1 Introduction

Wormholes have been known to be solutions of Einstein’s equations from the early of General Relativity (GR) in the works of Flamm in [1] and Einstein and Rosenin [2] that is also called the Einstein–Rosen Bridge (ERB). It became popular after the seminal works of Morris and Thorne in [3, 4]. The newly popularized wormholes were also the direct solutions of Einstein’s field equations, however, unlike the former one they were traversable [3, 4]. These traversable wormholes in the standard GR are supported by an exotic energy-momentum tensor [3,4,5,6]. Such exotic matters do not satisfy the null energy condition (NEC) which is known to be respected by all types of known normal or physical matters. In recent developments, the requirement of an exotic matter in traversable wormhole spacetimes has been removed in some specific modified theories of gravity. For more details, we refer to [7,8,9,10,11,12,13,14,15,16,17].

Apart from the traditional wormhole which is directly the solution of Einstein’s field equations, there is another type of wormhole that is not a direct solution to Einstein’s field equations. Such wormholes have been introduced by Matt Visser in [18, 19], using the Israel junction formalism [20]. The throat of this new traversable wormhole is a thin shell connecting the two geodesically incomplete spacetimes which are cut on a timelike hypersurface from an identical or two individual solutions of Einstein’s equations. At the timelike boundary hypersurface where the two spacetimes are joined, there exists an exotic surface fluid such that the throat is effectively a physical thin shell instead of a mathematical concept. Therefore such wormholes are known as thin-shell wormholes (TSWs). The fundamental difference between the wormholes and TSW is that while wormholes are supported by exotic matter that fills the entire spacetime, TSWs are powered by exotic matters that are accumulated at only their throat, and off the throat, everything is as normal as the corresponding bulk spacetime. This should be added that, in analogy with the standard wormholes, here for TSW the surface fluid present on the throat may also be physical in some modified theories of gravity [21,22,23].

Conceptually, one of the reasons that makes TSWs important is their similar physical structure to the bulk spacetime where they are constructed. For instance, the Schwarzschild TSW [24] possesses common features with the Schwarzschild black hole. Particularly, the null or timelike geodesics of the two, in most of the cases, are the same which makes it difficult to distinguish one from the other. To be more specific, we refer to the paper of Diemer and Smolarek in [25] where the dynamics of test particles in the STSW as well as the rotating Kerr TSW have been investigated. It can be seen in [25] that the geodesics of a null or a timelike particle that doesn’t cross the throat and remains on one side of the throat is the same as the geodesics of an identical particle with identical initial conditions in the Schwarzschild black hole. Furthermore, those particles that due to their initial conditions cross the throat and transmit to the other side of the throat (other universe) make a symmetric geodesics with respect to the throat (see Fig. 2 and Fig. 3 in [25] ). The geodesics on each side of the throat are also identical to the geodesics of an identical particle in the Schwarzschild black hole, up to the radial distance from the horizon coinciding with the location of the throat in the STSW. Such particles, after this point, continue their geodesics toward the horizon and beyond. Now suppose that the throat of the STSW is very close to the event horizon of the Schwarzschild black hole. In such a case, from a distant observer’s frame, the two geodesics are not easily distinguishable. Therefore, hypothetically one may consider a static black hole to be a TSW whose event horizon is replaced by a throat and the falling particles move to the other universe instead of the central singularity. This could also solve the problem of the singularity of the black holes. Let’s add that in the latter case where the throat approaches the event horizon the corresponding TSW may not be considered traversable anymore as the tidal force near the throat becomes unbearable for a traveler.

Furthermore, similar to the black hole spacetimes, the mechanical stability of TSWs has also attracted attention from the beginning of its introduction [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. Varela in [42] studied the instability of the Schwarzschild thin-shell wormhole (STSW) with the radius of the throat at \(a_{0}=3M\) which was introduced by Poisson and Visser in [24]. This instability was known from [24] where the STSW was supported by a surface fluid with a generic barotropic Equation of State (EoS). In [42] by virtue of the concept of variable EoS the STSW with \(a_{0}=3M\) became stable. Let us add that while the generic barotropic EoS implies \(p=p\left( \sigma \right) ,\) the variable EoS (see [43]) is described by \( p=p\left( \sigma ,a\right) \) where \(\sigma ,\) p,  and a are the surface-energy density, the surface transverse pressure, and the radius of the throat in a generic TSW, respectively.

It is well known that for the Schwarzschild black hole the radius of the innermost circular null geodesics is \(r_{c}=3M\) while its event horizon is located at \(r=r_{s}=2M.\) Referring to the original work of Poisson and Visser [24], we observe that the mechanical stability of the STSW is divided into two distinct regions at \(a_{0}=3M\) (see Eqs. (31) and (32) as well as Fig. 1 of Ref. [24]). This seems not to be a coincidence that the definite instability radius of the STSW and the innermost photonsphere are both at \(a_{0}=3M\) and \(r_{c}=3M.\) Is there a deeper connection between the innermost photonsphere of generic spherically symmetric black holes and the instability of the corresponding TSW? Answering this question is our main motivation for performing this current research. In the next Section, first of all, we show that for a generic asymptotically flat static spherically symmetric black hole there always exists a circular null geodesics that is called the innermost light ring. In [44] the existence of such a circular orbit in arbitrary dimensions has been proved. It was also shown that such an innermost null circular ring is unstable [44]. More recently Peng in [45] has also proved the existence of such photon ring for the extremal spherically symmetric asymptotically flat hairy black holes. In a different approach, Hod in [46] has also proved the same statement while for non-extremal cases he proved the existence of circular orbit in [47] (see also [48]). The null geodesics and light rings outside static black holes are the indirect sources of information about the corresponding black holes. This is because of the strong gravity of the black holes such that even light cannot escape from them. Therefore direct effects of black holes cannot be observed. Some of the important works on the indirect effects of black holes can be found in [44,45,46,47,48,49] (also see [50,51,52,53,54,55,56,57,58,59,60,61,62,63] and the references therein).

2 Generic spherical thin-shell wormhole with its throat at the photonsphere of the bulk spacetime

Inspired by the great works in [44,45,46,47,48,49], we start with a generic static spherically symmetric bulk spacetime described by

$$\begin{aligned} ds^{2}=-\psi \left( r\right) dt^{2}+\frac{dr^{2}}{\psi \left( r\right) } +r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}\right) , \end{aligned}$$
(1)

in which \(\psi \left( r_{+}\right) =0\) introduces the possible event horizon located at \(r=r_{+}.\) Let us add that (1) is not the most generic spherically symmetric line element as we imposed \(g_{tt}g_{rr}=-1\) which corresponds to a particular matter field with the energy-momentum tensor \( T_{\mu }^{\nu }=diag\left[ -\rho ,p,q,q\right] \) in which \(\rho +p=0.\)

The outer photonsphere of the metric (1) is defined as the circular geodesics of the null particles, namely photons. To find the location of such photonsphere, we write down the Lagrangian of the null particle which is given by

$$\begin{aligned} 2{\mathcal {L}}=-\psi \left( r\right) {\dot{t}}^{2}+\frac{{\dot{r}}^{2}}{\psi \left( r\right) }+r^{2}\left( {\dot{\theta }}^{2}+\sin ^{2}\theta {\dot{\varphi }} ^{2}\right) , \end{aligned}$$
(2)

where a dot implies a derivative with respect to an affine parameter. Having the Lagrangian defined in (2), we obtain the components of the generalized momentum which are given as follows

$$\begin{aligned} P_{t}=-E=\frac{\partial {\mathcal {L}}}{\partial {\dot{t}}}=-\psi \left( r\right) {\dot{t}}, \end{aligned}$$
(3)
$$\begin{aligned} P_{\varphi }=L=\frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}}=r^{2}\sin ^{2}\theta {\dot{\varphi }}, \end{aligned}$$
(4)
$$\begin{aligned} P_{r}=\frac{\partial {\mathcal {L}}}{\partial {\dot{r}}}=\frac{{\dot{r}}}{\psi \left( r\right) }, \end{aligned}$$
(5)

and

$$\begin{aligned} P_{\theta }=\frac{\partial {\mathcal {L}}}{\partial {\dot{\theta }}}=r^{2}{\dot{\theta }}. \end{aligned}$$
(6)

As it is seen from (2), the Lagrangian is independent of t and \( \varphi \) implying that the energy E and the angular momentum L of the null particle, are conserved. Following the definition of the Hamiltonian of the system i.e., \({\mathcal {H}}=\Sigma P_{\mu }{\dot{x}}^{\mu }-{\mathcal {L}},\) we obtain the Hamiltonian given by

$$\begin{aligned} 2{\mathcal {H}}=-E{\dot{t}}+\frac{{\dot{r}}^{2}}{\psi \left( r\right) }+r^{2}{\dot{\theta }}^{2}+L{\dot{\varphi }}. \end{aligned}$$
(7)

Furthermore, for a null particle \({\mathcal {H}}=0\) which consequently yields

$$\begin{aligned} {\dot{r}}^{2}=\psi \left( r\right) \left( E{\dot{t}}-r^{2}{\dot{\theta }}^{2}-L{\dot{\varphi }}\right) . \end{aligned}$$
(8)

Next, we set \(\theta =\frac{\pi }{2}\) that after introducing

$$\begin{aligned} {\dot{t}}=\frac{E}{\psi \left( r\right) }, \end{aligned}$$
(9)

and

$$\begin{aligned} {\dot{\varphi }}=\frac{L}{r^{2}}, \end{aligned}$$
(10)

in (8) we obtain

$$\begin{aligned} {\dot{r}}^{2}=\psi \left( \frac{E^{2}}{\psi }-\frac{L^{2}}{r^{2}}\right) . \end{aligned}$$
(11)

Finally, for circular geodesics, we impose \({\dot{r}}^{2}=0\) as well as \(\frac{ d}{dr}\left( {\dot{r}}^{2}\right) =0.\) Hence, we end up with the equation

$$\begin{aligned} 2\psi =r\psi ^{\prime }, \end{aligned}$$
(12)

and the ratio of the energy over angular momentum is given by

$$\begin{aligned} \left( \frac{E}{L}\right) ^{2}=\frac{\psi }{r^{2}}. \end{aligned}$$
(13)

In (12) a prime notation stands for the derivative with respect to r. We note that (12) is not a differential equation but an ordinary equation for identifying the radius of the photonsphere i.e., \(r=r_{c}.\) Considering the bulk spacetime to be an asymptotically flat black hole, we impose \(\psi \left( r_{+}\right) =0\) and \(\psi \left( r\rightarrow \infty \right) \rightarrow 1\). Moreover, Hod in [64] has proved that the circular orbits on the equatorial plane are the orbits with locally the shortest period (as measured by an asymptotic observer). Herein, the period of a null particle on the circular motion measured by an asymptotic observer is simply found by setting \(\Delta s=\Delta r=\Delta \theta =0\) and \(\Delta \varphi =2\pi \) in (1) which yields

$$\begin{aligned} T=\left. \frac{2\pi r}{\sqrt{\psi }}\right| _{r=r_{c}}. \end{aligned}$$
(14)

To find the minimum of T,  we apply

$$\begin{aligned} \left. T^{\prime }\left( r\right) \right| _{r=r_{c}}=0, \end{aligned}$$
(15)

which explicitly becomes

$$\begin{aligned} \left. 2\psi =r\psi ^{\prime }\right| _{r=r_{c}}, \end{aligned}$$
(16)

that is the same equation as (12). For Eq. (16) there exists at least one solution due to the behavior of the period of the circular orbit in Eq. (14) where we observe that \(T\left( r_{c}\rightarrow r_{+}\right) \rightarrow \infty \) and \(T\left( r_{c}\rightarrow \infty \right) \rightarrow \infty \). This implies that the period T at least for one critical \(r_{c}\) becomes minimum which is the solution of the Eq. (16) too. Having known that (16) admits at least one solution, we conclude that (12) also admits at least one solution which is, in fact, the radius of the innermost circular orbit. In summary, we have briefly shown that the innermost photonsphere exists and its radius satisfies Eqs. (12) and (16).

Next, we construct a thin-shell wormhole in the bulk spacetime (1) by applying the standard Israel junction formalism [20]. Without going through the details, we refer to the rich literature on the subject and only use their results [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. Considering the throat of the TSW to be at \(r=a>r_{+}\) the induced line element of the throat surface is given by

$$\begin{aligned} ds^{2}=-d\tau ^{2}+a\left( \tau \right) ^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}\right) , \end{aligned}$$
(17)

in which \(\tau \) is the proper time on the throat. To join the spacetimes of either side of the TSW at the throat smoothly, one has to apply the Israel junction conditions [20] upon which a surface energy-momentum tensor is required. This surface energy-momentum tensor is given by \(S_{\mu }^{\nu }=diag\left( -\sigma ,p,p\right) \) in which the surface energy density \(\sigma \) and the surface transverse pressure p respectively are given by

$$\begin{aligned} \sigma =-\frac{\sqrt{\psi \left( a\right) +{\dot{a}}^{2}}}{2\pi a}, \end{aligned}$$
(18)

and

$$\begin{aligned} p=\frac{\sqrt{\psi \left( a\right) +{\dot{a}}^{2}}}{8\pi }\left( \frac{2\ddot{a }+\psi ^{\prime }\left( a\right) }{\psi \left( a\right) +{\dot{a}}^{2}}+\frac{2 }{a}\right) , \end{aligned}$$
(19)

where a dot stands for a derivative with respect to the proper time \(\tau .\) Moreover, the energy density \(\sigma \) and the pressure p satisfy the energy conservation equation given by

$$\begin{aligned} \frac{d\sigma }{da}+\frac{2}{a}\left( \sigma +p\right) =0. \end{aligned}$$
(20)

Next, we assume that \(a=r_{c}\) is an equilibrium radius for the throat such that \({\dot{a}}=\ddot{a}=0.\) The equilibrium surface energy density and transverse pressure are then found to be

$$\begin{aligned} \sigma _{c}=-\frac{\sqrt{\psi \left( r_{c}\right) }}{2\pi r_{c}}, \end{aligned}$$
(21)

and

$$\begin{aligned} p_{c}=\frac{\sqrt{\psi \left( r_{c}\right) }}{8\pi }\left( \frac{\psi ^{\prime }\left( r_{c}\right) }{\psi \left( r_{c}\right) }+\frac{2}{r_{c}} \right) . \end{aligned}$$
(22)

Considering, \(r_{c}\) the radius of the innermost photonsphere, Eq. (16 ) implies

$$\begin{aligned} \frac{\psi ^{\prime }\left( r_{c}\right) }{\psi \left( r_{c}\right) }=\frac{2 }{r_{c}} \end{aligned}$$
(23)

and consequently

$$\begin{aligned} p_{c}=-\sigma _{c}. \end{aligned}$$
(24)

Now we study the stability of the TSW against a linear radial perturbation about the equilibrium radius of the throat. To do so, we rewrite Eq. (18) to get

$$\begin{aligned} {\dot{a}}^{2}+V\left( a\right) =0, \end{aligned}$$
(25)

in which the effective potential of the one-dimensional motion (25) is given by

$$\begin{aligned} V\left( a\right) =\psi \left( a\right) -\left( 2\pi a\sigma \right) ^{2}. \end{aligned}$$
(26)

For a linear perturbation, we expand the effective potential about its equilibrium radius such that

$$\begin{aligned} V\left( a\right)= & {} V\left( r_{c}\right) +V^{\prime }\left( r_{c}\right) \left( a-r_{c}\right) \nonumber \\{} & {} +\frac{1}{2}V^{\prime \prime }\left( r_{c}\right) \left( a-r_{c}\right) ^{2}+{\mathcal {O}}\left( \left( a-r_{c}\right) ^{3}\right) , \end{aligned}$$
(27)

in which while \(V\left( r_{c}\right) =\psi \left( r_{c}\right) -\left( 2\pi r_{c}\sigma _{c}\right) ^{2}\) is zero, one calculates

$$\begin{aligned} V^{\prime }\left( a\right) =\psi ^{\prime }\left( a\right) +8\pi ^{2}a\sigma \left( \sigma +2p\right) . \end{aligned}$$
(28)

where we used the energy-conservation equation (20). Please note that at the equilibrium where \(a=r_{c}\), \(\sigma _{c}\) is given by (21) which clearly yields from (26) that \(V\left( r_{c}\right) =0.\) Considering a generic barotropic EoS for the fluid present at the throat in the form of \(p=p\left( \sigma \right) \), we further calculate

$$\begin{aligned}{} & {} V^{\prime \prime }\left( a\right) =\psi ^{\prime \prime }\left( a\right) \nonumber \\{} & {} -8\pi ^{2}\left\{ \left( \sigma +2p\right) ^{2}+2\sigma \left( \sigma +p\right) \left( 1+2p^{\prime }\right) \right\} . \end{aligned}$$
(29)

In (29) we used the relation \(\frac{dp\left( \sigma \right) }{da}= \frac{dp\left( \sigma \right) }{d\sigma }\frac{d\sigma }{da}\) or equivalently \(\frac{dp\left( \sigma \right) }{da}=p^{\prime }\sigma ^{\prime }=-\frac{2}{a}\left( \sigma +p\right) p^{\prime }\) where we used (20). At \(a=r_{c}\), we impose \(p_{c}=-\sigma _{c}\) from (24) such that

$$\begin{aligned} V^{\prime }\left( r_{c}\right) =\psi ^{\prime }\left( r_{c}\right) -8\pi ^{2}r_{c}\sigma _{c}^{2} \end{aligned}$$
(30)

which after applying (23) and considering (21) it becomes identically zero. Furthermore from (29) at \(a=r_{c}\) and \( p_{c}=-\sigma _{c}\) one obtains

$$\begin{aligned} V^{\prime \prime }\left( r_{c}\right) =\psi ^{\prime \prime }\left( r_{c}\right) -8\pi ^{2}\sigma _{c}^{2}, \end{aligned}$$
(31)

which upon considering (21), it reduces to

$$\begin{aligned} V^{\prime \prime }\left( r_{c}\right) =\psi ^{\prime \prime }\left( r_{c}\right) -2\frac{\psi \left( r_{c}\right) }{r_{c}^{2}}. \end{aligned}$$
(32)

Taking into account the above results in the equation of motion (25), up to the first nonzero term for the effective potential, the equation of motion becomes

$$\begin{aligned} {\dot{a}}^{2}+\frac{1}{2}V_{c}^{\prime \prime }\left( a-r_{c}\right) ^{2}\simeq \varepsilon , \end{aligned}$$
(33)

in which \(V_{c}^{\prime \prime }=V^{\prime \prime }\left( r_{c}\right) \) and \(\varepsilon >0\) is the initial small mechanical energy of the throat after the perturbation. A derivative of (33) with respect to the proper time gives

$$\begin{aligned} \ddot{a}+\frac{1}{2}V_{c}^{\prime \prime }a\simeq \frac{1}{2}V_{c}^{\prime \prime }r_{c}. \end{aligned}$$
(34)

The latter implies that with \(V_{c}^{\prime \prime }>0\), the throat’s radius oscillates about its equilibrium radius i.e., \(a=r_{c}\) and consequently the TSW is stable against the radial linear perturbation. Otherwise, the throat’s radius expands or shrinks exponentially, and in either case, the corresponding TSW is not stable. Therefore it is important to identify the sign of \(V_{c}^{\prime \prime }.\) To do so, once more we refer to the properties of \(r_{c}\) obtained from Eq. (16). As we have mentioned before, \(r_{c}\) is the radius of the innermost null circular geodesics, and as was proved in [64], the corresponding orbit possesses the smallest period. Equation (16) admits the critical point(s) for the period function (14) but to be the minimum period one should impose

$$\begin{aligned} \left. T^{\prime \prime }\left( r\right) \right| _{r=r_{c}}>0. \end{aligned}$$
(35)

Using the explicit expression of \(T\left( r\right) \) in (14), we find

$$\begin{aligned} T^{\prime \prime }\left( r_{c}\right) =-\frac{2\pi r_{c}}{\sqrt{\psi \left( r_{c}\right) }}\left( \frac{\psi ^{\prime \prime }\left( r_{c}\right) }{\psi \left( r_{c}\right) }-\frac{2}{r_{c}^{2}}\right) , \end{aligned}$$
(36)

where we used \(\psi ^{\prime }\left( r_{c}\right) =\frac{2}{r_{c}}\psi \left( r_{c}\right) \) from (16) or (23). From (36) for the period to be minimum at \(r=r_{c}\) which is the radius of the innermost circular null geodesics, we apply \(T^{\prime \prime }\left( r_{c}\right) >0\) and consequently get the condition

$$\begin{aligned} \left( \frac{\psi ^{\prime \prime }\left( r_{c}\right) }{\psi \left( r_{c}\right) }-\frac{2}{r_{c}^{2}}\right) <0. \end{aligned}$$
(37)

The latter condition directly implies

$$\begin{aligned} V^{\prime \prime }\left( r_{c}\right) =\psi ^{\prime \prime }\left( r_{c}\right) -2\frac{\psi \left( r_{c}\right) }{r_{c}^{2}}<0, \end{aligned}$$
(38)

which completes our proof that the TSW with its throat at the innermost photonsphere is unstable against a radial linear mechanical perturbation. Before we finish this section it is remarkable to compare the equation of motion of the throat i.e., Eq. (25) with the circular motion of a null particle outside the black hole (1). As we have proved in a very recent work [65], the motion of photons on the innermost photonsphere is unstable (see also [44]). Furthermore, if Eq. (15 ) yields more than one critical point since the smallest and the largest radii correspond to the local minimum period due to the fact that \(T\left( r_{c}\rightarrow r_{+}\right) \rightarrow \infty \) and \(T\left( r_{c}\rightarrow \infty \right) \rightarrow \infty \), the number of critical points is odd. In such cases, there is at least one local maximum for the period and therefore \(T^{\prime \prime }\left( r_{c}\right) <0\) and consequently \(V^{\prime \prime }\left( r_{c}\right) >0\) which implies the corresponding TSW is stable. It should be added that there are special cases where the two light rings can be degenerate that is characterized by \( T^{\prime }\left( r_{c}\right) =T^{\prime \prime }\left( r_{c}\right) =0\) which from (16) and (36) it becomes \(r_{c}\psi ^{\prime \prime }\left( r_{c}\right) =\psi ^{\prime }\left( r_{c}\right) .\)

3 Conclusion

In this study, we have analytically proved that the instability of the STSW supported by a barotropic surface fluid with the equilibrium throat at \( a_{0}=3M\) is not a coincidence. In doing so, we started from a generic static spherically symmetric black hole and proved that the corresponding TSW constructed by applying the cut-and-paste method and powered by a generic barotropic surface fluid is unstable when its radius is the same as the innermost circular orbit of the black hole. This instability is also connected to the instability of the innermost photonsphere of the black hole as well (see [44, 65]). It should be add that the instability issue of the static spherically symmetric TSW supported by a barotropic perfect surface fluid can be solved by considering a variable EoS for the matter present on the throat. This is what Varela in [42] suggested for STSW. In this research, we studied and presented the connection between the stability of the photon sphere and a thin-shell wormhole in the static spherically symmetric black hole. The existence of a similar connection in the rotational black hole is an open problem that we shall address in a separate work.