Introduction

The study of AdS/CFT duality1,2and the fundamental characteristics of BHs frequently refer to (2 + 1)-dimensional gravity. It has been demonstrated that in (3 + 1)-dimensions, classical field perturbations cause a curvature singularity at the Cauchy horizon in the case of rotating BHs and spherically symmetric, electrically charged BHs, with or without a positive cosmological constant. The Banados-Teitelboim-Zanelli (BTZ) BH3is the most well-known and the first BH solution in (2+1) dimensions. It is a precise solution of the (2+1)-dimensional Einstein equation without a matter source. This spacetime is a regular solution with a non-singular center comprising the Ricci scalar, a negative cosmological constant, and the electromagnetic invariant, which can be coupled to gravity either linearly or non-linearly. The BTZ BH with chargedspinning variations were discovered in4-6. Including matter sources other than the Maxwell field and observing whether new BH solutions arise and its charged rotating versions (such as the duality relationships) continue to hold is both appealing and difficult. The history of studying the hairy expansions of the BTZ BH is probably as old as the BTZ BH itself. There are currently numerous explicitly hairy extensions of BTZ BHs that can be either static, charged, or rotating. The extra hair is typically a scalar field that couples to gravity either minimally or non-minimally. The scalar may or may not couple to itself via a scalar potential. Hairy BHs present an intriguing answer for theories involving gravity. Over the past few years, they have been the subject of great interest, mostly about the no-hair theorem. Since the scalar field can diverge on the horizon and BHs can become unstable, four-dimensional BHs coupled to scalar fields in asymptotically flat spacetimes are often excluded by the no-hair theorem. Such scenarios are not physically consistent7-10. The hairy BH solutions in asymptotically flat spacetimes are simply nonexistent in higher-dimensional conjectures \((D> 4)\)11,12. Hairy BH minimally connected to a scalar field in asymptotically dS spacetimes has been presented in13, but it was discovered to be dynamically unstable. Also, the exact BH solutions in dS spacetime with a nonminimally coupled scalar field have been evaluated and those solutions were also unstable14-17.

In the asymptotic flat situation, the interior of four and five-dimensional rotating BHs with scalar hair has been explored in18, where it is discovered that the inner horizon does not exist. Rotating BTZ BHs are known to exist in three dimensions3, and they have been considered potential candidates for consistent quantum gravity models19,20. Naturally, it should be interesting to investigate any modification of this toy model. It is noteworthy that rotating BHs in three-dimensional (2+1) Einstein gravity with nonminimally21and minimally22 connected scalar hairs have also been observed. In three-dimensional gravity, Zou et al.23derived an exact expression of the revolving BH solution nonminimally connected with the scalar field and evaluated the thermodynamics. They found that the thermodynamical conduct of this revolving hairy BH is the same as those of the revolving BH nonminimally connected with scalar hair. Xu and Zou24obtained the exact BH solutions with nonminimally connected scalar field in (2+1)-dimensional metric in the framework of Einstein-Power-Maxwell theory. Similarly, in (2+1)-dimensional metric, Karakasis and his collaborators25-28 discussed some aspects of BHs as well as rotating BHs connected with scalar fields in different frameworks. Gao et al.29 constructed the hairy rotating BH solutions in (2+1)-dimensional Einstein gravity coupled to a complex scalar field. Bakopoulos et al.30 presented explicit BHs in the context of beyond Horndeski theories that are equipped with fundamental scalar hair. They used the variable scalar charge to examine the features and attributes of their solutions. Lei et al.31 analyzed the quasinormal modes and superradiant instability of a rotating hairy BH, which possesses a Horndeski hair as deviation from Kerr BH with the perturbation of a massive scalar field.

The inspection of the wormhole (WH) compositions and their solutions is currently among the most exciting research areas of general relativity (GR) which is attracting the interest of numerous scholars. By GR, the occurrence of these enigmatic structures is made conceivable by the deformable metric generated by energy or matter. The WHs serve as the bridges joining two distinct locations on a manifold. The WHs have an asymptotically flat structure and this notion was first introduced by Flamm32. Then, Einstein and Rosen33introduced the WH structure acknowledged as the “Einstein-Rosen bridge” also recognized as the Einstein field equation’s solution in vacuum. The initial traversable WH structure was demonstrated by Morris and Thorne34. The presence of exotic matter plays a remarkable role in making a traversable WH and such type of matter contents violate the null energy constraints34-37. The presence of such matter contents is the major issue in developing the traversable WH structure. A most suitable approach to reduce such matter contents at the edges and corners of WH throat is presented in38,39. The components of matter contents located at thin-shell (TS) can be evaluated through the Darmois-Israel formalism40,41. A range of universe-wide WH configurations can be explored by examining stability, either by employing radial oscillations or by accounting for the appearance of an EoS for the exotic material in the WH throat. If the traversable WHs show stable constructions against linear oscillations, then they are more important. To preserve real symmetries, several authors have studied the thin-shell WHs (TSWHs) and the WH stable structures using linear perturbations. Poisson and Visser42examined the stable Schwarzschild TSWH structure. Lobo and Crawford43studied the stable composition of TSWHs with the cosmological constant and found that the viable findings appear merely for the positive options of the cosmological constant. The stable WH results with the influence of an electric field were identified by Eiroa and Romero44, who also demonstrated that WHs have stable geometries, but only for specific charge factors. According to Bronnikov et al.45, the regular black hole (BH) and TSWH for a phantom scalar field have a stable structure. The researchers also examined the stability of the shell around exact WH solutions in different modified theories of gravity46,47,48,49,50,51,52,53. The stable system of TSWH coupled with the charged regular BH for a non-linear electrodynamics field was demonstrated by Sharif and his co-author54. In the same way, other investigators inspected the stability of TSWH structures in the presence of various circumstances like electric fields, multiple EoSs, and other physical factors55-64. Several authors examined the stability and dynamics of various TS structures by using different BH solutions65,66,67,68,69,70,71.

As we have mentioned above, the various BHs have been utilized in the discussion of TSWH geometries. Rotating BH coupled with a scalar field also plays an important role in analyzing the stability of TSWHs. Recently, Javed et al.72 discussed the stability of TSWHs constructed through a (2+1)-dimensional revolving BH and exhibited that the occurrence of a scalar field raises the stability of the counter-revolving TSWHs. In this research article, we construct a TS from a rotating (2+1)-dimensional BH with a nonminimally coupled scalar field. The analysis of a rotating TS is more complex than that of a spherically symmetric TS developed from spherical symmetric geometry. Mazharimousavi et al.73examined the stability of (2+1)-dimensional rotating TSWHs using the Israel TS formalism. They developed a rotating TS model by positing that the sides of the throat are counter-spinning, concluding that this counter-rotational behavior indicates a stable arrangement. Tsukamoto and Kokubu74investigated the linear stability of a (2+1)-dimensional rotating TSWH by considering the opposing sides of the throat as corotating. They noted that WHs exhibit increased stability with higher angular momentum values. Sharif and Javed examined the collapse and expansion of scalar TS for different categories of BHs75,76,77.

Boson stars are compact objects made up of large and massless scalar fields, and their geometry and dynamics have garnered a lot of attention. Regardless of any particular equation of state, Ruffini and Bonazzola78investigated the stability of self-gravitating stars affected by a scalar field. By determining the scalar field values that result in stable configurations and offering insights into their genesis and stability, Seidel and Suen79examined the dynamics of boson stars. Jetzer80 investigated how gravitational instability contributes to the development of boson stars, examining the relationship between the scalar field’s mass and the stars’ size and mass. Compact stars with a massless scalar field either collapse or oscillate, according to Siebel et al.81. Núñez82 presented a cosmological model in 2016 that included a compact object encased in a spherical shell. The study demonstrated that the impacts of external and self-gravitational forces are balanced by inner tangential pressure. A thin scalar field layer created by the interaction of two Schwarzschild black holes was later studied by Núñez et al.83, who used motion analysis to comprehend the growth and contraction of the layer. The dynamic behavior of such a shell surrounding a charged black hole was investigated by Sharif and Abbas84. In order to provide important insights on boson stars and associated phenomena, Sharif and colleagues expanded their study85-87 to encompass BTZ, charged BTZ, spinning, and standard black holes.

We aim to observe the behavior of such fields with the inclusion of rotation and obtain the results between scalar fields and gravity in lower-dimensional scenarios. This manuscript is depicted as follows: The next section displays the formation of TS from rotating BH coupled with scalar hair. Section III presents the dynamics of TS for massive and massless scalar shells. The final section presents the comparison as well as the graphical interpretation of our results.

Counter-rotating TS formalism from rotating BH with scalar hair

A nonminimally coupled scalar field introduces a new term linking the scalar field to the spacetime curvature. This coupling enables an interaction between the scalar field and gravity, leading to intriguing phenomena such as gravitational waves generated by the scalar field’s dynamics. Research on rotating hairy black holes in three dimensions is particularly fascinating, as it offers a simpler mathematical framework compared to higher-dimensional spacetimes. The reduced complexity of three-dimensional spacetime facilitates easier calculations and provides new insights into the behavior and features of black holes. The action for a nonminimally coupled scalar field in three-dimensional spacetime can be written as21

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

with \(\mathcal {R}\) symbolizes the Ricci scalar, g depicts the determinant of metric tensor \(g^{\mu \nu }\) and \(\mathcal {V}(\psi )\) depicts potential functions of scalar field (\(\psi\)). To illustrate the coupling strength between gravity and the scalar field, \(\xi\)can be assumed to be equal to 1/8. Thus, the respective form of scalar field potential yields21

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

in which the real constants of integration are specified by \(\beta\), a and \(\mathcal {B}\). Also, \(\Lambda =\frac{1}{\mathcal {L}^2}\) presents the cosmological constant and \(\mathcal {L}\)depicts the length factor. The corresponding solution of the rotating BH nonminimally coupled scalar field is now taken into consideration as21

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

Moreover, the scalar field provides

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

Interestingly, by taking into account \(a=\mathcal {J}/6\) and \(\beta =-\mathcal {M}/3\), the above (2+1)-dimensional revolving BH solution can be turned into the revolving BTZ BH in the absence of scalar field for \(\mathcal {B}\)approaches to zero3. Furthermore, the lapse function can be re-written as21

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

The action is not constant when the thermodynamical parameters \(\mathcal {M}\) and \(\mathcal {J}\) of this revolving hairy BH are present in the scalar potential \(\mathcal {V}(\psi )\). We examine the event horizon’s location in Figs. (1) and (2) using a visual illustration of the metric function \(\mathcal {F}(r)\) for different values of the physical parameters. It is seen that by raising the angular momentum and mass of the spinning BH, the event horizon’s location shifts farther from the center of the revolving BH. The event horizon’s position is also impacted by the existence of a scalar field see Figs. (1) and (2).

Fig. 1
Fig. 1
Full size image

Study of the lapse function for distinct choices of physical parameters such as for BTZ BH \(\mathcal {J}=0,\mathcal {B}=0,\mathcal {L}=10\) (left plot) and rotating BTZ BH \(\mathcal {J}=0.3,\mathcal {B}=0,\mathcal {L}=10\) (right plot) by varying mass of the BH.

Fig. 2
Fig. 2
Full size image

Study of the lapse function for distinct choices of physical parameters such as for rotating BTZ BH with a scalar field \(\mathcal {J}=0.3,\mathcal {B}=0.8,\mathcal {L}=10\) (left plot) and \(\mathcal {J}=0.3,\mathcal {B}=1.2,\mathcal {L}=10\) (right plot) by varying mass of the BH.

Here, we use Visser’s cut-and-paste method to create the composition of TSWH inside the context of (2+1)-revolving hairy BH. In this approach, the event horizon and singularity of the BH structure can be eliminated from the developed TS structure. To develop the TS structure, we choose two revolving hairy BH configurations, one external and one internal using the angular momentum \(\mathcal {J}_{-}\) and \(\mathcal {J}_{+}\), respectively. We remove the following regions from the inner and outside spacetimes:

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

with \(\vartheta\) as the shell’s radius while \(r_e\) exhibits the event horizon’s locations. Timelike hypersurfaces are depicted by

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

The respective timelike hypersurfaces are \(\partial \Sigma _{\pm }\equiv \partial \Sigma _{+}=\partial \Sigma _{-}\). Additionally, we match the external and inner manifolds at the hypersurface \(\partial \Sigma\). To create the WH composition that is the most feasible for stellar transit, we implement the exact absolute values of angular momentum for both the internal and external eras of the throat but in opposite directions, such that \(\mathcal {J}_-+\mathcal {J}_+=0\). Thus, the internal and external eras of the throat are counter-revolving which leads to the counter-rotation of the lower and upper shells. To examine the dynamic role of the shell radius (\(\vartheta (t)\)), we propose an azimuthal component \(\Psi _{\pm }\) expressed as

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

Thus, the associated metric (3) can be displayed with the counter-rotation on the timelike hypersurface (\(\Sigma\)) as

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

Furthermore, the shell radius is dependent on proper time \(\tau\), and the hypersurface constituents are represented by the notation \(\xi ^i=(\tau ,\theta )\). The induced metric is expressed by

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

Now, we match the inner and outer spacetime at the hypersurface to develop the geometry of the shell. To calculate the stress-energy tensor components of matter contents located at the shell, we determine the extrinsic curvature (\(K_{ij}^{\pm }\)) provided by

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

with \(n_{\alpha }^{\pm }\) acts as the outwardly directed unit normal vectors. For proposed manifolds, it can be mathematically displayed as

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

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

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

The respective expression of energy-momentum tensor for the shell is expressed as39,40

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

in which stress-energy tensor is illustrated by \(S_{ij}\), \(\eta _{ij}\) describes the induced metric and \(K=tr[K_{ij}]=[K^{i}_{i}]\), where \([K_{ij}]=K^{+}_{ij}-K^{-}_{ij}\). For perfect fluid, we have \(S_{ij}=\left( \sigma +p\right) U_iU_j+p\eta _{ij}\), here energy density is represented with \(\sigma\), surface pressure as p, and velocity is denoted with \(U_i\). The Lanczos equations associated with the rotating TS become

$$\begin{aligned} \sigma =-\frac{1}{\pi }[K^{\psi }_{\psi }], \quad p=\frac{1}{\pi }[K^{\tau }_{\tau }]. \end{aligned}$$
(15)

For a BH spacetime, the respective internal and external pressures are described as \(p_+=\frac{1}{\pi }K^{+\tau }_{\tau }\) and \(p_-=\frac{1}{\pi }K^{-\tau }_{\tau }\), respectively. Thus, the pressures of both metrics are not the same. By inserting Eq.(13) in (15), the following expressions can be attained as

$$\begin{aligned} & \sqrt{\frac{\mathcal {J}^2 (3 \vartheta +2 \mathcal {B})^2}{36 \vartheta ^4}+\frac{\vartheta ^2}{\mathcal {L}^2}-\mathcal {M}_+\left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) +\dot{\vartheta }^2}-\sqrt{\frac{\mathcal {J}^2 (3 \vartheta +2 \mathcal {B})^2}{36 \vartheta ^4}+\frac{\vartheta ^2}{\mathcal {L}^2}-\mathcal {M}_- \left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) +\dot{\vartheta }^2}+\pi \vartheta \sigma =0, \end{aligned}$$
(16)
$$\begin{aligned} & \frac{d}{d\vartheta }\left( \sqrt{\frac{\mathcal {J}^2 (3 \vartheta +2 \mathcal {B})^2}{36 \vartheta ^4}+\frac{\vartheta ^2}{\mathcal {L}^2}-\mathcal {M}_+\left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) +\dot{\vartheta }^2}-\sqrt{\frac{\mathcal {J}^2 (3 \vartheta +2 \mathcal {B})^2}{36 \vartheta ^4}+\frac{\vartheta ^2}{\mathcal {L}^2}-\mathcal {M}_- \left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) +\dot{\vartheta }^2}\right) -\pi p=0. \end{aligned}$$
(17)

These equations can be expressed as

$$\begin{aligned} \frac{d\sigma }{d\vartheta }+\frac{1}{\vartheta }(p+\sigma )=0, \end{aligned}$$
(18)

providing the law of energy conservation of TS as

$$\begin{aligned} \dot{M}+p\dot{A}=0, \end{aligned}$$
(19)

where \(M=\sigma A\) indicates the mass and \(A=2\pi \vartheta\) depicts the shell’s area. For \(M=0\), the form \(M=2\pi \vartheta \sigma\) yields \(\sigma =0\), that is not feasible for a TS. Hence, \(M=0\) does not present a viable situation for TS because its non-vanishing nature exhibits the occurrence of such substances that generates discontinuances in the extrinsic curvature.

Now, we consider the barotropic EoS \(p=k\sigma\), in which k is a constant. The respective results of Eq.(18) are

$$\begin{aligned} \sigma =\sigma _{0}\left( \frac{\vartheta _{0}}{\vartheta }\right) ^{1+k}, \end{aligned}$$
(20)

where \(\vartheta _0\) and \(\sigma _{0}\) manifest the equilibrium shell’s location and shell’s surface density at \(\vartheta _0\), respectively. The mass of the shell then provides

$$\begin{aligned} M=2\pi \sigma _{0}\left( \frac{\vartheta _{0}^{k+1}}{\vartheta ^{k}}\right) . \end{aligned}$$
(21)

Equation (16) yields the EoM of the shell as

$$\begin{aligned} \dot{\vartheta }^2 + \mathcal {H}(\vartheta )=0, \end{aligned}$$
(22)

where

$$\begin{aligned} \mathcal {H}(\vartheta )= & \frac{1}{2} \left( \frac{ (3 \vartheta +2 \mathcal {B})^2}{18 \mathcal {J}^{-2}\vartheta ^4}+\frac{2 \vartheta ^2}{\mathcal {L}^2}-\mathcal {M}_+\left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) -\mathcal {M}_- \left( \frac{2 \mathcal {B}}{3 \vartheta }+1\right) \right) -\frac{(3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{36 \pi ^2 \vartheta ^4 \sigma ^2}-\frac{\pi ^2 \vartheta ^2 \sigma ^2}{4}. \end{aligned}$$
(23)

The energy conservation law is satisfied by equation (22), which manifests that the total kinetic \((\dot{\vartheta }^2)\) and potential \(\mathcal {H}(\vartheta )\) energies components disappear at any time. We briefly observe the dynamics of TS by considering the scalar field.

Dynamics of counter-rotating thin-shell

In cosmology, string theory, and condensed matter physics, among other branches of physics, TS made of scalar fields have been examined. Scalar fields have a remarkable role in exploring the dynamics in string theory, whereas they are utilized in cosmology to explain the universe’s accelerated expansion88,89. A fundamental application of the TS formalism in astrophysics, particularly in the study of objects analogous to boson stars, was introduced by N\(\acute{u}\tilde{n}\)ez et al.82,83. A complex scalar field shows stable configurations of the TS model82,83. Consequently, they examined the toy model of a scalar shell, observing the collapsing and expansion phenomena closely. They also examined the requirements for a stable configuration and entertained several unique configurations, such as the potential for an oscillating shell82,83. Among all things mentioned above, TS composed of scalar fields are intriguing physical systems that may help to resolve significant questions about the nature of the cosmos and its laws. Thin-shell matter contents play a significant role in examining both stable and dynamic configurations. Our focus is on studying the impact of a scalar field on the dynamic of TS. The purpose of this section is to examine the conduct of the effective potential, which is useful in discussing the dynamical characteristics of TS, including saddle points, expansion, and collapse. All of the parameters that are used in this research have been utilized in the literature that investigates the scalar shell dynamics of various BHs82,83. The effective potential for the rotating BH is characterized by

$$\begin{aligned} \mathcal {H}(\vartheta )=\frac{\frac{36 \vartheta ^6}{\mathcal {L}^2}-12 \vartheta ^3 \mathcal {B} (\mathcal {M}_++\mathcal {M}_-)+9 \vartheta ^2 \mathcal {J}^2+12 \vartheta \mathcal {J}^2 \mathcal {B}+4 \mathcal {J}^2 \mathcal {B}^2}{36 \vartheta ^4}-\frac{M^2}{16}-\frac{ \mathcal {M}_++\mathcal {M}_-}{2}-\frac{ (3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \vartheta ^2 M^2}. \end{aligned}$$
(24)

The respective EoM becomes

$$\begin{aligned} \dot{\vartheta }=\pm \sqrt{ \frac{1}{4} \left( \frac{4 (3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \vartheta ^2 M^2}+2 \left( -\frac{\mathcal {J}^2 (3 \vartheta +2 \mathcal {B})^2}{18 \vartheta ^4}-\frac{2 \vartheta ^2}{\mathcal {L}^2}+\frac{2 \mathcal {B} \mathcal {M}_+}{3 \vartheta }+\frac{2 \mathcal {B} \mathcal {M}_-}{3 \vartheta }+\mathcal {M}_++\mathcal {M}_-\right) +\frac{M^2}{4}\right) }, \end{aligned}$$
(25)

here ± represents the expansion and collapse of TS.

Here, we examine the TS’s dynamical in the light of a self-gravitating scalar field composition. To do this, we employ the transformation \(\left( u_{a}=\frac{\Phi _{,a}}{\sqrt{\Phi _{,b}\Phi ^{,b}}}\right)\), which associates the pressure and energy density of a perfect fluid with the differential of the scalar field and the potential function \(V(\Phi )\)82,83. The associated surface density as well as pressure is computed as82,83

$$\begin{aligned} P=\frac{1}{2}\left[ \Phi _{,b}\Phi ^{,b}-2V(\Phi )\right] ,\quad \rho =\frac{1}{2}\left[ \Phi _{,b}\Phi ^{,b}+2V(\Phi )\right] , \end{aligned}$$
(26)

in which \(V(\Phi )\)is the effective potential of scalar field. The stress-energy tensor in the form of scalar field is specified by82,83

$$\begin{aligned} S_{ij}=\nabla _{i}\Phi \nabla _{j}\Phi +\eta _{ij}\left[ V(\Phi )-\frac{1}{2}(\nabla \Phi )^2\right] . \end{aligned}$$

As, the induced line element is considered as a function of proper time \(\tau\), therefore, \(\Phi\) is based on \(\tau\). Hence, Eq.(26) leads to82,83

$$\begin{aligned} \rho =\frac{1}{2}\left[ \dot{\Phi }^2+2V(\Phi )\right] , \quad P=\frac{1}{2}\left[ \dot{\Phi }^2-2V(\Phi )\right] . \end{aligned}$$
(27)

In the form of scalar field, the shell’s total mass is expressed as

$$\begin{aligned} M=2\pi \vartheta \rho = \pi \vartheta [\dot{\Phi }^2+2V(\Phi )]. \end{aligned}$$
(28)

On plugging Eqs.(27) and (28) in (19), we get

$$\begin{aligned} \ddot{\Phi }+\frac{\dot{\vartheta }}{\vartheta }\dot{\Phi }+\frac{\partial V}{\partial \Phi }=0. \end{aligned}$$
(29)

This manifests a famous Klein-Gordon (KG) equation.

We simultaneously solve the energy conservation and KG equations for \(\Phi (\tau )\) and \(\vartheta (\tau )\) to investigate the dynamics of the scalar shell. The effective potential for considered BH with the scalar field is

$$\begin{aligned} \mathcal {H}(\vartheta )= & -\frac{ (3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \pi ^2 \vartheta ^4 \left( 2 V(\Phi )+\dot{\Phi } ^2\right) }-\frac{1}{16} \pi ^2 \vartheta ^2 \left( 2 V(\Phi )+\dot{\Phi } ^2\right) -\frac{1}{2} (\mathcal {M}_++\mathcal {M}_-)\\+ & \frac{\frac{36 \vartheta ^6}{\mathcal {L}^2}-12 \vartheta ^3 \mathcal {B} (\mathcal {M}_++\mathcal {M}_-)+9 \vartheta ^2 \mathcal {J}^2+12 \vartheta \mathcal {J}^2 \mathcal {B}+4 \mathcal {J}^2 \mathcal {B}^2}{36 \vartheta ^4}. \end{aligned}$$

In this article, we examine the EoM of a shell made of a scalar field to analyze its dynamics. The EoM \(\dot{\vartheta }^2+\mathcal {H}(\vartheta )=0\) can be analyzed by using two approaches. In the first, the pressure of the shell is considered as an explicit form of its radius, while in the second, the presence of an EoS links the pressure and density of the shell. As we previously discussed, the energy conservation law is represented by the equation \(\dot{\vartheta }^2+\mathcal {H}(\vartheta )=0\), which says that the total of the “kinetic contribution” \(\dot{\vartheta }^2\) and “potential contribution” \(\mathcal {H}(\vartheta )\) yields zero at all times. This is more of a condition than a dynamical equation because it states that the system can only evolve under conditions that meet the conservation equation. The shell cannot propagate because of \(\mathcal {H}(\vartheta )>0\). Consequently, only solutions with a negative or zero effective potential, that is, \(\mathcal {H}(\vartheta )<0\) or \(\mathcal {H}(\vartheta )=0\), are permitted. The second scenario, denoted by \(\mathcal {H}(\vartheta )=0\), can be attributed to either a stationary structure or critical points that result in \(\dot{\vartheta }^2=0\), which means orbits with an extremal radius \(\vartheta\). Initially, we explore the dynamics of the scalar shell in the first scenario (\(\dot{\vartheta }\ne 0\)) for both massive and massless scalar fields. By providing an initial radius, for example, \(\vartheta (\tau )\), the differential equations, such as the KG equation and conservation equation can be numerically integrated.

The constituents of the scalar field are not interconnected in any specific way by the EoS. The energy density of the scalar field is divided into potential and kinetic energies, rendering any single value of P insufficient to adequately represent the energy density. We utilize a specific formulation of \(\mathcal {\mathcal {V}}(\Phi )\) to establish a connection between each component of the stress-energy tensor and to ascertain the effective mass. Subsequently, we will examine two separate values of the potential function, referred to as:

  • Massless scalar field: \(V(\Phi )=0\),

  • Massive scalar field: \(V(\Phi )=M^2\Phi ^2\).

In the following subsections, we are interested in investigating the impact of massless and massive scalar fields on the expansion and collapsing behavior of a TS.

Massless scalar shell

Initially, we study the shell dynamics in the case of a massless scalar field or the absence of a scalar potential function. A theoretical physical field without a rest mass is called a massless scalar field. A scalar particle, which lacks spin and moves like a wave, represents this field. In quantum field theory, massless scalar fields are significant and vital to our comprehension of the basic forces of existence. Since pressure and surface density (\(P=\rho\)) have a direct relationship as a result of the vanishing \(V(\Phi )\), we do not require separate EoS in this case. Employing the constraint \(V(\Phi )=0\), the KG equation leads to

$$\begin{aligned} \ddot{\Phi }+\frac{\dot{\vartheta }}{\vartheta }\dot{\Phi }=0, \end{aligned}$$

whose integration provides

$$\begin{aligned} \dot{\Phi }=\frac{\lambda }{\vartheta }, \end{aligned}$$

where \(\lambda\) acts as the constant of integration. Hence, we get

$$\begin{aligned} \dot{\vartheta }^2= & \frac{\pi ^2 \lambda ^4}{16 \vartheta ^2}-\frac{\frac{36 \vartheta ^6}{\mathcal {L}^2}-12 \vartheta ^3 \mathcal {B} (\mathcal {M}_++\mathcal {M}_-)+9 \vartheta ^2 \mathcal {J}^2+12 \vartheta \mathcal {J}^2 \mathcal {B}+4 \mathcal {J}^2 \mathcal {B}^2}{36\vartheta ^4}+\frac{ (3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \pi ^2 \lambda ^4}\nonumber \\+ & \frac{1}{2} (\mathcal {M}_++\mathcal {M}_-). \end{aligned}$$
(30)

The corresponding effective potentials are

$$\begin{aligned} \mathcal {H}(\vartheta )= & \frac{1}{4} \left( -\frac{\pi ^2 \lambda ^4}{4 \vartheta ^2}+\frac{\frac{36 \vartheta ^6}{\mathcal {L}^2}-12 \vartheta ^3 \mathcal {B} (\mathcal {M}_++\mathcal {M}_-)+9 \vartheta ^2 \mathcal {J}^2+12 \vartheta \mathcal {J}^2 \mathcal {B}+4 \mathcal {J}^2 \mathcal {B}^2}{9 \vartheta ^4}\right. \nonumber \\- & \left. \frac{4 (3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \pi ^2 \lambda ^4}-2 (\mathcal {M}_++\mathcal {M}_-)\right) . \end{aligned}$$
(31)
Fig. 3
Fig. 3
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Effective potential related to the massless scalar shell for \(\mathcal {B}=0\) (left plot) and \(\mathcal {B}=0.4\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {J}=1,\mathcal {L}=10,\lambda =1.\).

Fig. 4
Fig. 4
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Effective potential related to the massless scalar shell for \(\lambda =0.3\) (left plot) and \(\lambda =0.7\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {J}=0.2,\mathcal {B}=1,\mathcal {L}=10.\).

Fig. 5
Fig. 5
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Effective potential related to the massless scalar shell for \(\mathcal {L}=5.5\) (left plot) and \(\mathcal {L}=8\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {J}=0.2,\mathcal {B}=1,\lambda =1.\).

Fig. 6
Fig. 6
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Effective potential related to the massless scalar shell for \(\mathcal {J}=0.5\) (left plot) and \(\mathcal {J}=1\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {L}=10,\mathcal {B}=1,\lambda =1.\).

Fig. 7
Fig. 7
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Massless scalar shell’s radius along the proper time for different values of  (left plot) and  with \(\mathcal {M}\text{ }_+=1,\mathcal {M}\text{ }_-=0,\mathcal {J}=0.2,\mathcal {L}=10\).

Fig. (3) describes the dynamics of the scalar shell via the shell’s radius for different values of \(\mathcal {B}\) by varying mass of BH. As the shell radius increases, the effective potential decreases, causing the shell to expand. If the shell’s radius increases along with its effective potential, this may indicate that, in the case of a collapsing structure, the shell is being confined by gravity. It is found that the scalar hair greatly affects the dynamics of the massless scalar shell of BTZ BH. Using the initial constraint (\(\dot{\vartheta }\ne 0\)) for a massless scalar field, we examine the effective potential and shell radius, as seen in Figs. (3)-(7). The massless scalar field’s effective potential behavior is consistent with a monotonically expanding shell (Figs. (3) and (5)), i.e., a shell that will expand see Figs. (7), beginning at a certain radius and moving at a positive initial velocity apart from the center \(\vartheta =0\). It is noted that the expansion rate reduces by increasing the integrating constant and scalar field parameters.

Massive scalar shell

The massive scalar field is a hypothesized physical field possessing a non-zero rest mass. This field is also characterized by a scalar particle that possesses no spin and exhibits wave-like behavior. Massive scalar fields manifest in various domains of physics, including electroweak theory, where they facilitate the interactions of the Higgs boson with other particles. Massive scalar fields are crucial in cosmology, as they can elucidate the presence of dark matter and enhance our comprehension of the early universe’s dynamics. Now, we observe the dynamics of the scalar shell with a massive scalar field, i.e., \(V(\Phi )=M^2\Phi ^2\)82,83. Equation (24) provides

$$\begin{aligned} 2V(\Phi )=\rho -P, \quad \dot{\Phi }^2=\rho +P. \end{aligned}$$
(32)

Here, we consider the specific expression of surface pressure as \(P=P_{0}e^{-\gamma \vartheta }\), in which \(\gamma\) and \(P_{0}\)are constants82,83. By taking the value of P and Eq. (18), we determine

$$\begin{aligned} \rho (\vartheta )=\frac{\beta }{\vartheta }+\frac{P_{0}e^{-\gamma \vartheta }}{\gamma \vartheta }, \end{aligned}$$
(33)

having \(\beta\) as a constant of integration. Putting this value in \(M=2\pi \vartheta \rho (\vartheta )\), the mass of massive scalar shell becomes

$$\begin{aligned} M=2\pi \beta +\frac{2\pi P_{0}e^{-\gamma \vartheta }}{\gamma }. \end{aligned}$$
(34)

Employing the forms of energy density and surface pressure in Eq. (32), we get

$$\begin{aligned} V(\Phi )= & \frac{\beta }{2\vartheta }+\frac{P_{0}e^{-\gamma \vartheta }}{2}\left( \frac{1}{\gamma \vartheta }-1\right) , \end{aligned}$$
(35)
$$\begin{aligned} \dot{\Phi }^2= & \frac{\beta }{\vartheta }+P_{0}e^{-\gamma \vartheta }\left( \frac{1}{\gamma \vartheta }+1\right) , \end{aligned}$$
(36)

which satisfies the KG equation. Hence, we obtain

$$\begin{aligned} \mathcal {H}(\vartheta )= & \frac{1}{4} \left( -\frac{(3 \vartheta +2 \mathcal {B})^2 (\mathcal {M}_+-\mathcal {M}_-)^2}{9 \pi ^2 \vartheta ^2 \left( \frac{P_0 e^{-\vartheta \gamma }}{\gamma }+\beta \right) ^2}+\frac{\frac{36 \vartheta ^6}{\mathcal {L}^2}-12 \vartheta ^3 \mathcal {B} (\mathcal {M}_++\mathcal {M}_-)+9 \vartheta ^2 \mathcal {J}^2+12 \vartheta \mathcal {J}^2 \mathcal {B}+4 \mathcal {J}^2 \mathcal {B}^2}{9 \vartheta ^4}\right. \\\nonumber- & \left. \pi ^2 \left( \frac{P_0 e^{-\vartheta \gamma }}{\gamma }+\beta \right) ^2-2 (\mathcal {M}_++\mathcal {M}_-)\right) . \end{aligned}$$

For massive scalar fields, Figs. (8) and (11) depict the effective potential behavior which can be obtained by solving the conservation and KG equations. The behavior of the effective potential \(\mathcal {H}(\vartheta )\) is dependent on the scalar field, as we can see from its explicit dependency in Fig. (8). For vanishing scalar field \(\mathcal {B}=0\), the expansion rate decreases. It is interesting to comment that the expansion enhances massive BHs. The integrating constant also plays a remarkable role in explaining the dynamic behavior of the shell. It is observed that the effective potential tends to zero as the shell radius enhances which provides the collapsing behavior of the shell see Fig. (9). Similarly, for greater values of \(p_0,\gamma\), the collapsing behavior also increases see Figs. (10) and (11). We also note that the expansion of the shell reduces as \(\mathcal {H}(\vartheta )\) goes to zero.

Fig. 8
Fig. 8
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Effective potential related to the massive scalar shell for \(\mathcal {B}=0\) (left plot) and \(\mathcal {B}=1\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {J}=1,\mathcal {L}=10,P_0=0.5,\beta =0.02,\gamma =1\).

Fig. 9
Fig. 9
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Effective potential related to the massive scalar shell for \(\beta =0.4\) (left plot) and \(\beta =1\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}\text{ }_-=0,\mathcal {J}=1,\mathcal {L}=10,P_0=0.5,\mathcal {B}=0.5,\gamma =1\).

Fig. 10
Fig. 10
Full size image

Effective potential related to the massive scalar shell for \(p_0=1\) (left plot) and \(p_0=2\) (right plot) by varying \(\mathcal {M}_+\) with \(\mathcal {M}_-=0,\mathcal {J}=1,\mathcal {L}=10,p_0=0.5,\beta =0.2,\mathcal {B}=0.5,\gamma =1\).

Fig. 11
Fig. 11
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Effective potential related to the massive scalar shell for \(\gamma =0.52\) (left plot) and \(\gamma =0.8\) (right plot) by varying \(\mathcal {M}\text{ }_+\) with \(\mathcal {M}_-=0,\mathcal {J}=1,p_0=0.5,\mathcal {L}=10,p_0=0.5,\beta =0.2,\mathcal {B}=0.5\).

Static composition

A static configuration shows that the forces of gravity and stress overcome each other, hence, \(\dot{\vartheta }=0\) must exist. Through the KG equation with \(\dot{\vartheta }=0\), we acquire

$$\begin{aligned} \ddot{\Phi }+\frac{\partial }{\partial \Phi }V(\Phi )=0, \end{aligned}$$
(37)

providing

$$\begin{aligned} \dot{\Phi }^2+2V(\Phi )=\mathcal {E}, \end{aligned}$$
(38)

in which \(\mathcal {E}\) exhibits the integration constant and represents a conserved quantity associated with energy. In this scenario, both the vanishing of the effective potential and its first differentiation are required by the EoM. Also, the oscillatory behavior of the scalar field composed at the shell is analyzed in Fig. (12). For static configuration, it is intriguing to mention that the scalar field expresses oscillatory behavior associated with the proper time. Also, the amplitude of oscillation decreases by increasing the scalar field mass. The radius \(\vartheta =\vartheta _0\) offers the maximal effective potential, and therefore, an unstable configuration, since \(\partial \mathcal {H}(\vartheta )/\partial \vartheta <0\) for \(\vartheta>\vartheta _0\), where \(\vartheta _0\) denotes the location of the static shell. This radius is based on the rest parameters in terms of \(\vartheta _0( \gamma ,\Omega ,m ,\mathcal {J})\) that is obtained on solving the two equations \(\mathcal {H}(\vartheta _0)=0\) and \(\frac{d\mathcal {H}(\vartheta )}{d\vartheta }|_{\vartheta =\vartheta _0}=0\), that can be turned into a fourth-order equation for \(\vartheta _0\) provided by

$$\begin{aligned} \frac{\mathcal {L} \left( 18 \pi ^4 \vartheta _0^6 \mathcal {E}^4+12 \pi ^2 \vartheta _0^4 \mathcal {E}^2 \mathcal {B} (\mathcal {M}_-+\mathcal {M}_+)+A_3\right) }{\vartheta _0 \mathcal {E}}=\frac{36 \pi ^2 (\vartheta _0+2) \vartheta _0^5 \mathcal {E}}{\mathcal {L}}+\frac{\mathcal {L} \left( 6 \vartheta _0 A_2 (2 \mathcal {B}-3)+A_1+A_4\right) }{\mathcal {E}}. \end{aligned}$$
(39)

Here

$$\begin{aligned} A_1= & 4 (\mathcal {B}-9) \mathcal {B} (\pi \mathcal {J} \mathcal {E}+\mathcal {M}_--\mathcal {M}_+) (\pi \mathcal {J} \mathcal {E}-\mathcal {M}_-+\mathcal {M}_+),\\ A_2= & (\pi \mathcal {J} \mathcal {E}+\mathcal {M}_--\mathcal {M}_+) (\pi \mathcal {J} \mathcal {E}-\mathcal {M}_-+\mathcal {M}_+),\\ A_3= & 9 \pi ^4 \vartheta _0^7 \mathcal {E}^4+18 \pi ^2 \vartheta _0^5 \mathcal {E}^2 (\mathcal {M}_-+\mathcal {M}_+)+9 \vartheta _0^3 (\mathcal {M}_--\mathcal {M}_+)^2+16 A_2 \mathcal {B}^2,\\ A_4= & 3 \pi ^2 \vartheta _0^2 \mathcal {E}^2 \left( 3 \mathcal {J}^2+4 \mathcal {B} (\mathcal {M}_-+\mathcal {M}_+)\right) . \end{aligned}$$

We also determine the values of \(\mathcal {E}\) for particular choices of physical parameters as shown in TABLE I. For specific choices of physical terms, we compute the respective expression of \(\mathcal {E}\) as follows :

  • For particular values of physical parameters \(\mathcal {M}_+=0,\mathcal {M}_-=1,\mathcal {B}=0,\mathcal {J}=0\)

    $$\begin{aligned} \mathcal {E}_\pm =\frac{\sqrt{\frac{\vartheta _0^3 \left( 2 \vartheta _0 (\vartheta _0+2)-\mathcal {L}^2\right) \pm 2 \sqrt{(\vartheta _0+2)^2 \vartheta _0^8-(\vartheta _0+2) \vartheta _0^7 \mathcal {L}^2+\vartheta _0^4 \mathcal {L}^4}}{\vartheta _0^4 (\vartheta _0+2) \mathcal {L}^2}}}{\pi }. \end{aligned}$$
    (40)

  • For \(\mathcal {M}_+=0,\mathcal {M}_-=1,\mathcal {B}=0\)

    $$\begin{aligned} \mathcal {E}_\pm =\frac{\sqrt{\frac{4 (\vartheta _0+2) \vartheta _0^4+\mathcal {L}^2 \left( (\vartheta _0-2) \mathcal {J}^2-2 \vartheta _0^3\right) \pm \sqrt{\mathcal {L}^4 \left( 16 \vartheta _0^4-4(\vartheta _0-2) \vartheta _0^3 \mathcal {J}^2+(\vartheta _0-2)^2 \mathcal {J}^4\right) +A_5}}{\vartheta _0^4 (\vartheta _0+2) \mathcal {L}^2}}}{\sqrt{2} \pi }, \end{aligned}$$
    (41)

    where

    $$\begin{aligned} A_5=16 \vartheta _0^8 (\vartheta _0+2)^2-8\vartheta _0^4 (\vartheta _0+2) \mathcal {L}^2 \left( 2 \vartheta _0^3-(\vartheta _0-2) \mathcal {J}^2\right) . \end{aligned}$$

  • For \(\mathcal {M}_+=0;\mathcal {M}_-=1\);

    $$\begin{aligned} \mathcal {E}_\pm =\frac{\sqrt{\frac{36 (\vartheta _0+2) \vartheta _0^6\pm \sqrt{1296 (\vartheta _0+2)^2 \vartheta _0^{12}+\mathcal {L}^4 \left( 1296 \vartheta _0^6 (\vartheta _0+\mathcal {B})^2+\mathcal {J}^4 (3 \vartheta _0+2 \mathcal {B})^2 (2 (\vartheta _0-4) \mathcal {B}+3 (\vartheta _0-2) \vartheta _0)^2-A_8\right) -72 (\vartheta _0+2) \vartheta _0^6 A_7 \mathcal {L}^2}+A_6 \mathcal {L}^2}{\vartheta _0^6 (\vartheta _0+2) \mathcal {L}^2}}}{3 \sqrt{2} \pi }, \end{aligned}$$
    (42)

    where

    $$\begin{aligned} A_6= & \mathcal {J}^2 (3 \vartheta _0+2 \mathcal {B}) (2 (\vartheta _0-4) \mathcal {B}+3 (\vartheta _0-2) \vartheta _0)-6 \vartheta _0^3 \left( 3 \vartheta _0^2+2 (\vartheta _0-1) \mathcal {B}\right) \mathcal {J}^2,\\ A_7= & 18 \vartheta _0^5+12 (\vartheta _0-1) \vartheta _0^3 \mathcal {B}-\mathcal {J}^2 (3 \vartheta _0+2 \mathcal {B}) (2 (\vartheta _0-4) \mathcal {B}+3 (\vartheta _0-2) \vartheta _0),\\ A_8= & 12 \vartheta _0^3 \mathcal {J}^2 (3 \vartheta _0+2 \mathcal {B}) (2 (\vartheta _0-4) \mathcal {B}+3 (\vartheta _0-2) \vartheta _0) \left( 3 \vartheta _0^2+2 (\vartheta _0-1) \mathcal {B}\right) . \end{aligned}$$

TABLE I presents the values of the conserved quantity \(\mathcal {E}\) for various configurations of BHs, specifically the BTZ and rotating BTZ geometries, in the presence of a scalar field. The parameters analyzed include the shell radius \(\vartheta _0\), the cosmological length scale \(\mathcal {L}\), and the static field parameter \(\mathcal {B}\). As observed, \(\mathcal {E}\) exhibits a significant dependence on these parameters, reflecting the complex interplay between the scalar field dynamics and the gravitational fields of BHs. For the BTZ BH with \(\mathcal {J} = 0\), increasing \(\vartheta _0\) leads to a gradual decrease in \(\mathcal {E}\), indicating that larger radii correspond to lower energy configurations, which may suggest enhanced stability under these conditions. In contrast, for the rotating BTZ BH, the introduction of angular momentum \(\mathcal {J}\) and varying values of \(\mathcal {B}\) dramatically affect the energy landscape, particularly with the presence of the scalar field. Notably, as \(\mathcal {B}\) increases, \(\mathcal {E}\) tends to increase as well, highlighting the strong influence of the scalar field’s energy contribution in rotating systems. Additionally, comparing the results with and without the scalar field, we observe that the presence of the field tends to stabilize configurations at certain radii, suggesting a nuanced relationship between scalar fields and BH dynamics. This behavior demonstrates the importance of parameter selection in understanding the stability and energy distribution in gravitational systems influenced by scalar fields.

Table 1 Values of \(\mathcal {E}\) by varying shell radius for different geometries.
Full size table
Fig. 12
Fig. 12
Full size image

Oscillatory behavior of scalar field for static configuration along the proper time for \(m=0.3\) (left plot) and \(m=0.8\) (right plot).

Concluding remarks

To sum up, this research explores in detail the dynamics of a TS consists of massive and massless scalar fields interacting with a rotating BH in 2+1 dimensions with nonminimal coupling of the scalar field. In lower-dimensional systems, the research explains the complex interactions between scalar fields and gravity by examining how these fields behave when rotation and coupling effects are present. The analysis shows that the dynamics of the shell are strongly affected by the nonminimal coupling of scalar fields which determine whether the shell expands or collapses depending on the scalar field parameters through effective potential. The scalar shell EoM is analyzed and two integration techniques are explained, highlighting the significance of energy conservation as a constraint in determining allowable solutions. The findings demonstrate that the existence of scalar hair has a crucial influence on the dynamics of the scalar shell. Massless scalar fields expand monotonically, whereas massive scalar fields collapse. These behaviors are contingent upon the effective potential, integrating constants, and initial conditions, respectively. The dynamics of the scalar shell along the shell’s radius for various values of \(\mathcal {B}\) are described by changing the mass of the BH in Fig. (3). It is discovered that the dynamic behavior of the massless scalar shell of the BTZ BH is significantly impacted by the scalar hair. Using the initial condition as (\(\dot{\vartheta }\ne 0\)) for a massless scalar field, we examined the effective potential and shell radius, as illustrated in Figs. (3)-(7). The massless scalar field’s effective potential behavior is consistent with a monotonically expanding shell (see Figs. (3) and (5)). This means that the shell will expand indefinitely from a given radius with a positive initial velocity away from the center \(\vartheta =0\); for example, see Figs. (7). It is observed that raising the scalar field parameter raises the expansion rate while raising the integrating constant decreases it. Figs. (8) and (11) show the effective potential behavior for enormous scalar fields, which can be obtained by solving the conservation and KG equations. Because of its explicit dependence on the scalar field, we can see in Fig. (8) that the behavior of the effective potential \(\mathcal {H}(\vartheta )\) depends on the scalar field. The expansion rate falls when the scalar field is absent. The dynamic behavior of the shell is also remarkably explained by the integrating constant. It is discovered that when the shell radius rises, the effective potential approaches zero, causing the shell to collapse (see Fig. (9). Similarly, the collapsing behavior of the shell also raises higher choices of \(p_0,\gamma\); see Figs. (10) and (11). To sum up, the results in TABLE I provide light on the complex interplay between BH dynamics and scalar fields. The research highlights the crucial roles that factors like the scalar field parameter, cosmological length scale, and shell radius play in forming the conserved quantity. Our findings show that angular momentum provides a complicated energy profile that is sensitive to the existence and strength of the scalar field, whereas greater shell radii in static configurations correspond to lower energy states and possible stability.

We conclude that the dynamical behavior of TS depends on the BH parameters and the presence of massless and massive scalar field. It is found that the effective potential expressed as effective potential increases negatively and shows collapsing behavior of TS if effective potential approaches zero.