Embedded class-I solution of compact stars in gravity with Karmarkar condition
Introduction
Compact objects are usually designated as neutron stars, white dwarfs and black holes in astrophysics. Neutron stars and white dwarfs are born in the result of gravitational collapse which generally happen due to degeneracy pressure of the relativistic objects. These objects possess huge densities values but are volumetrically smaller. Although, we do not know the precise characteristics of these kinds of compact stars, but these objects are presumed to be heavy stars having tiny radius. Every kinds of stellar objects are mostly acknowledged as degenerate stars, except black holes. In general relativity () the study of configured equilibrium of compact stars is valuable because of the massive nature and high densities of these objects. The study of compact objects in the framework of and modified gravitational theories has always been considered as a topic of great interest in astrophysics. To investigate celestial compact structure modeling, we need an exact solution to Einstein field equations (EFE). In , Schwarzschild [1] obtained the EFE solution for the interior structure of compact objects. In this regard, various modified gravitational theories have been presented in order to obtain the complicated exact solutions of EFE. In the framework of observational data, Tolman [2] and Oppenheimer [3] investigated some realistic models of non-transversable stellar objects and claimed that the physical characteristics of these objects represent the relationship between the internal pressure and the gravitational force which eventually leads in a state of equilibrium structures. In the study of stars internal configuration, this phenomenon has considerable importance and provides realistic results in various occasions. Further, compact celestial structure was examined by Baade and Zwicky [4]. According to their study the supernova might transform into smaller compact objects after the observation of strongly magnetized spinning neutrons. Moreover, Ruderman [5] identifies for the first time that at the center of the stellar object the nuclear density exhibits anisotropic behavior.
A perfect fluid was assumed to be a source of celestial structures for the creation of stellar objects. These objects are commonly defined as ultra-dense, isotropic, celestial bodies that seem to be spherically symmetric. Whereas, isotropy is assumed as a desired attribute, still it does not have a general feature of compact stars. The concept of non-zero anisotropy in a stellar configuration was first presented by Bowers and Liang [6]. Moreover, Ruderman [5] investigated the possibility that stars might come with huge densities , when the nuclear matter is anisotropic in nature. In fact, anisotropic matter distribution plays a crucial role in narrating the inner representation and development phase of the relativistic stellar objects. In anisotropic fluid distribution background, a lot of literature is available [7], [8], [9], [10], [11]. In anisotropic distribution, the pressure component of the fluid sphere divides into two components, radial and transverse.
One of the most promising topics of modern research in cosmology is the accelerated expansion of the universe. Cosmologists argue that the accelerating expansion of the universe depends on dark energy and dark matter which retains negative pressure. As a substitute of , different gravitational theories have been presented to unfold the mystery behind dark energy issues. These gravitational theories are recognized as modified theories of gravity. Some of these modified theories are . Among these valuable theories, is one of the most simplest and popular theory, obtained as an arbitrary function of Ricci scalar. This theory was proposed by Buchdahl [12]. Later on, Nojiri and Odintsov [13] demonstrated some models of theory of gravity by placing curvature as a function of Ricci scalar and the outcomes of their considered models are quite viable and stable. Further, Starobinsky [14] presented an interesting class of theory of gravity models that showed the physically acceptable results in laboratory testing of the solar structure. Some gravity models were proposed by Hu and Sawicki [15] by ignoring the cosmological constant and their study evident some interesting results in regard to accelerating expansion phenomena. The viability of physical attributes of compact stars by taking exponential type models of theory of gravity was discussed by Cognola et al. [16]. In modified theories of gravity beyond and its Hilbert-Einstein action, diffeomorphism invariance and the Bianchi identities violation have attracted a lot of interest. Hamity and Barraco [17] derived the generalized Bianchi identities for the non-linear gravity to throw some light on the issue that theory of gravity generates higher than second order equations of motion and violates Bianchi identities. Further, Wang et al. [18] confirmed the local energy–momentum conservation of Bianchi identities by establishing the equivalence relation between Palatini and the Brans–Dicke gravity. Moreover, Koivisto [19] explored a composition of gravity and the generalized Brans–Dicke gravity and claimed covariant conservation from both the metric tensors and the Palatini variational techniques.
For physically stable models, one could use an analytical approach of EFE and assume the family of a four dimensional manifold and transform it into Euclidean space. The embedding family of curved geometry into geometries of higher dimensions is considered to create various new exact solutions in astrophysical stellar systems. Schlai [20] designated the embedding issue on geometrically important spacetimes for the very first time. In this regard, Nash [21] presented the isometric embedding theorem. Several features of anisotropic compact objects by employing embedding class one approach have been discussed in literature [22], [23], [24], [25], [26], [27], [28], [29]. A class of non-static fluid distribution along with non-vanishing acceleration was studied by Gupta and Gupta [30]. Further, Gupta and Sharma [31] assumed plane symmetric metric to explore the embedding class-I solutions of non-static perfect fluid. The embedding class constraint yields a differential equation in static spherically symmetric geometry relating the two components of metric potentials, known as Karmarkar condition [32]. A lot of work [33], [34], [35], [36] has been done related to Karmarkar condition, defined as . The Karmarkar condition develops a connection between two parts of metric tensors and for a spherical static symmetric fluid distribution. Maurya et al. [37] and Bhar et al. [38] studied the EFE by adopting the Karmarkar condition and composing numerous classes of embedded class-I solutions. They also observed that these outcomes exhibit stable nature and might be helpful in exploring the internal structure of the stellar objects. Later, in the background of gravity, Sharif and Saba [39] investigated the charged anisotropic solutions and their derived solutions are physically consistent and stable. In this regard, a class of embedded solutions by adopting Karmarkar condition is recently presented by Upreti et al. [40].
In the background of , Maurya et al. [41] probe an anisotropic compact star using embedding class-I approach. They studied different features of compact stars in the presence of anisotropic fluid distribution and claimed that obtained outcomes describe the internal core of stellar objects. Further, Bhar et al. [42] presented a new relativistic anisotropic compact star model which is physically acceptable for embedding spacetime and can be used to describe the interior solution of stellar objects. Moreover, the relativistic model for anisotropic compact stars in background was studied by Prasad et al. [43]. Their results indicate that by adopting embedding class-I condition the obtained relativistic stellar structure is physically reasonable. Recently, Mustafa et al. [44] introduced new exact solutions of EFE in the framework of Bardeen black hole spacetime by using the well known Karmarkar condition. Their chosen model demonstrated the well-behaved nature under the particular values of the parameters. By getting motivated from literature, in this particular paper, we extended the concept of Bhar et al. [42] in the background of theory of gravity and analyzed the physical attributes of the compact stars, namely, , , , and . To meet our goal, we assume two different theory of gravity models by employing the Karmarkar condition in the framework of anisotropic pressure. We consider a particular model for one of the metric potential and by adopting the Karmarkar condition, we construct the second metric potential of spherically symmetric spacetime.
The outline of the current article is organized as follows: In the next segment, the field equations have been developed by using Karmarkar condition with an anisotropic matter source. Two of viable gravity models along with boundary constraints have been introduced in Section 3. In Section 4, we compute the constant values by using the matching conditions. In Section 5, we discuss the physical attributes of the compact stars in detail. In the last portion, we provide the final verdict and conclusion of our study.
Section snippets
Modified field equations
First, we are proceeding to develop the field equations in gravity context. For this purpose, we consider the action of gravity [45] defined as Here, is a function of Ricci scalar and is the Lagrangian matter. Varying the action identified in (1) regarding the metric potential , we obtain the succeeding gravity field equation Thus, . Although, and indicate covariant derivative and D’Alembertian notation,
Viable gravity models
Here, we consider the two realistic and simple viable gravity models to analyze the modified field equations presented in Eqs. .
Matching conditions
For the stellar compact objects, the intrinsic boundary metric irrespective of the geometry (interior or exterior) will remain the same. This phenomena verifies that whatever the coordinates system covering the surface of the boundary, the components of the metric tensor will remain continuous. For stellar compact objects, the Schwarzschild solution is observed as the most suitable possibility to choose from the various choices of the matching conditions in the context of . Jebsen-Birkhoff’s
Physical characteristics of the compact stars models
Here, we discuss the graphical behavior of the considered compact stars in the context of gravity models. We studied the graphical illustration of energy density, pressure components, stability and equilibrium condition, energy conditions, mass function, adiabatic index, etc. For plotting the graphs, we consider the same values of constants mentioned in Table 1. Moreover, for Model-1, we fix the constant parameter and the variation of , for different compact stars is given as:
Conclusion
In order to explore a new family of embedded class-I solutions in the anisotropy background, we assume two realistic gravity models by using the Karmarkar condition. Since the Karmarkar condition reduced the solution-generating method of EFE to a single metric potential by providing a connection between and . For this purpose, we assume the metric potentials and by employing Karmarkar condition, we obtain second metric potential given as , where
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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