Charged gravastars admitting conformal motion

We propose a new model of a {\it gravastar} admitting conformal motion. While retaining the framework of the Mazur-Mottola model, the gravastar is assumed to be internally charged, with an exterior defined by a Reissner-Nordstr{\"o}m rather than a Schwarzschild line element. The solutions obtained involve (i) the interior region, (ii) the shell, and (iii) the exterior region of the sphere. Of these three cases the first case is of primary interest since the total gravitational mass vanishes for vanishing charge and turns the total gravitational mass into an {\it electromagnetic mass} under certain conditions. This suggests that the interior de Sitter vacuum of a charged gravastar is essentially an electromagnetic mass model that must generate the gravitational mass. We have also analyzed various other aspects such as the stress energy tensor in the thin shell and the entropy of the system.


I. INTRODUCTION
By extending the concept of Bose-Einstein condensate to gravitational systems, Mazur and Mottola [1,2] have proposed a new solution for the endpoint of a gravitational collapse in the form of cold, dark, compact objects known as gravastars. These contain an isotropic de Sitter vacuum in the interior, while the exterior is defined by a Schwarzschild geometry, separated by a thin shell of stiff matter of arbitrary total mass M . This model implies that the space of a gravastar has three different regions with different equations of state (EOS) [3,4,5,6,7,8,9,10,11,12,13,14,15], as defined as follows: I. Interior: 0 ≤ r < r 1 , p = −ρ, II. Shell: r 1 < r < r 2 , p = +ρ, III. Exterior: r 2 < r, p = ρ = 0. Here r 2 − r 1 = δ is the thickness of the shell. The presence of matter on the thin shell is required to achieve * Electronic address: anisul@iucaa.ernet.in † Electronic address: farook rahaman@yahoo.com ‡ Electronic address: saibal@iucaa.ernet.in § Electronic address: kamalnandi1952@yahoo.co.in ¶ Electronic address: kuhfitti@msoe.edu the crucial stability of such systems under expansion by exerting an inward force to balance the repulsion from within.
Building on this background, we propose a new model of a gravastar admitting conformal motion by assuming a charged interior but with an exterior defined by a Reissner-Nordström line element instead of Schwarzschild. The basic motivation for this model is that, in general, compact stars tend to have a net charge on the surface [16,17,18,19,20]. This is an essential consideration in the study of the stability of a fluid sphere; in fact, it has been argued [16,17] that a spherical fluid distribution of uniform density with a net surface charge is more stable than a surface without charge. According to de Felice et al. [21], the inclusion of charge inhibits the growth of spacetime curvature and which therefore plays a key role in avoiding singularities. It has also been argued that gravitational collapse can be averted in the presence of a charge since the gravitational attraction may be counter-balanced by the electrical repulsion (in addition to the pressure gradient [22,23].) In searching for a natural relationship between geometry and matter for such stars through the Einstein field equations, we take into account the well-known inheritance symmetry. This symmetry is contained in the set of conformal killing vectors (CKV) Here L is the Lie derivative operator and ψ is the conformal factor. It is supposed that the vector ξ generates the conformal symmetry and the metric g is conformally mapped onto itself along ξ. Neither ξ nor ψ need to be static, even in the case of a static metric [24,25]. Due to this and other properties, CKVs have provided a deeper insight into the spacetime geometry connected to the astrophysical and cosmological realm [26,27,28,29,30,31,32,33]. It has been shown by Ray et al. [34] that, under the conformal killing-vectors approach, charged fluid spheres provide electromagnetic mass (EMM) models in which gravitational mass and other physical parameters originate solely from the electromagnetic field. So one of the motivations in the present investigation is to include CKVs and see to what extent conformal motion admits EMM models along with other relevant physical features. This paper is organized as follows: in Sec. II the Einstein-Maxwell field equations are provided, along with the CKVs for a charged gravastar. The solutions are obtained in Sec. III for a charged gravastar with conformal motion in connection with (A) the interior region, (B) the shell, and (C) the exterior region of the sphere. Secs. IV, V, and VI deal with the stress energy tensor in the thin shell, entropy within the shell, and unknown constants, respectively. In Sec. VII we conclude.

II. THE EINSTEIN-MAXWELL FIELD EQUATIONS
The Einstein field equations for the case of a charged perfect fluid source are where the energy-momentum tensor components for the matter source and electromagnetic field, respectively, are given by Here ρ, p and u i are the matter-energy density, the fluid pressure, and the velocity four-vector of a fluid element (with u i u i = 1), respectively. The corresponding Maxwell electromagnetic field equations are where the electromagnetic field tensor F ij is related to the electromagnetic potentials through the relation F ij = A i,j − A j,i and is equivalent to Eq. (6). In the above equations, J i is the current four-vector satisfying J i = σu i , where σ is the charge density and κ = 8π, using relativistic units G = c = 1.
Here and in what follows a comma denotes the partial derivative with respect to the coordinates. Next, given the static spherically symmetric spacetime (7) the Einstein-Maxwell field equations may be written as and Equation (11) may be expressed for the electric field E in the following equivalent form: where q(r) is the total charge of the sphere.

III. THE CHARGED GRAVASTAR WITH CONFORMAL MOTION
Equation (1) implies the following: with ξ i = g ik ξ k . Here 1 and 4 stand for the spatial and temporal coordinates r and t, respectively. Equations (13) yield the following expressions [34]: where C 1 , C 2 , and C 3 are integration constants. Given solutions (14) and (15), Eqs. (8), (9), and (10) take the following form [34]: 1 From the above equations, one may easily obtain values for E, ρ and p [34]: A. Interior region of the charged gravastar Observe next that Eqs. (21) and (22) provide an essential relationship between the metric potentials and the physical parameters ρ and p: 2ψ So given the ansatz ρ+ p = 0, it follows from the Eq. (23) that the value of ψ turns out to be either ψ = 0 or ψ = ψ 0 r, where ψ 0 is a dimensionless integration constant. This leads to the following exact analytical forms for all of the parameters: Here ψ 0 = ψ 0 /C 3 is a constant, which has the inverse dimension of r. The reason is that under the condition illustrates a case of positive density and negative pressure, resulting in an outward push from the interior region, which is consistent with the physics of a gravastar. On the other hand, 1/2r 2 − 3 ψ 2 0 < 0 (or ψ 2 0 < 1/6r 2 ) represents a collapsing case with negative density and positive pressure, which is not the subject of concern here. Thus for the purpose of gravastar physics, our above solutions are assumed to obey the condition 0 < ψ 2 0 < 1/6r 2 . For ψ 2 0 = 0, we find that both p and ρ are inversely proportional to r 2 but with opposite signs for their proportionality constants. A nonzero value of ψ 2 0 leads to a translational shift of this form with magnitude 3 ψ 2 0 , as can be seen from from Eq. (24). The electric field E is found to be inversely proportional to r and is independent of ψ 0 . On the other hand, σ is inversely proportional to r. Its value is zero for ψ 0 = 0, which suggests that the above power law behavior of p and ρ is independent of σ. Both e ν and e −λ are proportional to r 2 and hence equal with equal proportionality constants.
The active gravitational mass M (r), by virtue of the field Eq. (8), may be expressed in the following form: Here the pressure and density fail to be regular at the origin, but the effective gravitational mass is always positive since ψ 2 0 < 1/6r 2 and will vanish as r → 0. In other words, the expression for M (r) does not lead to a singularity.
Let us now match the interior solution to the exterior Reissner-Nordström solution at the boundary in the customary manner. To do so, we have to keep in mind that instead of a solid sphere, we are dealing here with a bubble-like hollow sphere of radius r 2 = r 1 +δ, so that for the limit δ → 0, we have the de Sitter spherical void of radius r 2 → r 1 . (According to Ref. [1], δ does not exceed Planck length by more than a few orders of magnitude.) For convenience of notation, let us denote the radius r 2 by a, the radius of the junction surface. Then following Ray et al. [34], the total gravitational mass m(r = a), which is obtained after matching the solution interior to r = a to the exterior Reissner-Nordström solution at the boundary, can be expressed as where M (a) is the total active gravitational mass and q(a) 2 /2a is the mass equivalence of the electromagnetic field. Observe from Eq. (29) that the vanishing charge q turns the total gravitational mass m into an EMM under the constraint ψ 0 < √ 3/(2 √ 2)q. It is interesting to note that the mathematical expressions and physics of the interior region resemble the EMM model of Ray et al. [34]. The apparent reason for this is the use of the Reissner-Nordström line element. In other words, the interior de Sitter void (p = −ρ) in a charged gravastar is the same as in the case already addressed by Ray et al. [34]. This implies that the interior de Sitter void of a charged gravastar must, in analogous fashion, generate the gravitational mass. This particular feature is a new one and was not obtainable in the non-charged case of Mazur and Mottola [1,2]. This mass provides the attractive force resulting from the collapse of the sphere and counter-balances the repulsive force due to electromagnetic field.
It is worth noting that the equation of state p = −ρ (known in the literature as a false vacuum, degenerate vacuum, or ρ-vacuum [35,36,37,38]) represents a repulsive pressure which in the context of an accelerating Universe may be related to the Λ-dark energy, an agent responsible for the second phase of the inflation [39,40,41,42,43]. So the charged gravastar seems to be connected to the dark star [44,45,46].

B. Shell of the charged gravastar
Using the EOS p = ρ, we get the solution where ψ 1 > 0 is an integration constant. Other parameters are where ψ 1 = ψ 1 /C 2 2 has the same dimension as r. Although the electric field E is inversely proportional to r, it depends on the integration constant ψ 1 , unlike the previous case. Eq. (35) suggests that the requirement of a real-valued σ may only be achieved with the condition ψ 1 < r/2. Combining this with the previous condition, ψ 1 > 0 (or ψ 1 > 0), we observe that the above solutions for the shell of the gravastar are valid within the range 0 < ψ 1 < r/2. It is obvious from Eq. (31) that the EOS p = ρ = 0 for the exterior de Sitter region corresponds to ψ 1 = r/3, which is within the upper limit of the above condition ψ 1 < r/2.
The proper thickness of the shell is obtained next: where R = r 1 − 2 ψ 1 /r. Thus a real value for the thickness confirms that the integration constant ψ 1 must be less than r/2. This condition also suggests a real σ.
Finally, using the symbol E for the energy, we get within the shell Thus E is exactly proportional to the coordinate thickness of the shell, (as opposed to the proper thickness).

C. Exterior region of the charged gravastar
For the exterior region (p = ρ = 0), the Reissner-Nordström space-time is where

IV. THE STRESS ENERGY TENSOR IN THE THIN SHELL
The second fundamental forms associated with the two sides of a thin shell (junction surface) are [16,17] where n ± ν are the unit normals to S: with n µ n µ = 1. Here ξ i are the intrinsic coordinates on the shell, f = 0 is the parametric equation of the shell S, and − and + corresponds to the interior and exterior regions, respectively.
Using the Lanczos equations [47,48,49,50,51], one can find the surface stress energy Σ and the surface tangential pressures p θ = p φ ≡ p t : The surface mass M shell of this thin shell may be defined as Here M can be interpreted as the total mass of the Reissner-Nordström gravastar. It takes the following form: where v θ and v φ are the surface tensions. Then if Eqs. (42) and (43) are the EOS becomes With the requirement of a positive density and positive pressure, the equation of state parameter ω(a) is always positive. The position of the thin shell (junction surface) plays a crucial role: if a is sufficiently large, then ω(a) ≈ 1. For some value of a in Eq. (43), we may get p t = 0, yielding a dust shell.

V. ENTROPY WITHIN THE SHELL
Following Mazur and Mottola [1,2], we now calculate the entropy by letting r 1 = b and r 2 = b + ǫ: Here s is the entropy density, which may be written as where α 2 is a dimensionless constant. Thus the entropy of the fluid within the shell is where x = r 2 − 5r ψ 1 + 6 ψ 2 1 1/2 . If ψ 1 → 0 and the thickness of the shell is negligibly small compared to its position from the center of the gravastar (i.e., if ǫ << b), then the entropy is given by S ≈ αkB G bǫ.

VI. THE UNKNOWN CONSTANTS ψ0 AND ψ1
In this section we determine the approximate values of the constants ψ 0 and ψ 1 . To this end we recall that the thin shell of a gravastar consists of a perfect stiff fluid [1]. Earlier we denoted the outer radius by r = a, the junction surface. So by Eq. (31), ρ = p = p t . Furthermore, for a sufficiently large, p t = −v ≈ Σ by Eq. (46). So we have These two equations then yield the approximate values of the unknown constants: Since Σ > 0, Eq. (42) also implies that ψ 0 > 1 a 1 − 2M/a + Q 2 /a 2 .

VII. CONCLUSIONS
This paper discusses a new model of a gravastar admitting conformal motion, within the framework of the Mazur-Mottola model. The gravastar is assumed to be internally charged with an exterior defined by a Reissner-Nordström rather than a Schwarzschild metric. The solutions obtained cover (i) the interior region, (ii) the shell, and (iii) the exterior region of the sphere. Of these three cases, the first case is of particular interest because the total gravitational mass m(r = a) = 1 2 √ 2 (3 − 8 ψ 2 0 q 2 )q, obtained by matching the interior solution to the exterior Reissner-Nordström solution at the boundary, vanishing for vanishing charge q and turning the total gravitational mass m into an EMM under the constraint ψ 0 < √ 3/(2 √ 2)q. This, in turn, suggests that the interior de Sitter void of a charged gravastar, having the same form as the EMM model, must generate the gravitational mass that provides the attractive force resulting from the collapse of the sphere and which counter-balances the repulsive force due to the charge. Moreover, the equation of state p = −ρ, known in the literature as a false vacuum or a ρ-vacuum, suggests that the charged gravastar is connected with the dark star.
An analysis of the stress energy tensor of the thin shell has shown that given the requirement of a positive density and positive pressure, the equation of state parameter ω(a) is always positive. Moreover, as the radius of the thin shell increases, w(a) → 1. For some value of r = a, we may have p t = 0, yielding a dust shell.
In calculating the entropy, it was found that if ψ 1 → 0 and if, in addition, the thickness of the shell is small compared to the radius, then the entropy is given by S ≈ αkB G bǫ, where b is the inner radius of the gravas-tar shell. The final calculations determined the approximate values of the constants ψ 0 and ψ 1 with ψ 0 > 1 a 1 − 2M/a + Q 2 /a 2 .