Gravastar configuration in non-conservative Rastall gravity

In the present article, we have presented the exact solutions of gravastar with Kuchowicz metric potential in the background of non-conservative Rastall gravity. Within the context of Mazur-Mottola’s [, ] concept of Bose–Einstein condensation to gravitational systems, the grvastar structure consists of three layers: interior part, intermediate part (i.e., thin shell) and exterior part. All the gravastar’s exact solutions have been derived with the aid of Kuchowicz metric potential by considering isotropic matter distribution. For the thin shell (filled with ultra-relativistic stiff fluid) with different parameters like length, energy, entropy and surface redshift have been calculated, which show the stability of our proposed model. Interestingly, all the thin shell results are obtained without taking shell approximation i.e., 0 < e −ν ≡ h ≪ 1. The exterior part, which is absolutely a vacuum is characterized by the Schwarzschild geometry and the interior part give output in the form of non-singular results. Generally, the main purpose of this work is to obtain the exact, non-singular, horizon free, stable model and we have achieved these goals in the presence of Rastall parameter.


Introduction
With the development of Einstein's General Theory of Relativity (GR), we have come to know about the existence of black hole(BH), which has been a challenging problem in modern astrophysics due to the presence of event horizon and singularity in its structure, where all physical laws break down. A lot of work on BH has been done in the literature [3][4][5][6][7][8][9][10][11][12][13]. But the drawbacks of BH (i.e., event horizon and singularity) stimulated the researchers to construct such a model that would deal with such types of challenges. In the continuation of such problems, Mazur and Mottola [1,2] have presented the magnificent, hypothetical concept of 'a Gravitational Vacuum Star' (gravastar) as a gravitational cold, dark, compact, and spherically symmetric object by extending the Bose-Einstein condensate idea. This gravastar is known as the alternative to BH that can resolve the issues related to BH. There is no singularity and event horizon in a gravastar model. A gravastar structure has three parts: i)-center of gravastar is the interior zone 0 < r < Q 1 , with an equation of state (EOS) p = − ρ, which consists of dark energy and isotropic fluid, it gives regular solutions and prevents the formation of singularity, ii)-the boundary of gravastar is the thin shell Q 1 < r < Q 2 with EOS p = ρ filled with ultrarelativistic perfect fluid, it is the membrane that separates the interior region from the exterior region and iii)-third part is the exterior region r > R 2 at p = ρ = 0, which is completely vacuumed region described by Schwarzschild geometry. Due to these aspects of the gravastar model, a lot of work has been done by many authors [14][15][16][17][18][19][20][21] to model the gravastar configuration.
Sengupta et al [22] have investigated the results for gravastar in the framework of Braneworld gravity. They have given a stable and non-singular model by investigating the different physical aspects of thin shell. Ray et al [23] have described the gravastar's salient features for lower and higher dimensions. They have also highlighted the different circumstances under which a gravastar is considered as an alternative to BH. Usmani et al [24] have determined the solutions for charged gravastar with conformal motion by considering the Reissner-Nordström spacetime in the exterior region. This work initiated the other researchers to do work more on gravastar models. Bhar [25] has generalized the work of Usmani et al [24] for the higher dimensional charged gravastar. She has used the Reissner-Nordström spacetime in the exterior region. These solutions satisfied the consequences of Usmani et al [24]. Further, Bhar [26] has modified the results of Usmani et al [24] in f (T) gravity. Sharif and Waseem [27] have formulated the solutions of gravastars in the presence of electric charge with conformal motion in f (R, T) gravity. They have used the linear model f (R, T) = R + 2βT and thin shell approximation. By replacing β as zero, all the outcomes returned to the findings of Usmani et al [24]. Bhatti et al [28] have obtained the gravastar solutions in the framework of modified Gauss-Bonnet gravity. They analyzed the various features of thin shell to discuss the stability of given model. Latter on, the same authors [29] have presented the charged gravastar model in which they have achieved the new non-singular solutions, to see the impact of charge on the gravastar structure presented in [28].
Rahman et al [30] have analyzed the solutions for (2 + 1)-dimensional gravastars by taking anti-de Sitter spacetime. Also, the same authors [31] have explicitly provided the new solutions for gravastars by including the charge term in the field equations. Ghosh et al [32] have generalized the work of Rahman et al [31] for Ddimensions with the impact of charge distribution, which fulfill the requirements for the existence of gravastar. After very short interval of time, Ghosh et al [33] have discussed the same results as in [32] without taking charge and gave the exact, non-singular solutions for gravastar in GR. The new results for gravastars with Karmarkar condition in GR satisfying the Mazur-Mottola concept has been presented by Ghosh et al [34]. Their solutions were stable, regular in the interior region, non-singular, and horizon free. The non-singular and exact solutions for gravastars in f (R, T) gravity have been discussed by Das et al [35]. They have determined the physical properties with graphical representation, which showed the stability and validity of gravastar in f (R, T). Shamir et al [36] have examined the new results for the gravastar model in f (G, T) gravity and also discussed the physical behavior of interior, exterior solutions, and thin shell. The non-singular, horizon-free solutions with an electric charge for gravastars in f (T) theory, have been presented by Debnath [37]. He used Reissner-Nordström spacetime rather than Schwarzschild spacetime for the exterior region. Yousaf [38] has investigated the solutions for cylindrical gravastar with the influence of electric charge in the framework of f (R, T) gravity. Ghosh et al [39] have studied the new results of gravastar in torsion trace gravity. By following Mazur-Mottola [1, 2] approach, the singular free solutions of charged gravastar in f (R, T) theory of gravity have been presented by Majeed et al [40].
Einstein's General Theory of Relativity has great importance than other physical theories as it revealed the many gravitational hidden pieces of evidence and objects of nature like gravitational lensing, gravitational time dilation, and black holes. But some issues related to dark energy, dark matter, and the accelerating universe are still to be solved, which motivated the researchers to modify this theory. So a number of modified theories have been proposed successively such as f (R), f (T), f (R, T), f (G), f (G, T) and f (R, G) theories etc. A lot of work has been done for the models of gravastar in these theories by many researchers in [41][42][43][44][45]. In 1972, an alternative of GR was proposed by Rastall [46], which recently has captured more attention in the literature due to some beneficial understandings, such as it is stable on gravitational lensing occurrence and deals with entropy and universe age problem in cosmic era. Further, Rastall theory may be regarded as a phenomenological theory of some quantum effects in the relativistic structure [47]. Recently, Rastall theory has been implemented in cosmology to discuss the nature of cosmological evolution. It was found [48,49] that Rastall theory is well consistent with various observational data and it gives some interesting results. For example in Rastall theory, the evolution of dark matter fluctuations is the same as that in the ΛCDM model [49]. But the dark energy is clustered in Rastall theory. This leads to generate the inhomogeneities in the evolution of dark matter, which is different from the standard cold dark matter (CDM) model. The inner region of the gravastar is the de-Sitter core with EOS p = − ρ, so it contains dark energy. It would be interesting to study the gravastar structure in Rastall theory, with the hope that one can get a more realistic model of gravastar in Rastall theory which would be consistent with the most recent astrophysical measurements for the detection of gravastar.
Rastall [46] modified the conservation law of GR i.e., ( into non-conservation law, which can be stated as where η is a Rastall parameter, which coupled the matter field and geometry in non-minimal manner that make it different from the conservation law of GR. The standard form of Rastall field equations is Here κ, R and T ςυ are coupling constant of Rastall theory, Ricci scalar and energy momentum tensor, respectively. These equations can be reformulated as The energy momentum tensor for an ideal fluid is In the literature, a lot of work has been done by many researchers in this modified theory of gravity for different stellar structures. Meng-Sen Ma and Ren Zhao [50] have studied the non-commutative geometry of black holes in Rastall theory of gravity by taking two different metrics ansatz. Debnath [51] has analyzed the gravastar solutions in Rastall-Rainbow gravity with electromagnetic effects. He took spherically symmetric spacetime and find out the non-singular, horizon-free solutions for gravastar. Salako et al [52] have constructed the compact stars model with anisotropic fluid in the framework of Rastall theory. They have determined the values of unknown constants in the form of mass and radii of the compact star and also discussed the stability of their solutions through physical properties like surface redshift and regularity condition. Abbas and Shahzad [53] have explored the new solutions for compact stars with a conformal motion by taking isotropic fluid. Das et al [54] have obtained the new generalized solutions in Rastall gravity with the use of its cosmological consequences. They have discussed universal thermodynamics by choosing the specific Rastall theory parameter. Shahzad and Abbas [55] have provided the comparative study of GR and Rastall theory of gravity for three different compact stars with different radii and they have checked the compactness and stability of their model. Further, Abbas and Shahzad [56] have used the Karori and Barua type metric by taking different anisotropic compact stars i.e., 4U 1820-30, Her X-1 and SAX J 1808.4-3658(SSI) with radii 10 km, 7.7 km and 7.07 km, respectively, in Rastall gravity. They have found the unknown constants and plotted the graphs for different physical conditions like energy condition, stability, energy density, and hydrostatic equilibrium and redshift that make sure that their model is compact, regular, and stable. The solutions for quintessence compact stars in Rastall theory have been provided by Abbas and Shahzad [57]. A study on thermodynamic geometry of a black hole by considering isotropic fluid in Rastall theory has been provided by Soroushfar et al [58]. Maurya et al [59] have investigated the solutions of decoupling gravitational field equation by minimal geometric deformation procedure in the framework of Rastall theory. Prihadi et al [60] have generalized the solutions of Kerr-Newman-NUT black hole in the presence of charge factor in Rastall gravity. Recently, Abbas and Majeed.
[61] have obtained the new results for gravastars in Rastall gravity and provided the stable, non-singular, event horizon free model for gravastar.
The main objectives of the present study are to find stable, regular, non-singular, and horizon-free results for gravastar by using the Kuchowicz type metric potential in Rastall theory of gravity. To achieve this aim, we have planned our work in the following manner: In section 2, we furnish the Rastall field equations with Kuchowicz metric potential and non-conservation law. In section 3 three layers of gravastar structure have been discussed i.e., i-Interior region, ii-Thin shell region, and iii-Exterior region and junction condition. Section 4, consists of thin shell physical properties i.e., length, energy, entropy, surface redshift, and equation of state. The last section deals with is the conclusion of the present work.

Rastall's field equations and non-conservation law
For an interior spacetime, the spherically symmetric metric is where μ(r) and λ(r) are the metric potentials. The explicit form of equation (3) in Rastall gravity for spherical symmetric spacetime (5) is given below We all are familiar with the remarkable achievements of GR by concealing many hidden aspects of nature and a lot of adorable relativistic models in the literature have been found. Among these models, some require the analytical solutions of the gravitating system like the gravastar model. In this regard, different approaches such as hydrostatic equilibrium equation, thin-shell approximation, conformal motion, Karmarkar conditions, and Kuchowicz metric potential have been implemented to solve the Einstein's field equations in GR. But in the present article, we suggest the metric potential e μ( r) as Kuchowicz type [62] with arbitrary constants B and C, where B has the dimension [L −2 ] and C is dimensionless. It is considered as a successful methodology to figure out various aspects of compactness and physical viability in the formation of different relativistic stellar objects.
The utilization of this particular type of metric function is exceedingly meaningful in the field equations to get the analytical and non-singular models of the gravitating. It provides the feasibility to explore more surreptitious aspects of nature in the gravitational field [63][64][65][66]. Ghosh et al [67] have investigated the exact and singularityfree solutions of gravastar with the implementation of the Kuchowicz metric function in GR. Here, we formulate a new exact model of gravastar structure in the framework of Rastall theory of gravity by adopting the Kuchowicz metric potential. The reason behind adoption of this type of metric function is that, this metric potential is completely free from any singularity, shows a regular behavior throughout the gravastar. Using this function we have studied different features of three different regions of gravastar. All the solutions for the interior region are regular at the centre and are horizon free, which are the consistency conditions for the existence of gravastar.
The expression for Kuchowicz metric function is given by r B r l n C 2 2 By implementing the above equation (9), the set of field equations (6)- (8) can be written in the following form The non-conservation law of energy momentum tensor in Rastall theory can be reported in the following way

Gravastar configuration
The gravastar composition appears with three regions that are discussed as under

Interior region of gravastar
With the conjecture of Mazur-Mottola [3,4], we assume the EOS with parameter ω = − 1 as follows From equations (13) and (14), we have where k 0 is a constant that is related to central density in the interior region, which is responsible to balance the inner influence of gravitation by the thin shell. By using equations (10) and (15), we get

Gravastar's thin shell
The thin shell is the non-vacuum part of gravastar (Q = Q 1 < r < Q 2 = Q + ò) with EOS p = ρ which consists of ultra-relativistic stiff fluid as defined by Zel'dovich [68] to model the cold byronic universe with very high density. These type of fluids have been used by many researchers for cosmological and astrophysical point of view. Thin shell is considered as very thin but finite and it links the two parts i.e., interior region and Schwarzschild exterior geometry. From equations (10)-(12) by using EOS p = ρ, one can achieve where k 2 is constant of integration. By implementing the condition p = ρ in equation (13), we get the shell density as where k 3 is an integration constant and b = ph ph --32 1 1 48 . The variation of the matter density as well as pressure throughout the shell is shown in figure 1.

Schwarzschild spacetime and boundary conditions
The exterior region is totally vacuum and we consider Schwarzschild spacetime obeying the EOS p = ρ = 0, which can be defined as where M is the total mass of the gravastar. In gravastar's structure there are two boundaries with co-relation of inner area, thin shell and exterior part. First one relates with inner zone and thin shell, second relates with thin shell and exterior part. We have used the way of matching metric function at the boundary to find the expressions for the unknown constants B and C involved in Kuchowicz metric potential (9) in terms of total mass of gravastar M and the exterior radius Q 2 . So, we have It is recalled that the gravastar model has been proposed by Mazur , the values of M and R may vary. By substituting these specific parametric values, we obtain B = 0.014991 km −2 and C 2 = 0.0558327. When these parametric values are implemented in physical variables, we get a physically stable, well-behaved and non-singular gravastar model.

Junction conditions
We know that gravastar's structure has three main regions: interior part, thin shell and exterior. The intermediate thin shell connects interior region to the exterior region at r = Q. The junction conditions are the continual matching of metric potentials at shell region but this condition is not guaranteed for the existence of their derivatives. We use the formalism given by Darmois [70] and Israel [71] to find the intrinsic surface stresses at junction surface Ω, which can be expressed by Lanczos equation [72] as: lm shows the discontinuity of second fundamental forms and the signs +,− are the indicators for exterior and interior boundaries, respectively. The second fundamental form is given by Here n  n depicts the normal unit vector to the surface Ω, which is given in the form with n ν n ν = 1. Here ζ l is the intrinsic coordinate on the shell and f (x l (ζ l )) = 0 is the parametric equation of the shell, which in the present case takes the form f (r) = r − Q = 0, implying r = Q. By using Lanczos equation [72], we find surface stress-energy tensor as S lm = diag[ñ, − χ, − χ, − χ], specifying ñ as surface energy density and χ as the surface pressure. These are given by Here, F corresponds to g 00 of the interior region (−) and exterior region (+). From the above equations (27) and (28), we get The thin shell mass of gravastar is By manipulating the above equation (31), we can get the total mass of gravastar, which is

Shell energy
The shell energy content is Variation of the energy throughout the thin shell has been shown in figure 3.

Shell entropy
The shell entropy is given by whereas the entropy density denoted by s(r) is given as where α is a dimensionless constant. In the present case, we apply some units as G = c = 1 and ξ = ÿ = 1 then entropy density takes the form

Surface redshift
The analysis of surface redshift of the gravastars may be regarded as one of significant aspect regarding the stability and detection of gravastars. Buchdahl [69] investigated that for the stability of static and isotropic matter configuration, the value of surface redshift must be less than 2,i.e., Z s < 2 [69,73]. Ivanov [74] have examined that for static anisotropic gravitating source, Z s can attain value upto 3.84. Barraco and Hamity [75] concluded that Z s 2 for an isotropic fluid in the absence of cosmological constant, while Z s 5 for anisotropic source in the presence of cosmological constant [73]. The surface redshift for thin shell can be calculated by the following formula [67] ( ) | | ( ) = -+ -Z r g 1 , 4 0 s tt 1 2 which gives the final result as The variation of surface redshift Z s has been plotted in figure 5. It is evident from this figure that the value of Z s lies within 1 throughout the thin shell. So, our proposed gravastar model is stable as well as physically acceptable.

Equation of state
We get the EOS parameter by using equations (29) and ( Hence, for the above value of EOS parameter, the gravastar model would be stable as the condition

Conclusion
The current study is about the consequences of the gravastar idea with Kuchowicz type metric potential in Rastall theory pursuing the Mazur-Mottola [1, 2] strategy. We have considered the isotropic fluid with spherically symmetric and static line-element in the interior region while a vacuum solution has been taken in the exterior region. In this work, we have designated the metric potential for the time coordinate as Kuchowicz type metric potential. The consideration of such a specific form of metric potential is beneficial in the following way: by taking Kuchowicz metric potential it is easy to determine the other metric potential from the system of a differential equation, it helps to find exact solutions and gives non-singular results in the interior of gravastar. According to the Mazur-Motolla [1,2] gravastar (an alternative to Black Hole) configuration comprises three main parts: i-Interior region, ii-Thin shell, and iii-Exterior region, which has the following consequences • I-Thin shell covered the interior sector (0 r < r 1 = Q) which is considered as dark energy. In this inner region, we have found the solutions of energy density, pressure, metric potential of the radial coordinate, and gravitational mass by considering EOS p = − ρ (negative isotropic pressure and positive density) with the help of Rastall field equations, non-conservation law, and Kuchowicz metric potential equation (9). We have obtained the results which are regular at r = 0 in the interior region.
• II-The second part of gravastar, which is thin shell (Q = r 1 < r < r 2 = Q + ò) is assumed as very thin but positive, obeys the EOS p = − ρ. It is filled with ultrarelativistic stiff-fluid. We have found exact solutions without thin shell approximation i.e., 0 < e − ν ≡ h = 1 in this region. With the help of Kuchowicz metric potential (9) and Rastall field equations (10)- (12), the other metric coefficient e μ( r) has been achieved. The shell density is obtained by the use of equation (19) in non-conservation law equation (13) and its physical behavior with numerical values η = 1, M = 2.75M e can be shown in figure 1. The behavior of density shows the stability of our model with positive and decreasing values which means that it is less denser at junction interface. With the help of these results, different properties of the thin shell such as proper length, shell energy, shell entropy, and surface redshift have been discussed with their graphical representation. The graph for proper length in figure 2 expresses the positive and increasing behavior relative to radial component r. From figure 3, we can observe that the variation of shell energy is finite, positive, and directly proportional to the radius r of gravastar. Figure 4 indicates the same behavior as in figure 3. To check the stability of gravastar, some researchers [31][32][33][34][35][36] have calculated the surface redshift for static, isotropic perfect fluids and they have examined that its value must be less than 2. In our work, surface redshift is calculated by using equation (40), and its graph is shown in figure 5, which shows that redshift has values less than 2 indicating the stability of our gravastar model.
• III-The third region is exterior of gravastar (r > r 2 = Q + ò) satisfying p = ρ = 0 that consists of vacuum only. This part is defined by Schwarzschild spacetime given in equation (23). To obtain the constants B and C in equation (9), first we have made expressions for them in terms of mass and exterior radius by matching the metric coefficients of interior and exterior spacetimes, which can be see in equations (22) and (23). Then final values for B is 0.014991 km −2 and for C 2 is 0.055 832 7, which make our model as physically stable.
• IV-Junction interface is the boundary at which the interior and exterior regions are connected to each other by thin shell. By using second fundamental law in Lanczos equation with unit normals on surface, we get the expressions for surface pressure χ, surface energy density ñ, thin shell mass and total mass of gravastar given in equations (29), (30), (31) and (32), respectively.
The gravastar is an amazing hypothetical idea introduced by Mazur and Mottola [1,2], which structurally looks like BH externally. We know that two main issues associated with BH are singularity and event horizon, all physics laws fail at singularity but the gravastar idea resolve these drawbacks of BH . In our present article, we have found new non-singular, horizon free solutions for gravastar with Kuchowicz type metric potential in Rastall gravity and the graphical study proves our model as stable and physically acceptable.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.