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 some known stellar objects such as the star remnants^3-5. The essence of these models show that the field equations are useful and applied as tools to provide results with astrophysical significance^6-18.

One of the groundbreaking developments in the theory of general relativity was made by Schwarzschild³ who derived the first solution to Einstein’s field equations. This solution has been crucial in understanding the behavior of massive objects and their interaction with gravity and has allowed to obtain Einstein's original cosmological solutions for a uniform distribution of fluid.

Modeling compact stellar objects has become a popular and important endeavor to explore various characteristics including their mass, charge, structure and stability¹⁹. 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. Sokolov¹⁸ states that phase transitions are determinants in the evolution of neutron stars. The presence of an electrical field is also a cause of anisotropy¹⁹. Bowers and Liang⁶ indicate that the presence of anisotropy can modify the structure of compact objects. Herrera¹¹ 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 Ragel³⁵ 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 Maharaj³¹, Thirukkanesh and Ragel³², Malaver^33-34, Thirukkanesh and Ragel³⁵, Mak and Harko³⁶, Malaver and Iyer³⁷).

Malaver and researches³⁸ have discussed the effect of electromagnetic fields on compact stellar bodies in a Buchdahl space time. Malaver, Iyer, and Khan³⁹ 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).

The main purpose of this study is to obtain a new variety of explicit solutions of Einstein field equations with a metric function proposed by Buchdahl⁴² and considering the existence of pressure anisotropy. In section 2 are shown the field equations and boundary conditions that describe the gravitational behavior of the astrophysical objects and the solutions for the Einstein field equations are given by in Section 3. In Section 4, are discussed the physical conditions that must have a charged star and the physical analysis of particular cases are given in Section 5. Finally in Section 6, we conclude that the proposed model describes a charged stable star and that the matter variables can contribute to the study of stellar structure.

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

with h and v functions of r only, as in the case of any static and spherical symmetric distribution of matter.  The Einstein-Maxwell field equations given by

The quantities P, P_r, P_t and E 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 ^-2h(r) and A² y² (x) =e^2v(r) with arbitrary constants A and c>0, suggested by Durgapal and Bannerji⁴³ . The metric (1) can be expressed as

and Einstein field equations are written as follows

o is the charge density and dots in system of equations (6)-(11) stand for derivatives with respect to x. With the transformations of Durgapal and Bannerji⁴³, the mass within a radius r for the realistic stellar body is given by for

Matching of the exterior and interior of the compact object at the boundary (r = R) is done by comparing the line element (1) with Reissner-Nordstrom exterior spacetime:

In this paper, we assume the following lineal equation of state

Materials and Methods

To be able to solve the system (6)-(11) we have taken the metric potential Z(x) of Buchdahl⁴² and for the electric field the proposal of Lighuda and researches⁴⁴ , respectively

where K is a parameter related to the geometry of the star and a is a real constant. The metric potential Z(x) is continuous at the centre of the star and well behaved in the interior of the star. The electric field is finite at the center of the star and remains continuous in the interior.

Substituting (15) and (16) in (6) we obtain

we have for the radial pressure

Using (17) in (12), the expression of the mass function is

With (15) and (16) in eq. (11), the charge density is

With (15), (16) and (18), the eq. (7) becomes

Integrating (21), we obtain

where for convenience

The metric functions can be written as

The anisotropy factor A is given by for

Physical Requirements

Any physically acceptable solution must satisfy the following conditions^32,45 :

The gravitational potentials e^2h and e^2v are functions that take finite and positive values along the radial coordinate and  are continuous throughout the stellar interior.

The energy density p decreases continuously from the centre r=0 and becomes zero at the surface r=R .

The radial pressure P_r must be finite at the centre and it must vanish at the surface of the sphere.

The radial pressure and density gradients  dP_r/dr^<0 and dP/dr^<0 for 0< r< R .

The anisotropy is zero at the center r=0, i.e. A(r=0) =0.

Any physically acceptable model must satisfy the causality condition, that is, for the radial sound speed V²_sr = dP_r/dP ,we should have  0< v²_sr <1.

(vii) At the surface of the star the charged interior solution should be matched with the Reissner–Nordström exterior spacetime (13).

Results and Discussion

For the new solutions, the gravitational potentials e^2hand e^2v have finite values and are continuous throughout the stellar interior, this is in agreement with the result by Sunzu, Maharaj and Ray²⁶. At the center  e^2h(0) and e^{2vh(0) =} = A² c²₁ K^2A e^2c . We show that in r=0 , (e^2h(r))’_r=0 = (e^2h(r))’ =(e^2v(r))’_r=0 = 0 and this makes is possible to verify that the gravitational potentials should be finite at the center and avoid the singularities within the stellar interior.

The energy density P and radial pressure P_r are decreasing functions with maximum values at the centre of the star. In the center p(r = 0) = 3c(K - 1)/K and P_r(r = 0) = c(K - 1)/K, therefore the energy density will be non-negative in r=0 and P_r(r = 0) > 0 . In the surface of the star P_r(r = R) = 0 and for the second fundamental form we have

Gradients dP/dr and dP_r/dr acquire negative values with the radial parameter. For 0< r < R

According equations (30) and (31) the pressure and density diminish in the stellar interior and vanishes on the surface of the star. The solution for r=R must match the Reissner–Nordström exterior space–time as:

and therefore, the continuity of e^v and e^h across the boundary r=R is

Then for the matching conditions, we obtain:

Table 1 contains the values of a, K and masses for different stars in r=R

Table 1: Values of  K, a and M(Mo) in r=R

M_o = sun’s mass

The figures 1,2,3,4,5,6,7,8 and 9 represent the plots of e^2h, e^2v, P , P_r , dp/dr, dP/dr, M , o² and A with the radial coordinate. For all the plots c=1.

Figure 1: Metric potential e2h versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 2: Metric potential e2v versus the  stellar radius for K=6  (solid line), K=8  (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 3: Energy density p versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 4: Radial pressure P versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 5: Energy density gradient versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 6: Radial pressure gradient versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 7: Mass function versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 8: Charge density O2 versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

Figure 9: Anisotropy versus the stellar radius for K=6 (solid line), K=8 (long- dash line) and K=10 (dash-dot line) with a=0.0002. Click here to view Figure

For different values of parameter K, the gravitational potentials e^2h (Figure 1) and e^2v (Figure 2) are monotonically increasing function with the radial distance, continuous throughout the stellar interior and takes higher values when K is increased. The energy density from Figure 3, is observed to be maximum at the star core. It decreases as the radial distance increases for all the values of K. This indicates that the model is physically realistic as we expect the maximum value of energy at the centre as observed by Sunzu, Maharaj and Ray²⁶. The radial pressure is also a decreasing function with radial coordinate with the maximum at the centre as a noted in Figure 4. The radial variation of energy density gradient has been shown in Figure 5, in which it is observed that dp/dr <0 in all the cases studied. In Figure 6 is also shown that the profile of dP_r/dr indicates that the radial pressure gradient is negative inside the star. In Figure 7, the mass of the stellar body increases monotonically from the centre to the surface for different values of K.  It is also physically realistic for well behaved models²⁴. The charge density is a continuously decreasing function as noted in Figure 8. The measure of pressure anisotropy A in Figure 9 shows that it is finite, regular, continuous, and increasing from the core of stellar object, reaches a maximum and then decreases near the surface. We can also note that A admits higher values with a growth of K.

We can compare the values calculated for the mass function with observational data of some astrophysical objects such as for PSR J1823-3021G, PSR J1748-2446an and PSR J1518+4904^46-48. The values of the stellar masses for these compact stars are tabulated in Table 2.

Table 2: The approximate values of the masses for the compact stars

The recently discovered pulsar PSR J1823 3021G is known to be part of a binary system and has the potential of being one of the most massive known pulsars⁴⁷. The same is noted with the binary pulsar PSR J1748-2446an which is a massive system that exceeds 2Mo⁴⁸. Iyer ^40-41 has outlined a method to quantize the gravitational field by examining the gradation of rank tensors within a metric wavefunction framework. Astrophysical regions would demonstrate a rank2 tensor, analogous to Schwarzschild metrics, by analyzing Einstein’s Field Equations and the Schwarzschild metrics to explain the gravitational interaction in terms of spacetime curvature. To validate theoretical framework of rank tensor gradation and metric wave functionality, many experimental approaches have been advanced to proposals ^48-60, including high-energy particle collisions, gravitational wave observations, quantum entanglement experiments, astrophysical observations, and laboratory simulations.

Conclusion

This study included a choice of generalized metric function which has regained some choices made by previous researchers³⁹. 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 proposed models can be compared with the pulsars PSR J1823-3021G, PSR J1748-2446an and PSR J1518+4904^43-44 and well behaved. 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^50-60.

Acknowledgment

The author would like to thank Magadh University for granting the Ph.D. research work. The Department of Environmental Science, A.N. College, Patna of the Magadh University, is highly appreciated for allowing the GIS laboratory work. The author is also profoundly grateful to the National Remote Sensing Center (NRSC), Indian Space Research Organisation (ISRO), Govt. of India for their guidance during the Satellite data procurement.

Funding Sources

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Conflict of Interest

The authors declares that there is no conflict of interest regarding the publication of this article.

Data Availability Statement

This statement does not apply to this article

Ethics Statement

This research did not involve human participants, animal subjects, or any material that requires ethical approval.

Informed Consent Statement

This study did not involve human participants, and therefore, informed consent was not required.

Authors’ Contribution

M.M (Manuel Malaver): Supervision and Writing – Original Draft.

R.I (Rajan Iyer): Review & Editing

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