Constraining study of charged gravastars solutions in symmetric teleparallel gravity

In recent years, gravitational collapse as well as stellar configuration are major astrophysical phenomena that have captured the interest of many researchers. White dwarfs, neutron stars, and black holes (BHs) are examples of new and massive compact celestial bodies that are produced as the consequence of gravitational collapse. Currently, several authors have suggested that the most dense celestial objects besides the BHs could be created by the gravitational collapse of a massive star. In this respect, by taking into account an expanded concept of Bose-Einstein condensation in the gravitational system, Mazur and Mottola 1 first proposed a novel theory of collapsing stellar objects dubbed as gravitational vacuum star (gravastar). It is considered that the gravastar model offers a solution to the problems with classic BHs and fulfills all theoretical requirements for a stable stellar configuration. In this conjecture, it is anticipated that quantum vacuum fluctuations contribute significantly to the dynamics of the collapse. A phase transition occurs leading to a repulsive de-Sitter core that balances the collapsing object and prevents the production of the horizon as well as the singularity 2 . Such transformation takes place extremely near to the bound $\frac{2m}{r}=1$, making it very challenging for an outside observer to identify the gravastar from a genuine BH.

The gravastar geometry is illustrated by three distinct zones where the internal geometry ($r\geq 0,r<\mathcal{R}_{1}$) is based upon the isotropic de Sitter core comprising the equation of state (EoS) $-\rho=p$. The external vacuum zone ($r>\mathcal{R}_{2}$) is demonstrated by the Schwarzschild geometry with EoS $p=\rho=0$. The internal and external regions are segregated through a thin shell ($\mathcal{R}_{1}<r<\mathcal{R}_{2}$) of stiff fluid having EoS $\rho=p$ in which $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ depict the internal and external radii of gravastar. After the proposal of Mazur and Mottola, an enormous discussion has been done on gravastar. Visser and Wiltshire 3 analyzed the stable structure of gravastars and examined that distinct EoSs yield the stable configuration of gravastars. Carter 4 presented novel exact gravastar solutions and observed the impact of EoS on different zones of gravastar geometry. Bilić et al. 5 obtained the gravastars solutions by considering the Born-Infeld phantom instead of de Sitter spacetime and observed that at the star’s core, their findings can manifest the dark compact configurations.

Horvat and Ilijić 6 investigated the role of energy bounds in the intermediate domain of the gravastar and checked the stable configuration via radial perturbations as well as the speed of sound. Several authors 7 -12 displayed the internal structure of gravastar by adopting distinct EoSs concerning different aspects. Lobo and Arellano 13 constructed various gravastar structures with the inclusion of nonlinear electrodynamics and discussed some specific features of their structures. Horvat et al. 14 extended the concept of gravastar by inducing an electric field and inspected the stability of the internal and external regions. On the same ground, Turimov et al. 15 observed the consequences of magnetic field on the gravastar geometry and determined the exact solutions of rotating gravastar. The detailed study of thin-shell developing from different inner and outer manifold as studied in fk1 -fk6 . Recently, Javed and Lin ff1 studied the exact gravastar solutions in general relativity by considering the effects of a cloud of strings and quintessence field.

In cosmological conjectures, the causes behind the cosmological expansion yields the formation of modified gravitational theories to general relativity (GR). In these theories, the notion of matter and curvature coupling yields distinct approaches like $f(R,T)$ gravity 16 , $f(R,T,R_{\alpha\beta}T^{\alpha\beta})$ gravity 17 and $f(\mathcal{G},T)$ theory 18 in which $R$ denotes the curvature invariant, $T$ depicts the trace of energy-momentum tensor (EMT) and $\mathcal{G}$ describes the Gauss-Bonnet invariant. There also exists some modified theories other than these theories and a lot of different scenarios have been discussed in these modified theories. Zubair et al. 18a discussed the thermodynamics of some newly developed $f(R,L_{m},T)$ theories with conserved energy-momentum tensor. Zubair and Farooq 18b presented the reconstruction as well as some dynamical aspects of bouncing scenarios in $f(T,\mathcal{T})$ gravity.

The exploration of gravastar geometries has prompted the astrophysicist to analyze the influences of extended gravitational conjectures on different modes of gravastar. In $f(R,T)$ background, Das et al. 19 discussed the gravastar geometry and observed its characteristics graphically corresponding to various EoSs. Shamir and his co-author 20 depicted the non-singular gravastar solutions and derived mathematical formulations of various physical terms in $f(\mathcal{G},T)$ framework. Sharif and Waseem 21 inspected the effects of Kuchowicz metric potential on gravastar geometry in $f(R,T)$ theory. Yousaf and his collaborators 22 ; 23 examined the features of gravastar concerning some specific constraints in different theories. Bhatti et al. 24 demonstrated some particular features of gravastar geometry with a specific EoS in $f(R,\mathcal{G})$ scenario. Sharif and Naz 25 ; 26 studied the salient aspects of gravastar in the absence and presence of electric field in $f(R,T^{2})$ gravity. Azmat et al. 26a developed the analytic form of gravastar with anisotropic and non-uniform characteristics through the gravitational decoupling procedure in $f(R,T)$ gravity. The inclusion of charge parameters plays an important role in discussing the solutions of ultracompact objects. Tello-Ortiz et al. 26b obtained the new hairy spherically symmetric and asymptotically flat-charged BH solution starting from the Reissner-Nordström manifold by means of extended geometric deformation approach. Azmat and Zubair 26c analyzed the impact of the charge on the anisotropic non-uniform gravastar model in $f(R,T)$ framework.

An inheritance symmetry with a collection of conformal Killing vectors (CKVs) is crucial for determining a natural consistent connection between the components of geometry and matter for giant celestial bodies via field equations. In contrast to previous analytical methods, these vectors are used to get precise analytical solutions in more suitable forms. Implementing these vectors, the large system of nonlinear partial differential equations can be converted into an ordinary one. Usmani et al. 27 discussed the different aspects of gravastar with electric charge and obtained solutions for its distinct eras associated with CKVs. Sharif and Waseem 28 investigated the impact of the charge on gravastar in $f(R,T)$ theory by considering these vectors. Bhar and Rej 29 presented the charged gravastar model conceding the CKV in the framework of $f(\mathcal{T})$ theory. Sharif and his collaborators 30 ; 31 explored the analytic solutions of uncharged as well as charged gravastar models by employing CKVs in $f(R,T^{2})$ theory. The study of thin-shell gravastars developed from inner de Sitter and outer various BH geometries is presented in fa7 -fa12 .

Recently, the $f(Q)$ gravity 32 as an extended gravitational theory gained a lot of attention. It falls under the category of symmetric teleparallel gravity (STG) in which gravity is linked to the non-metricity and $f(Q)$ displays the generic function of the non-metricity scalar $Q$. The basic difference between GR and teleparallel gravity is the part of an affine connection instead of the physical manifold. The accelerated cosmic expansion is characterized by $f(Q)$ gravity at least to the same statistical precision as the other well-known extended gravitational theories 33 . Different aspects like energy conditions 34 and Newtonian limit 35 have been discussed in this context. Hassan et al. 36 inspected the solution of wormholes (WHs) by considering the Lorentzian as well as the Gaussian configurations of exponential and linear models. Mustafa et al. 37 determined the WH solutions in this framework with the Karmarkar condition. They presented that such conditions yield the possible existence of traversable WHs obeying the energy bounds. Sharma and his collaborators 38 explored the WH solutions in the light of STG. They observed that some models of the $f(Q)$ gravity by employing the exact shape and redshift function may generate such solutions that satisfy the energy constraints. The detailed study of wormhole structure via thin-shell in different modified theories of gravity is presented in fa1 -fa6 . Also, the study of observational constraints in the framework of accelerated emergent $f(Q)$ gravity model is explored in fa13 . D’Ambrosio et al. 39 obtained the static spherically symmetric BH solutions in this gravity. Javed and his collaborators 40 ; 41 examined the thermodynamics of perturbed BHs as well as charged BHs in this framework. Classical works comprising the modified theories of gravity on compact objects by different authors can be found in the literature Jr1 ; Jr2 ; Jr3 ; Jr4 ; Jr5 ; Jr6 ; Jr7 ; Jr8 ; Jr9 ; Jr10 ; Jr11 .

In this manuscript, we construct the non-singular solutions of charged gravastar in the light of $f(Q)$ gravity with CKV. This work explores the effect of charge on gravastars, a novel astronomical object proposed as a black hole substitute. Based on the hypothesis developed by Mazur and Mottola in the context of general relativity, this study focuses on the implications of $f(Q)$ gravity. Three separate domains make up the gravastar: the exterior, intermediate shell, and internal sections. Using a particular $f(Q)$ gravity model with conformal Killing vectors, the paper shows that pressure is equal to negative energy density and that the inner domain pushes against the spherical shell. The intermediate shell balances the repulsive force from the interior domain with ultrarelativistic plasma and pressure proportionate to energy density. Two methods are used in the exterior region: first, the vacuum precise solution is calculated, and then the Reissner-Nordström metric is taken into account. We investigate stability requirements for both scenarios and match these spacetimes through junction conditions. The results imply that charged gravastar solutions are physically realistic. The manuscript is managed in the following pattern. The next section is dedicated to presenting the fundamentals of $f(Q)$ gravity. Section 3 manifests the physical aspects of charged gravastars with CKVs. The physical characteristics like the EoS parameter and proper length of the intermediate thin shell region are discussed in section 4. The outcomes of this manuscript are elaborated in the last section.

In STG, we assume that the gravastar being studied is presented within a differentiable Lorentzian manifold known as $\mathcal{M}$. This manifold can be sufficiently described by the metric $g_{ij}$, the determinant $g$ and the affine connection $\Gamma$ which is defined as:

$$g=g_{ij}dx^{i}\otimes dx^{j},$$

(1)

Herein $\Gamma^{\epsilon}_{\,\,\,\,\varepsilon}$ indicates the connection that can be regenerated in the form of disformation, contortion tensor and Levi-Civita 42 :

$$\Gamma^{\epsilon}_{\,\,\,\,\varepsilon}=w^{\epsilon}_{\,\,\,\,\varepsilon}+K^{\epsilon}_{\,\,\,\,\varepsilon}+L^{\epsilon}_{\,\,\,\,\varepsilon}.$$

(2)

The above-said relation can be rewritten as follows:

$$\Gamma^{\epsilon}_{\,\,\,\,ij}=\gamma^{\epsilon}_{\,\,\,\,ij}+K^{\epsilon}_{\,\,\,\,ij}+L^{\epsilon}_{\,\,\,\,ij}.$$

(3)

In the above Eq. (3), the expressions $L$, $\gamma$, and $K$, define the disformation tensors, affine connection, and contortion respectively. Now, the non-metricity term and it’s associated tensor in $f(Q)$ background are provided by

$$Q^{\epsilon}_{\,\,\,\,\varepsilon}=\Gamma_{(ab)},\quad Q_{\epsilon ij}=\nabla_{\epsilon}g_{ij}.$$

(4)

In the aforementioned relation, the tensor symmetric part satisfies the relation is given by

$$F_{(ij)}=\frac{1}{2}\bigg{(}F_{ij}+F_{ji}\bigg{)}.$$

(5)

It is noted from the above discussion that if contortion disappears, then only the disformation tensor is expressed as:

$$Q_{\epsilon ij}=-L^{\varepsilon}_{\,\,\,\,\epsilon i}g_{\varepsilon j}-L^{\varepsilon}_{\,\,\,\,\epsilon j}g_{\varepsilon i},$$

(6)

The disformation tensor is defined by the following formula:

$$L^{\epsilon}_{ij}=\frac{1}{2}Q^{\epsilon}_{ij}-Q_{(ij)}^{\epsilon}.$$

(7)

In the STG formalism, the non-metricity scalar within the scope of superpotential is provided as

$$Q=-P^{\epsilon ij}Q_{\epsilon ij},$$

(8)

where $-P^{\epsilon ij}$ define the superpotential, which is revealed as:

$$P^{\epsilon}_{\,\,\,\,ij}=\frac{1}{4}\bigg{[}2Q^{\epsilon}_{\,\,\,\,(ij)}-Q^{\epsilon}_{\,\,\,\,ij}+Q^{\epsilon}g_{ij}-\delta^{\epsilon}_{(i}Q_{j)}-\overline{Q}^{\epsilon}g_{ij}\bigg{]}.$$

(9)

Here, $Q^{\epsilon}=Q^{j}_{\,\,\,\,\epsilon j}$ and $\overline{Q}_{\epsilon}=Q^{i}_{\,\,\,\,\epsilon i}$ are important unconstrained traces for $Q_{\epsilon ij}=\nabla_{\epsilon}g_{ij}$. After the necessary calculations, we have the following modified action integral for $f(Q)$ gravity 43 for STG formalism:

$$\mathcal{S}=\int d^{4}x\sqrt{-g}f(Q)+[\mathcal{S}_{\mathrm{M}}+L_{e}].$$

(10)

Here, $\mathcal{S}_{\mathrm{M}}$ defines the action integral and $L_{e}$ is a candidate for electric field. The modified field equations for $f(Q)$ gravity can be obtained by varying the Eq. (10), which are calculated as:

$$\frac{2}{\sqrt{-g}}\nabla_{\gamma}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{ij}\right)+\frac{1}{2}g_{ij}f+f_{Q}\left(P_{i\gamma i}\,Q_{j}\;^{\gamma i}-2\,Q_{\gamma ii}\,P^{\gamma i}\;_{j}\right)=-T_{ij},$$

(11)

where $f_{Q}\equiv\frac{df}{dQ}$ and $T_{ij}$ represents the energy momentum tensor, which is given as:

$$T_{ij}+E_{ij}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{M}})}{\delta g^{ij}}.$$

(12)

Here $\mathcal{L}_{\mathrm{M}}$ manifests the Lagrangian density of matter field and it obeys the identity $\int\sqrt{-g}\mathcal{L}_{\mathrm{M}}d^{4}x=\mathcal{S}_{\mathrm{M}}[g,\Gamma,\Psi_{i}]$. Further, on changing the action within the scope of affine connection $\Gamma^{\epsilon}_{\,\,\,\,ij}$, we obtain the following relation:

$$\nabla_{i}\nabla_{j}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{ij}\right)=0.$$

(13)

Corresponding to the spherically symmetric metric, the line element is provided as:

$$ds^{2}=e^{a}dt^{2}-e^{b}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}).$$

(14)

Further, for isotropic matter configuration, the energy-momentum tensor becomes:

$$T_{ij}=diag(e^{a}\rho,e^{b}p,r^{2}p,r^{2}psin^{2}\theta).$$

(15)

Now the other involved tensors like electromagnetic stress energy tensor $E_{i\upsilon}$ is given by

$$E_{ij}=2(F_{i\zeta}F_{j\zeta}-\frac{1}{4}g_{ij}F_{\zeta\chi}F^{\zeta\chi}),$$

(16)

where

$$F_{ij}=\mathcal{A}_{i,j}-\mathcal{A}_{j,i}.$$

Further, $F_{ij}$ denotes the tensor for electromagnetic field provided by

	$\displaystyle F_{ij,\zeta}+F_{\zeta i,j}+F_{j\zeta,i}=0,$		(17)
	$\displaystyle(\sqrt{-g}F^{ij})_{,j}=\frac{1}{2}\sqrt{-g}j^{i}.$		(18)

The electromagnetic four potentials are denoted by $\mathcal{A}_{i}$, and all components of electromagnetic field tensor become zero except the radial component $F_{01}$. Equation (80) is fulfilled when $F_{01}$ is antisymmetric, i.e., $F_{01}=-F_{10}$. Equation (16) can be used to obtain the electric field ($E$).

$$E(r)=\frac{1}{2r^{2}}e^{a(r)+b(r)}\int_{0}^{r}\sigma(r)e^{b(r)}r^{2}dr=\frac{q(r)}{r^{2}},$$

(19)

where $q(r)$ and $\delta$ manifest the total charge and charge density of the system, respectively. Now, we will derive the field Eqs. (11) and (13) for spherically symmetric spacetime, which is calculated as:

	$\displaystyle 8\pi T_{tt}$		$\displaystyle=\frac{e^{a-b}}{2r^{2}}[2rf_{QQ}Q^{\prime}(e^{b}-1)+f_{Q}[(e^{b}-1)(2+ra^{\prime})+(1+e^{b})rb^{\prime}]+fr^{2}e^{b}],$		(20)
	$\displaystyle 8\pi T_{rr}$		$\displaystyle=-\frac{1}{2r^{2}}[2rf_{QQ}Q^{\prime}(e^{b}-1)+f_{Q}[(e^{b}-1)(2+ra^{\prime}+rb^{\prime})-2ra^{\prime}]+fr^{2}e^{b}],$		(21)
	$\displaystyle 8\pi T_{\theta\theta}$		$\displaystyle=-\frac{r}{4e^{b}}[-2rf_{QQ}Q^{\prime}a^{\prime}+f_{Q}[2a^{\prime}(e^{b}-2)-ra^{\prime 2}+b^{\prime}(2e^{b}+ra^{\prime})-2ra^{\prime\prime}]+2fre^{b}],$		(22)

Now, by using the Eqs. (15) and (16) into Eqs. (20-22), the modified form of $f(Q)$ field equation becomes

	$\displaystyle 8\pi\rho+E^{2}$		$\displaystyle=\frac{1}{2r^{2}e^{b}}[2rf_{QQ}Q^{\prime}(e^{b}-1)+f_{Q}[(e^{b}-1)(2+ra^{\prime})+(1+e^{b})rb^{\prime}]+fr^{2}e^{b}],$		(23)
	$\displaystyle 8\pi p-E^{2}$		$\displaystyle=-\frac{1}{2r^{2}e^{b}}[2rf_{QQ}Q^{\prime}(e^{b}-1)+f_{Q}[(e^{b}-1)(2+ra^{\prime}+rb^{\prime})-2ra^{\prime}]+fr^{2}e^{b}],$		(24)
	$\displaystyle 8\pi p+E^{2}$		$\displaystyle=-\frac{1}{4re^{b}}[-2rf_{QQ}Q^{\prime}a^{\prime}+f_{Q}[2a^{\prime}(e^{b}-2)-ra^{\prime 2}+b^{\prime}(2e^{b}+ra^{\prime})-2ra^{\prime\prime}]+2fre^{b}],$		(25)
	$\displaystyle\sigma$		$\displaystyle=\frac{e^{-b(r)}\frac{\partial\left(r^{2}E\right)}{\partial r}}{4\pi r^{2}}.$		(26)

in which the non-metricity scalar is illustrated by

$$Q=\frac{1}{r}(a^{\prime}+b^{\prime})(e^{-b}-1).$$

(27)

For the current analysis, the linear model of $f(Q)$ gravity is taken into account which is exhibited as:

$$f=\alpha Q+\Phi,$$

(28)

where $\alpha$ and $\Phi$ are the model parameters. Now, by plugging Eq. (27) and Eq. (28) into Eqs. (23-26), one can get the following revised final version of field equations:

	$\displaystyle 8\pi\rho+E^{2}$		$\displaystyle=\frac{2\alpha+2\alpha e^{-b(r)}\left(rb^{\prime}(r)-1\right)+r^{2}\Phi}{2r^{2}},$		(29)
	$\displaystyle 8\pi p-E^{2}$		$\displaystyle=-\frac{-2\alpha e^{-b(r)}\left(ra^{\prime}(r)+1\right)+2\alpha+r^{2}\Phi}{2r^{2}},$		(30)
	$\displaystyle 8\pi p+E^{2}$		$\displaystyle=\frac{e^{-b(r)}\left(2\alpha ra^{\prime\prime}(r)+\alpha\left(ra^{\prime}(r)+2\right)\left(a^{\prime}(r)-b^{\prime}(r)\right)-2r\Phi e^{b(r)}\right)}{4r},$		(31)

The study of charged gravastars in $f(Q)$ gravity is an important endeavor that combines fundamental motives in gravitational physics with the discovery of unique astronomical phenomena. The study’s major goal is to understand the complex interplay between gravastars’ charged nature and the changes introduced to the gravitational sector by $f(Q)$ gravity. This project is motivated by the theoretical richness of $f(Q)$ gravity, which goes beyond general relativity by including a function $f(Q)$ of the non-metricity scalar $Q$. Symmetries are very important to developing the inherent connection between geometry and matter through Einstein’s equations. The CKVs among the well-known symmetries, offer superior outcomes as they provide a more profound understanding of geometry. The equation that governs CKVs is defined as:

$\displaystyle\pounds_{\xi}g_{ij}=\xi~{}g_{ij},$

(33)

For a four-dimensional metric, where $i$ and $j$ range from 1 to 4, the equation on the left represents the Lie derivative associated with the metric tensor along with the vector field $\xi$. It is important to note that the function $\Phi$ acts as an arbitrary function of the radial $r$ as well as time $t$ coordinate, even in the static scenario. Now, by applying Eq. (33) on the spacetime via Eq. (14), the following form of conformal Killing equations can be obtained:

$\displaystyle\xi^{1}a^{\prime}=\Psi,~{}~{}~{}~{}~{}\xi^{4}=W,~{}~{}~{}~{}~{}\xi^{1}=\frac{\Psi~{}r}{2},~{}~{}~{}~{}~{}\xi^{1}b^{\prime}+2{\xi^{1}}_{1}=\Psi,$

(34)

that lead to a simultaneous solution given by

$\displaystyle e^{a}=X^{2}r^{2},~{}~{}~{}~{}~{}e^{b}=\big{(}\frac{Y}{\Psi}\big{)}^{2},~{}~{}~{}~{}~{}\xi^{i}=W~{}{\delta^{i}_{4}}+\big{(}\frac{r\Psi}{2}\big{)}{\delta^{i}_{1}},$

(35)

in which $W$, $X$ and $Y$ are considered as the arbitrary constants. Now, by plugging Eq. (35) into Eqs. (29-LABEL:eq:26), we have the following field equations under the effect of conformal symmetry:

	$\displaystyle 8\pi\rho+E^{2}$		$\displaystyle=-\frac{\alpha\Psi(r)^{2}}{Y^{2}r^{2}}+\frac{\alpha}{r^{2}}-\frac{2\alpha\Psi(r)\Psi^{\prime}(r)}{Y^{2}r}+\frac{\Phi}{2},$		(36)
	$\displaystyle 8\pi p-E^{2}$		$\displaystyle=\frac{3\alpha\Psi(r)^{2}}{Y^{2}r^{2}}-\frac{\alpha}{r^{2}}-\frac{\Phi}{2},$		(37)
	$\displaystyle 8\pi p+E^{2}$		$\displaystyle=\frac{\alpha\Psi(r)^{2}}{Y^{2}r^{2}}+\frac{2\alpha\Psi(r)\Psi^{\prime}(r)}{Y^{2}r}-\frac{\Phi}{2},$		(38)

On solving this field equation simultaneously, the expressions for the energy density, pressure, and electric field can be attained as follows:

	$\displaystyle\rho$		$\displaystyle=\frac{\alpha}{16\pi r^{2}}-\frac{3\alpha\Psi(r)\Psi^{\prime}(r)}{8\pi Y^{2}r}+\frac{\Phi}{16\pi},$		(40)
	$\displaystyle p$		$\displaystyle=\frac{\alpha\Psi(r)^{2}}{4\pi Y^{2}r^{2}}-\frac{\alpha}{16\pi r^{2}}+\frac{\alpha\Psi(r)\Psi^{\prime}(r)}{8\pi Y^{2}r}-\frac{\Phi}{16\pi},$		(41)
	$\displaystyle E^{2}$		$\displaystyle=-\frac{\alpha\Psi(r)^{2}}{Y^{2}r^{2}}+\frac{\alpha}{2r^{2}}+\frac{\alpha\Psi(r)\Psi^{\prime}(r)}{Y^{2}r},$		(42)

This section is devoted to analyzing the some of physical attributes of charged gravastars such as the EoS parameter, the proper length, energy and entropy admitting the CKVs. First, we are going to explore the EoS parameter.