Manuel Malaver

doi:10.19080/AJPN.2024.01.555552

Research Article

Some New Models for Quark Stars with Strange Matter Equation of State

Manuel Malaver*

Maritime University of the Caribbean, Department of Basic Sciences, Catia la Mar, Venezuela

Submission:August 21, 2024;Published: November 11, 2024

*Corresponding author:Manuel Malaver, Maritime University of the Caribbean, Department of Basic Sciences, Catia la Mar, Venezuela

How to cite this article: Manuel M. Some New Models for Quark Stars with Strange Matter Equation of State. Academic Journal of Physics Research. 2024; 1(1): 555552.DOI:10.19080/AJPN.2024.01.555552

Abstract

In order to describe the properties and behavior of anisotropic charged compact stars, in this study we presented new exact solutions to the Einstein-Maxwell field equations. We assume a linear equation of state and a metric function defining a gravitational potential dependent on an adjustable parameter n. We used a relationship between gravitational potential and the electric field that helps to solve the field equations. The class of solution obtained undergoes several physical analyses for the realistic compact star models. The analysis reveals that both gravitational potentials and matter variables are well behaved and the model satisfies important physical conditions as the measure of anisotropy and matching.

Keywords:Compact Stars; Gravitational Potential; Linear Equation of State; Adjustable Parameter; Einstein-Maxwell System

Introduction

The phenomena of supernovae stars giving birth to strange stars through gravitational collapse has motivated a number of researchers to explore the geometry of stellar inner portions [1,2]. In general relativity the Einstein field equations are useful in examining the physical characteristics and gravitating behaviours of compact stellar bodies such as the white dwarfs, neutron stars and black holes are examples of star remnants [3-5]. The Einstein-Maxwell field equations have been used to produce a number of models that have been applied to investigate several properties of compact stellar objects [6-18]. The essence of these models show that the field equations are useful and applied as tools to provide results with astrophysical significance.

One of the groundbreaking developments in the theory of general relativity was made by Schwarzschild [3] who derived the first solution to Einstein’s field equations known as the Schwarzschild solution. This solution describes the gravitational field outside a spherical mass by assuming that the electric field, angular momentum of the mass and cosmological constant is all zero. This solution has been crucial in understanding the behavior of massive objects and their interaction with gravity.

Modeling compact stellar objects has become a popular and important endeavor to explore various characteristics including their mass, charge, structure and stability [19]. Some reasonable physical stellar models can be proposed with various state equations as the linear equation of state [20-26], quadratic equation of state [27-30], polytropic equation of state [31-32] and Van der Waals equation of state [33-34].

The study of anisotropy pressure in stellar objects in the presence of strong gravitational fields is a topic of fundamental importance for many researchers in astrophysics. Sokolov [18] states that phase transitions are determinants in the evolution of neutron stars. The presence of an electrical field is also a cause of anisotropy [19]. Bowers and Liang [6] indicate that the presence of anisotropy can modify the structure of compact objects. Herrera [11] concludes that the pressure anisotropy influences matter stability due to the appearance of radial forces of different sign in the stellar interior causing a disturbance in the system balance. Thirukkanesh and Ragel [35] state that anisotropy influences the structure and some physical parameters of compact stars such as mass and compactness. Moreover, there exist a number of research studies that have come up with anisotropic models (Takisa and Maharaj [31], Thirukkanesh and Ragel [32], Malaver [33-34], Thirukkanesh and Ragel [35], Mak and Harko [36], Malaver and Iyer [37]).

Malaver and researches [38] have discussed the effect of electromagnetic fields on compact stellar bodies in a Buchdahl space time. Malaver, Iyer, and Khan [39] have determined some physical characteristic in the framework of Einstein-Gauss-Bonnet gravity for compact stellar objects with the metric potential proposed by Buchdahl. Iyer [40-41] has recently published many papers as well as presentations on the importance of Rank-n tensor time quantifying gravity in quantum states with gravity and tensor time metrics. In these studies, the gradation of time tensors from rank-6 to rank-1 vectors in spacetime presents a novel approach to unifying General Relativity (GR) and Quantum Relativity (QR).

In this study we formulated a new mathematical model for a compact star using a linear equation of state with a specific metric function and considering the existence of pressure anisotropy. This paper is structured as follows: in Section 2, we present Einstein´s field equations that describe the gravitational behavior of the astrophysical objects. In Section 3, we used the proposed metric potential that allows solving analytically the field equations. In Section. 4, we discuss the physical conditions that must have a charged star. Finally in Section 5, we conclude that the proposed model describes a charged stable star and that the matter variables as radial pressure and energy density have maximum values at the center of the star.

Einstein Field Equations

We considered a distribution of matter with spherical symmetry whose stress tensor is locally anisotropic. The metric in a star in Schwarzschild coordinates will be described by the simple form of line element

Where v(r) and λ(r) are variables defining the gravitational potentials derived from the Einstein-Maxwell field equations given by for

The quantities , p , p and E r t ρ refer to as energy density, radial pressure, tangential pressure and electric field, respectively. The basic field equations (2)-(4) are transformed to find the solution to the Einstein field equations with the transformations, x = cr², Z(x)=e^−2λ(r), and A² y² (x) = e^2v(r) with arbitrary constants A and c>0, suggested by Durgapal and Bannerji [42]. The metric (1) can be expressed as

and the Einstein field equations can be written as

σ is the charge density and dots denote differentiation with respect to x. With the transformations of [42], the mass within a radius r of the sphere take the for

In this paper, we assume the following strange matter equation of state within the framework of MIT-Bag model

The New Obtained Models

Following Feroze and Siddiqui [27] and Malaver [21], we take the particular form of the gravitational potential, Z ( x) as

where a is a real constant and n is an adjustable parameter. According Lighuda et al. [43] we also assume for the electric field intensity

We have considered the particular cases for n=1, 2. For case n=1, using Z ( x) in eq. (6) we obtain

Substituting (15) in eq. (12), the radial pressure can be written in the form

Using (15) in (11), the expression of the mass function is

and for charge density

Substituting (13), (14) and (16) with n=1 in eq. (7), we have

Integrating (19), we obtain

and for the anisotropy Δ we have

Physical Properties of the New Models

Any physically acceptable solutions must satisfy the following conditions [32]:
a) Regularity of the gravitational potentials in the origin.
b) Radial pressure must be finite at the center.
c) P_r>0 and ρ > 0 in the origin.

The new models satisfy the system of equations (6) - (9) and constitute another new family of solutions for a charged quark star with anisotropy. The metric functions e^2λ and e^2v can be written in terms of polynomial functions, and the variables energy density, pressure and charge density also are represented analytical.

When n =1, matching conditions for r = R can be written as

In all the new classes of solutions, the mass function is continuous and behaves well inside of the star and the charge density not present singularities in the origin.

In the two cases the anisotropy vanishes at the centre of the star, i.e. Δ(r=0) =0. The solutions (15)-(32) for charged quark stars are physically acceptable and not present singularities in the origin, as it established Jotania and Tikekar [44].

Conclusion

This study included a choice of generalized metric function which has regained some choices made by previous researchers [21,43]. Moreover, the developed model was observed to be regular. That is, the potentials are above zero at the centre of a star showing that the model is regular. The matter variables as the energy density and radial pressure decrease with the increase of the radial distance. However, these variables have maximum values at the centre of the star. This indicates that the matter variables are well behaved within the stellar interior. Quantifying gravity in quantum states with gravity and tensor time metrics presents a novel approach to unifying General Relativity and Quantum Relativity. Astrophysical regions would demonstrate a rank2 tensor using Einstein’s Field Equations and the Schwarzschild metrics to explain the gravitational interaction in terms of spacetime curvature. Many experimental approaches such as high-energy particle collisions, gravitational wave observations, quantum entanglement experiments, astrophysical observations, and laboratory simulations have promising advances to find signatures of the quantum interior with astro exterior of these compact stars, especially pulsars [45-49].

We show as a modification of the parameter n of the gravitational potential affects the electrical field, charge density and the mass of the stellar object. The models presented in this article may be useful in the description of relativistic compact objects with charge, strange quark stars and configurations with anisotropic matter.