Abstract
The motivation for constructing a thinshell wormhole from a (2+1)dimensional rotating black hole arises from the desire to study the effects of a nonminimally coupled scalar field in this particular spacetime. By investigating the behavior of such a field in the presence of rotation, we can gain insights into the interplay between gravity and scalar fields in lowerdimensional systems. Additionally, this construction allows us to explore potential connections between black hole physics and exotic phenomena like traversable wormholes. The radial perturbation around the equilibrium throat radius is considered to explore the stable configuration for specific values of physical parameters. Then, the equations of state, specifically the phantomlike and generalized Chaplygin gas model for exotic matter is used to conduct an extensive investigation into the stability of the counterrotating thinshell wormholes. Our results show that the presence of a scalar field enhances the stability of the counterrotating thinshell wormholes.
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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 faroff cosmological locations. In spherically symmetric static spacetimes, the most intriguing cosmic entity that can be produced using general relativity is the timelike thinshells. These cosmological object models have been used to examine several astrophysical events, including supernova explosions and gravity collapse. In 1966, Israel^{1} published a significant study that offered a concrete formalism for building timelike shells in general by joining two distinct manifolds at the thinshell 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 selfgravitating thinshells. Finding out if pertinent thinshell 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 thinshell formalism^{1}, determine the presence of these exotic matters near the WH’s throat. According to Visser^{2}, 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.
Thinshell 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 phantomlike 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 models^{3,4,5,6,7,8,9,10,11,12,13}. By examining minor radial perturbations around the equilibrium throat radius, the stability of thinshell WHs has been studied. By examining radial perturbations, Poisson and Visser^{14,15} investigated the linear stability of WHs. Taking into account the existence of a cosmological constant, Lobo and Crawford^{16} examined the stable construction of spherically symmetric thinshell WHs. Numerous studies^{17,18,19,20,21,22,23,24,25,26,27,28,29,30} have examined the stability of thinshell WHs in both stable and unstable forms.
A gravitational vacuum star, or gravastar, is a wellknown thinshell geometrical structure having outside black hole (BH) geometry and inside de Sitter geometry^{31}. Numerous investigators created this geometric arrangement against the backdrop of various BH spacetimes and also looked into its stability with various EoS^{32,33,34,35,36}. In^{37}, the prototype gravastar model with internal de Sitter spacetime and external Schwarzschild BH is examined, while in^{38}, a gravastar with phantom energy is examined. For different matter distributions at thinshell, 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 Schwarzschildde Sitter and RN spacetimes with interior de Sitter region with specific EoS^{39,40}. The gravastar model in the background of noncommutative geometry is also explored in^{41,42}. 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 in^{43,44}. The study of thinshell gravastars developed from inner de Sitter and outer various BH geometries is presented in^{45,46,47,48,49}.
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 higherdimensional or (3+1)dimensional BHs. In contrast to their higherdimensional 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 EinsteinMaxwell theory^{51} and the EinsteinMaxwelldilaton theory^{52}. By imposing a traceless constraint on the energymomentum 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 higherdimensional BHs. In the framework of charged rotating (2+1)dimensional BH, the stable configurations of WH structure are discussed in^{54}.
Hassaine and Martinez^{55} 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 source^{56}. 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 EinsteinpowerMaxwell (EPM) theory without the use of a traceless condition. The authors study the properties of BHs in a threedimensional spacetime named HMTZ BH^{58}. They consider the dynamics that are introduced to the system when gravity is coupled to a scalar field^{58}. 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 threedimensional physics of BHs after looking at the solutions to the field equations. The BH solution minimally coupled with the scalar field is also presented in^{59}. This research advances our knowledge of the intricate interplay between scalar fields and gravity in lowdimensional spacetimes^{58,59}. Thermodynamical properties of (2+1)dimensional rotating BHs with hair is investigated in^{60}. 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 thinshell 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 counterrotating thinshell 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 counterrotating thinshell WHs is developed in Section “Formalism of counterrotating thinshell wormholes“. The stability of counterrotating thinshells are examined in Section “Stability analysis“ by using linearized radial perturbation and EoS (phantomlike 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 higherdimensional spacetimes, it allows for simpler mathematical models. Three dimensions are simpler, making computations easier to handle and providing new information on BH properties. For threedimensional spacetime, the action with a nonminimally coupled scalar field can be written as^{60}
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 be^{60}
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 as^{60}
where
Also, the scalar field becomes
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 as^{60}
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.
In the subsequent sections, we developed the counterrotating thinshell WHs from the considered rotating BHs through wellknown cut and paste approach.
Formalism of counterrotating thinshell wormholes
With implications for multiple gravitational events and dualities with other theories like KalbRamond gravity, nonminimal scalar couplings to curvature are important in gravitational theories. Several important themes emerge when these characteristics are extended to lowerdimensional systems, especially when considering thinshell 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 nonminimal scalar fields in lowerdimensional 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 thinshell WH in the framework of (2+1)rotating hairy BH by adopting Visser’s cutandpaste method. The geometry of thinshell 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 faroff 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:
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
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 counterrotating. This represents that the lower and upper shells are counterrotating, 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 by^{62}
Hence, the respective line element (3) can be expressed with the counterrotating frame on the timelike hypersurface (\(\Sigma \)) as
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
The development of a WH throat, which can be seen as a onedimensional 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 nonminimally 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 wellestablished 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 thinshell WH barrier, giving a consistent framework for analyzing the system’s stability and attributes. While adjustments may be required in gravity theories with nontrivial couplings or extra fields, the conventional junction conditions’ simplicity and consistency makes them an appropriate choice for analyzing the counterrotating thinshell 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 stressenergy tensor components in this WH throat have nonzero values and can be computed using the second fundamental form referred as extrinsic curvature (\(K_{ij}^{\pm }\)) given as
where \(n_{\alpha }^{\pm }\) denotes the outward directed unit normal vectors. Mathematically, for considered manifolds, it can be expressed as
with \(\dot{\Delta }=d\Delta /d\tau \). Hence, we get
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
here stressenergy 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
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
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 3acceleration, which mathematically captures this phenomenon. In this regard, the observer 3acceleration 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 nonzero component of 3acceleration is given as
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
In the next section, we explore the stability of the counterrotating thinshell WHs through linearized radial perturbation about equilibrium shell radius and specific choices of EoS.
Stability analysis
The stability of counterrotating hairy thinshell 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 thinshell WHs are stable or unstable.
Linearized radial perturbation
To examine the stability of counterrotating hairy thinshell 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 thinshell 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
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 uptoorder terms
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
with
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
Figures 2, 3, 4, 5, 6 and 7 are used to explore the stability of counterrotating thinshell 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 counterrotating thinshell 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 counterrotating thinshell 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 counterrotating thinshell WHs.
Equations of state
In recent years, there has been growing interest in studying the stability of counterrotating thinshell WHs under different equations of state. One such EoS that has gained attention is the phantomlike equation of state, which violates the null energy condition. Additionally, researchers have also explored the stability of counterrotating thinshell 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 phantomlike 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 infinite^{63}. 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 need^{64}. This work explored the energy states of static spherically symmetric traversable MorrisThorne WHs in a recently proposed feasible f(R) gravity model^{64}.
The energy conservation law for counterrotating 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
which yields \(\frac{d\sigma }{d\tau }+\frac{\dot{\Delta }}{\Delta }(\sigma +p)=0.\) For the phantomlike EoS as \(p=\chi \sigma \), we get
which leads to
By using conservation equation (25), we have
it can be expressed as
where \(\Upsilon =\sigma _0^2 \Delta _0^{2 (\chi +1)}\).
At \(\Delta =\Delta _0\), the corresponding second derivative of the potential function concerning \(\Delta \) become
We assess the stability of counterrotating thinshell 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 concaveup 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 counterrotating thinshell 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 counterrotating thinshell WHs for phantom energy (see Fig. 10).
A recent suggestion presented a family of simple cosmological models based on special perfect fluids^{65}. 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 sloweddown cosmic expansion. Another option is the inhomogeneous Chaplygin gas, which can play a dual role, possessing properties of both dark matter and dark energy^{66}. For investigating the stability of counterrotating thinshell WHs, the generalized Chaplygin gas EoS is a suitable choice due to its unique properties. Its peculiar, negative pressureproducing 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 twoparameter generalized Chaplygin gas model, which is given as^{67,68}
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 models^{68}. Using the conservation Eq. (15), we get
By using \(\dot{\Delta }=0=\ddot{\Delta }\) at \(\Delta =\Delta _0\), above equation becomes
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.
For this EoS, the respective conservation equations become
hence we get
The respective effective potential function becomes
We presently intend to investigate how the stability of the counterrotating thinshell 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 counterrotating thinshell WHs are intriguing and could have significant effects on our comprehension of spacetime and the nature of the cosmos. Our concept of spacetime could be completely altered by research on counterrotating thinshell 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 nohair theorem when examining the creation of a thinshell 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 counterrotating thinshell 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 counterrotating thinshell 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 counterrotating thinshell 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 counterrotating thinshell WHs filled with generalized Chaplygin gas shows stable behavior (Fig. 11).
The study investigates the stability and behavior of a thinshell 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 counterrotating thinshell 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 nohair 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 counterrotating thinshell 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 thinshell WHs and revolving BHs. Although lower dimensions are simpler, our results offer significant insight that can be applied to higherdimensional 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.
Data availability
All data generated or analysed during this study are included in this published article.
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Acknowledgements
Faisal Javed acknowledges Grant No. YS304023917 to support his Postdoctoral Fellowship at Zhejiang Normal University. This research is also supported by Researchers Supporting Project number RSPD2024R650, King Saud University, Riyadh, Saudi Arabia.
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Javed, F., Fatima, G., Ashebo, M.A. et al. Stability of lower dimensional counterrotating thinshell wormholes with scalar hair. Sci Rep 14, 17277 (2024). https://doi.org/10.1038/s41598024625902
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DOI: https://doi.org/10.1038/s41598024625902
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