and only if \(V''(a_0)>0\) where the motion of the throat is oscillatory with angular frequency \(\omega =\sqrt{V''(a_{0})/2}\). Note that \(V(a_{0})\) has a local minimum at \(a_{0}\). To carry out the analysis, we can quantify the conditions to obtain stable wormholes. In our case, we find that these parameters need to satisfy

$$\begin{aligned} 0< & {} \frac{1}{2} f''(a_0) + \frac{\mathrm{d} p}{\mathrm{d} \sigma } \bigg ( -2 G (p + \sigma ) \pi \Lambda \nonumber \\&- 16 G^2 \pi ^2 \sigma (p + \sigma ) + \frac{4 G \pi \gamma (p + \sigma )}{a_0} \bigg ) \nonumber \\&- 16 G^2 p^2 \pi ^2 + 4 G p \pi \Lambda - \frac{1}{4} \Lambda ^2. \end{aligned}$$
(57)

Next we assume four fluid models for studying the stability of the dRGT wormhole: (1) the linear model, (2) the Chaplygin gas model, (3) the generalized Chaplygin gas model and (4) the logarithm model. It has been shown in the previous section that we have only one free parameter (\(\gamma \)) in the dRGT theory by using \(\zeta = 0\). The parameters in this work in natural units are given by

$$\begin{aligned} G= & {} 6.72 \times 10^{-57} ~\mathrm{eV}^{-2},\quad m_g = 1.22 \times 10^{-22} ~\mathrm{eV},\nonumber \\ \Lambda= & {} 4.33 \times 10^{-66}~ \mathrm{eV}^2, \end{aligned}$$
(58)

where the gravitational constant G, the cosmological constant, \(\Lambda \) are taken from review of particle physics [60] and the graviton mass is the upper bound values from the LIGO-VIRGO gravitational wave observations [2]. Moreover, the free parameter, \(\gamma \) of the dRGT massive gravity has been fixed by fitting rotational curves of the galaxies in several data [51] and we will use this value for the following study. The \(\gamma \) parameter and the thin-shell wormhole mass read

$$\begin{aligned} \gamma= & {} 6.05 \times 10^{-34} ~\mathrm{eV},\quad M = 3.36 \times 10^{66} ~\mathrm{eV}, \end{aligned}$$
(59)

where we have assumed that M is roughly equal to the lower bound mass of black hole which is three times of the solar mass. Using the above numerical values, we can estimate the event horizon and the cosmological horizon via Eq. (10) by setting \(f(r)=0\).

However, it is much more convenient to use the dimensionless values of the physical parameters given below:

$$\begin{aligned} \Lambda = 0.0001,\quad \gamma = 0.001,\quad M = 1. \end{aligned}$$
(60)

Note that one can easily change the units from this dimensionless parameters to the others, e.g. SI units or natural units, and vice versa. Now we consider the energy density given in Eq. (44) and write for the static case at the throat

$$\begin{aligned} \sigma = \frac{1}{8\pi G} \left( \frac{2\gamma }{a} - \Lambda \right) - \frac{2}{8\pi G a} \sqrt{f}. \end{aligned}$$
(61)

Here we can solve the above equation to write \(\sigma \) in terms of a when substituting a function f(a). It was found that the stability of transparent spherically symmetric thin shells to linearized spherically symmetric perturbations about static equilibrium has been examined Ref. [61].

4.1 Linear model

We begin our stability analyses by considering the pressure which is proportional the energy density [56]:

$$\begin{aligned} p(\sigma ) = \epsilon _0 \sigma . \end{aligned}$$
(62)

It is easy to show that

$$\begin{aligned} \frac{\mathrm{d} p}{\mathrm{d} \sigma } = \epsilon _0. \end{aligned}$$
(63)

Notice that the change in the pressure on the energy density is a constant. Moreover, the throat of the wormhole basically locates between the event horizon and the cosmological horizon. After substituting the above results into the stability condition (57), we find

$$\begin{aligned} 0< & {} \frac{1}{2} f''(a_0) - \frac{1}{4} \bigg ( \Lambda ^2 + 8 \pi ( \epsilon _0 - 1) \epsilon _0 \Lambda \sigma + 64 \pi ^2 \epsilon _0 (1 + 2 \epsilon _0) \sigma ^2 \bigg )\nonumber \\&+ \frac{4 \pi \gamma \epsilon _0 (\epsilon _0 + 1)}{a} . \end{aligned}$$
(64)

In order to visualize the stability region of the model, we plot the stability contour in terms of \(\epsilon _{0}\) and \(a_{0}\). Our result is illustrated in Fig. 2 for the linear model. We notice that in order to satisfy the stability condition (57) the constant \(\epsilon _{0}\) has negative values in the throat radius \(a_{0}\).

Fig. 2
figure 2

The stable region of the linear model \(p(\sigma ) = \epsilon _0 \sigma \). The contour shows that the constant \(\epsilon _{0}\) has negative values in the throat

Full size image

4.2 Chaplygin gas model

We next consider the Chaplygin gas model for the exotic matter. The pressure is already given in Ref. [56]:

$$\begin{aligned} p(\sigma ) = -\epsilon _0 \bigg ( \frac{1}{\sigma } - \frac{1}{\sigma _0} \bigg ). \end{aligned}$$
(65)

It is trivial to show that

$$\begin{aligned} \frac{\mathrm{d} p}{\mathrm{d} \sigma } = \frac{\epsilon _0}{\sigma _0^2}, \end{aligned}$$
(66)

where \(\sigma _0\) is a constant. After substituting the above results into the stability condition (57), we find in this case

$$\begin{aligned} 0< & {} \frac{1}{2} f''(a_0) + \frac{1}{4 a \sigma ^3 \sigma _0^2} \bigg ( 16 \pi \gamma \epsilon _0 \sigma _0 (\epsilon _0(\sigma - \sigma _0) \nonumber \\&+\, \sigma ^2 \sigma _0) - a \big ( \Lambda ^2 \sigma ^3 \sigma _0^2 \nonumber \\&+\, 64 \pi ^2 \epsilon _0 \sigma ^2 (\epsilon _0 (\sigma - \sigma _0) + \sigma \sigma _0^2 )\nonumber \\&+\, 8 \pi \epsilon _0 \Lambda \sigma _0 ( \epsilon _0 (\sigma - \sigma _0) + \sigma ^2 (3 \sigma _0 - 2 \sigma ) ) \big ) \bigg ). \end{aligned}$$
(67)

Here we plot the stability contour in terms of \(\epsilon _{0}\) and \(a_{0}\) for this model. The result is illustrated in Fig. 3. The stable region for this case is represented in Fig. 3. We notice that in order to satisfy the stability condition (57) \(\epsilon _{0}\) can have both negative values and positive ones in the throat with radius \(a_{0}\).

Fig. 3
figure 3

The stable region of the Chaplygin gas model \(p(\sigma ) = -\epsilon _0 ( \frac{1}{\sigma } - \frac{1}{\sigma _0})\). The result shows that \(\epsilon _{0}\) can have both negative values and positive ones in the throat with radius \(a_{0}\)

Full size image

4.3 Generalized Chaplygin gas model

In addition, the Chaplygin gas model given in the previous subsection can be generalized where the relation between \(p(\sigma )\) and \(\sigma \) takes the form [56]

$$\begin{aligned} p(\sigma ) = - \bigg ( \frac{\sigma _0}{\sigma } \bigg )^{\epsilon _0} \end{aligned}$$
(68)

and

$$\begin{aligned} \frac{\mathrm{d} p}{\mathrm{d} \sigma } = \epsilon _0 \frac{\sigma _0^{\epsilon _0}}{\sigma ^{\epsilon _0 + 1}}. \end{aligned}$$
(69)

After substituting the above results into the stability condition (57), we find in this case

$$\begin{aligned} 0< & {} \frac{1}{2} f''(a_0) + \frac{1}{4a} \sigma ^{-1-\epsilon _0} \bigg ( -64 a \pi ^2 \bigg ( \frac{\sigma _0^{2 \epsilon _0}}{\sigma ^{\epsilon _0 - 1}} \bigg )\nonumber \\&+ 8 \pi \bigg ( \frac{\sigma _0}{\sigma } \bigg )^{\epsilon _0} (-2 a \Lambda \sigma ^{1 + \epsilon _0} + \epsilon _0 ( (-2 \gamma + a \Lambda ) \nonumber \\&+ 8 a \pi \sigma ) \sigma _0^{\epsilon _0}) + \sigma \bigg (- 8 \pi \epsilon _0 ((-2 \gamma + a \Lambda ) + 8 a \pi \sigma ) \sigma _0^{\epsilon _0}\bigg ) \bigg ) \nonumber \\&- \frac{1}{4} \Lambda ^2. \end{aligned}$$
(70)

Here we display the stability contour in terms of \(\epsilon _{0}\) and \(a_{0}\) illustrated in Fig. 4. The stable region for this case is represented in Fig. 4 for the generalized Chaplygin gas model. We find that in order to satisfy the stability condition (57) \(\epsilon _{0}\) has positive values in the throat with radius \(a_{0}\).

Fig. 4
figure 4

The stable region of the generalized Chaplygin gas model \(p(\sigma ) = ( \frac{\sigma _0}{\sigma })^{\epsilon _0}\). The result shows that \(\epsilon _{0}\) has positive values in the throat with \(a_{0}\)

Full size image

4.4 Logarithm model

We provide the last example in which the pressure \(p(\sigma )\) and the energy density \(\sigma \) are related via [56]

$$\begin{aligned} p(\sigma ) = \epsilon _0 \text {log}\bigg ( \frac{\sigma }{\sigma _0} \bigg ) \end{aligned}$$
(71)

and

$$\begin{aligned} \frac{\mathrm{d} p}{\mathrm{d} \sigma } = \frac{\epsilon _0}{\sigma }. \end{aligned}$$
(72)

After substituting the above results into the stability condition (57), we find in this particular case

$$\begin{aligned} 0< & {} \frac{1}{2} f''(a_0) + \frac{4 \pi \gamma \epsilon _0 \bigg (\sigma + \epsilon _0 \text {log}\big ( \frac{\sigma }{\sigma _0} \big ) \bigg ) }{\sigma a} \nonumber \\&- \frac{ \Lambda ^2 + 8 \pi \Lambda \big ( \epsilon _0 + \epsilon _0 (\frac{\epsilon _0}{\sigma } - 2) \text {log} \big ( \frac{\sigma }{\sigma _0} \big ) \big ) }{4 a}\nonumber \\&- \frac{64 \pi ^2 \epsilon _0 \bigg ( \sigma + \epsilon _0 \text {log} \big ( \frac{\sigma }{\sigma _0} \big )\big ( 1 + \text {log} \big ( \frac{\sigma }{\sigma _0} \big ) \big ) \bigg ) }{4 a}. \end{aligned}$$
(73)

Here we display the stability contour in terms of \(\epsilon _{0}\) and \(a_{0}\) illustrated in Fig. 5. The stable region for this case is represented in Fig. 5 for the logarithm model. We observe that in order to satisfy the stability condition (57) \(\epsilon _{0}\) can have both negative values and positive ones in the throat with radius \(a_{o}\).

Fig. 5
figure 5

The stable region of the linear model \(p(\sigma ) = \epsilon _0 \text {log}( \frac{\sigma }{\sigma _0})\). The result shows that \(\epsilon _{0}\) can have both negative values and positive ones in the throat with radius \(a_{0}\)

Full size image

5 Energy conditions

In this section, we shall analyze the energy conditions for the thin-shell wormholes in the dRGT massive gravity. We check the null, weak, and strong conditions at the wormhole throat for all existing models present in the previous section.

  1. I.

    Null energy condition is expressed in terms of energy density and pressure as follows:

    $$\begin{aligned} \rho _{\mathrm{eff.}}+P_{\mathrm{eff.}} \ge 0, \end{aligned}$$
    (74)

    which yields

    $$\begin{aligned} \rho _{\mathrm{eff.}}+P_{\mathrm{eff.}}= & {} \sigma - \frac{1}{8\pi G}\left( \frac{2\gamma }{a} - \Lambda \right) \nonumber \\&+ p + \frac{1}{8\pi G}\left( \frac{\gamma }{a} - \Lambda \right) \nonumber \\= & {} \sigma +p - \frac{1}{8\pi G}\frac{\gamma }{a}\ge 0. \end{aligned}$$
    (75)
  2. II.

    Weak energy condition is given by

    $$\begin{aligned} \rho _{\mathrm{eff.}}\ge 0,\,\,\,\,\rho _{\mathrm{eff.}}+P_{\mathrm{eff.}} \ge 0, \end{aligned}$$
    (76)

    which gives the following result for the thin-shell wormholes in the dRGT massive gravity:

    $$\begin{aligned} \rho _{\mathrm{eff.}}=\sigma - \frac{1}{8\pi G}\left( \frac{2\gamma }{a} - \Lambda \right) \ge 0, \end{aligned}$$
    (77)
  3. III.

    Strong energy condition is governed by

    $$\begin{aligned} \rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}} \ge 0,\,\,\,\,\rho _{\mathrm{eff.}}+P_{\mathrm{eff.}} \ge 0, \end{aligned}$$
    (78)

    which gives the following result for the thin-shell wormholes in the dRGT massive gravity:

    $$\begin{aligned} \rho _{\mathrm{eff.}}+3P_{\mathrm{eff.}}= & {} \sigma - \frac{1}{8\pi G}\left( \frac{2\gamma }{a} - \Lambda \right) \nonumber \\&+ 3p + \frac{3}{8\pi G}\left( \frac{\gamma }{a} - \Lambda \right) \nonumber \\= & {} \sigma +3p + \frac{1}{8\pi G}\left( \frac{\gamma }{a} - 2\Lambda \right) \ge 0. \end{aligned}$$
    (79)

5.1 Linear model

When substituting the pressure and the energy density of this model into Eqs. (75), (77) and (79), we find

$$\begin{aligned} \rho _{\mathrm{eff.}} + P_{\mathrm{eff.}}= & {} \frac{( (1 + 2 \epsilon _0)\gamma - a (1 + \epsilon _0) \Lambda ) - 2 (1 + \epsilon _0) \sqrt{f(a)} }{8 \pi G a} \nonumber \\\ge & {} 0, \end{aligned}$$
(80)
$$\begin{aligned} \rho _{\mathrm{eff.}}= & {} -\frac{\sqrt{f(a)}}{4 \pi G a} \ge 0, \end{aligned}$$
(81)
$$\begin{aligned} \rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}= & {} \frac{3 ( (1 + 2 \epsilon _0)\gamma - a (1 + \epsilon _0) \Lambda ) - 2 (1 + \epsilon _0) \sqrt{f(a)} }{8 \pi G a} \nonumber \\\ge & {} 0. \end{aligned}$$
(82)
Fig. 6
figure 6

The variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a of the linear model with \(p(\sigma ) = \epsilon _0 \sigma \)

Full size image

In order to analyze the energy conditions, we will choose the values of \(\epsilon _{0}\) in the stable regions shown in Fig. 2 and then verify the energy conditions. Figure 6 shows the variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a in the linear model \(p(\sigma ) = \epsilon _0 \sigma \). We observe that all energy conditions are violated in this model.

5.2 Chaplygin gas model

We substitute the pressure and the energy density of this model into Eqs. (75), (77) and (79) and then we obtain

$$\begin{aligned}&\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}} = \frac{1}{8}\bigg ( \frac{(\gamma - a \Lambda )}{\pi G a} \nonumber \\&\qquad - \frac{8 \epsilon _0 ( (-2\gamma + a \Lambda ) + 8 a G \pi \sigma _0 + 2 \sqrt{f(a)} )}{\sigma _0 ((2 \gamma - a \Lambda ) - 2 \sqrt{f(a)} )} \nonumber \\&\qquad - \frac{2 \sqrt{f(a)}}{a G \pi } \bigg ) \ge 0, \end{aligned}$$
(83)
$$\begin{aligned}&\rho _{\mathrm{eff.}} = -\frac{\sqrt{f(a)}}{4 \pi G a} \ge 0, \end{aligned}$$
(84)
$$\begin{aligned}&\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}} = \frac{1}{8}\bigg ( \frac{3 (\gamma - a \Lambda )}{\pi G a} \nonumber \\&\qquad - \frac{24 \epsilon _0 ( (-2\gamma + a \Lambda ) + 8 a G \pi \sigma _0 + 2 \sqrt{f(a)} )}{\sigma _0 ((2 \gamma - a \Lambda ) - 2 \sqrt{f(a)} )} \nonumber \\&\qquad - \frac{2 \sqrt{f(a)}}{a G \pi } \bigg ) \ge 0. \end{aligned}$$
(85)
Fig. 7
figure 7

The variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a of the Chaplygin gas model with \(p(\sigma ) = -\epsilon _0 ( \frac{1}{\sigma } - \frac{1}{\sigma _0})\)

Full size image

In order to quantify the energy conditions, we will choose the values of \(\epsilon _{0}\) in the stable regions shown in Fig. 3 and then examine the energy conditions. Figure 7 shows the variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a in the linear model \(p(\sigma ) = -\epsilon _0 ( \frac{1}{\sigma } - \frac{1}{\sigma _0})\). We observe that all energy conditions are violated for \(a<100\) in this model.

5.3 Generalized Chaplygin gas model

When substituting the pressure and the energy density of this model into Eqs. (75), (77) and (79), we find

$$\begin{aligned}&\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}} \nonumber \\&\quad = \frac{1}{8 \pi G a} \bigg ( (\gamma - a \Lambda ) - 2 \sqrt{f(a)} \nonumber \\&\qquad - G a (8 \pi )^{1 + \epsilon _0} \bigg ( - \frac{G a \sigma _0}{(-2\gamma + a \Lambda ) + 2 \sqrt{f(a)}} \bigg )^{\epsilon _0} \bigg ) \nonumber \\&\quad \ge 0, \end{aligned}$$
(86)
$$\begin{aligned}&\rho _{\mathrm{eff.}} = -\frac{\sqrt{f(a)}}{4 \pi G a} \ge 0, \end{aligned}$$
(87)
$$\begin{aligned}&\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}} \nonumber \\&\quad = \frac{1}{8 \pi G a} \bigg ( 3 (\gamma - 3 a \Lambda ) - 2 \sqrt{f(a)} \nonumber \\&\qquad - 3 G a (8 \pi )^{1 + \epsilon _0} \bigg ( - \frac{G a \sigma _0}{(-2\gamma + a \Lambda ) + 2 \sqrt{f(a)}} \bigg )^{\epsilon _0} \bigg ) \nonumber \\&\quad \ge 0. \end{aligned}$$
(88)
Fig. 8
figure 8

The variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a of the generalized Chaplygin gas model with \(p(\sigma ) = ( \frac{\sigma _0}{\sigma })^{\epsilon _0}\)

Full size image

We here quantify the energy conditions by choosing the values of \(\epsilon _{0}\) in the stable regions shown in Fig. 4 and then examine the energy conditions. Figure 8 shows the variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a in the generalized Chaplygin gas model \(p(\sigma ) = (\frac{\sigma _0}{\sigma })^{\epsilon _0}\). We observe that all energy conditions are violated for positive values of \(\epsilon _{0}\).

5.4 Logarithm model

We substitute the pressure and the energy density of this model into Eqs. (75), (77) and (79) and then we obtain

$$\begin{aligned} \rho _{\mathrm{eff.}} + P_{\mathrm{eff.}}= & {} \frac{(\gamma - a \Lambda ) - 2 \sqrt{f(a)}}{8 \pi G a}\nonumber \\&+ \epsilon _0 \text {log} \bigg ( \frac{(2 \gamma - a \Lambda ) - 2 \sqrt{f(a)}}{8 \pi G a \sigma _0} \bigg ) \nonumber \\\ge & {} 0, \end{aligned}$$
(89)
$$\begin{aligned} \rho _{\mathrm{eff.}}= & {} -\frac{\sqrt{f(a)}}{4 \pi G a} \ge 0, \end{aligned}$$
(90)
$$\begin{aligned} \rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}= & {} \frac{3(\gamma - a \Lambda ) - 2 \sqrt{f(a)}}{8 \pi G a}\nonumber \\&+ 3 \epsilon _0 \text {log} \bigg (\frac{(2 \gamma - a \Lambda ) - 2 \sqrt{f(a)}}{8 \pi G a \sigma _0} \bigg ) \nonumber \\\ge & {} 0. \end{aligned}$$
(91)
Fig. 9
figure 9

The variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a of the log model with \(p(\sigma ) = \epsilon _0 \text {log}( \frac{\sigma }{\sigma _0})\)

Full size image

We here quantify the energy conditions by choosing the values of \(\epsilon _{0}\) in the stable regions shown in Fig. 5 and then examine the energy conditions. Figure 9 shows the variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a in the logarithm model \(p(\sigma ) = \epsilon _0 \text {log}( \frac{\sigma }{\sigma _0})\). We observe that all energy conditions are violated for positive values of \(\epsilon _{0}\) with \(a>100\).

6 Embedding diagram

In this section, we construct the wormhole geometry via the embedding diagrams to represent a thin-shell wormhole in the dRGT massive gravity and extract some useful information by considering an equatorial slice, \(\theta =\pi /2\) and a fixed moment of time, \(t = const.\,\). Therefore the metric reduces to

$$\begin{aligned} \mathrm{d}s^2 = \frac{\mathrm{d}r^2}{f(r)} + r^2 \mathrm{d}\phi ^2, \end{aligned}$$
(92)

where

$$\begin{aligned} f(r) = 1 - \frac{2M}{r} - \frac{\Lambda r^2}{3} + \gamma r + \zeta . \end{aligned}$$
(93)

Next we embed the metric from Eq. (93) into 3-dimensional Euclidean space to visualize this slice and hence the spacetime can be written in cylindrical coordinates as

$$\begin{aligned} \mathrm{d}s^2= & {} \mathrm{d}z^{2}+\mathrm{d}r^{2}+ r^2 \mathrm{d}\phi ^2\nonumber \\= & {} \Big (1+\Big (\frac{\mathrm{d}z}{\mathrm{d}r}\Big )^{2}\Big )\mathrm{d}r^{2}+r^2 \mathrm{d}\phi ^2. \end{aligned}$$
(94)

Comparing Eq. (92) with Eq. (94) generates the expression for the embedding surface, which is given by

$$\begin{aligned} \frac{\mathrm{d}z}{\mathrm{d}r}= \pm \sqrt{\frac{1-f(r)}{f(r)}}. \end{aligned}$$
(95)

where f(r) is given in Eq. (93). However, the integration of the above expression cannot be solved analytically. Performing a numerical technique allows us to illustrate the wormhole shape as in Fig. 10.

7 Discussions and conclusions

In this work, we have studied the thin-shell wormholes in dRGT massive gravity. In order to construct the thin-shell wormhole, two bulks of the spacetime geometry are glued together via the cut-and-paste procedure [52]. Moreover, the junction conditions of dRGT spacetime are also derived in this work. The massive graviton correction term of the dRGT theory, \(m_g^2\mathcal {U}(g,\phi ^2)\), in the Einstein equation is represented in terms of effective anisotropic pressure fluid. However, if there is only this correction term, without invoking exotic fluids, we have found that the thin-shell wormholes cannot be stabilized. We have also quantified the dynamics of the spherical thin-shell wormholes in the dRGT theory. We then examined the stability conditions of the wormholes by introducing four existing models of the exotic fluids at the throat. In addition, we analyzed the energy conditions for the thin-shell wormholes in the dRGT massive gravity by checking the null, weak, and strong conditions at the wormhole throat.

We have quantified the energy conditions of the four models by choosing the values of \(\epsilon _{0}\) in the stable regions shown in Sect. 4. We have considered the variation of \(\rho _{\mathrm{eff.}} + P_{\mathrm{eff.}},\,\rho _{\mathrm{eff.}}\) and \(\rho _{\mathrm{eff.}} + 3P_{\mathrm{eff.}}\) as a function of a in all models: (1) the linear model \(p(\sigma ) = (\frac{\sigma _0}{\sigma })^{\epsilon _0}\), (2) the Chaplygin gas model \(p(\sigma ) = -\epsilon _0 ( \frac{1}{\sigma } -\frac{1}{\sigma _0})\), (3) the generalized Chaplygin gas model \(p(\sigma ) = ( \frac{\sigma _0}{\sigma })^{\epsilon _0}\) and (4) the logarithm model \(p(\sigma ) = \epsilon _0 \text {log}( \frac{\sigma }{\sigma _0})\). Choosing the values of \(\epsilon _{0}\) in the stable regions, we have observed that in general the classical energy conditions are violated by introducing all existing models of the exotic fluids. Moreover, we have quantified the wormhole geometry by using the embedding diagram to represent a thin-shell wormhole in the dRGT massive gravity.

Fig. 10
figure 10

The wormhole shape obtained via the embedding diagrams in 3-dimensional asymptotic flat spacetime. We have used \(M=1, \Lambda =0.0001, \gamma =0.001\) and set \(\zeta =0\)

Full size image

However, there are some limitations in the present work—for example, the construction of the shadow cast by the thin-shell wormhole in the dRGT massive gravity is worth investigating. Regarding this, we can evaluate the test particle geodesics and determine the trajectories of photons around the wormhole. This can be straightforwardly done by following the work of Refs. [16, 21, 62]. Additionally, we can elaborate our work by studying the gravitational lensing effect in the spacetime of the wormhole metric (10). This allows us to determine the deflection angle of the photon due to the presence of the wormhole in the dRGT massive gravity. It is worth mentioning that gravitational lensing and particle motions around non-asymptotically flat black hole spacetime in dRGT massive gravity have been addressed in Ref. [63].