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Generalised isothermal models with strange equation of state

We consider the linear equation of state for matter distributions that may be applied to strange stars with quark matter. In our general approach the compact relativistic body allows for anisotropic pressures in the presence of the electromagnetic field. New exact solutions are found to the Einstein-Maxwell system. A particular case is shown to be regular at the stellar centre. In the isotropic limit we regain the general relativistic isothermal universe. We show that the mass corresponds to values obtained previously for quark stars when anisotropy and charge are present.


Introduction
In an early and seminal treatment the existence of quark matter in a stellar configuration in hydrostatic equilibrium was suggested by Itoh 1

Basic equations
It is our intention to model the interior of a dense realistic star with a general matter distribution. On physical grounds we can take the gravitational field to be static and spherically symmetric. Consequently, we assume that the gravitational field of the stellar interior is represented by the line element in Schwarzschild coordinates (x a ) = (t, r, θ, φ). We consider the general case of a matter distribution with both anisotropy and charge. Therefore we take the energy momentum tensor for the interior to be an anisotropic charged imperfect fluid; this is represented by the form where ρ is the energy density, p r is the radial pressure, p t is the tangential pressure and E is the electric field intensity. These physical quantities are measured relative to the comoving fluid velocity u i = e −ν δ i 0 . The line element (2) and the imperfect matter distribution (3) generate the Einstein field equations; the field equations can be written in the form where primes denote differentiation with respect to r and σ is the proper charge density.
We are utilising units where the coupling constant 8πG c 4 = 1 and the speed of light c = 1. The Einstein-Maxwell system of equations (4)-(7) describes the gravitational behaviour for an anisotropic charged imperfect fluid. For matter distributions with p r = p t (isotropic pressures) and E = 0 (no charge) we regain Einstein's equations for an uncharged perfect fluid from (4)- (7).
An equivalent form of the field equations is obtained if we introduce new variables: the independent variable x and new functions y and Z. These are given by x = Cr 2 , Z(x) = e −2λ(r) and A 2 y 2 (x) = e 2ν(r) , which was earlier used by Durgapal and Bannerji 21 to describe neutron stars. In terms of the new variables the line element (2) becomes where A and C are arbitrary constants. The transformation (8) simplifies the field equations, and we find that the system (4)-(7) can be written as 4Zẏ where dots denote differentiation with respect to the variable x.
The definition represents the mass contained within a radius r which is a useful physical quantity. The mass function (14) has the form in terms of the new variables introduced in (8).
On physical grounds we expect that the matter distribution for realitic stellar matter should satisfy a barotropic equation of state p r = p r (ρ). For the investigations in this paper we assume the particular equation of state where α and β are constants. This is a simple linear relationship with desirable physical features and contains models investigated previously. Now it is possible to rewrite (10)-(13) as the system where the quantity ∆ = p t −p r is defined as the measure of anisotropy. The Einstein-Maxwell equations as expressed in (17)-(22) is a system of six nonlinear equations in terms of eight variables (ρ, p r , p t , ∆, E, σ, y, Z). The system (17)- (22) is under-determined so that there are different ways in which we can proceed with the integration process. Here we show that it is possible to specify two of the quantities and generate an ordinary differential equation in only one dependent variable in the integration process. This helps to produce a particular exact model.

New solutions
In this paper, we choose physically reasonable forms for the gravitational potential Z and electric field intensity E and then integrate the system (17)- (22) to generate exact models.
We make the specific choices where a, b, d, n and k are real constants. The potential Z is regular at the origin and continuous in the stellar interior for a wide range of values for the parameters a, b and n. The electric field intensity E is a bounded and decreasing function from the origin to the surface of the sphere. Therefore the forms chosen in (23)-(24) are physically acceptable. These specific choices for Z and E simplify the integration process. Equation (21) can be written aṡ where we have used (23) and (24). This has the advantage of being a first order linear equation in the gravitational potential y.
Equation (25) can be integrated in closed form to give where we have defined and D is a constant of integration. Now from (23), (24) and (26) we can generate an exact model for the system (17)- (22) as follows The equations (27) When k = 0 then the mass function (15) has the form .
The expression of the mass function given in (35)

Isotropic models
It is possible to consider the special case of isotropic pressures with p r = p t in the uncharged limit for neutral matter. When k = 0 (E = 0) equation (32) becomes From the above equation we easily observe that when a = 1 and a = 1 + 4α (1+α) 2 the measure of anisotropy ∆ vanishes. When a = 1 we note from (29) that ρ = 0 since b = k = 0.
The above solution was obtained by Saslaw et al 25 in their investigation of general relativistic isothermal universes. Since ρ ∝ r −2 we may interpret (37) as a relativistic cosmological metric since there is an analogy that can be made with the well known Newtonian solution as pointed out by Chandrasekhar 26

Physical analysis
We observe that the exact solution (27) where . It is clear from (39) that the gravitational potentials are nonsingular at the origin for all values of n. If n = 1 then there is no singularity in the energy density ρ and the model is nonsingular throughout the stellar interior. However if n = 1 then the gravitational potential potentials (39) may continue to be well behaved but singularities may appear in the matter variables at the origin r = 0. By considering a particular example we can demonstrate graphically that the matter variables are well behaved outside the origin. Figures 1-5 represent the energy density, the radial pressure, the tangential pressure, the electric field intensity and the measure of anisotropy, respectively. Note that solid lines represent uncharged matter (E = 0) and dashed lines include the effect of charged matter E = 0. To plot the graphs we choose the parameters n = 2, a = 1, b = 40, k = 2, d = 1, α = 1 3 and β = 0.3569, and the stellar boundary is set at r = 1. From Figures 1-2 we see that both the energy density ρ and the radial pressure p r are continuous throughout the interior, increasing from the centre to r = 0.32 and then decreasing. Note that the radial pressure is zero at the boundary r = 1 for the uncharged case E = 0. We observe from Figure 3 that the tangential press p t is continuous and well behaved in the interior regions. From Figure 4 we observe that the electric field intensity E is decreasing smoothly throughout the stellar interior. We can observe from Figure 5 that the measure of anisotropy is continuous throughout the stellar interior. The behaviour of ∆ outside the centre is likely to correspond to physically realistic matter in the presence of the electromagnetic field. Figure 5 has a profile similar to the anisotropic boson stars studied by Dev and Gleiser 33 and the compact anisotropic relativistic spheres of Chaisi and Maharaj 34 .
From the figures we can see that the effect of electric field intensity E is to produce lower values for ρ, p r , p t and ∆. where we have set C = 1 and y = r 2 R 2 . Then the mass contained within a radius s has the form For simplicity we setbd = 2 so that these expressions reduce to which are simple forms. It is now easy to calculate the density and mass for particular parameter values from (40)  We now comment on some physical aspects of the solutions found which are of interest.
Firstly, there are a number of free parameters in the solution which may be determined by imposing boundary conditions. The solutions may be connected to the Reissner-Nordstrom exterior spacetime with the interior spacetime (39) across the boundary r = s where M and Q represent the total mass and the charge of the star, respectively. This gives the conditions Therefore the continuity of the metric coefficients across the boundary r = s is maintained.
The number of free parameters (A, C, D, b and n) ensures that these necessary conditions are satisfied. Secondly, from the graphs generated for the energy density and the pressure we observe that these quantities are negative in regions close to the centre. We believe that this feature arises because of the strange equation of state that we have imposed on the model.
To avoid negative matter variables we would need to use another solution to describe the core with our solution serving as the envelope. This is the situation that arises in the gravastar model of Mazur and Mottola where the singular core is replaced by a de Sitter condensate through a phase transition near the location of the event horizon. This idea has been utilised by Lobo 18 to demonstrate that dark energy stars are stable. Thirdly, the example of this paper and other exact solutions found for the Einstein-Maxwell system suggests that the negativity of the energy density and the pressure close to the centre may be a generic feature of charged stellar models. It appears that the existence of the electric field always makes the energy density negative close to the centre of the sphere. This suggests the conjecture that a charged fluid cannot satisfy the energy conditions near the stellar centre is possible. This will be the topic of a separate investigation.