Anisotropic stellar models admitting conformal motion

Anisotropic stellar models admitting conformal motion

Ayan Banerjee ayan˙ Department of Mathematics, Jadavpur University, Kolkata-700032, India    Sumita Banerjee Department of Mathematics, Budge Budge Institute of Technology, West Bengal 700137, India    Sudan Hansraj Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa    Ali Ovgun Physics Department , Eastern Mediterranean University, Famagusta, Northern Cyprus, Mersin 10, Turkey
July 15, 2019

We address the problem of finding static and spherically symmetric anisotropic compact stars in general relativity that admit conformal motions. The study is framed in the language of gravity theory in order to expose opportunity for further study in the more general theory. Exact solutions of compact stars are found under the assumption that spherically symmetric spacetimes which admits conformal motion with matter distribution anisotropic in nature. In this work, two cases have been studied for the existence of such solutions: first, we consider the model given by f(R) = R and then f(R) = aR+b. Finally, specific characteristics and physical properties have been explored by analytically along with graphical representations for conformally symmetric compact stars in f(R) gravity.

General Relativity; f(R ) gravity; conformally symmetric; compact stars

I Introduction:

From the time of Sir Isaac Newton our understanding of the nature of gravity has advanced but mysteries in physics still remain. The process is still continuing after the first formulation of Einstein’s theory of General Relativity (GR) Einstein (), which is one of the most successful fundamental theories of gravity in modern physics. Despite its success, many extensions of the original Einstein equations have been investigated to accommodate present observational data on both astrophysical and cosmological scales. Observational data has confirmed that the Universe is undergoing a phase of accelerated expansion. Direct support is provided by the high precision data from Type Ia supernovae Supernova (), the Cosmic Microwave Background (CMB) anisotropies Dunkley (), baryonic acoustic oscillations Eisenstein () and from gravitational lensing Contaldi (). In particular, within the context of General Relativity, the observed cosmic acceleration of the Universe cannot be explained without resorting to additional exotic matter components (such as Dark Energy (DE) or dark mater (DM)). Until now, no consistent model of dark energy has been proposed which can yield a convincing self-consistent picture of the observed Universe, both in the field of fundamental physics and in that of astrophysics. Currently experimental tests for DE and DM are under design and it is believed that with precision instrumentation such as the Square Kilometre Array radio telescopes such ideas will either be confirmed or ruled out. For this reason modifications of General Relativity have been proposed and are currently receiving a great deal of attention.

Alternative gravity theories generally appear to exhibit good agreement between theory and observation. and the introduction of additional geometrical degrees of freedom may assist us in abandoning the concepts of DE or DM. Many alternative theories have been proposed such as f (R) gravity Capozziello (), scalar-tensor theories Setare (), braneworld models Dvali (), Gauss-Bonnet gravity Nojiri (), Galileon gravity Nicolis () have gained a lot of attention in recent years. Additionally Ellis ellis1 (); ellis2 () revived an idea proposed by Weinberg wein (), that of trace-free Einstein gravity, which has also gone by the name unimodular unimod1 (); unimod2 () gravity. In this framework an attempt was made to rectify the discrepancy in the value of the cosmological constant as predicted by quantum gravity in contrast with what is actually observed. Dadhich dad-hans () has also presented strong arguments for using the Lovelock polynomial terms as the generator of the Lagrangian density. The remarkable feature of the Lovelock theory is that while it is polynomial in the Riemann tensor, Ricci tensor and Ricci scalar, it nevertheless results in at most second order equations of motion. Moreover, the zeroth case corresponds to the cosmological constant while the linear case yields the familiar Einstein gravity theory. The effects of the higher curvature terms are only felt in dimensions greater than 4 so that the theory is a natural generalisation of GR to higher dimensions. Investigations into higher dimensional gravity is strongly motivated by string theoretic ideas which are supposed to be compatible with gravity theories. It should be mentioned here that many of these modifications have provided a number of significant results in cosmology Maeda (); Ananda () as well as astrophysics Capozziello ().

In the present article we have endeavoured to frame the core problem in the language and formalism of the f(R) theory although the specific cases under consideration reduce to the standard Einstein theory which contains an action principle that is linear in the Ricci scalar. The reason for this choice of formalism is that it opens the way for further investigations of more general functional forms of the Ricci scalar such as the Starobinksy model in subsequent work. In f(R) gravity theories the Einstein-Hilbert action is generalized with a function of the Ricci scalar R, (see the reviews Faraoni (); Buchdahl ()) have also been extensively analyzed recently. Earlier interest in f(R) theories occurred in connection with the early Universe Faraoni (), and a number of viable f(R) models have proposed that satisfy the observable constraints - see e.g., Refs. Amendola (). It is noted that f(R) theory was contemplated long ago by Buchdahl Buchdahl () who introduced f(R) gravity using non-linear Lagrangians. One of the main reasons for increased interest in f(R) gravity was motivated by inflationary scenarios, in the Starobinsky model, where f(R) = R-+ gravity are considered in Starobinsky (). The Cosmology of generalized modified gravity models have been studied by Carroll et al Carroll (). In this context, many authors have demonstrated interest by exploring some of their interesting properties and characteristics, such as: structure formation and stability conditions of neutron stars, which was studied by Santos (); Tsujikawa () and the gravitational collapse of spherically symmetric perfect fluids by Sharif ().

Apart from the cosmological solutions, gravitational theories must be tested also in the astrophysical Level to get a more realistic picture of the gravitational fields. In this sense, strong gravitational regimes found in relativistic stars could discriminate General Relativity from its generalizations Psaltis (). However, it should be remembered that present day astronomical observational data suggests that neutron stars can be used to investigate possible deviations from GR and act as probes for any modified gravity model. These ideas will recent added impetus in the coming decade when the Square Kilometre Array experiment is fully functional. At the same time, the structure of compact stars in perturbative f(R) gravity has been studied in Arapoglu (). Another class of interesting models on hydrostatic equilibrium of stellar structure that have been investigated in the framework of f(R)-gravity by Stabile (). The primary motivation for finding exact interior solutions to Einstein’s Field equations in f(R) gravity for their static and spherically symmetric case is that they can be used to model physically relevant compact objects. With the estimate values for masses and radii from different compact objects, the recent observational prediction fails to be compatible with the standard neutron star models because densities of nuclear matter is much lower than the densities of such compact objects. This situation can theoretically be explained by the existence of two different kinds of interior pressures namely, radial and transverse pressure. In this regard, the study of models of anisotropic stellar distributions has received great interest in GR as well in modified theories of gravity. It is required of such models that elementary requirements are met in order to be physically viable. For example, it is expected that the energy conditions are satisfied, the interior and exterior spacetimes are smoothly matched across a suitable hypersurface and that causality is not violated. Having these in mind, the motivation is to study models of the anisotropic compact stars in the framework of f(R) gravity by admitting non-static conformal symmetry using the diagonal tetrad method. Indeed the model considered here corresponds to the standard Einstein theory in view of the choice of an action principle that is linear in the Ricci scalar. The consequence is that the equations of motion are the usual second order variety. If the more general Starobinksy model were selected then the equations of motion would involve derivatives of the order of four. This case will be pursued in another study. At first, Maartens and Maharaj Maartens () have studied the static spherical distributions in charged imperfect fluids admitting conformal symmetry and now its become a popular model to established a natural relationship between geometry and matter source, with the help of the Einstein field equations. In the same context Das et al. Das (), obtained a set of solutions describing the interior of a compact star under f(R, T) gravity and Ray et al. Ray () have studied electromagnetic mass models for static spherical distribution admitting CKV.

The study of anisotropic spherically symmetric distributions have a long history. Such models are heavily under-determined and admit a large number of solutions. Bowers and Liang bowers () presented a simple anisotropic model with constant density. The mass implications in anisotropic distributions were revisited by Baracco ıet al barraco () and it was shown that the well known Buchdahl mass limit applicable to isotropic spheres may be violated. Ivanov ivanov () established mass-radius bounds for anisotropic fluids with spherical symmetry. Mathematical prescriptions such as conformal flatness were studied by Stewart sterwart () who postulated forms for the mass function to generate exact models. Nonstatic spheres admitting a one parameter group of conformal motions were studied by Herrera and Ponce de Leon Herrera () but without specifying a linear equation of state.

Motivated by the above discussion, we study the compact stars solutions with their interesting properties and characteristics in the context of f(R) theory of gravity admitting initially non-static conformal symmetry and with a specified equation of state. The paper is organized as follows: in Section II, we review f(R) modified theories of gravity whereas in Section III, the CKVs are formulated. In Section IV, we formulate the system of gravitational field equations, then in Section V we describe the star interior, using the spherically symmetric line element and CKVs. In Section VI, we have discussed various physical features of the model such as energy conditions, equilibrium condition by using Tolman-Oppenheimer-Volkoff (TOV) equation and stability issue. Finally in Section VII, we conclude with a summary and discussions.

Ii Basic Mathematical Formulation in f(R) Gravity

Let us consider the following action in f(R) gravity of the form


where is the determinant of the space-time metric assuming that, and is the matter Lagrangian. Here f(R) is a general function of the Ricci scalar, R. Within the context of the metric formalism, variation of the action Equation (1) with respect to generates the field equations


where the term represents the energy momentum tensor of the matter scaled by a factor of and is an extra stress-energy tensor contribution by the f(R) modified gravity theory and is given by

In the above equation, and primes denote derivatives with respect to Ricci scalar . Now, considering a static spherically symmetric distribution of matter, the line element may be expressed in the form


where the metric functions and are functions of the radial coordinate which may be utilised to compute the the surface gravitational redshift and the mass function, respectively. From the metric it is possible to construct asymptotically flat spacetimes when and as .

To obtain some particular strange star models we assume that the material composition filling the interior of the compact object is anisotropic in nature and accordingly we express the energy-momentum tensor in the form


where depicts energy density, is the radial pressure, to be the tangential pressure, respectively. In the expression is the four-velocity of the fluid and corresponds to the radial four vector. Note that with the assumption of an anisotropic particle pressure, the field equations may be solved by an infinite number of metrics. Additional restrictions based on physical grounds must be introduced to model realistic objects. Customarily an equation of state relating the pressure and the density may be imposed, however, this approach runs into sever mathematical problems making the equation of pressure isotropy impractical to solve. In this work, we elect to utilise a geometric prescription. New classes of anisotropic star solutions admitting conformal motions are sought and investigated.

Iii Conformal Killing Vector

To establish a natural relationship between geometry and matter through the Einstein’s GR, and searching for an exact solutions by convincing approach we use a non-static conformal symmetry. Following the idea a work has recently been considered in Das (). For instance, one may specify the vector as the generator of this conformal symmetry, and the metric is conformally mapped onto itself along so that the following relationship


is obeyed. Here represents the Lie derivative operator of the metric tensor whereas is the conformal killing vector.

As for the choice of the vector , generates the conformal symmetry within the framework of the standard GR, then the metric is conformally mapped onto itself along . According to Bhmer et al. Bohmer (), for the choice of static metric, neither nor needs be static. It should be noted that if then Eq. (6) gives the Killing vector, if then it yields a homothetic vector and if then it gives conformal vectors. Furthermore, the spacetime becomes asymptotically flat when , which also implies that the Weyl tensor will vanish. All such conformally flat spacetimes have been found. They are either the Schwarzschild interior solution in the case of no expansion or the Stephani stephani () stars if expansion is occurring. Accumulated all of this properties of CKV have been effectively provide a deeper insight Accumulated all of this properties of CKV have effectively provided a comprehensive picture of the underlying spacetime geometry. Here Eq. implies that


with . From Eq. and , the following expressions Bohmer () are obtained as

where and stand for spatial and temporal coordinates and respectively. Then Eq. (6) provides the following relationship by using the metric (2), we have


where , and are all integration constants.

Iv The Field equations

The tetrad matrix for the metric is defined as


Therefore, the Ricci scalar is determined as


Using the space time metric given by Eq. (4), the modified field Eq. (2) generates the equations


governing the evolution of the fluid. In the following we focus on several physically relevant f(R) gravity models and seek interior solutions of the Einstein field equations with conformal Killing vector, that can model compact objects. The field equations of f(R) modified theories are highly non-trivial to solve, so the entire calculations are highly dependent on the equation ansatz. We choose to solve the continuity equation for decoding the system of equations for our purpose.

V Solutions:

As a first step in our analysis we consider solutions for f(R)=R and f(R) = aR + b, where and are purely constant quantities. Note that this prescriptions is essentially the general relativity case without and with a cosmological constant. We further assume that the standard matter for a fluid distribution obeys the equation of state


where the state parameter is a constant and , for causality. That is the speed of sound remains subluminal provided that . Now we determine the solutions under the above conditions.

v.1 Case I:

4.1.1: When

Indeed, by using Eqs. , we obtain the solution of this ordinary differential equation in the form


while the conformal factor, Ricci scalar and mass functions are given by


where is an integration constant. In this case, the velocity of sound is given by


On account of the assumption of isotropic particle pressure, we have not invoked the relationship so as not to over-determine the system. However, the solution above does indeed exhibit an equation of state of the form consistent with perfect fluid distributions. Observe that setting gives the Saslaw et al. saslaw () isothermal fluid model with both the energy density and pressure obeying the inverse square law and obviously the barotropic linear equation of state . That the metric potential is not constant in this model may be attributed to the presence of the conformal symmetry.

Figure 1: Variation of density and pressure with Energy conditions of the first model: We have plotted the energy density and pressures, respectively in the left panel whereas in the right panel we have drawn the behavior of different energy conditions when the constant term ’B = 0.8’ assuming a positive value. Here we see that null energy conditions (NEC) is satisfied but strong energy conditions (SEC) is violated for isotropic case.

4.1.2: When

Using the Eqs. , we obtain the solution when matter distribution is anisotropic in nature and it given by




and the other set of expressions are given by


where is integration constant. The sound velocity is determined as


Note that in view of the latitude afforded by the assumption of pressure anisotropy the field equations were augmented by the equation of state .

Figure 2: Variation of density and pressure with Energy conditions of the second model: In the left panel the functions of energy density and pressures have been plotted with the radial coordinate and in the right panel we have plotted all the energy conditions inside the star when f(R) = R and . The parametric values have been taken for the graphs are shown in the legend.
Figure 3: Illustrative plot of square of sound speed (in the left panel), whereas we have plotted variation of different forces in the right panel so that the system is subjected to the equilibrium position for the f(R) = R solution. The parameter values have been taken for the graphs are shown in the legend.

v.2 Case II: f(R)= aR + b

Now we are going to consider the second case where the effects of the cosmological constant enter the picture through the constant . The initial case is that of

4.2.1: When

Using Eqs. and assuming the isotropic condition, provides the following general solution


and the conformal factor, Ricci scalar and mass functions are given by


where is an integration constant. The sound velocity is then obtained as


which is the extreme value for the sound speed ordinarily associated with a stiff fluid.

Figure 4: Plots for variation of density and pressure with Energy conditions when f(R)= aR + b: In the left panel the functions of energy density and pressures have been plotted with the radial coordinate and in the right panel we have plotted all the energy conditions inside the star for isotropic model of the fluid sphere. The parameter values have been taken for the graphs are shown in the legend.
Figure 5: Illustrative plot of square of sound speed (in the left panel), whereas we have plotted variation of different forces in the right panel, so that the system is subjected to the equilibrium position for the f(R) = aR+b solution. The parameter values have been taken for the graphs are shown in the legend.

4.2.2: When

Using Eqs. , and Eq. the differential equations when solved give


with other functions are given by




The sound velocity can be found out as


where is an integration constant.

Figure 6: Radial variation of energy density and pressures have been plotted in the left panel, whereas all the energy conditions inside the star for anisotropic model of the fluid sphere when f(R) = aR+b solution.
Figure 7: In the left panel the variation of the radial and transverse sound speed have been plotted, whereas in the right panel the equilibrium condition concerning TOV equation have been considered for anisotropic f(R) = aR+b solution. The parameter values have been taken for the graphs are shown in the legend.

Vi Physical features of the model

vi.1 Energy conditions

For physically acceptable model, we are going to verify whether the anisotropic fluid sphere satisfies all the energy conditions or not, namely: (i) Null energy condition (NEC), (ii) Weak energy condition (WEC) and (iii) Strong energy condition (SEC) at all points in the interior the star. We therefore attempt to write down the following inequalities as follows:


In figs. 1, 2, 4 and 6 we plot variation of all the energy conditions inside the compact objects for a given with different values of constant parameters. In the case of isotropic pressure when f(R) = R, plotted in Fig. 2, it is evident that null energy conditions (NEC) is satisfied but strong energy conditions (SEC) is violated for our parametric choice however for other cases the energy conditions are valid inside the compact star.

vi.2 TOV Equation

The success of this model lies in its stability under the different forces namely gravitational, hydrostatic and anisotropic forces. To examine these we consider the generalized Tolman-Oppenheimer-Volkov (TOV) equation for anisotropic fluid distribution is given by leon ()


where the effective gravitational mass is defined by


Substituting Eq. (46) into Eq. (45), we obtain the simple expression


Summarising, this expression is arranged in terms of gravitational mass and hence gives the equilibrium condition for the compact star, involving the gravitational, hydrostatic and anisotropic forces for stellar objects. This equation can be expressed in the more compact form as


where , and represents the gravitational, hydrostatic and anisotropic forces, respectively. In order to illustrate this qualitatively, we use graphical representation for both cases when f(R) = R and f(R) = aR+b, which are shown in Figs. 3, 5 and 7 (right panel) by assigning the value of = 0.58. In spite of this model we see that in every cases and takes the negative value while is positive. As a result, it is clear that gravitational force is counterbalanced by the combined effect of hydrostatic and anisotropic forces to hold the system in static equilibrium. Models to explain the static equilibrium in-depth have been extensively studied by Rahaman et al. rah16 () and Rani & Jawad jawad ().

vi.3 Stability Analysis

We are interested in checking the sound speed, using the concept of Herrera’s cracking (or overtuning) Herrera (), which lies within the range i.e., according to this procedure the radial and transverse velocity of sound lies within the proposed range. We have shown graphically the radial and transverse velocity of sound for both cases when f(R) = R and f(R) = aR + b, which are shown in Figs. 3, 5 and 7 (left panel) and observe that these parameters satisfy the inequalities and everywhere within the stellar object except the isotropic case when f(R) = R.

Vii Conclusion

In this work, we have studied analytical solutions for compact stellar objects with a general static interior source in the framework of f(R) gravity satisfying the conformal Killing vectors equations. The investigation has been performed for two specific cases when f(R) = R and f(R) = aR + b; further the stars are assumed to be anisotropic in their internal structure. In order to get exact solutions we have considered a systematic approach by assuming the spherical symmetry and interior of the dense star admitting non-static conformal symmetry.

For a simplest form of the fluid sphere, we have studied the two specific arguments for both isotropic as well as non isotropic cases. Interestingly, we have identified that among the two cases ( case 4.1 and 4.2 ), only the solution for isotropic condition when f(R) = R is not physically valid because the behavior of different energy conditions does not attribute the regularity conditions at the interior of the star. Furthermore, it has been found that our solution satisfies all the energy conditions and that pressures are positive and finite throughout interior of the stars which are needed for physically possible configurations have been discussed in detail. For a stable configuration, we have shown that the generalized TOV equation which describes the equilibrium condition for an anisotropic fluid subject to by the different forces, viz. gravitational force (), hydrostatic force () and anisotropic force () in Figs. 3, 5 and 7 (left panel). Another interesting result of this paper is related with checking the stability of our model by adapting Herrera’s cracking concept Herrera () and it has been found that the radial and transverse speeds of sound lies within the limit of which are shown in Figs. 3, 5 and 7 (reght panel). Hence we concluded that our solution might have astrophysical relevance by this theory through fine tuning. Therefore, it would be an interesting task to verify our solution with sample data for more satisfactory features in the realm of physical reality which will be our next venture in this line of study. Moreover, we propose to consider more general functional forms of to go beyond the standard model of general relativity as considered in this work.


AB, SB are thankful to the authorities of Inter University Centre for Astronomy and Astrophysics, Pune for giving them an opportunity to visit IUCAA where a part of this work was carried out.


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