Introduction

In the study of general relativity, traversable wormholes (WHs) have received a lot of attention. They have a stable arrangement that permits the mobility of the observer in both directions between far-off cosmological locations. In spherically symmetric static spacetimes, the most intriguing cosmic entity that can be produced using general relativity is the time-like thin-shells. These cosmological object models have been used to examine several astrophysical events, including supernova explosions and gravity collapse. In 1966, Israel1 published a significant study that offered a concrete formalism for building time-like shells in general by joining two distinct manifolds at the thin-shell position. The solutions of a given gravitational theory that describe two locations divided by an infinitely thin zone where the matter is contained are known as self-gravitating thin-shells. Finding out if pertinent thin-shell designs are dynamically and thermodynamically stable is crucial. Exotic material is necessary to maintain this stability. Energy conditions, which may be assessed using the Israel thin-shell formalism1, determine the presence of these exotic matters near the WH’s throat. According to Visser2, using particular geometric structures for the WH can reduce the violation of energy constraints. This shows that the requirement for large quantities of exotic matter to ensure stability can be reduced by using specific geometrical concepts.

Thin-shell WH stability is an important topic in cosmology and astrophysics because it allows for the investigation of feasible WH solutions. The equation of state (EoS) for the matter distribution at the WH throat is critical in determining the stable structure. There are several exotic matter models, one of which is the phantom-like EoS, described as \(p=\chi \sigma \), where \(\chi <0\) is a constant. Depending on the value of \(\chi \), this model depicts a variety of matter distributions. The EoS corresponds to the phantom energy, quintessence, and dark energy states for \(\chi <-1\), \(\chi >-1/3\), and \(\chi <-1/3\), respectively. The generalized Chaplygin gas defined as \(p=-\mathcal {G}/\sigma ^\lambda \), where \(0<\lambda \le 1\) and \(\mathcal {G}\) is a positive constant, which is another prominent EoS. Several researchers have investigated using these models3,4,5,6,7,8,9,10,11,12,13. By examining minor radial perturbations around the equilibrium throat radius, the stability of thin-shell WHs has been studied. By examining radial perturbations, Poisson and Visser14,15 investigated the linear stability of WHs. Taking into account the existence of a cosmological constant, Lobo and Crawford16 examined the stable construction of spherically symmetric thin-shell WHs. Numerous studies17,18,19,20,21,22,23,24,25,26,27,28,29,30 have examined the stability of thin-shell WHs in both stable and unstable forms.

A gravitational vacuum star, or gravastar, is a well-known thin-shell geometrical structure having outside black hole (BH) geometry and inside de Sitter geometry31. Numerous investigators created this geometric arrangement against the backdrop of various BH spacetimes and also looked into its stability with various EoS32,33,34,35,36. In37, the prototype gravastar model with internal de Sitter spacetime and external Schwarzschild BH is examined, while in38, a gravastar with phantom energy is examined. For different matter distributions at thin-shell, it is discovered that the formed structure can be a BH, stable, unstable, or “bounded excursion” gravastar. Later, this work was extended for the choice of exterior Schwarzschild-de Sitter and RN spacetimes with interior de Sitter region with specific EoS39,40. The gravastar model in the background of noncommutative geometry is also explored in41,42. The stable configurations of the gravastar model are investigated with interior de Sitter and exterior regular as well as charged quintessence BHs with variable EoS in43,44. The study of thin-shell gravastars developed from inner de Sitter and outer various BH geometries is presented in45,46,47,48,49.

The choice of a (2+1)-dimensional BH provides a reduced model while retaining all of the key features of BH physics. Lower dimensionality preserves important characteristics like the existence of an event horizon and the BHs spinning characteristics while making the computations less difficult. In this reduced situation, examining the behavior of a nonminimally coupled scalar field offers important new perspectives on the interaction between scalar fields and gravity. Current research has focused a lot of interest on the study of BH structures in (2+1)-dimensions. The event horizons, thermodynamic features, and Hawking temperatures of these BHs are comparable to those of higher-dimensional or (3+1)-dimensional BHs. In contrast to their higher-dimensional counterparts, however, (2+1)-dimensional BHs give a more straightforward mathematical model. The first description of a (2+1)-dimensional BH is given by Banados, Teitelboim, and Zanelli (BTZ)50. Subsequent additions included the Einstein-Maxwell theory51 and the Einstein-Maxwell-dilaton theory52. By imposing a traceless constraint on the energy-momentum tensor and concentrating on a particular power of the Maxwell scalar, \((F_{\mu \nu }F^{\mu \nu })^{3/4}\). Cataldo et al.53 created another (2+1)-dimensional BH. It is important to point out that the simplicity of (2+1)-dimensional BHs makes them an appealing subject for research, providing insights into the basic characteristics of BHs without requiring the mathematical complexity of higher-dimensional BHs. In the framework of charged rotating (2+1)-dimensional BH, the stable configurations of WH structure are discussed in54.

Hassaine and Martinez55 investigated higher dimensional BHs with a conformally invariant Maxwell source and accounting for an action for an Abelian gauge field. They also produced charged BH solutions in a variety of dimensions using a nonlinear electrodynamics source56. They conducted their inquiry using a matter lagrangian with arbitrary power (k) of the Maxwell invariant \((F_{\mu \nu }F^{\mu \nu })^{k}\). Gurtug et al.57 constructed a (2+1)-dimensional BH in the Einstein-power-Maxwell (EPM) theory without the use of a traceless condition. The authors study the properties of BHs in a three-dimensional spacetime named HMTZ BH58. They consider the dynamics that are introduced to the system when gravity is coupled to a scalar field58. The study concentrates on the behavior of these BHs in the vicinity of their asymptotic regions, where the effects of the scalar field are most prominent. The authors discuss how their findings impact our understanding of the three-dimensional physics of BHs after looking at the solutions to the field equations. The BH solution minimally coupled with the scalar field is also presented in59. This research advances our knowledge of the intricate interplay between scalar fields and gravity in low-dimensional spacetimes58,59. Thermodynamical properties of (2+1)-dimensional rotating BHs with hair is investigated in60. Bueno et al.61 present the regular (2+1)-dimensional BHs coupled with a scalar field.

In addition to introducing new dynamics, the nonminimal coupling can result in some fascinating occurrences, such as the development of thin-shell WHs. We may gain a better understanding of the role of nonminimally coupled scalar fields in the setting of BHs by examining the behavior of the scalar field near the rotating BH. In this study, the stability of counter-rotating thin-shell WHs developed from (2+1)-dimensional rotating BHs nonminimally coupled scalar fields are examined. The paper is structured as shown below. Section “ROTATING BLACK HOLES with scalar hair“ presents the exact (2+1)-dimensional rotating BHs nonminimally coupled scalar fields solution and a foundational formalism for the creation of counter-rotating thin-shell WHs is developed in Section “Formalism of counter-rotating thin-shell wormholes“. The stability of counter-rotating thin-shells are examined in Section “Stability analysis“ by using linearized radial perturbation and EoS (phantom-like and generalized Chaplygin gas). The final section presents the summary of our findings.

ROTATING BLACK HOLES with scalar hair

The term “hair” describes fields or physical attributes that a BH may have in addition to its mass, charge, and angular momentum. Research on these characteristics is still ongoing and the details are not fully known. The shape of the BH would be altered by hair, which might also have an impact on its observational characteristics. The nonminimally coupled scalar field introduces a new term in the action that couples the scalar field to the curvature of spacetime. This coupling allows for interactions between the scalar field and gravity, leading to interesting phenomena such as gravitational waves generated by the scalar field’s dynamics. Rotating hairy BH research is particularly fascinating in three dimensions because, in contrast to higher-dimensional spacetimes, it allows for simpler mathematical models. Three dimensions are simpler, making computations easier to handle and providing new information on BH properties. For three-dimensional spacetime, the action with a nonminimally coupled scalar field can be written as60

$$\begin{aligned} \mathcal{I}=\frac{1}{2}\int {d^3x\sqrt{-g}\left( \mathcal {R}-g^{\mu \nu }\nabla _{\mu }\psi \nabla _{\nu }\psi -\xi \mathcal {R}\psi ^2-2\mathcal {V}(\psi )\right) }, \end{aligned}$$
(1)

where g is the determinant of metric tensor \(g^{\mu \nu }\), \(\mathcal {R}\) is the Ricci scalar, \(\mathcal {V}(\psi )\) denotes the potential functions of scalar field (\(\psi \)). To present the signification of coupling strength among the scalar field and gravity, \(\xi \) can be taken as equal to 1/8. Hence, the respective expression of scalar field turns out to be60

$$\begin{aligned} \mathcal {V}(\psi )=-\frac{1}{\mathcal {L}^2}+\frac{a^2\psi ^{10}}{512\mathcal {B}^4}\frac{\left( \psi ^6-40\psi ^4 +640\psi ^2-4608\right) }{(\psi ^2-8)^5}+\frac{1}{512}\left( \frac{1}{\mathcal {L}^2}+\frac{\beta }{\mathcal {B}^2}\right) \psi ^6, \end{aligned}$$
(2)

the real integrating constants are denoted with \(\beta \), a and \(\mathcal {B}\). Here, \(\Lambda =\frac{1}{\mathcal {L}^2}\) which denotes the cosmological constant and \(\mathcal {L}\) is the length parameter. Now, we consider the respective solution of the rotating BH nonminimally coupled scalar field as60

$$\begin{aligned}{} & {} ds^2=-\mathcal {F}(r)dt^2+\mathcal {F}(r)^{-1}dr^2+r^2\left( \omega (r)dt+d\Phi \right) ^2, \end{aligned}$$
(3)

where

$$\begin{aligned}{} & {} \mathcal {F}(r)=3\beta +\frac{2\mathcal {B}\beta }{r}+ \frac{r^2}{\mathcal {L}^2}+\frac{\left( 3r+2 \mathcal {B}\right) ^2a^2}{r^4}, \quad \omega (r)=-\frac{\left( 3r+2\mathcal {B}\right) a}{r^3}. \end{aligned}$$
(4)

Also, the scalar field becomes

$$\begin{aligned} \psi (r)=\pm \sqrt{\frac{8\mathcal {B}}{r+\mathcal {B}}}. \end{aligned}$$
(5)

It is interesting to mention that the above (2+1)-dimensional rotating BH solution is reduced to the rotating BTZ BH in the absence of scalar field as \(\mathcal {B}\Rightarrow 0\) by considering \(a=\mathcal {J}/6\) and \(\beta =-\mathcal {M}/3\)50. Also, the above presented lapse function can be rewritten as60

$$\begin{aligned} \mathcal {F}(r)=-\mathcal {M}\left( 1+\frac{2\mathcal {B}}{3r}\right) +\frac{r^2}{\mathcal {L}^2} +\frac{\left( 3r+2\mathcal {B}\right) ^2J^2}{36r^4},\quad \omega (r)=-\frac{\left( 3r+2\mathcal {B}\right) \mathcal {J}}{6r^3}. \end{aligned}$$
(6)

In the presence of thermodynamical quantities \(\mathcal {M}\) and \(\mathcal {J}\) of this rotating hairy BH in the scalar potential \(\mathcal {V}(\psi )\), the action is not invariant. In Fig. (1), we study the position of the event horizon through a graphical behavior of metric function \(\mathcal {F}(r)\). It is noted that the position of the event horizon moving away from the center of the rotating BH by increasing its angular momentum. The presence of a scalar field also affects the position of the event horizon see Fig. 1.

Figure 1
figure 1

Graphical analysis of metric function for different values of \(\mathcal {B}\) along radial coordinate and angular momentum with \(\mathcal {M}=0.5,\mathcal {L}=1\). Here, we use \(\mathcal {B}=0\) for left plot \(\mathcal {B}=0.5\) for middle plot and \(\mathcal {B}=1\) for right plot. Also, the light green region shows \(\mathcal {F}(r)>0\) and light gray region depicts \(\mathcal {F}(r)<0\). The boundary line between these regions shows the position of the event horizon \(\mathcal {F}(r)=0\).

Full size image

In the subsequent sections, we developed the counter-rotating thin-shell WHs from the considered rotating BHs through well-known cut and paste approach.

Formalism of counter-rotating thin-shell wormholes

With implications for multiple gravitational events and dualities with other theories like Kalb-Ramond gravity, non-minimal scalar couplings to curvature are important in gravitational theories. Several important themes emerge when these characteristics are extended to lower-dimensional systems, especially when considering thin-shell WHs and revolving black holes in (2+1) dimensions. These include the resulting mathematical simplicity in lower dimensions, the exploration of dualities between gravitational models, insights into the fundamental interactions between gravity and scalar fields, and the relevance of these couplings to exotic phenomena such as traversable WHs. Analysing the effects of non-minimal scalar fields in lower-dimensional systems offers an innovative perspective on the basic interactions in gravitational physics and creates new avenues for comprehending the connections between various concepts and occurrences. Here, we focus on developing the geometry of thin-shell WH in the framework of (2+1)-rotating hairy BH by adopting Visser’s cut-and-paste method. The geometry of thin-shell WHs in this formalism can avoid the event horizon and singularity of the BH structure. The fundamental attraction of this approach is to lessen the amount of exotic matter, which is extremely important for keeping the WH throat open for stellar transit across far-off universes. With angular momentum \(\mathcal {J}_{+}\) and \(\mathcal {J}_{-}\), respectively, we select two rotating hairy BH geometries, one inside and one exterior. We eliminate the subsequent areas from both the interior and outer spacetimes:

$$\begin{aligned} \Sigma _{\pm }\equiv \{r_{\pm }<\Delta |\Delta >r_e\}. \end{aligned}$$
(7)

where \(\Delta \) represents the radius of shell radius and \(r_e\) denotes the positions of the event horizon. The boundary of remaining manifolds are timelike hypersurfaces given as follows

$$\begin{aligned} \partial \Sigma _{\pm }\equiv \{r_{\pm }=\Delta |\Delta >r_e\}. \end{aligned}$$
(8)

The corresponding timelike hypersurfaces are \(\partial \Sigma _{\pm }\equiv \partial \Sigma _{+}=\partial \Sigma _{-}\). Further, we match the inner and outer manifold at the hypersurface \(\partial \Sigma \). To develop the WH structure which is the most suitable for stellar travel, we consider the angular momentum of both the inner and outer sides of the WH throat have the same absolute value of angular momentum but opposite in direction, i.e., \(\mathcal {J}_++\mathcal {J}_-=0\). Hence, in this regard, the inner and outer sides of the throat are counter-rotating. This represents that the lower and upper shells are counter-rotating, i.e., they rotate in opposite directions. To investigate dynamically behavior of the shell radius (\(\Delta (t)\)), we introduce an azimuthal coordinate \(\Psi _{\pm }\) given by62

$$\begin{aligned} d\Psi _{\pm }=d\Phi _{\pm }-\frac{\left( 3\Delta (t_{\pm }) +2\mathcal {B}\right) \mathcal {J}_{\pm }}{6\Delta (t_{\pm })^3} dt_{\pm }. \end{aligned}$$
(9)

Hence, the respective line element (3) can be expressed with the counter-rotating frame on the timelike hypersurface (\(\Sigma \)) as

$$\begin{aligned} ds^2_{\pm }=-F_{\pm }(r_{\pm })dt^2_{\pm }+\frac{dr^2_{\pm }}{\mathcal {F}_{\pm } (r_{\pm })}+r^2_{\pm }\left\{ d\Psi _{\pm }+\frac{\mathcal {J}_{\pm }}{2}\left( \frac{1}{\Delta ^2(t_{\pm })}-\frac{1}{r^2_{\pm }}\right) +\frac{2\mathcal {B}}{3}\left( \frac{1}{\Delta ^3(t_{\pm })}-\frac{1}{r^3_{\pm }}\right) dt_{\pm }\right\} ^2. \end{aligned}$$
(10)

Also, the hypersurface coordinates are denoted with \(\xi ^i=(\tau ,\theta )\) and the shell radius depends on proper time \(\tau \). The line element of the induced metric is written as

$$\begin{aligned} ds^2=-d\tau ^2+\Delta ^2(\tau )d\Psi ^2. \end{aligned}$$
(11)

The development of a WH throat, which can be seen as a one-dimensional ring of matter is the consequence of two spacetimes smoothly matching. The junction condition provides a consistent and mathematically rigorous method for determining the behavior of the spacetime across the event horizon of a BH. This is important for understanding the physical properties of BHs and for making predictions about their behavior. Additionally, the junction condition can be used to study the effects of the non-minimally coupled scalar field on the geometry of the BH spacetime, providing insights into the interaction between gravity and the scalar field. This can lead to a better understanding of the dynamics of BHs in these theories and potentially lead to new insights or predictions about their behavior. General relativity is a well-established and thoroughly verified gravity theory that accurately describes the behavior of spacetime and matter in the presence of gravitational fields. General relativity’s junction conditions are widely accepted and applied in a variety of gravitational circumstances, including BHs and WHs. Despite the presence of a nonminimally linked scalar field and the investigation of exotic phenomena such as traversable WHs in the research, the fundamental laws of general relativity are still applicable to the gravitational interactions inside the system. The standard junction conditions ensure that energy and momentum are conserved across the thin-shell WH barrier, giving a consistent framework for analyzing the system’s stability and attributes. While adjustments may be required in gravity theories with non-trivial couplings or extra fields, the conventional junction conditions’ simplicity and consistency makes them an appropriate choice for analyzing the counter-rotating thin-shell WHs in the research article. The standard junction conditions remain a legitimate and dependable tool for analyzing the dynamics of the gravitational system under examination since they adhere to the recognized principles of general relativity and have proven to be successful in gravitational research. The stress-energy tensor components in this WH throat have non-zero values and can be computed using the second fundamental form referred as extrinsic curvature (\(K_{ij}^{\pm }\)) given as

$$\begin{aligned} K_{ij}^{\pm }=-n_{\alpha }^{\pm }\left( \frac{d^2x_{\pm }^\alpha }{d\xi ^id\xi ^j} +\Gamma ^\alpha _{\mu \nu }\frac{dx^{\mu }_{\pm }}{d\xi ^{i}}\frac{ dx^{\nu }_{\pm }}{ d\xi ^{j}}\right) ,\quad i,j=0,2, \end{aligned}$$
(12)

where \(n_{\alpha }^{\pm }\) denotes the outward directed unit normal vectors. Mathematically, for considered manifolds, it can be expressed as

$$\begin{aligned} n_{\alpha }^{\pm }=(n_{0},n_{1},n_{2})=\left( -\dot{\Delta }^2, \frac{\sqrt{\dot{\Delta }^2 +\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}}{\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}},0\right) , \end{aligned}$$

with \(\dot{\Delta }=d\Delta /d\tau \). Hence, we get

$$\begin{aligned} K^{\tau \pm }_{\tau }=\pm \frac{2 \ddot{\Delta } -\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{9 \Delta ^5}+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})}{6 \Delta ^4}+\frac{2 \mathcal {B} \mathcal {M}}{3 \Delta ^2}+\frac{2 \Delta }{\mathcal {L}^2}}{2 \sqrt{\dot{\Delta }^2 +\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}},\quad K^{\theta \pm }_{\theta }=\pm \frac{\sqrt{\dot{\Delta }^2+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}}{\Delta }, \end{aligned}$$
(13)

Due to the presence of a matter thin layer, there is a discontinuity in the components of extrinsic curvature given as \(\textit{k}_{ij}=K_{ij}^{+}-K_{ij}^{-}\). The respective components of such matter, contents can be evaluated as

$$\begin{aligned} S_{ij}=-\frac{1}{\pi }\left\{ [K_{ij}]-\eta _{ij}K\right\} , \end{aligned}$$
(14)

here stress-energy tensor is depicted with \(S_{ij}\), \(\eta _{ij}\) represents induced metric and \(K=tr[K_{ij}]=[K^{i}_{i}]\) where \([K_{ij}]=K^{+}_{ij}-K^{-}_{ij}\). For perfect fluid content with surface pressure (p) energy density (\(\sigma \)) and velocity component \(U_i\) is defined as \(S_{ij}=\left( \sigma +p\right) U_iU_j+p\eta _{ij}\). Hence, we have the following expressions

$$\begin{aligned} \sigma =-\frac{2 \sqrt{\dot{\Delta }^2+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}}{\pi \Delta },\quad p=\frac{2 \ddot{\Delta }-\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{9 \Delta ^5}+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})}{6 \Delta ^4}+\frac{2 \mathcal {B} \mathcal {M}}{3 \Delta ^2}+\frac{2 \Delta }{\mathcal {L}^2}}{\pi \sqrt{\dot{\Delta }^2+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}}. \end{aligned}$$
(15)

We are interested in discussing the equilibrium position of the shell. Now, we assume the equilibrium position of the shell is \(\Delta _0\). In this regard, the proper time differential of equilibrium shell radius vanishes. Hence, it can be written as \(\dot{\Delta _0}=0=\ddot{\Delta _0}\). Also, we get

$$\begin{aligned} \sigma _0 =-\frac{2 \sqrt{\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{36 \Delta ^4_0}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta _0 }+1\right) +\frac{\Delta _0 ^2}{\mathcal {L}^2}}}{\pi \Delta _0 },\quad p_0=\frac{-\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{9 \Delta ^5_0}+\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})}{6 \Delta ^4_0}+\frac{2 \mathcal {B} \mathcal {M}}{3 \Delta ^2_0}+\frac{2 \Delta _0 }{\mathcal {L}^2}}{\pi \sqrt{\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{36 \Delta ^4_0}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta _0 }+1\right) +\frac{\Delta ^2_0}{\mathcal {L}^2}}}. \end{aligned}$$
(16)

Two types of gravitational fields can exist in WHs. To resist being drawn in by an attractive WH, which draws items towards it, observers must exert an outward force. A repulsive WH, on the other hand pulls objects away, and forces the observer to pull themselves closer to prevent being pushed even further away. Depending on how the gravitational field of the WH is configured, a certain force may be necessary. While a repellent WH necessitates an internal force, an attracting WH calls for an outside force. Positive values indicate attraction, while negative values indicate repulsion, in the radial component of the 3-acceleration, which mathematically captures this phenomenon. In this regard, the observer 3-acceleration can be calculated as \(a^\varpi =u^{\varpi };_{\beta }u^{\beta }\), where \(u^\varpi =\frac{dx^\varpi }{d\tau }=(\frac{dt}{d\tau },0,0)\). For considered BH solution, the respective non-zero component of 3-acceleration is given as

$$\begin{aligned} a^\alpha =\Gamma ^\alpha _{tt}\left( \frac{dt}{d\tau }\right) ^2=\frac{1}{2} \left( -\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{9 \Delta ^5}+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})}{6 \Delta ^4}+\frac{2 \mathcal {B} \mathcal {M}}{3 \Delta ^2}+\frac{2 \Delta }{\mathcal {L}^2}\right) . \end{aligned}$$
(17)

The Einstein field equations are used to deduce the equation of motion for a thin shell WH. It depicts how the form and position of the WH change when it interacts with matter and energy. This equation takes the curvature of spacetime and the distribution of matter within the WH into account, offering insights into its behavior and probable traversability. It can be calculated from the Eq. (15) as \(\dot{\Delta }^2+ \mathcal {H}(\Delta )=0,\) where potential function is written as

$$\begin{aligned} \mathcal {H}(\Delta )=-\frac{1}{4} \pi ^2 \Delta ^2 \sigma ^2+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}. \end{aligned}$$
(18)

In the next section, we explore the stability of the counter-rotating thin-shell WHs through linearized radial perturbation about equilibrium shell radius and specific choices of EoS.

Stability analysis

The stability of counter-rotating hairy thin-shell WHs develops from a (2+1)-dimensional rotating hairy BH examined by linearized perturbation about the equilibrium shell radius. Then, we consider the EoS to explore the relationship between pressure and energy density within the WH, whereas linearized perturbation analysis investigates tiny changes near the equilibrium configuration. These properties can help determine whether thin-shell WHs are stable or unstable.

Linearized radial perturbation

Figure 2
figure 2

Stability of counter-rotating thin-shell WHs (\(\mathcal {B}=0\)) using linearized radial perturbation for \(\mathcal {J}=0.1\) (left plot) and \(\mathcal {J}=0.2\) (right plot) for \(\mathcal {M}=1,\mathcal {L}=1\). Here, the red curve represents the graphical behavior of the metric function of rotating BH spacetime.

Full size image
Figure 3
figure 3

Stable configurations (\(\mathcal {B}=0\)) for \(\mathcal {J}=0.5\) (left plot) and \(\mathcal {J}=0.7\) (right plot) for \(\mathcal {M}=1,\mathcal {L}=1\).

Full size image
Figure 4
figure 4

Stability of counter-rotating thin-shell WHs with nonminimally coupled scalar field (\(\mathcal {B}\ne 0\)) using linearized radial perturbation for \(\mathcal {J}=0.1\) (left plot) and \(\mathcal {J}=0.2\) (right plot) for \(\mathcal {B}=0.1,\mathcal {M}=1,\mathcal {L}=1\).

Full size image
Figure 5
figure 5

Stable configurations (\(\mathcal {B}\ne 0\)) for \(\mathcal {J}=0.5\) (left plot) and \(\mathcal {J}=0.7\) (right plot) for \(\mathcal {B}=0.1,\mathcal {M}=1,\mathcal {L}=1\).

Full size image

To examine the stability of counter-rotating hairy thin-shell WHs, the linearized radial perturbation approach has been used. With the help of this technique, a thorough investigation of these structures behavior and dynamics under minor disturbances is possible, revealing details about their stability as well as their potential for collapse or expansion. Researchers can find out new things about physics and the behavior of thin-shell WHs by investigating the linearized radial perturbations to better understand the aspects that affect the stability of these structures. For this purpose, we determine the second derivative of the potential function as

$$\begin{aligned} \mathcal {H}''(\Delta )= \frac{2}{\mathcal {L}^2}-\frac{4 \mathcal {B}^3 (\mathcal {E}+1)}{\Delta ^3 \mathcal {L}^2}-\frac{1}{2} \pi ^2 \left\{ \mathcal {W}_0^2 \sigma (p+\sigma )+p^2\right\} , \end{aligned}$$
(19)

where \(\mathcal {W}_0^2\) denotes the EoS parameter written as \( \mathcal {W}_0^2=\frac{\partial p}{\partial \sigma }\). Also, the expansion of effective potential about \(\Delta =\Delta _0\) is written For stable configurations, we expand the potential function (\(\mathcal {H}(\Delta )\)) about \(\Delta _0\) by considering Taylor series up-to-order terms

$$\begin{aligned} \mathcal {H}(\Delta )=\mathcal {H}(\Delta _0)+\mathcal {H}'(\Delta _0) (\Delta -\Delta _0)+\frac{1}{2}\mathcal {H}''(\Delta _0) (\Delta -\Delta _0)^2+O[(\Delta -\Delta _0)^3]. \end{aligned}$$
(20)

It is interesting to mention that \(\mathcal {H}(\Delta _0)=0=\frac{d\mathcal {H}}{d\Delta }\mid _{\Delta =\Delta _0}\). Hence, stable configuration of the shell is directly linked with 2nd derivative of potential function. It is stable if \(\frac{d^2\mathcal {H}}{d\Delta ^2}\mid _{\Delta =\Delta _0}>0\), otherwise unstable. Hence, we have

$$\begin{aligned} \frac{d^2\mathcal {H}}{d\Delta ^2}\mid _{\Delta =\Delta _0}=-\frac{27 A_1 \Delta _0 ^4+9 \Delta _0 ^2 \mathcal {B} \left( A_3 \Delta _0 +2 A_2 \mathcal {B}\right) +A_5+\mathcal {J}^4 \mathcal {B} \mathcal {L}^2 \left( 27 (3 \mathcal {W}_0^2 -4) \Delta _0 ^3+8 (\mathcal {W}_0^2 -2) \mathcal {B}^3\right) }{3 \Delta _0 ^6 \left( A_4+4 \mathcal {J}^2 \mathcal {B}^2 \mathcal {L}^2\right) }, \end{aligned}$$
(21)

with

$$\begin{aligned} A_1= & {} (\mathcal {W}_0^2 -1) \mathcal {J}^4 \mathcal {L}^2+4 \Delta _0 ^4 \left( (\mathcal {W}_0^2 -3) \mathcal {J}^2+2 \mathcal {W}_0^2 \mathcal {L}^2 \mathcal {M}^2\right) -6 (\mathcal {W}_0^2 -1) \mathcal {J}^2 \Delta _0 ^2 \mathcal {L}^2 \mathcal {M}-8 (\mathcal {W}_0^2 -1) \Delta _0 ^6 \mathcal {M},\\ A_2= & {} (5 \mathcal {W}_0^2 -8) \mathcal {J}^4 \mathcal {L}^2+4 \mathcal {J}^2 \Delta _0 ^2 \left( (\mathcal {W}_0^2 -5) \Delta _0 ^2+(7-4 \mathcal {W}_0^2 ) \mathcal {L}^2 \mathcal {M}\right) +4 (2 \mathcal {W}_0^2 -1) \Delta _0 ^4 \mathcal {L}^2 \mathcal {M}^2,\\ A_3= & {} 2 \mathcal {J}^2 \Delta _0 ^2 \left( 10 (\mathcal {W}_0^2 -4) \Delta _0 ^2+3 (10-7 \mathcal {W}_0^2 ) \mathcal {L}^2 \mathcal {M}\right) +8 \Delta _0 ^4 \mathcal {M} \left( (5 \mathcal {W}_0^2 -2) \mathcal {L}^2 \mathcal {M}-3 (\mathcal {W}_0^2 -2) \Delta _0 ^2\right) ,\\ A_4= & {} 9 \left( \mathcal {J}^2 \Delta _0 ^2 \mathcal {L}^2+4 \Delta _0 ^6-4 \Delta _0 ^4 \mathcal {L}^2 \mathcal {M}\right) +12 \Delta _0 \mathcal {B} \mathcal {L}^2 \left( \mathcal {J}^2-2 \Delta _0 ^2 \mathcal {M}\right) ,\\ A_5= & {} 4 \mathcal {J}^2 \Delta _0 \mathcal {B}^3 \mathcal {L}^2 \left( (11 \mathcal {W}_0^2 -20) \mathcal {J}^2-18 (\mathcal {W}_0^2 -2) \Delta _0 ^2 \mathcal {M}\right) . \end{aligned}$$

The stability criteria can be further characterized as \(\mathcal {W}_0^2>\mathcal {K}_0^2(\Delta _0)\) if \(3 \Delta _0 ^6 \left( A_4+4 \mathcal {K}^2 \mathcal {B}^2 \mathcal {L}^2\right) >0\) and \(\mathcal {W}_0^2<\mathcal {K}_0^2(\Delta _0)\) if \(3 \Delta _0 ^6 \left( A_4+4 \mathcal {J}^2 \mathcal {B}^2 \mathcal {L}^2\right) <0\) where

$$\begin{aligned}{} & {} \mathcal {K}_0^2(\Delta _0)=-\frac{6 \mathcal {L}^2 (\Delta _0 +\mathcal {B}) \left( 3 \mathcal {J}^2 \Delta _0 +2 \mathcal {J}^2 \mathcal {B}-6 \Delta _0 ^3 \mathcal {M}\right) }{A_4+4 \mathcal {J}^2 \mathcal {B}^2 \mathcal {L}^2}+\frac{6 \mathcal {J}^2 (\Delta _0 +\mathcal {B})}{3 \mathcal {J}^2 \Delta _0 +2 \mathcal {J}^2 \mathcal {B}-6 \Delta _0 ^3 \mathcal {M}}+\frac{\mathcal {B}}{\Delta _0 +\mathcal {B}}+1. \end{aligned}$$
(22)
Figure 6
figure 6

Stable configurations (\(\mathcal {B}\ne 0\)) for \(\mathcal {J}=0.1\) (left plot) and \(\mathcal {J}=0.2\) (right plot) for \(\mathcal {B}=0.5,\mathcal {M}=1,\mathcal {L}=1\).

Full size image
Figure 7
figure 7

Stable configurations (\(\mathcal {B}\ne 0\)) for \(\mathcal {J}=0.5\) (left plot) and \(\mathcal {J}=0.7\) (right plot) for \(\mathcal {B}=0.5,\mathcal {M}=1,\mathcal {L}=1\).

Full size image

Figures 2, 3, 4, 5, 6 and 7 are used to explore the stability of counter-rotating thin-shell WHs developed from rotating BTZ BH (\(\mathcal {B}=0\)) and rotating hairy BH (\(\mathcal {B}\ne 0\)). First, we consider the choice \(\mathcal {B}=0\) with minimum values of angular momentum of rotating BTZ BH as shown in Figs. 2 and 3. It is noted that the stability regions of the counter-rotating thin-shell WHs increase as the angular momentum of the rotating BTZ BH increases. We also observe the behavior of the metric function and the position of the event horizon through red curves. It is noted that the position of the event horizon moves away from the center as the angular momentum of the rotating BH increases see Figs. 2 and 3. It is found the more stable regions above and below the axis are found as the angular momentum of the rotating BH approaches 1. In Fig. 4, we explore the effects of a scalar field on the stability of counter-rotating thin-shell WHs with minimum values of \(\mathcal {B}\) and \(\mathcal {J}\). It can be seen from Figs. 2 and 4, the stable regions become larger in the presence of scalar field parameters. Similarly, we discuss the stability regions for \(\mathcal {B}=0.1\) and \(\mathcal {J}\Rightarrow 1\) see Fig. 5 and comparatively effects of the scalar field can be analyzed by comparing Fig. 3 and 5. Then, we discuss the stability of the WH structure when \(\mathcal {B}\Rightarrow 1\) with minimum values of \(\mathcal {J}=0.1,0.2\) as shown in Fig. 6 and higher values of angular momentum \(\mathcal {J}=0.5,0.7\) as shown in Fig. 7. It is interesting to mention that the developed structure shows maximum stable regions for the choice of higher values of scalar field parameter smaller values of angular momentum see Fig. 6. Hence, the scalar field and rotation of the BH structure greatly affect the geometrical configurations of the counter-rotating thin-shell WHs.

Equations of state

In recent years, there has been growing interest in studying the stability of counter-rotating thin-shell WHs under different equations of state. One such EoS that has gained attention is the phantom-like equation of state, which violates the null energy condition. Additionally, researchers have also explored the stability of counter-rotating thin-shell WHs using the generalized Chaplygin gas EoS, which has been proposed as a unified model for dark matter and dark energy. Investigating the stability of these WHs under these equations of state. In this regard, we begin the analysis by considering the phantom-like EoS which is written as \(p=\chi \sigma \) with \(\chi <0\). Cosmological evidence supports the hypothesis that the Universe’s accelerated expansion is powered by an exotic fluid with an equation of state written as \(\chi = p/\sigma < -1\). This scenario is referred to as phantom energy. This special fluid violates the null energy criterion that is necessary for stable traversable WHs to exist. The idea of phantom energy has renewed interest in the study of WHs and provided exciting new directions for future measurements of distant supernovae and the Cosmic Microwave Background (CMB). Lobo et al.63 presented the new exact WH solutions backed by phantom energy. They determined the metric, energy density, and pressure distribution required to support these WH constructions by carefully crafting a particular shape function, and also examined important aspects of the resulting spacetime. Notably, depending on which parameter choices are used, it is shown that the WH’s mass function can be either finite or infinite63. The presence of exotic matter sources that violate the null energy condition (NEC) is necessary for traversable WHs to exist in General Relativity; however, modified gravity theories may be able to overcome this need64. This work explored the energy states of static spherically symmetric traversable Morris-Thorne WHs in a recently proposed feasible f(R) gravity model64.

The energy conservation law for counter-rotating WH components is founded on the notion that energy may only be transported or changed, rather than generated or destroyed. This means that the entire energy of matter and fields entering one end of a WH must be equal to the total energy departing the other. Furthermore, any oscillations in energy within the WH must follow this conservation equation for it to remain stable and intact. The respective expression of the energy conservation law was given as

$$\begin{aligned} p\frac{d}{d\tau }(2\pi \Delta )+\frac{d}{d\tau }(2\sigma \pi \Delta )=0, \end{aligned}$$
(23)

which yields \(\frac{d\sigma }{d\tau }+\frac{\dot{\Delta }}{\Delta }(\sigma +p)=0.\) For the phantomlike EoS as \(p=\chi \sigma \), we get

$$\begin{aligned} \frac{d\sigma }{d\Delta }+\frac{\sigma }{\Delta }(1+\chi )=0, \end{aligned}$$
(24)

which leads to

$$\begin{aligned} \sigma (\Delta )=\sigma (\Delta _0) \left( \frac{\Delta _0}{\Delta }\right) ^{(1+\chi )}. \end{aligned}$$
(25)

By using conservation equation (25), we have

$$\begin{aligned} \mathcal {H}(\Delta )=-\frac{1}{4} \pi ^2 \Delta _0 ^2 \sigma _0^2 \left( \frac{\Delta _0 }{\Delta }\right) ^{2 \chi }+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}, \end{aligned}$$
(26)

it can be expressed as

$$\begin{aligned} \mathcal {H}(\Delta )=-\frac{1}{4} \pi ^2\Upsilon \Delta ^{-2 \chi }+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}, \end{aligned}$$
(27)

where \(\Upsilon =\sigma _0^2 \Delta _0^{2 (\chi +1)}\).

Figure 8
figure 8

Plots of potential function of counter-rotating thin-shell WH filled with quintessence-type fluid distribution for different values of angular momentum as \(\mathcal {J}=0.4\) (left plot) and \(\mathcal {J}=1\) (right plot) with \(\mathcal {A}=10,\chi =-0.1,\mathcal {M}=1,\mathcal {L}=1\).

Full size image
Figure 9
figure 9

Plots of potential function of counter-rotating thin-shell WH filled with dark energy-type fluid distribution for for different values of angular momentum as \(\mathcal {J}=0.4\) (left plot) and \(\mathcal {J}=1\) (right plot) with \(\mathcal {A}=10,\chi =-0.5,\mathcal {M}=1,\mathcal {L}=1\).

Full size image

At \(\Delta =\Delta _0\), the corresponding second derivative of the potential function concerning \(\Delta \) become

$$\begin{aligned} \mathcal {H}''(\Delta _0)= & {} \frac{1}{2} \pi ^2 \Upsilon \chi (-2 \chi -1) \Delta _0 ^{-2 \chi -2}+\frac{\mathcal {J}^2}{2 \Delta _0 ^4}+\frac{5 \mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{9 \Delta _0 ^6}-\frac{4 \mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})}{3 \Delta _0 ^5}-\frac{4 \mathcal {B} \mathcal {M}}{3 \Delta _0 ^3}+\frac{2}{\mathcal {L}^2}. \end{aligned}$$
(28)

We assess the stability of counter-rotating thin-shell WHs for various values of the EoS parameter, which correspond to the various kinds of matter contents. Assuming \((\chi >-1/3)\), we first examine the quintessence type matter configuration at the shell. Fig. 8 illustrates the consistent behavior of the constructed structure by graphical analysis of potential function, which is an intriguing observation. Unstable configurations result from the potential function’s concave-up tendency. The derived structure for the dark energy type matter configuration \((\chi <-1/3)\) exhibits concave down behavior, leading to an unstable configuration of the counter-rotating thin-shell WHs Fig. 9. Next, we take into account the fluid content \((\chi <-1)\), which represents the phantom energy type and illustrates the potential function’s concave downward feature. In light of this, we obtain the unstable configuration (\(\mathcal {H}''(\Delta _0)<0\)) of the counter-rotating thin-shell WHs for phantom energy (see Fig. 10).

A recent suggestion presented a family of simple cosmological models based on special perfect fluids65. According to one of these ideas, the cosmos is made up of a material called Chaplygin gas, a perfect fluid with the equation of state \(p = -\mathcal {G}/\sigma \), where \(\mathcal {G}\) is a positive constant. This model represents a transition from an accelerated de Sitter stage to a slowed-down cosmic expansion. Another option is the inhomogeneous Chaplygin gas, which can play a dual role, possessing properties of both dark matter and dark energy66. For investigating the stability of counter-rotating thin-shell WHs, the generalized Chaplygin gas EoS is a suitable choice due to its unique properties. Its peculiar, negative pressure-producing nature makes it possible to look into unconventional energy sources that might be responsible for the existence of WHs. Its application in cosmology to explain the universe’s fast expansion also provides a foundation to its successful usage in examining the stability of these intriguing formations. A sort of dark energy with peculiar properties, such negative pressure can be explained mathematically by the generalized Chaplygin gas EoS. The ability of this EoS to explain the universe’s rapid expansion has led to extensive research in the field of cosmology. Furthermore, there is the two-parameter generalized Chaplygin gas model, which is given as67,68

$$\begin{aligned} p=-\frac{\mathcal {G}}{\sigma ^\lambda }, \quad 0<\lambda \le 1. \end{aligned}$$
(29)

These cosmological models have the potential to provide a cohesive macroscopic phenomenological explanation of dark energy and dark matter, expanding upon the conventional \(\Lambda \)CDM models68. Using the conservation Eq. (15), we get

$$\begin{aligned} 0= & {} \mathcal {G} \Delta -2^{\lambda } \pi ^{-\lambda -1} \left( 2 \ddot{\Delta }+\frac{-9 \mathcal {J}^2 \Delta ^2-8 \mathcal {J}^2 \mathcal {B}^2+6 \Delta \mathcal {B} \left( 2 \Delta ^2 \mathcal {M}-3 \mathcal {J}^2\right) +\frac{36 \Delta ^6}{\mathcal {L}^2}}{18 \Delta ^5}\right) \\&\quad \times&\left( -\frac{1}{\Delta }\sqrt{\dot{\Delta }^2+\frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}+\mathcal {M} \left( -\frac{2 \mathcal {B}}{3 \Delta }-1\right) +\frac{\Delta ^2}{\mathcal {L}^2}}\right) ^{\lambda -1}. \end{aligned}$$

By using \(\dot{\Delta }=0=\ddot{\Delta }\) at \(\Delta =\Delta _0\), above equation becomes

$$\begin{aligned} 0= & {} \mathcal {G} \Delta _0 -\frac{2^{\lambda -1} \pi ^{-\lambda -1}}{9 \Delta ^5_0} \left( -9 \mathcal {J}^2 \Delta _0 ^2-8 \mathcal {J}^2 \mathcal {B}^2+6 \Delta _0 \mathcal {B} \left( 2 \Delta ^2_0 \mathcal {M}-3 \mathcal {J}^2\right) +\frac{36 \Delta ^6_0}{\mathcal {L}^2}\right) \nonumber \\\times & {} \quad \left( -\frac{1}{\Delta _0 }\sqrt{\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{36 \Delta ^4_0}+\mathcal {M} \left( -\frac{2 \mathcal {B}}{3 \Delta _0 }-1\right) +\frac{\Delta ^2_0}{\mathcal {L}^2}}\right) ^{\lambda -1}. \end{aligned}$$
(30)

Now, we consider the left side of the above equation as \(G(\Delta _0)=0\) and observe the graphical behavior to determine the respective position of the equilibrium throat. For different values of \(\mathcal {B}\), we determine the position of equilibrium shell radius. It is noted that \(\Delta _0= 0.3, 0.33, 0.36, 0.39\) for \(\mathcal {B}=0.1, 0.3,0.5,0.7\) with specific values of physical parameters as shown in the left plot of Fig. 11.

Figure 10
figure 10

Plots of potential function of counter-rotating thin-shell WH filled with phantom energy type fluid distribution for for different values of angular momentum as \(\mathcal {J}=0.4\) (left plot) and \(\mathcal {J}=1\) (right plot) with \(\mathcal {A}=10,\chi =-2.5,\mathcal {M}=1,\mathcal {L}=1\).

Full size image
Figure 11
figure 11

Study of equilibrium throat radius through \(G(\Delta _0)=0\) as shown in the left plot with \(\mathcal {M}=0.1,\mathcal {L}=0.5,\mathcal {J}=0.5,\mathcal {G}=0.5,\lambda =1\). The right plot represents the potential function of the counter rotating thin-shell WH for respective calculated values of equilibrium throat radius \(\Delta _0= 0.3, 0.33, 0.36, 0.39\) for \(\mathcal {B}=0.1, 0.3,0.5,0.7\) respectively with \(\mathcal {M}=0.1,\mathcal {L}=0.5,\mathcal {J}=0.5,\mathcal {G}=0.5,\lambda =1\).

Full size image

For this EoS, the respective conservation equations become

$$\begin{aligned} \dot{\sigma }+\frac{\dot{\Delta }}{\Delta } \left( \frac{\sigma ^{\lambda +1}-\mathcal {G}}{\sigma ^\lambda }\right) =0, \end{aligned}$$
(31)

hence we get

$$\begin{aligned} \sigma =\left( \left( \frac{\Delta _0}{\Delta }\right) ^{\lambda +1} \left( \sigma _0^{\lambda +1}-\mathcal {G}\right) +\mathcal {G}\right) {}^{\frac{1}{\lambda +1}}. \end{aligned}$$
(32)

The respective effective potential function becomes

$$\begin{aligned} \mathcal {H}(\Delta )= & {} -\frac{1}{4} \pi ^2 \Delta ^2 \left( \mathcal {G}+\left( \frac{\Delta _0 }{\Delta }\right) ^{\lambda +1} \left( \left( \frac{\pi }{2}\right) ^{-\lambda -1} \left( -\frac{\sqrt{\frac{\mathcal {J}^2 (3 \Delta _0 +2 \mathcal {B})^2}{36 \Delta _0 ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta _0 }+1\right) +\frac{\Delta _0 ^2}{\mathcal {L}^2}}}{\Delta _0 }\right) ^{\lambda +1}-\mathcal {G}\right) \right) ^{\frac{2}{\lambda +1}}\\+ & {} \quad \frac{\mathcal {J}^2 (3 \Delta +2 \mathcal {B})^2}{36 \Delta ^4}-\mathcal {M} \left( \frac{2 \mathcal {B}}{3 \Delta }+1\right) +\frac{\Delta ^2}{\mathcal {L}^2}. \end{aligned}$$

We presently intend to investigate how the stability of the counter-rotating thin-shell WHs are affected by generalized Chaplygin gas. To plot the aforementioned potential function, we utilize the equilibrium throat radius calculations for particular values of \(\mathcal {B}\). It should be noted that, for the computed values of equilibrium throat radius, the potential function represents the local minima (see Fig. 11. As a result, the potential function exhibits minimal behavior in this position, signifying the concave up behavior. As a result, our constructed structure becomes stable for these kinds of matter contents, as the Fig. 11 right plot illustrates.

Concluding remarks

In general, theoretical physics research on counter-rotating thin-shell WHs are intriguing and could have significant effects on our comprehension of space-time and the nature of the cosmos. Our concept of space-time could be completely altered by research on counter-rotating thin-shell WHs, which could also lead to new directions in theoretical physics. Nevertheless, the existence of a hairy WH solution complicates the simplicity suggested by the no-hair theorem when examining the creation of a thin-shell WH using a (2+1)-dimensional rotating BH and incorporating a nonminimally linked scalar field. Introducing a scalar field as a nonminimally coupled component complicates the system and causes deviations from the traditional parameters that usually define BHs. The presence of additional fields or features in the WH solution indicates its hairy nature, which surpasses the conventional description of black holes. This emphasizes the complexities of gravitational systems in spacetimes with lower dimensions. This paper is devoted to studying the counter-rotating thin-shell WHs developed from two equivalent copies of rotating BHs coupled scalar field with opposite rotation. For this purpose, we considered the azimuthal coordinate transformation to match these spacetimes at the hypersurface. Then, by using Lanczos equations, the components of matter contents are calculated. We are interested in exploring the effects of the scalar field on the dynamical configurations of the shell through linearized radial perturbation and specific choice of EoS, i.e., phantomlike EoS and generalized Chaplygin gas. A detailed discussion of both choices is given below:

  • Linearized Radial Perturbation: First, we discuss the effects of low and higher values of angular momentum on the stability of counter-rotating thin-shell WHs developed from rotating BTZ BHs (\(\mathcal {B}=0\)) (Figs. 2 and 3). It is found that position of the event horizon moves away from the center and stability regions increase as the angular momentum of rotating BTZ BH increases. Then, we explore the effects of the scalar field on the stability of counter-rotating thin-shell WHs by considering (\(\mathcal {B}\ne 0\)) with different values of \(\mathcal {B}\) and \(\mathcal {J}\) (Figs. 4, 5, 6 and 7). The maximum stability regions are found for smaller values of angular momentum and higher value of scalar field parameter (Fig. 6).

  • Phantomlike EoS: For different ranges of phantomlike EoS parameters represent the different types of matter contents. It is noted that the developed structure shows stable behavior only for the choice of quintessence type fluid distribution and depicted unstable behavior for both dark energy as well as phantom energy type matter contents (Figs. 8, 9 and 10).

  • Generalized Chaplygin Gas: For such type of matter contents, the potential function shows concave up behavior for the calculated values of equilibrium shell radius for specific values of scalar field parameter. Hence, it denoted that the developed structure of counter-rotating thin-shell WHs filled with generalized Chaplygin gas shows stable behavior (Fig. 11).

The study investigates the stability and behavior of a thin-shell WH that is constructed with a scalar field and is subjected to rotation. This research provides insights into the interaction between gravity, scalar fields, and unusual events in lower dimensions. The study of the impact of the scalar field on the stability of counter-rotating thin-shell WHs offers valuable insights into how supplementary fields can strengthen the structural stability of these unusual spacetime formations. Hence, the study not only enhances our comprehension of intricate gravitational systems in lower dimensions but also questions the validity of the no-hair theorem by demonstrating the presence and importance of hairy WH solutions in these distinct spacetime configurations. Finally, it is concluded that the presence of a scalar field enhances the stability of the developed structure for a smaller value of angular momentum. The results obtained from this study will clarify the relationship between rotating BHs, nonminimally linked scalar fields, and the development as well as maintenance of WH structures, in addition to further our knowledge of the stability of counter-rotating thin-shell WHs. This information might ultimately affect how we see the basic structure of spacetime and the viability of interstellar travel. In summary, our study in (2+1) dimensions clarify the complex interaction between scalar fields and gravity in the framework of thin-shell WHs and revolving BHs. Although lower dimensions are simpler, our results offer significant insight that can be applied to higher-dimensional systems, allowing us to gain a better understanding of the dynamics of scalar fields and gravity in 4D spacetime. This research highlights the important ramifications of these connections across multiple dimensions, going beyond simple mathematical exploration.