Anisotropic compact stars in general relativity

Anisotropic compact stars in general relativity


We present model of an asisotropic fluid sphere under Einstein’s general theory of relativity. For this we employ the Tolman-Kuchowicz (1); (2) metric which provides a class of singularity free solutions. The set of solutions satisfies all the physical requirements of the realistic stars and eventually indicates the possibility to describe compact objects. To make the model consistent with the compact stars, bounds on the model parameters have been obtained under two specific case studies which provide theoretical supports in favour of neutron stars as well as strange stars. Consequences of mathematical modeling of compact stars, through several physical parameters involving in the stars, have been analyzed.

General Relativity; metric potentials; compact stars


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1 Introduction

Einstein’s general theory of relativity (GTR) (3) represents a grand tool of gravitation to understand uniquely the febric of the space-time and therefore the behavioural pattern of the cosmic bodies as well as phenomena. The first successful field theoretical application of GTR to black hole solution by Schwarzschild (4) and to the structure of the universe by Einstein himself (5) made pavement for a new avenue in the research fields of astrophysics and cosmology.

As a consequence of gravitational collapse we get a panorama of stellar formations which include white dwarf, neutron star and black hole (in the observational level) and also quark star (in the theoretical level). All these stellar objects are commonly known as compact stars which are basically considered as spherically symmetric and isotropic. However, isotropy may be a favoured feature but need not be a general characteristics of the stellar objects. Anisotropic factor () which deals with the inhomogeniety in pressures into two components - radial pressure and tangential pressure, is actually a function to consider the internal situation of the star compared to the idealized isotropic case.

The concept of anisotropy was originally proposed by Ruderman (6) and later on by several other scientists (7); (8); (9). As the reasons behind pressure anisotropy different factors are thought to be responsible, such as very high density region in the core region, various condensate states (like pion condensates, meson condensates etc.), superfluid 3A, mixture of fluids of different types, rotational motion, presence of magnetic field, phase transition etc. The basic ideas involved in the anisotropy and their applications in the diverse fields are available in the literature (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23).

It is argued by Maurya (24) that the chances of having anisotropy is much higher in compact star because the interaction among the particles is too relativistic and they become too random to maintain any uniform distribution throughout the region. This relativistic nature of particles in compact star could be one of the possible reason for giving birth of significant anisotropy in the compact star. It is therefore understood that the anisotropic force inside the star makes the compact object more compact than the isotropic condition which eventually makes possible transition of a neutron star to strange star.

Under the above background our motivation in the present paper is to investigate for a relativistic stellar model with anisotropic fluid sphere. The outline of this study is as follows: We provide the Einstein field equations for anisotropic stellar source and their solutions in Sect. 2 whereas the junction conditions are discussed in Sect. 3. In Sect. 4 we have performed a comparative study for distinct cases, viz., (i)  0,  = 0 and (ii) , where and are positive real numbers. Several interesting properties of the physical parameters which include the metric potentials, density, pressures, anisotropy, cosmological parameter, energy conditions, stability (via conservation equation and Herrera condition), compactification factor and redshift under two special case studies have been conducted in Sect. 5. The last Sect. 6 is devoted as a platform for providing some salient features and concluding remarks.

2 Basic equations of Einstein’s field and their solutions

To describe the spacetime of a compact stellar objects, we consider the spherically symmetric line element as


Literature survey (25) shows that some interesting static spherically-symmetric perfect fluid solutions in canonical coordinates for the above metric were proposed by Tolman (1), Patwardhan and Vaidya (26) and Mehra (27) where , and by Kuchowicz (2) and Leibovitz (28) where with , , and as positive constants. The metric, therefore, with the mentioned metric potentials will be referred later on simply as the Tolman-Kuchowicz (TK) metric.

Now we are assuming that the stellar system is composed of anisotropic fluid and hence the energy-momentum tensor can be expressed in the following standard form:


where , and represent the radial pressure, tangential pressure and matter-energy density, respectively.

We have considered the erstwhile Einsteinian cosmological constant as radially dependent, i.e. and hence the Einstein field equations for the metric (1) with energy-momentum tensor are obtained as


where ’ denotes the differentiation with respect to the radial coordinate . In above field equations we have considered geometrized units .

Now to describe the interior structure of stellar system we assume the radial pressure is directly related to the energy density by the relation


where and are positive real numbers.

By using the metric (1) and Eqs. (4) to (6), we can write the expressions for radial pressure , tangential pressure and energy density in the following form as


Then the EOS parameters corresponding to the radial and tangential pressures are given by


Hence the anisotropy parameter can be defined as


3 Junction conditions

The first fundamental form of the boundary surface inviolved in the metric (1) should be the same whether obtained from the interior or exterior space-time and this guarantees that for some coordinate system the metric components will be continuous across the surface. Therefore, we consider that the exterior system is equivalent to the metric


which is the modified Schowarzschild metric with the cosmological constant and hence turns into the usual Schowarzschild metric for vanishing .

The continuity of the metric functions, involved in the interior and the exterior metrics, at the outer boundary of the fluid sphere (), as well as the requirement of matching condition for the radial pressure, i.e. at the surface, immediately provides the constants , , and in the following forms


In the later part of the article it can be observed that tuning of these constants are very important to determine different physical properties of compact stars.

4 Comparative case studies for the proposed stellar model

It is clear from Eqs. (15)-(18) that in general the metric coefficients in Eq. (1) depend on the factors , , , and . Now, following Deb et al. (23) we have maximized the anisotropic stress () at the surface where and using the observed values of the different compact stellar candidates as presented in Table 1 and Table 2 we get several solutions for . However, among those derived values of we have chosen only one specific value, which is consistent with the Buchdahl condition (29) and physically valid. It can be observed that the whole system basically depends on the choice of the parameters and . Therefore, we consider the following two cases for different choice of and which eventually shows interesting physical features of the present model as is evident from tables and graphs through the subsequent text. One can note that for the two chosen cases, viz., (i)  0,  = 0 and (ii) , the obtained value of for the same stellar candidate differs due to the assumed values of and .

4.1 Case 1: 0,  = 0

This case can be treated as a special case as we are assuming and hence the EOS (6) turns into the ordinary one with the form , where takes exactly the role of the equation of state parameter so that from Eq. (11) one can easily get .

4.2 Case 2:

For this general case there is possibility to get a class of physical situations for different values of and . However, for practical purposes in the present case study we have assigned specific values only for these constants.

\arraybackslashStrange Stars \arraybackslashObserved Mass () \arraybackslashPredicted Radius (Km) \arraybackslash \arraybackslash \arraybackslashSurface Redshift \arraybackslash
\arraybackslash \arraybackslash (30) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (32) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (33) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (34) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
Table 1: Numerical values of physical parameters for the different compact stars for , and
\arraybackslashStrange Stars \arraybackslashObserved Mass () \arraybackslashPredicted Radius (Km) \arraybackslash \arraybackslash \arraybackslashSurface Redshift \arraybackslash
\arraybackslash \arraybackslash (30) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (32) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (33) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (34) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
\arraybackslash \arraybackslash (31) \arraybackslash \arraybackslash \arraybackslash \arraybackslash \arraybackslash
Table 2: Numerical values of physical parameters for the different compact stars for , and

5 Discussions on the physical properties of the stellar model

5.1 Regularity and Reality Conditions

Metric potentials

The solution should be free from physical and geometrical singularities, i.e. at the centre the metric potentials and should have non-zero positive values in the range . At origin Eq. (1) provides and which is also evident from Fig. 1.

Figure 1: Behavior of the potential for Case 1 (upper left panel) and Case 2 (upper right panel) whereas behavior of the potential for Case 1 (lower left panel) and Case 2 (lower right panel) as a function of the fractional radial distance for the strange star candidate are shown.

Matter-energy density

The energy density inside the star should be positive as is evident from Eq. (19) as follows:

Figure 2: Behavior of density for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

This same feature can be observed from Fig. 2 where the density function is maximum at the centre and decreases monotonically to become minimum at the surface to be valid physically. For the Case 1 density is zero at the surface unlike the Case 2. Also, for Case 2 density is much higher than the Case 1 inside the stellar system.

Fluid pressures

Like the energy density the pressure also should be positive inside the star as is eveident from Eqs. (20) and (21) which are given by


Fig. 3 features that the pressure is maximum at the centre and decreases gradually to become zero at the surface. This shows physical validity of the achieved solution.

Figure 3: Behavior of the radial pressure for Case 1 (upper left panel) and Case 2 (upper right panel) whereas behavior of the tangential pressure for Case 1 (lower left panel) and Case 2 (lower right panel) as a function of the fractional radial distance for the strange star candidate are shown.

In this connection the anisotropic plot (Fig. 4) is also convincing for physical viability. Fig. 4 features that anisotropy is minimum at the centre and increases gradually inside the stellar system to reach the maximum value at the surface as predicted by Deb et al. (23) in their work.

Figure 4: Behavior of the pressure anisotropy for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

5.2 Energy conditions

The energy conditions, viz., Null Energy Condition (NEC), Weak Energy Condition (WEC) and Strong Energy Condition (SEC) will be valid only when the following inequalities hold simultaneously for our system given as

Figure 5: Behavior of the energy conditions for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

The energy conditions are plotted in Fig. 5 which provides favourable situation for our model.

5.3 Stability of the stellar model

TOV equation

The Tolaman-Oppenhiemer-Volkoff (TOV) (1); (35) equation for the anisotropic matter with variable cosmological constant () in radial direction are given as


It is Obviously that the above generalized TOV equation describes the equilibrium condition for the strange star subject to the gravitational () and hydrostatic () plus another anisotropic force () due to the anisotropic nature of the stellar object. Now, the above equation can be written via different forces as




Fig. 6 shows the counterbalancing pattern of different forces to attain equilibrium of the spherical configuration. The left panel of Fig. 6 shows that is counter balanced by the combined forces and , whereas the right panel of Fig. 6 features that counterbalances the combined effect of and .

Figure 6: Behavior of different forces for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

Herrera condition

To verify stability of our anisotropic model we have plotted the radial () and transverse () sound speeds in Fig. 7. It is observed that the inequalities and are satisfied everywhere within the stellar system and obeys the causality condition. Now the radial and transverse sound speeds are given by




Figure 7: Behavior of the square of the sound speed for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

On the other hand, following the cracking concept (36); (37) we check stability of the local anisotropic configuration by finding out the potentially stable region where the radial speed of sound is greater than the transverse speed of sound, i.e. . From Fig. 8 one can note that and so that . Thus the present compact stellar model provides a stable configuration of the spherical system by satisfying both the causality condition and the Herrera cracking concept.

Figure 8: Behavior of the difference of square of the sound speed for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the strange star candidate are shown.

Cosmological parameter

The cosmological parameter as expressed in Eq. (10) is drawn in Fig. 9 which show regular behaviour for both the Cases 1 and 2.

Figure 9: Behavior of the cosmological parameter for Case 1 (left panel) and Case 2 (right panel) as a function of the fractional radial distance for the compact star are shown.

5.4 Compactfication factor and surface redshift

The effective mass of the spherical stellar system can be given as



Figure 10: Behavior of the compactification factor for Case 1 (upper left panel) and Case 2 (upper right panel) whereas behavior of the redshift for Case 1 (lower left panel) and Case 2 (lower right panel) as a function of the fractional radial distance for the strange star candidate are shown.

In terms of the effective mass, let us now define the compactification factor as follows


Hence the surface redshift is given by


6 Conclusion

In the present study we have considered an anisotropic fluid sphere in the framework of Einstein’s general theory of relativity. The spherical space-time is assumed to be of the Tolman-Kochowicz type and solutions of the Einstein field equations are found by using this metric. One can observe that all the parameters involved in the solutions set are viable within the specified physical conditions. Moreover, we have done two case studies under different numerical choices of and which are positive real numbers of the EOS (6). It is interesting to note from tables and figures that both the cases represent solutions of the compact stars. The first choice, viz., Case 1 ( 0,  = 0) provides neutron stars whereas Case 2 () provides strange stars.

However, we would like to high light some of the salient features of the present models which are as follows:

1. Density and pressure: We have shown the profile of density in Fig. 2. For the first case the left panel of Fig. 2 features that for and the density of the stellar system is bellow than the normal nuclear density (  ). In the second case for and the right panel of Fig. 2 shows that nuclear density inside the compact stellar model is higher than . Hence, it confirms that the stellar system for the first case is compatible for the neutron star and for the second case the system corresponds to the ultra dense strange stars. In Fig. 3 we have shown the variation of and in the upper and lower panel, respectively. We have shown variation of the anisotropic stress in the left and right panel of Fig. 4, which feature that for both the cases the anisotropy for our system is minimum at the centre and maximum at the surface.

2. EOS: For the chosen value of and the EOS in the second case resembles with that of the MIT Bag model and turns the compact stellar system into the strange quark star. However, in the first case due to the chosen value of and we find a simple linear form of the EOS which turns the spherically symmetric and compact stellar system into a less dense neutron star.

3. Energy conditions: We find for the both chosen cases the energy conditions, viz., NEC, WEC and SEC are consistent with our system. We have shown variation of the energy conditions with respect to the radial coordinate in Fig. 5, which confirms physical acceptability of the obtained solutions.

4. Forces: In Fig. 6 we observe that the left panel for Case 1 shows a positive and hence repulsive gravity which indicates towards a less dense compact stellar system (e.g. neutron stars). On the other hand, the right panel for Case 2 shows a negative and hence attractive gravity which indicates formation of highly dense compact stars (e.g. strange stars).

5. Stability: To discuss stability of the system we have examined both the causality condition and the Herrera cracking condition. Figs. 7 and 8 exhibit that for both the cases the inequalities , and are valid simultaneously and hence confirms stability of the stellar system in terms of the causality condition and the Herrera cracking condition.

6. Compactification factor and redshift: We have featured variation of the compactification factor and redshift function in the upper and lower panel in Fig. 10, respectively for both the cases. For the chosen values of , and we have derived the factor and surface redshift for the different compact star candidates and presented their values in Tables 1 and 2. We find from these tables that the values of and are higher for the Case 2 compared to the Case 1. For both the Cases as , so our system is consistent with the Buchdahl condition (29).

In summary, in this article employing the Tolman-Kuchowicz (1); (2) metric and using parametric values of , and we have successfully presented singularity free and completely stable stellar system which simultaneously as well as uniquely describes the neutron stars for and the strange stars for constraints.


SR is thankful to the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India and The Institute of Mathematical Sciences, Chennai, India for providing all types of working facility and hospitality under the Associateship scheme. A part of this work was completed while D.D. was visiting IUCAA and the author gratefully acknowledges the warm hospitality and facilities in the library there.


  1. journal: Eur. Phys. J. C


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