Compact star models in class I spacetime
Abstract
In the present article, we have presented completely new exact, finite and regular class I solutions of Einstein’s field equations i.e. the solutions satisfy the Karmarkar condition. For this purpose needfully we have introduced a completely new suitable \(g_{rr}\) metric potential to generate the model. We have investigated the various physical aspects for our model such as energy density, pressure, anisotropy, energy conditions, equilibrium, stability, mass, surface and gravitational redshifts, compactness parameter and their graphical representations. All these physical aspects have ensured that our proposed solutions are wellbehaved and hence represent physically acceptable models for anisotropic fluid spheres. The models have satisfied causality and energy conditions. The presented models are also stable by satisfying Bondi condition and Abreu et al. condition, in equilibrium position and static by satisfying TOV equation, Harrison–Zeldovich–Novikov condition, respectively. For the parameters chosen in the paper are matching in modeling Vela X1, Cen X3, EXO 1785248 and LMC X4. The M–R graph generated from the solutions is matching the ranges of masses and radii for the considered compact stars. This work also estimated the approximate moment of inertia for the mentioned compact stars.
1 Introduction
After the pioneering work on the anisotropic relativistic compact star by Bowers and Liang [1], many researchers have investigated the possible origin of anisotropy in compact stars. Ruderman [2] proposed that at high density \(\sim 10^{15}\) g/cc the matter starts interacting relativistically that arises the anisotropy in pressure. It is also suggested that due to the formation of superfluid neutrons inside neutron stars, pressure anisotropy may also arise [3]. As a results of research by many authors pressure anisotropy can be trigger by types of phase transition [4], pioncondensation [5], slow rotation [6], strong magnetic field [7] etc. Letelier and his coauthor [8, 9, 10] have shown that such anisotropic matters can be considered as the composition of two perfect fluids, or a perfect fluid and a null fluid, or two null fluids.
Many researchers are also interested to investigate the properties of compact stars in higher dimensions. The concepts of extra dimensions were first proposed by Kaluza [11] and Klein [12] independently when they unify the gravity and EMforce. Liddle et al. [13] have analyzed the effect of extra dimensions in the maximum mass of neutron stars (NS). They have assumed an equation of state for noninteracting cold neutrons and found that the maximum mass of NS was reduces by the presence of extra dimensions. Chattopadhyay and Paul [14] have considered the Vaidya–Tikekar spacetime in ndimensional Einstein’s field equations to analyze the properties of compact stars. Bhar et al. [15] incorporated the conformal Killing equations in ndimensional Einstein’s field equation and discussed anisotropic compact stars.
Various extended theories of gravity have been used to investigate the physical properties of compact stars. Pani et al. [16] have used general class of alternative theories which includes scalartensor theories, a scalar field coupled to quadratic curvature invariant and indirectly f(R) theories as special cases. In their work, they have developed a systematic tool to rule out physical theories that are incompatible with observations. The concept of embedding 4dimensional spacetime in 5dimensional hyperspace was used by Castro et al. [17]. Here they have embedded four dimensional spacetime into five dimensional braneworld. It is shown that on choosing equation of state for hadronic, hybrid and quark stars, the maximum mass is controlled by brane tension \(\lambda \) which lies in the range \(3.89\times 10^{36} (\equiv 1.44M_\odot )< \lambda < 10^{38}\) dyne/cm\(^2\). It is important to remind that constructing exact interior solutions representing nonuniform stellar distributions is nearly impossible in the context of the braneworld. This is because the nonlocality and nonclosure of the braneworld equations, produced by the projection of the bulk Weyl tensor on the brane, lead to a very complicated system of equations which make the study of nonuniform distributions very hard [18]. It is yet to discover the criteria about what restriction should be imposed on braneworld equations to obtain a closed system [19]. To solve this problem, it is necessary to understand the bulk geometry and how a 4D spacetime cab be embedded. On the other hand, Karmarkar [20] embedded 4dimensional spacetime into 5dimensional Euclidean space known as class I. This method implies an equation that links the metric coefficients \(g_{tt}\) and \(g_{rr}\) thereby simplifying to solve the field equations. A similar concept was also used in string theory on embedding branes e.g. in the Randall–Sundrum model [21].
Assuming anisotropic fluid distribution, Mak and Harko [22] derived the condition for lower limit of mass for any anisotropic compact stars which was strongly depends on degree of anisotropy. Dev and Gleiser [23] derived the critical condition for compactness parameter 2M / R for anisotropic relativistic stars and was also found strongly depends on nature and degree of anisotropy. They have also shown that for anisotropic compact stars the redshift can go arbitrary large. In there second paper, Dev and Gleiser [24] stressed on stability of anisotropic compact stars. Here they have found that, depends on the nature of anisotropy there can exist stable anisotropic star even at \(\Gamma < 4/3\) while its isotropic counterpart isn’t. Herrera et al. [25] obtained an algorithm to generate all spherically symmetric anisotropic fluid distributions from two generators. One of the generators was linked with anisotropy and other with reshift function. On the other hand, Lake [26] used Newtonian hydrostatic equation for isotropic fluid distribution and generate infinite number of anisotropic solutions simply by assuming density.
Recently, Ivanov [27] derived a condition which is similar to Karmarkar condition, for conformally flat spacetime. However, the solutions resulting from the two theories are completely different. In Karmarkar spacetime there are no physical solutions describing isotropic fluid distributions, however, physical solution exist if electric charge or anisotropy or both are incorporated [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. In this article also we are exploring new physical solutions satisfying field equations under class I category and discuss the solutions to model compact stars.
The present article has been designed as follows: In Sect. 2, we have written the Einstein field equations for static and spherically symmetric matter distribution. The Embedding class I spacetime satisfying the Karmarkar condition has described in Sect. 3. Section 4 contains our new class I solutions along with mass, compactness parameter, surface redshift and gravitational redshift. We have explored the acceptance of our solutions in Sect. 5. The determination of constants by using the matching condition and analysis of all energy conditions have been done in Sects. 6 and 7, respectively. The equilibrium and stability are analyzed in the subsections of Sect. 8: Sect. 8.1 displayed the equilibrium situation and the stability analysis has done in Sect. 8.2. The generating functions for our model have calculated in Sect. 9. Moment of inertia and Mass relationship are analyzed in Sect. 10. Finally, The result and discussions have been done in Sect. 11.
2 Einstein’s field equations
3 The Karmarkar condition
From Eq. (10) one can see that the pressure anisotropy \(\Delta (r)\) is zero throughout the fluid sphere if either first or second or both the factors on righthand side of Eq. (10) are zero. The vanishing of first factor on the right side of Eq. (10) yields the Kohlar–Chao solution [52] whereas if the second factor is zero then the corresponding solution will be the Schwarzschild interior solution [53].
Finally, we arrive at a point to find out class I solutions but we have four independent equations (3)–(5) and (9) with five unknowns, namely \(\lambda (r),~\nu (r),~\rho (r),~p_r(r)\) and \(p_t(r)\). Therefore, it is impossible to find the exact solutions, for this purpose we shall consider a new metric potential function to generate our models.
4 New embedding class I solutions
5 The central values and physical analysis
6 The matching condition and determination of constants
7 The energy conditions
8 The equilibrium and stability analysis
The equilibrium position and stable state are most important situations for the noncollapsing compact object within our Universe. In this section, we are going to check the equilibrium and stability of the fluid distributions represented by our solutions.
8.1 The equilibrium condition
The behaviors of these three forces are shown in Fig. 8 and it is ascertained that our solutions representing fluid distributions are in equilibrium positions.
8.2 The stability condition
Here, we are willing to analysis the stability condition for our models with the help of (1) stability factor, (2) Adiabatic index and (3) Harrison–Zeldovich–Novikov criterion.
8.2.1 Causality condition
In General Theory of Relativity, the maximum velocity is the velocity of light, which is equal to 1 in the gravitational unit.
 (i)
The condition for potentially stable region is \(1< \{v_t(r)\}^2  \{v_r(r)\}^2 < 0\).
 (ii)
The condition for potentially unstable region is \( 0< \{v_t(r)\}^2  \{v_r(r)\}^2< 1\).
8.2.2 Adiabatic index
8.2.3 Harrison–Zeldovich–Novikov criterion
9 Generating functions
Masses and radii of the four celestial compact stars with their approximate moment of inertia
Stars  \(M/M_{\odot }\)  R (km)  Refs.  \(M/M_{\odot }\)  R (km)  I (g cm\(^2\)) 

Observed  Observed  Approx.  
Vela X1  1.77 ± 0.08  9.560 ± 0.08  [64]  1.77  9.56  1.543 \(\times 10^{45}\) 
Cen X3  1.49 ± 0.08  9.170 ± 0.13  [64]  1.49  9.17  1.221 \(\times 10^{45}\) 
LMC X4  1.04 ± 0.09  8.301 ± 0.2  [64]  1.04  8.30  0.707 \(\times 10^{45}\) 
EXO 1785248  1.30 ± 0.02  8.849 ± 0.4  [65]  1.30  8.80  0.964 \(\times 10^{45}\) 
10 Moment of inertia and mass relationship
Numerical values of constants for the four celestial compact stars
Stars  a (km\(^{2}\))  c (km\(^{2}\))  A  B 

Vela X1  0.001  0.00836  − 0.84901  0.03182 
Cen X3  0.001  0.00748  − 0.68739  0.03108 
EXO 1785248  0.001  0.00712  − 0.61514  0.03088 
LMC X4  0.001  0.00642  − 0.47473  0.03016 
Estimated values of the central and surface densities, cental pressure, surface redshift at the boundary and twice of compactness parameter with buchdahl limit for four celestial compact stars corresponding to the values of constants given in Table 2
Stars  \(\rho _c~ (10^{14}) \)  \(\rho _s(10^{14})\)  \(p_c(10^{34})\)  Z(R)  \(2u_s\)  Buchdahl 

\(\mathrm{gm}/\mathrm{cm}^3\)  \(\mathrm{gm}/\mathrm{cm}^3\)  \(\mathrm{dyne}/\mathrm{cm}^2\)  limit [66]  
Vela X1  9.5460  5.1018  6.1901  0.2602  0.3703  \(< \frac{8}{9}\) 
Cen X3  8.5432  5.0550  4.5353  0.2171  0.3250  \(< \frac{8}{9}\) 
EXO 1785248  8.1240  5.1071  3.7548  0.1914  0.2955  \(< \frac{8}{9}\) 
LMC X4  7.3335  5.0386  2.7293  0.1552  0.2506  \(< \frac{8}{9}\) 
11 Result and discussions

Metric potentials: The finite and nonsingular metric potential functions are necessary to generate the physical viable models of anisotropic compact stars. In our models, \(e^{\lambda (0)}\) = 1 and \(e^{\nu (0)} = \frac{1}{4a^2\pi }\left[ 2Aa\sqrt{\pi }+B\sqrt{c}\left\{ e^{1}+\sqrt{\pi } g_0\right\} \right] ^2 =\) positive constant for nonzero a. i.e. the metric potential functions are wellbehaved at the centre of the stars. The graphical representations of \(e^{\lambda (r)}\) and \(e^{\nu (r)}\) indicate that they are finite and regular throughout the radius of stars (see Fig. 1 (above)) and hence they are suitable to generate the models for anisotropic compact stars.

Energy density and pressures: The energy density \(\rho (r)\), radial pressure \(p_r(r)\) and transverse pressure \(p_t(r)\) should remain positively finite throughout the interior of fluid spheres. The energy density and radial pressure are maximum at the centre and decreasing in nature towards the surface. Also, the radial pressure should vanish at the surface of fluid sphere. Figures 1 (below) and 2 approve that our obtained energy density and pressures (radial and transverse) are good in behavior as they have satisfied all those conditions. To compare with observational data we have calculated the numerical values of central, surface densities and central pressure for different compact stars given in Table 3. The central and surface densities both are of order \(10^{14}\) and the central pressure is of order \(10^{34}\), which are almost same with observational data.

Equation of state parameters: For real matter distribution the equation of state(EoS) parameters \(\omega _r(r) = \frac{p_r(r)}{\rho (r)}\) and \(\omega _t(r) = \frac{p_t(r)}{\rho (r)}\) should lie in \(0<\omega _r(r), \omega _t(r) <1 \) [63]. In our present models, both the EoS parameters are within the region \(0<\omega _r(r), \omega _t(r) <1\) (see Fig. 4), which is another important testimony of our wellbehaved models.

Anisotropy: The anisotropic factor \(\Delta (r) =\) \( p_t(r)  p_r(r)\) should vanishes at the stellar centre. If \(\Delta (r) > 0\) then the anisotropic force \(F_a(r) = \frac{2\Delta (r)}{r}\) is outward directed i.e. play as a impulsive force, which can support more compact construction. And if \(\Delta (r) < 0\) then the anisotropic \(F_a(r)\) is inward directed. In our models, \(\Delta (0) = 0\) and positively increasing within the stellar interiors, clear from Fig. 3 (above) and consequently the anisotropic force is repulsive in nature, assist to construct more compact stars. The similar result is seen in the behavior of \(\eta (r)\), it indicates the anisotropy as a outward force.

Energy conditions: To illustrate the present models are of physical matter distributions we have plotted the L.H.Ss of all inequalities in Eq. (37) and evidently we can see from Fig. 6 along with Fig. 1 (below) that our solutions satisfied all the energy conditions.

Mass function and compactness parameter: The profiles of mass function m(r) and compactness parameter u(r) are shown in Fig. 5 for four compact stars. The m(r) and u(r) tends to zero when r tends to zero and monotonically increasing toward the surfaces. According to Buchdahl[] the mass to radius ration \(\frac{M}{R} < \frac{4}{9}\) or equivalently \(2u_s = 2u(R) < \frac{8}{9}\). We have computed the numerical values \(2u_s\), provided in Table 3 and all these values indicate that our solutions satisfied the Buchdahl limit.

Equilibrium: The anisotropic compact stars are in equilibrium under the action of gravitational, hydrostatic and anisotropic forces. The matter distributions represented by our solutions are in equilibrium positions, clear from Fig. 8. Here, hydrostatic and anisotropic forces are repulsive and gravitational force is attractive in nature.

Stability: We have analyzed the stability situation with the help of (1) Causality condition, (2) Adiabatic index, (3) Harrison–Zeldovich–Novikov criterion. Figures 9 and 10 (above) indicate that the sound velocities (radial \(v_r(r)\) and transverse \(v_r(r)\)) are positive and less than 1 and the stability factor \(\{v_r(r)\}^2  \{v_r(t)\}^2\) is negative i.e. our solutions represent physical matter distributions, which are potentially stable. The profiles of adiabatic index (Fig. 3 (below)) and mass in terms of central density (Fig. 11) are another two evidences for stable configuration represented by our solutions.

M–RGraph: The profile of masssurface radius relationship is shown in Fig. 12. It is found that for the ranges of masses given in Table 1 have the same corresponding radii provided on the same table. These means that our solutions yield the ranges of masses and radii very closed to that the observed values. Therefore, the solutions may represent the chosen compact stars.

I–MGraph: Since the M–R graph was in good agreement with the observed masses and radii of the chosen stars, we can estimate the corresponding moment of inertia I from the I–M graph (Fig. 13). For Vela X1, \(I\approx 1.543\times 10^{45}~\mathrm{g\,cm}^2\); Cen X3, \(I\approx 1.221\times 10^{45}~\mathrm{g~cm}^2\); EXO 1785248, \(I\approx 0.964\times 10^{45}~\mathrm{g~cm}^2\) and LMC X4, \(I\approx 0.707 \times 10^{45}~\mathrm{g~cm}^2\).

Surface redshift and gravitational redshift: The variations of surface redshift and gravitational redshift are shown in Fig. 7. From that figure, we can see that surface redshift \(Z_s(r) \rightarrow 0\) as \(r \rightarrow 0\) and thereafter monotonically increasing unto the surfaces of stellar spheres. The gravitational redshift \(Z_g(r)\) has opposite behavior, it is maximum at the centres and decreasing toward the surfaces of the fluid configurations. Moreover, \(Z_s(r)\) and \(Z_g(r)\) are coincided at the surfaces \( r = R\), i.e. \(Z(R) = Z_s(R) = Z_g(R)\). Further, we have calculated the numerical values of surface redshift \(Z_s(r)\) at the boundaries of the celestial stars i.e. the maximum values, given in Table 3 and all these maximum values of \(Z_s(r)\) are within the range suggested by Ivanov [68].
Notes
Acknowledgements
Farook Rahaman would like to thank the authorities of the InterUniversity Centre for Astronomy and Astrophysics, Pune, India for providing research facilities. Nayan Sarkar and Susmita Sarkar are grateful to CSIR (Grant no.: 09/096(0863)/2016EMRI.) and UGC (Grant no.: 1162/(sc)(CSIRUGC NET , DEC 2016)), Govt. of India, for financial support, respectively.
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