Generalized anisotropic models for conformal symmetry
Abstract
We find a new family of exact solutions to the Einstein system of equations with an anisotropic fluid distribution for a spherically symmetric spacetime containing a conformal Killing vector. Simple analytic expressions describe the matter variables and the metric functions which are regular at the centre and the interior of the body. We demonstrate that the new class of exact solutions is physically reasonable and may be utilized to model a compact object. A detailed graphical analysis of the matter variables shows that the criteria for physical acceptability are satisfied. The energy conditions are satisfied, causality is not violated, and the body is stable in terms of cracking, the Harrison–Zeldovich–Novikov stability criterion, and the adiabatic index inequality. It is, therefore, possible to geometrically describe a compact object with a conformal symmetry for an astrophysical application.
1 Introduction
It is necessary to solve the nonlinear Einstein field equations to describe physical systems in relativistic astrophysics. One approach to solve the field equations systematically is to utilize the underlying symmetries and group structure of the equations. The Lie symmetry generators that leave an equation invariant to generate a Lie algebra and facilitate finding an exact solution. Some recent examples of this approach in describing stellar systems are contained in the works of Govinder and Govender [1], Abebe et al. [2, 3] and Mohanlal et al. [4, 5]. Therefore, the group theoretic approach is useful in solving nonlinear equations for relativistic stars. A second approach in solving the field equations systematically is to utilize the presence of symmetries on the spacetime manifold. A practical example of a spacetime symmetry is a conformal symmetry. Conformal Killing vectors generate constants of the motion along null geodesics. The presence of a conformal symmetry restricts the metric functions and often leads to new exact solutions. Early examples of relativistic systems with a conformal Killing vector are the models of Herrera and Ponce de Leon [6] and Herrera et al. [7]. Esculpi and Aloma [8] and Rahaman et al. [9] constructed spheres with anisotropic pressures with a one-parameter group of conformal motions. Manjonjo et al. [10, 11] showed that the two metric functions in static spherically symmetric spacetimes are related if a conformal Killing vector exists. Mafa Takisa et al. [12] and Kileba Matondo et al. [13, 14] have shown that this relationship between the potentials may be exploited to model relativistic stars with anisotropy and electric charge. Therefore, the conformal symmetry approach is also useful in solving the nonlinear field equations for gravitating objects.
Solutions of the Einstein field equations may be categorised and classified using a number of approaches. A well-known approach is to use the Petrov classification which utilizes the algebraic symmetries of the Weyl tensor. Another approach is to utilize spacetime symmetries to classify and categorise exact solutions as emphasized by Stephani et al. [15]. The symmetry approach can lead to new exact solutions, e.g. in spherically symmetric spacetimes cosmological solutions have been found by Dyer et al. [16], Havas [17] and Sussman [18] with homothetic and conformal vectors. This symmetry approach will also be useful in describing gravitational phenomena in astrophysical situations. Only a few results have been found for compact objects, and only for the special case of a conformal Killing vector. Shee et al. [19], and Newton Singh et al. [20] analytically obtained new exact spherically symmetric solutions with conformal geometry for compact stars in strong gravitational fields. We expect that a comprehensive analysis of conformal symmetries will provide deeper insights into many other astrophysical situations. Other types of spacetime symmetries such as curvature collineations need to be considered in the study of compact stars.
We now make some comments relating to the physical features of local anisotropy and conformal symmetries. Firstly, local anisotropy has been discussed by several authors most notably in the comprehensive treatment of Herrera and Santos [21] and references therein. Some recent references for anisotropy are contained in the analysis of Kileba Matondo et al. [13] who regain observational parameters for astronomical objects such as PSR J1614-2230 and SAX J1808.4-3658. Possible sources for anisotropy include the variation of the magnetic field intensity in neutron stars in the period of post-main sequence development, the presence of a very high density region in the core, several condensate states (such as pion condensates, meson condensates, etc.), a type 3A superfluid, fluid mixtures of various types and rotational motion. The concept of cracking was introduced in [21] which is directly related to the presence of anisotropy. Cracking is essential to describe the stability of the matter distribution close to equilibrium. Secondly, the presence of conformal motions has been related to physical processes for stars. Important references in this regard are the pioneering treatments of Herrera and coworkers [6, 7]. Several stellar models with spherical symmetry in general relativity have been generated with a conformal Killing vector, see for example the recent paper of Kileba Matondo et al. [13]. Conformal motions are related to the property of self-similarity which was emphasized in the study of Herrera and Di Prisco [22] for static axially symmetric relativistic fluids. Self-similarity is important in systems with no characteristic length scale, and for systems close to the critical point with infinite correlation length. Density fluctuations then occur at all length scales as seen in critical opalescence arising in regions of continuous phase transitions. In particular conformal motions, extending the concept of geometric similarity, have applied to models of wormholes in general relativity. Most analyses of conformal motions have been restricted to spherically symmetric models. e.g. the stellar model in [12, 13]. Recently, Ivanov [23] has proposed a conformally flat realistic anisotropic compact star model by using only analytical approach. The study of Herrera and Di Prisco [22] is related to the important physically geometry system of static axial symmetry. Axial symmetry should be an area that is considered in future work using the approach of this paper; this should produce new results not present in models with spherical symmetry.
Finding an exact solution to the Einstein field equations do not always lead to a physically acceptable model for a relativistic star. Physical criteria such as causality and regularity at the centre should be satisfied. In addition, the predictions of the model should be consistent with observations relating to quantities much as the stellar radius, mass and redshift. This has become more important in recent times since there has been a substantial improvement in the observations for highly compact relativistic stars. In this paper, we impose a conformal symmetry in static spherical spacetimes and utilize the results of Manjonjo et al. [11]. The matter distribution is anisotropic. It is then possible to solve the Einstein field equations. A detailed and comprehensive analysis shows that the model is regular, well behaved and satisfies the criteria for physical acceptability. Our analysis in this paper shows that the conformal geometry produces a model consistent with observations. This is true for the obtained masses and radii of compact objects in particular. Our analysis shows that a conformal Killing vector may be associated with the real life astronomical objects.
The article starts with the introduction in Sect. 1. In Sect. 2, we present the appropriate Einstein field equations with an anisotropic fluid distribution for a spherically symmetric spacetime containing a conformal Killing vector. The exact solution for conformal anisotropic models by considering gravitational metric potential \(e^{\lambda }\) is presented in Sect. 3. In Sect. 4, We introduce the matching conditions and determine the constant parameters A, B and b with the total mass of star \(m_s\). We present all the physical conditions for well behaved anisotropic conformal models in Sect. 5. In Sect. 6, we performed a detailed physical analysis (analytical and graphical) of the spherical conformal model via. physical conditions as mentioned in Sect. 5. The Conclusion of the paper has been made in Sect. 7.
2 Metric and the Einstein field equations
3 Exact solution for conformal anisotropic models
4 Matching conditions
5 Physical conditions for anisotropic conformal models
Any realistic stellar model includes the following set of physical conditions:
A3. Causality condition: It is well known that the radial and tangential sound velocity should not be more than the light velocity in any medium. The radial and tangential velocities of sound are denoted as \(v^2_r=\frac{dp_r}{d\rho }\) and \(v^2_t=\frac{dp_t}{d\rho }\). Therefore both velocities should satisfy the inequality \(0<\frac{dp_r}{d\rho } \le 1\) and \(0<\frac{dp_t}{d\rho } \le 1\) everywhere inside the star.
A4. Energy conditions: The anisotropic conformal solution should satisfy the following energy conditions namely dominant energy condition (DEC) \(\rho \ge p_r\) and \(\rho \ge p_t\). From (A2) it is obvious that the null energy condition (NEC) should be satisfied as well. furthermore the solution must hold for the strong energy condition (SEC) \(\rho + p_r+2\,p_t \ge 0\) and trace energy condition (TEC) or strong dominant energy condition \(\rho \ge p_r+2\,p_t\) (it is not necessary) at the interior of the star.
A6. The Harrison–Zeldovich–Novikov stability criterion: The Harrison–Zeldovich–Novikov criterion for the stability of the self-gravitating compact star states that the mass of the star increases with the central density (\(\rho _0\)) which implies that \(dM_{\rho _0}/d \rho _0 > 0\) for the stable region where \(M_{\rho _0}\) is the mass function in terms of the central density.
6 physical analysis of the spherical conformal model
The physically realistic model should satisfy the conditions (A1)–(A9) which arise in Sect. 5. These conditions depend upon the choice of parameters k and \(\alpha \). We have to choose the values of these parameters so that the conformal symmetric anisotropic solution is well behaved and satisfies the conditions (A1)–(A9) everywhere within the star . Therefore we have chosen following set of values of the parameters: (i). \(k=0\) and \(\alpha =1\) (Conformally flat), (ii). \(k=-2\) and \(\alpha =-1\) (Nonconformally flat). Then the values the physical parameters for conformally and nonconformally flat solution at the centre (\(r = 0\)) are given as:
Physical parameters for different values of n for the star with mass = 2.01 (\(M_{\theta }\)) and predicted radius (R) = 7 km
Value of n | Central density (g/cm\(^{3}\)) | Surface density (g/cm\(^{3})\) | Central pressure (dyne/cm\(^{2})\) | A (km\(^{-2})\) | B | b (km\(^{-2})\) | a (km\(^{-2})\) |
---|---|---|---|---|---|---|---|
4 | 6.43163 \(\times 10^{15}\) | 1.52630 \(\times 10^{15}\) | 2.40258 \(\times 10^{36}\) | 0.0008160955886 | 0.4975795805 | 0.002233 | 0.02 |
12 | 6.43163 \(\times 10^{15}\) | 1.52630 \(\times 10^{15}\) | 2.36172 \(\times 10^{36}\) | 0.001116104435 | 0.3646778882 | 0.000772 | 0.02 |
20 | 6.43163 \(\times 10^{15}\) | 1.52484 \(\times 10^{15}\) | 2.40247 \(\times 10^{36}\) | 0.001173649996 | 0.3465538989 | 0.000467 | 0.02 |
500 | 6.43163 \(\times 10^{15}\) | 1.52276 \(\times 10^{15}\) | 2.40244 \(\times 10^{36}\) | 0.001254046405 | 0.3240557905 | 0.0000189 | 0.02 |
1000 | 6.43163 \(\times 10^{15}\) | 1.52271 \(\times 10^{15}\) | 2.35984 \(\times 10^{35}\) | 0.001255318967 | 0.3238321587 | 0.00000945 | 0.02 |
B7. The behaviour of the adiabatic index \((\Gamma )\) is given by Fig. 6 which shows that it is greater than 4 / 3 throughout inside the star. On the other hand, the ratio of radial pressure and density is also decreasing outwards (see Fig. 2) which indicates that our model is stable and the temperature is decreasing away from the centre.
B8. From the Fig. 3 we observe that the causality condition should be satisfied within the star [24], i.e. the square of the radial \(\left( v^2_{r}\right) \) and tangential \(\left( v^2_{t}\right) \) sound velocity are lying within the interval [0, 1]. We also observe from Fig. 7 that the inequality \(0 \le |v^2_t - v^2_r| \le 1\) is satisfied throughout the stellar model which implies that our model is stable.
B9. Figure 8 shows the behaviour of the TOV equation which describes the equilibrium condition of the solution. From Fig. 8, we observe that the gravitational force is counter-balanced by the joint action of the hydrostatic force and anisotropic force. It is also noted that the hydrostatic force is more than the anisotropic force when \(r/r_s \in (0, 7.6)\) and has less value when \(r/r_s\in [7.6, 1)\).
For Table 1, we used the numerical values of the gravitational constant \(G=6.673\times 10^{-8} cm^3/gs^2\), velocity of light \(c=2.997\times 10^{10} cm/s\), solar mass \(M_{\theta }=1.475\) km while we have chosen the radius of the star at boundary \(r_r=R\) in the graphical representation. For plotting of the figures, the numerical values of physical parameters have been obtained for various values of n for the compact star which are presented in Table 1. We Note that the value of surface density for the compact star is approx. 6.63 times higher than nuclear saturation density (\(2.3 \times 10^{14}\)).
7 Conclusion
We have pursued the solution of the Einstein field equations with anisotropies by imposing a conformal Killing vector. A parallel treatment and associated methodology have been developed in anisotropic gravity and supergravity models using external fields. This has been pursued by Mateos and Trancanelli [36] for strong coupled anisotropic plasmas, by Jain et al. [37] for anisotropic fluids from dilaton driven holography, and Giataganas et al. [38] for Einstein-axion-dilaton systems. These results provide a path to obtain such gravity solutions which have been useful in explaining properties of strongly coupled plasmas in the papers of Giataganas [39] and Chernicoff et al. [40]. We similarly expect that this alternate approach may also lead to useful insights in studying stellar systems that have been analyzed in this paper with conformal motions.
Here we have presented a new class of exact solutions to the Einstein system of equations for an anisotropic fluid distribution in the presence of a conformal Killing vector. This new class of solutions is physically acceptable in an astrophysical scenario and can be used to model relativistic compact objects. We have achieved this by producing analytic expressions for the matter variables and the metric functions. A detailed graphical analysis of the matter variables was performed and we checked the criteria for physical acceptability. In particular, we demonstrated that the energy conditions were satisfied, causality is obeyed, and the star is stable in terms of cracking, the Harrison–Zeldovich–Novikov stability criterion, and the adiabatic index inequality. In this paper, we have highlighted the role of conformal symmetries in generating exact solutions of the Einstein field equations. We have shown the exact solution may be used to model a star with anisotropic distributions. In this sense, our solution extends the earlier treatments of Mafa Takisa et al. [12] and Kileba Matondo et al. [13, 14]. The advantage of this investigation is that we obtain a model which satisfies all physical criteria: regularity, casuality, stability and energy conditions for a realistic matter distribution. In advantage, we can characterise our model geometrically with a conformal Killing vector. This paper is part of our endeavour to extend the geometrical characterisation of cosmological solutions to astrophysical solutions for compact objects.
To present the numerical analysis of the achieved solution we consider the mass of the compact stellar object is \(2.01~M_{\odot }\) and the chosen parametric values of n are given as \(n=4, 12, 20, 500\) and 1000. In the left and right panel of Fig. 1 we have shown variation of metric potentials viz., \(e^\lambda \) and \(e^\nu \), respectively. The variation of \(p_t\), \(p_r\) and \(\rho \) are shown in the left, middle and right top panel of Fig. 2 which features that they all have finite value at the centre and they decrease monotonically to reach their minimum values at the surface. Further, we present the change of anisotropy (\(\Delta \)), radial (\(\omega _r\)) and tangential (\(\omega _t\)) equation of state parameter with respect to the fractional radial coordinate r / R in the left, middle and right bottom panel of Fig. 2. Based on the Figs. 1 and 2 it is worthwhile to mention that our system is free from any sort of singularity whether geometrical or physical. Interestingly, we find that the anisotropy in the present system is minimum, i.e., zero at the centre and increases throughout the interior of the stellar system to reach it’s maximum value at the surface as predicted by Deb et al. [41]. The profile of \(v^2_r\) and \(v^2_t\) are shown in the left and right panel of Fig. 3, respectively and the profile of the difference of the square of the sound velocities is shown in Fig. 7 which confirm the stability of the present model as it is consistent with both the causality condition and Herrera cracking condition by following the inequalities \(0<v^2_r, v^2_t<1\) and \(0<\mid v^2_t - v^2_r \mid < 1 \). Figure 4 features that as all the energy condition is valid for this model and confirms the physical validity of the achieved solution. In Fig. 5 we present the profile for compactification factor and redshift in the left and right panel, respectively. The variation of mass with respect to central density has been shown in the left panel of Fig. 6 which features that for our system \(dM/d{\rho _c}>1\) and indicates stability of the system by following the Harrison–Zeldovich–Novikov stability criterion. In the right panel of Fig. 6 we have shown the variation of the adiabatic index \(\Gamma \) and find that as in the present case \(\Gamma >4/3\) our system is dynamically stable against the infinitesimal radial pulsation. Finally, The Fig. 8 shows that the present system is stable under the hydrostatic equilibrium which represents the variation of the different forces viz., anisotropic (\(F_a\)), hydrostatic (\(F_h\)) and gravitational (\(F_g\)) with respect to the radial coordinate r / R. We find that the present system is in hydrostatic equilibrium as resultant of the forces is zero, ie., \(F_a+F_h+F_g=0\). In Table 1 we have predicted values of the different physical parameters and constants due to the chosen parametric values of n and assumed value of the gravitational mass of the stellar system. In a nutshell, in this article by imposing a conformal Killing vector we present a singularity free and physically acceptable model which is suitable to study anisotropic and spherically symmetric compact stars.
We believe that this geometrical approach may provide deeper insights into the spacetime geometry, and may, in fact, lead to new exact solutions which are not easily obtainable via other methods. In the future, we intend to consider other spacetime symmetries such as curvature collineations in the astrophysical context.
Notes
Acknowledgements
The author SKM acknowledges continuous support and encouragement from the administration of University of Nizwa. SDM thanks the University of KwaZulu-Natal for its continued support. SDM acknowledges that this work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology and the National Research Foundation. We all are grateful to the anonymous referee for several useful suggestions which have enabled us to modify the manuscript substantially.
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