# Generating physically realizable stellar structures via embedding

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## Abstract

In this work we present an exact solution of the Einstein–Maxwell field equations describing compact charged objects within the framework of classical general relativity. Our model is constructed by embedding a four-dimensional spherically symmetric static metric into a five-dimensional flat metric. The source term for the matter field is composed of a perfect fluid distribution with charge. We show that our model obeys all the physical requirements and stability conditions necessary for a realistic stellar model. Our theoretical model approximates observations of neutron stars and pulsars to a very good degree of accuracy.

## 1 Introduction

Einstein’s general theory of relativity has successfully accounted for various observations or cosmological scales as well as in astrophysical contexts [1, 2]. The golden age of cosmology has seen the theory fine-tuned to a high degree of accuracy in explaining the Hubble rate, matter content, baryogenesis, nucleosynthesis, as well as the possible origin and subsequent evolution of the Universe. General relativity, as an extension of Newtonian gravity, is especially useful in describing compact objects in which the gravitational fields are very strong. Some of these objects include neutron stars, pulsars and black holes where densities are of the order of \(10^{14}\) g cm\(^{-3}\) or greater. The first exact solution of the Einstein field equations representing a bounded matter distribution was provided by Schwarzschild [3]. This solution described a constant density sphere with the exterior being empty. The constant density Schwarzschild solution was a toy model which cast light on the continuity of the gravitational potentials and the behaviour of the pressure at the surface of the star. However, the interior Schwarzschild solution was noncausal in the sense that it allowed for faster than light propagation velocities within the stellar interior. This prompted the search for physically viable solutions of the Einstein field equations describing realistic stars. Now we are a century later and we have thousands of exact solutions of the field equations describing a multitude of stellar objects ranging from perfect fluids, charged bodies, anisotropic matter distributions, higher-dimensional stars and exotic matter configurations. Spherical symmetry is the most natural assumption to describe stellar objects. However, there is a wide range of stellar solutions exhibiting departure from sphericity. These solutions include the Kerr metric which describes the exterior gravitational field of a rotating stellar object [4]. In the limit of vanishing angular momentum, the Kerr solution reduces to the exterior Schwarzschild solution. There have also been numerous attempts at extending the Kerr metric to allow for dissipation and rotation [5, 6, 7].

In order to generate exact solutions of the Einstein field equations, researchers have employed a wide range of techniques to close this system of highly nonlinear, coupled partial differential equations. In the quest to obtain exact solutions describing static compact objects one imposes (i) symmetry requirements such as spherical symmetry, (ii) an equation of state relating the pressure and energy density of the stellar fluid, (iii) the behaviour of the pressure anisotropy or isotropy, (iv) vanishing of the Weyl stresses, (v) space-time dimensionality, to name just a few [8, 9, 10, 11, 12, 13, 14]. These assumptions render the problem of finding exact solutions of the field equations mathematically more tractable. There is no guarantee that the resulting stellar model actually describes a physically realizable stellar structure. In the case of nonstatic, radiating stars, various exact solutions are known in the literature ranging from acceleration-free collapse, Weyl-free collapse, vanishing of shear, collapse from/to an initial/final static configuration as well as anisotropic collapse models.

The Randall–Sundrum braneworld scenario has generated an intense interest in higher-dimensional gravity and modified theories of gravity [15]. Braneworld stars were shown to have nonunique exteriors due to radiative-type stresses arising from five-dimensional graviton effects emitting from the bulk [16]. Govender and Dadhich showed that the gravitational collapse of a star on the brane is accompanied by Weyl radiation [17]. They concluded that a collapsing sphere on the brane is enveloped by the brane generalised Vaidya solution which is in turn matched to the Reissner–Nordstrom metric. The mediation of the Vaidya envelope is a unique feature of the braneworld collapse which is not present in standard 4-d Einstein gravity. A recent model by Banerjee et al. showed that Weyl stresses lead naturally to anisotropic pressures within the core of a braneworld gravastar [18]. In their model the Mazur and Mottola gravastar picture [19] is considered within the Randall–Sundrum II type braneworld scenario. Recently, Dadhich et al. demonstrated the universality of the constant density Schwarzschild solution in general Einstein–Lovelock gravity and the universality of the isothermal sphere for pure Lovelock gravity when \(d \ge 2N + 2\). In a recent paper Chakrabory and Dadhich ask a pertinent question: “Do we really live in four dimensions or higher?” This question arises from the fact that while gravity is free to propagate in higher dimensions while all other matter fields are confined to four dimensions, gravity cannot distinguish between 4-d Einstein or in particular, 7-d pure Gauss–Bonnet dynamics [20].

The idea of embedding a purely gravitational field represented by a four-dimensional Riemannian metric into a flat space of higher dimensions has resurrected interest in so-called class one space-times. Karmarkar derived the necessary condition for a general spherically symmetric metric to be of class one [21]. In general, if the lowest number of dimensions of flat space in which a Riemannian space of dimension *n* can be embedded in \(n + p\), then the Riemannian space is referred to as class *p*. Class one space-times have been successfully utilised to model compact objects such as strange star candidates, neutron stars and pulsars [22, 23, 24, 25, 26, 27, 28]. These theoretical models accurately predict and agree with observations regarding the masses, radii, compactness and densities of these objects within experimental error. On the other hand Momeni et al. [29, 30, 31] have obtained the realistic compact objects for Tolman–Oppenheimer–Volkoff equations in *f*(*R*) gravity in different context.

In this work we use the condition arising from embedding a 4-d spherically symmetric static metric in Schwarzschild coordinates into a 5-d flat space-time to model a charged compact object. By choosing one of the metric potentials on physical grounds, the embedding condition gives us the second metric potential which then completely describes the gravitational behaviour of the model. This paper is structured as follows: In Sect. 2 we introduce the 4-d Einstein space-time and provide the necessary and sufficient condition for embedding this space-time into a 5-d flat space-time. The Einstein–Maxwell field equations describing the gravitational behaviour of our stellar model are presented in Sect. 3. In Sect. 4 we derive an exact solution of the Einstein–Maxwell equations describing a charged static sphere by making use of the embedding condition derived in the previous section. The boundary conditions required for the smooth matching of the interior of the star to the vacuum Schwarzschild exterior solution is given in Sect. 5. The physical viability of our model is considered in Sect. 6. We conclude with a discussion in Sect. 7.

## 2 Class one condition for spherical symmetric metric

*r*.

*K*is a positive constant. On inserting the components \( z^1,z^2, z^3, z^4 \) and \(z^5\) into the metric (2), we obtain

## 3 Einstein–Maxwell field equations

*p*is the isotropic pressure and \(v^\nu \) is the fluid four-velocity given as \(e^{-\nu (r)/2}v^\nu =\delta ^\nu _\mu \). We are using geometrized units and thus take \(\kappa =8\pi \) and \(G=c=1\).

*A*is the four-potential and \(J^\mu \) is the four-current. Since the charge is at rest there will be no magnetic field generated in the rest frame (at each interior point of the star) and the four-potential and four-current are given, respectively, as

*m*(

*r*) is the mass function for an electrically charged compact star model then it can be defined in terms of the metric function \(e^{\lambda }\) and electric charge

*q*as

## 4 New class of general solutions for a charged compact star

*m*(

*r*) and electric charge

*q*, we assume the following form for the metric function \(e^{\nu }\) (Fig. 1):

*A*and

*B*are positive constants and \(n \le -1\). This form of the metric function is well motivated and has been utilised by Maurya et al. [34] to model charged compact stars arising from the Karmarkar condition. In these models they took \(n > 2\). The parameter

*n*acts as a ’switch’ and characterises various well-known models available in the literature. It is clear from Eq. (22) that \(n = 0\) renders the space-time flat, which is meaningless in the present context of this paper. It was first pointed out by Tikekar and more recently by Maurya et al. [35] that the Karmarkar condition together with isotropic pressure (in the case of neutral fluids) admits two solutions: the Schwarzschild interior solution and the Kohler–Chao–Tikekar solution [40, 41, 42]. The Schwarzschild solution is conformally flat, ie., the Weyl tensor vanishes at each interior point of the sphere. The Kohler–Chao–Tikekar solution is not conformally flat and furthermore represents a cosmological solution. This is to say that there is no finite radius for which the pressure vanishes in the Kohler–Chao–Tikekar solution. We regain the Kohlar–Chao–Tikekar solution when we set \(n = 1\) in Eq. (22). Furthermore, we observe from Table 3 that the product

*nA*is approximately constant for large

*n*. As pointed out here that as \(n \rightarrow -\infty \) the metric function \(\nu = Cr^2 + \ln {B}\) where we have defined here \(C = - nA\). This form of the metric function \(\nu \) has been already used to construct electromagnetic mass (EMMM) models by Maurya et al. [35]. These models have the peculiar feature of vanishing electromagnetic field, mass, pressure and density when the parameter \(n = 0\). In addition, the fluid obeys an equation of state of the form \(p + \rho = 0\) implying that the pressure within the bounded configuration is negative. In this study we will consider solutions for \(n < 0\). We should point out that the solution describes a physical viable compact when \(n \le -2.7\). For \(n > -2.7\), causality is violated within the stellar fluid as the sound speed exceeds unity. We have started our physical analysis with \(n =-6.5\), since there are no physically realizable stars between \(n = -2.7\) to \(-6.5\) as observed by Gangopadhyay et al. [36] In the limiting case \( n= -2.7 \) one expects low mass stars.

## 5 Boundary conditions

*M*is a constant representing the total mass of the charged compact star.

(ii) The radial pressure \(p_{\mathrm{r}}\) must vanish at the boundary (\(r = R\)) of the star (i.e. the continuity of \(\frac{\partial g_{tt}}{\partial r}\) across the boundary of the star) [50], which is known as the second fundamental form.

*B*can be determined by using the condition \(e^{\nu (R)}=e^{-\lambda (R)}\), which yields

*A*can be determined from the expression

## 6 Physical properties of the solution

### 6.1 Regularity

- (i)
Metric functions at the centre, \(r=0\): we observe from Eqs. (22) and (23) that the metric functions at the centre \(r=0\) assume the values \(e^{\nu (0)}=B\) and \(e^{\lambda (0)}=1\). This shows that metric functions are free of a singularity and positive at the centre (since

*B*is positive). Also, the two metric functions \(e^{\nu }\) and \(e^{\lambda }\) are monotonically increasing function of*r*(Figs. 1 and 2). - (ii)
Pressure at the centre \(r=0\): From Eq. (26), we obtain the pressure

*p*at the centre \(r=0\) as \(p_0=-A\,(D+2n)/8\,\pi \). Since*A*and*D*are positive, it follows that the central pressure is positive provided that \(D < - 2n\). - (iii)
Matter density at the centre \(r=0\): We require that the matter density be positive at central point of the star. Inspection of Eq. (26) gives us \(\rho _{0}=(3\,A\,D/8\,\pi )\). Since

*A*and \(D(=A\,B\,n^{2}\,K)\) are positive due to positivity of*A*,*B*, \(n^{2}\) and*K*, the central density \(\rho _c\) is positive.

### 6.2 Causality

### 6.3 Energy conditions

#### 6.3.1 Equilibrium condition

#### 6.3.2 Stability through adiabatic index

#### 6.3.3 Harrison–Zeldovich–Novikov static stability criterion

*n*for low density stars. It is clear that \(\mathrm{d}M/\mathrm{d}\rho _0\) decreases as |

*n*| increases for high density stars.

### 6.4 Electric charge

*n*| with the difference becoming indistinguishable at the stellar surface for very large |

*n*|. We may then interpret

*n*as a ’stabilizing’ factor. The variation of charge with

*n*suggests that lower values of

*n*imply lower charge which in turn means smaller electromagnetic repulsion. Figure 3 shows that larger |

*n*| leads to greater surface charge thus indicating greater electromagnetic repulsion here. This would mean that the surface layers of the charged body is more stable than the inner core. The onset of collapse of such a body could proceed in an anisotropic manner or the collapse could lead to the cracking of the object thus avoiding the formation of a black hole. As pointed out by Ray et al. [45] the charge can be as high as \(10^{20}\) coulombs and hydrostatic equilibrium may still be achieved, however, these equilibrium states are unstable. Bekenstein [46] argued that high charge densities will generate very intense electric fields. This will in turn induce pair production within the star, thus destabilizing the core. As an illustration we calculate the amount of charge at the boundary in units of coulombs for the compact star \(4U1608-52\) as follows: (i) \(8.90468 \times 10^{19}\) coulomb for \(n = - 6.5\), (ii) \(9.52895\times 10^{19}\) coulomb for \( n = -10 \), (iii) \(1.0370\times 10^{20}\) coulomb for \(n = - 50\), (iv) \(1.05471\times 10^{20}\) coulomb for \(n= - 500\), (v) \(1.05645\times 10^{20}\) coulomb for \(n= - 5000\), (vi) \(1.05662\times 10^{20}\) coulomb for \(n = - 50000\). However, the amount of charge in coulombs throughout the star can be determined by multiplying every recorded value in Table 1 by a factor of \(1.1659\times 10^{20}\).

The electric charge for compact star \(4U 1538-52\) for different values of *n* in the relativistic unit (km)

| \(n = - 6.5\) | \(n = - 10\) | \(n = - 50\) | \(n = - 500\) | \(n = -5000\) | \(n = -50{,}000\) |
---|---|---|---|---|---|---|

0.0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.2 | 0.004136 | 0.005083 | 0.006337 | 0.006603 | 0.006630 | 0.006632 |

0.4 | 0.035663 | 0.042614 | 0.051869 | 0.053831 | 0.054026 | 0.054045 |

0.6 | 0.133453 | 0.153669 | 0.180723 | 0.186445 | 0.187010 | 0.187067 |

0.8 | 0.353352 | 0.391594 | 0.442957 | 0.453784 | 0.454852 | 0.454958 |

1.0 | 0.763760 | 0.817304 | 0.889463 | 0.904631 | 0.906125 | 0.906273 |

### 6.5 Effective mass and compactness parameter for the charged compact star

*M*/

*R*) ratio as proposed by Buchdahl [49] for static spherically symmetric isotropic fluid models is given by \(2M/R\le 8/9 \). On the other hand, [51] proved that for a compact charged fluid sphere there is a lower bound for the mass–radius ratio,

*u*(

*r*) is defined by

### 6.6 Redshift

*u*, which implies that the surface redshift for any star cannot be arbitrarily large because the compactness

*u*satisfies the Buchdahl maximally allowable mass–radius ratio (Fig. 11). However, the surface redshift will increase with increase of compactness

*u*. Also, from Table 5. we observe that the surface redshift decreases with an increase in |

*n*|.

### 6.7 Equation of state

*B*is the bag constant. This equation of state has been successfully used to model compact objects in general relativity ranging from neutron stars through to strange star candidates. A recent model of a radiating star in which the collapse proceeds from an initial static configuration obeying a linear equation of state of the form \(p_\mathrm{r} = \alpha (\rho - \rho _\mathrm{s})\) where \(p_\mathrm{r}\) is the radial pressure, \(\rho _\mathrm{s}\) is the surface energy density and \(\alpha \) is the EoS parameter showed that the variation of \(\alpha \) affects the temperature profile of the collapsing body. Figure 12 shows the variation of the ratio \(p/\rho \) with

*r*/

*R*. We note that the pressure is less than the density at each interior point of the configuration. This ratio is also positive everywhere inside the star. As |

*n*| increases, the ratio \(p/\rho \) decreases with the differences tending to zero towards the surface layers of the star (Fig. 12).

Comparison between estimated and observed values of mass and radius for different compact stars [36]

Compact star | \(M/M_{\odot }\) (estimated) | | | \(M/M_{\odot }\) (observed) | \(R\,\) (Km) (observed) |
---|---|---|---|---|---|

\(4U 1538-52\) | 0.87 | 7.866 | 0.162938 | 0.87 ± 0.07 | 7.866 ± 0.21 |

SAX J1808.4-3658 | 0.90 | 7.951 | 0.166756 | 0.9 ± 0.3 | 7.951 ± 1.0 |

PSR J1903+327 | 1.667 | 9.4265 | 0.26052 | 9.438 ± 0.03 | 1.667 ± 0.021 |

Numerical data of \(AR^2\) corresponding to observed mass and radius with reference to Table 1 for different values of *n*

Compact stars | \( n=-6.5\) | \(n=-10\) | \(n=-50\) | \(n=-500\) | \(n=-5000\) | \(n=-50{,}000\) |
---|---|---|---|---|---|---|

\(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | |

\(4U1538{-}52\) | 0.0334 | 0.021731 | 0.004353 | 0.00043544 | 0.000043545 | 0.0000043545 |

SAX J1808.4-3658 | 0.034453 | 0.02242 | 0.0044913 | 0.0004493 | 0.00004493 | 0.000004493 |

Numerical data of \(AR^2\) corresponding to observed mass and radius with reference to Table 1 for different values of *n*

Compact stars | \( n=-45\) | \(n=-100\) | \(n=-1000\) | \(n=-10{,}000\) | \(n=-100{,}000\) |
---|---|---|---|---|---|

\(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | \(A{R}^{2}\) | |

PSR J1903+327 | 0.009804 | 0.004415 | 0.00044175 | 0.000044178 | 0.0000044178 |

Numerical data of physical parameters \(AR^2\), *A*, *B*, *D*, *K* and *nA* for different values of *n* for \(4U 1538-52\)

| \(AR^2\) | | | | | |
---|---|---|---|---|---|---|

\(-6.5\) | 0.033400 | \(5.3980 \times 10^{-4} \) | 0.548101 | 10.38552 | \(8.30822\times 10^{2} \) | \(-0.0035087\) |

\(-10\) | 0.021731 | \(3.5121\times 10^{-4} \) | 0.549818 | 16.26344 | \(8.422215\times 10^{2} \) | \(-0.0035121\) |

\(-50\) | 0.004353 | \(7.0350\times 10^{-5}\) | 0.552284 | 83.46073 | \(8.592418 \times 10^{2} \) | \(-0.0035175\) |

\(-500\) | 0.00043544 | \(7.03753\times 10^{-6}\) | 0.552841 | \(8.39474\times 10^{2}\) | \(8.630715 \times 10^{2} \) | \(-0.003518765\) |

\(-5000\) | 0.000043545 | \(7.03753\times 10^{-7}\) | 0.552899 | \(8.39963\times 10^{3}\) | \(8.634831 \times 10^{2}\) | \(-0.003518765\) |

\(-50{,}000\) | 0.0000043545 | \(7.03753\times 10^{-8}\) | 0.552905 | \(8.40010\times 10^{4}\) | \(8.635231 \times 10^{2}\) | \(-0.003518765\) |

Numerical data of physical parameters \(AR^2\), *A*, *B*, *D*, *K* and *nA* for different values of *n* for \(SAX J1808.4-3658\)

| \(AR^2\) | | | | | |
---|---|---|---|---|---|---|

\(-6.5\) | 0.034453 | \(5.4496\times 10^{-4} \) | 0.538534 | 10.3087854 | \(8.313872\times 10^{2} \) | \(-0.00354224\) |

\(-10\) | 0.02242 | \(3.5464\times 10^{-4} \) | 0.540283 | 16.1520157 | \(8.42981\times 10^{2} \) | \(-0.0035464\) |

\(-50\) | 0.0044913 | \(7.1039\times 10^{-5}\) | 0.552284 | 83.4607306 | \(8.509081 \times 10^{2} \) | \(-0.00355195\) |

\(-500\) | 0.0004493 | \(7.1070\times 10^{-6}\) | 0.543421 | \(8.345640\times 10^{2}\) | \(8.643643 \times 10^{2} \) | \(-0.0035535\) |

\(-5000\) | 0.00004493 | \(7.1070\times 10^{-7}\) | 0.5434891 | \(8.350693\times 10^{3}\) | \(8.647794 \times 10^{2}\) | \(-0.0035535\) |

\(-50{,}000\) | 0.000004493 | \(7.1070\times 10^{-8}\) | 0.5434959 | \(8.351198\times 10^{4}\) | \(8.648210 \times 10^{2}\) | \(-0.0035535\) |

The central density, surface density and central pressure for compact star candidate \(4U 1538-52\)

Value of | Central density (gm/cm\(^{3}\)) | Surface density (gm/cm\(^{3}\)) | Central pressure (dyne/cm\(^{2}\)) | Surface redshift |
---|---|---|---|---|

\(-6.5\) | \(9.0314\times 10^{14} \) | \(7.52234\times 10^{14} \) | \(6.82219\times 10^{34}\) | 0.20954368 |

\(-10\) | \(9.2018\times 10^{14} \) | \(7.38531\times 10^{14} \) | \(6.34375\times 10^{34}\) | 0.208319814 |

\(-50\) | \(9.4589\times 10^{14} \) | \(7.18798\times 10^{14} \) | \(5.62454\times 10^{34}\) | 0.206572451 |

\(-500\) | \(9.5175\times 10^{14} \) | \(7.1447\times 10^{14} \) | \(5.4610\times 10^{34}\) | 0.206180642 |

\(-5000\) | \(9.5230\times 10^{14} \) | \(7.14015\times 10^{14} \) | \(5.4444\times 10^{34}\) | 0.206139815 |

\(-50{,}000\) | \(9.5236\times 10^{14} \) | \(7.1397\times 10^{14} \) | \(5.4427\times 10^{34}\) | 0.206135287 |

The central density, surface density and central pressure for compact star candidate \(PSR J1903+327\)

Value of | Central density (gm/cm\(^{3}\)) | Surface density (gm/cm\(^{3}\)) | Central pressure (dyne/cm\(^{2}\)) | Surface redshift |
---|---|---|---|---|

\(-45\) | \(1.0896\times 10^{15} \) | \(7.18066\times 10^{14} \) | \(1.5310\times 10^{35}\) | 0.39816 |

\(-100\) | \(1.09906\times 10^{15} \) | \(7.13153\times 10^{14} \) | \(1.50569\times 10^{35}\) | 0.39714 |

\(-1000\) | \(1.10608\times 10^{15} \) | \(7.09568\times 10^{14} \) | \(1.48703\times 10^{35}\) | 0.39641 |

\(-10{,}000\) | \(1.1065\times 10^{15} \) | \(7.09028\times 10^{14} \) | \(1.48481\times 10^{35}\) | 0.39634 |

\(-100{,}000\) | \(1.10657\times 10^{15} \) | \(7.08991\times 10^{14} \) | \(1.4846\times 10^{35}\) | 0.39633 |

## 7 Discussion

In this paper we attempted to obtain electromagnetic mass models (EMMM) which were first addressed by Lorentz. The Lorentz electromagnetic mass models had the distinguishing feature that vanishing charge density is accompanied by the simultaneous vanishing of all other thermodynamical quantities. In addition, the equation of state of these models is of the form \(\rho + p = 0\), giving rise to negative pressure. The solution obtained in this work relaxes this particular equation of state, allowing for positive pressure. The gravitational and thermodynamical behaviour of our model is controlled by a parameter *n*. Switching off *n* results in the vanishing of the charge density and all other thermodynamical quantities such as density and pressure. We use a novel approach: embedding a spherically symmetric, static metric in Schwarzschild coordinates into a five-dimensional flat metric. This embedding is equivalent to the Karmarkar condition: the requirement for a spherically symmetric metric to be of embedding class 1. The condition obtained from this embedding relates the gravitational potentials thus reducing the problem of finding an exact solution of the Einstein–Maxwell field equations to a single-generating function. By specifying one of the gravitational potentials on physical grounds, we obtain the second potential which completely describes the gravitational behaviour of the compact object. The junction conditions required for the smooth matching of the interior space-time to the exterior Reissner–Nördstrom space-time fixes the constants in our solution and determines the mass contained within the charged sphere. Our model displays many salient features which bode well for describing a compact, self-gravitating object. Graphical analysis of the solution shows that the density and pressure are monotonically decreasing functions of the radial coordinate. The pressure vanishes at some finite radius. This indicates that our solution can be utilised to describe a bounded object unlike the Kohler–Chao solution, which arises from the imposition of the Karmarkar condition together with pressure isotropy. Causality is obeyed at each interior point of the configuration. Stability analysis via the adiabatic index and the Harrison–Zeldovich–Novikov static stability criterion indicate that our model is stable. Analysis of the variation of charge with the radial coordinate reveals an interesting characteristic of our model. The charge increases with the parameter |*n*|. This increase is largest towards the surface layers of the charged object becoming simultaneously indistinguishable for very large values at the surface. This implies that the surface layers are more stable (larger repulsive forces here) than the inner core layers. This ’differentiated’ stability may lead to anisotropic collapse or the subsequent cracking of the sphere should this object starts to collapse. This phenomenon has not been discussed elsewhere in the literature. The influence of the parameter *n* is clearly drawn out in Tables 2, 3, 4. Table 2 shows that our theoretical model describes compact objects to a very good degree of accuracy with regards to observed masses and radii of stars. Tables 5 and 6 clearly show that variations in the model parameters stabilise for very large *n*. Tables 7 and 8 illustrate the influence of the parameter *n* on the central density, surface density, central pressure and surface redshift for the stars 4*U* 1538–52 and SAX J1808.4–3658. It is clear that for very large *n* variations in these physical quantities tend to zero. This feature indicates that the parameter *n* can be viewed as a ’building’ constant, that is to say, an increase in *n* is accompanied by an increase in mass, radius and charge which builds up the star from \(r = 0\) through to the surface. In this work we have utilised \(n < 0\) and the case \(n \ge 0\) was studied by [34]. Future work has been initiated to consider the case of general *n*.

## Notes

### Acknowledgements

The author S. K. Maurya acknowledges authority of University of Nizwa for continuous support and encouragement to carry out this research work. Also the authors are indeed grateful to a honorable referee for pointing out some important and substantial remarks/comments which were required for the manuscript to meet the standard of the esteemed journal.

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