Charged Gravastar in a Dark Energy Universe

Here we constructed a charged gravastar model formed by an interior de Sitter spacetime, a charged dynamical infinitely thin shell with an equation of state and an exterior de Sitter-Reissner-Nordstr\"om spacetime. We find that the presence of the charge is crucial to the stability of these structures. It can as much favor the stability of a bounded excursion gravastar, and still converting it in a stable gravastar, as make disappear a stable gravastar, depending on the range of the charge considered. There is also formation of black holes and, above certain values, the presence of the charge allows the formation of naked singularity . This is an important example in which a naked singularity emerges as a consequence of unstabilities of a gravastar model, which reinforces that gravastar is not an alternative model to black hole.


Introduction
Although we have strong theoretical and experimental evidences in favor of the existence of black holes, lots of paradoxical problems about them do exist [1]. Besides, it was shown recently that observational data could give strong arguments in the existence of event horizons, but we can not prove it directly [2].
We also have the fact that the picture of gravitational collapse provided by Einstein's General Relativity cannot be completely correct since in the final stages of collapse quantum effects must be taken into account at high curvature values, or short distances, compared with the Planck length scale, as pointed out by Chapline [3] and other researchers.
There are many other additional models proposed as black holes mimickers, but among the alternative models to these compact objects, the gravitational vacuum stars (gravastars) [12] received special attention, partially due to the tight connection between the cosmological con-stant and our accelerated expanding universe, although it is very difficult to distinguish these objects from black holes.
In the original model of Mazur and Mottola (MM) [12], gravastars consist of five layers. To study the dynamical stability of such compact object, Visser and Wiltshire (VW) [13] argued that such five-layer models can be simplified to three-layer ones. They also pointed out that there are two different types of stable gravastars which are stable gravastars and "bounded excursion" gravastars. The first one represents a stable structure already formed, while the second one is a system with a shell which oscillates around a equilibrium position which can loose energy and stabilize at the end.
Recently we have studied the stability of some three layer gravastar models [14][15][16][17][18][19]. The first model [14] consisted of an internal de Sitter spacetime, a dynamical infinitely thin shell of stiff fluid, and an external Schwarzschild spacetime, as proposed by VW [13]. We have shown explicitly that the final output can be a black hole, a "bounded excursion" stable gravastar, a Minkowski, or a de Sitter spacetime, depending on the total mass of the system, the cosmological constant m  and the initial position of the dynamical shell. We 0 have concluded that although it does exist a region of the space of the initial parameters where it is always formed stable gravastars, it still exists a large region of this space where we can find black hole formation. In the sequence, considering after an equation of state the shell [15], instead of only using a stiff fluid   0   , we concluded that gravastar really is not an alternative model to black hole. We also have generalized the former one considering an interior constituted by an anisotropic dark energy fluid [16,19]. We have again confirmed the previous results, i.e., that both gravastars and black holes can be formed, depending on the initial parameters. It is remarkable that for this case we have an interior fulfilled by a physical matter, instead of a de Sitter vacuum. Thus, it is similar to dark energy star models.
Recently, Carter [20] studied spherically symmetric gravastar solutions which possess an (anti) de Sitter interior and a (anti) de Sitter-Schwarzschild or Reissner-Nordström exterior, separately. He followed the same approach that Visser and Wiltshire took in their work [13] assuming a potential and then founding the equation of state of the shell. He found a wide range of parameters which allow stable gravastar solutions, and presented the different qualitative behaviors of the equation of state for these parameters, for both cases, those are, a (anti) de Sitter-Schwarzschild or Reissner-Nordström exterior.

  V R
Differently from Carter's work [20], we consider here a different approach, as in the previous works. In a first step, we generalized our second work on gravastars [15], introducing an external de Sitter-Schwarzschild spacetime [17]. The aim was to study how the cosmological constant affects the gravastar formation, and we found that the exterior cosmological constant imposes a limit on the gravastar formation since the dark energy density inside the gravastar has to be greater than the surrounding spacetime. Now we are interested in the influence of the charge, combined with the influence of an exterior cosmological constant, considering a de Sitter-Reissner-Nordström exterior spacetime. For this configuration we showed that the presence of the charge can change considerably the stability conditions of these structures. It can as much favor the stability of a bounded excursion gravastar, converting it in a stable gravastar, as make disappear a stable gravastar or even to allow a naked singularity formation.
In a previous work [18] we have already considered the exterior of the gravastar is a Reissner-Nordström spacetime, but with zero total mass and, depending on the parameter 1 of the equation of state of the shell, and the charge, a gravastar structure can be formed. We have found that the presence of the charge contributes to the stability of the gravastar, if the charge is greater than a critical value.
The paper is organized as follows: In Section 2 we present the metrics of the interior and exterior spacetimes, with their extrinsic curvatures, the equation of motion of the shell and the potential of the system. In Section 3 we analyze the influence of the presence of the charge in the gravastar model confirming the existence of naked singularity formation and we investigate the formation of gravastar from numerical analysis of the general potential. Finally, in Section 4 we present our conclusions.

Formation of Gravastars in a De
Sitter-Reissner-Nordström Spacetime This gravastar model is described by an interior spacetime with a cosmological constant , given by the de Sitter metric, The thin shell is characterized by the hypersurface   and given by the metric  is the proper time.
where In order to find the mass of the shell, and then its potential, it is necessary to consider the junction conditions.
The continuity of the first fundamental form imposes that where the dot represents the differentiation with respect to  .
Thus, the interior and exterior normal vector are given by and e  The interior and exterior extrinsic curvature are given by Following Lake [21], we have where M is the mass of the shell. Thus Then, substituting Equations (4) and (5) into (15) we get Solving the Equation (16) for we obtain the potential [15]. In order to keep the ideas of our work [15] as much as possible, we consider the thin shell as consisting of a fluid with a equation of state, , where  and denote, respectively, the surface energy density and pressure of the shell and p  is a constant. The equation of motion of the shell is given by [21]   where  is the four-velocity. Since the interior region is constituted by a fluid with cosmological constant and the exterior corresponds to a charged spacetime characterized by the Reissner-Nordström with exterior cosmological constant, we get since [22]   we can solve Equation (18) giving (20) where is an integration constant. Substituting Equation (20) into , we obtain the general expression for the potential, 2 1 Redefining the Schwarzschild mass , the charge , the cosmological constants and and the radius as Copyright © 2013 SciRes. JMP Therefore, for any given constants , , i , e and m  , Equations (21) or (27)   The gravastar model constructed here shows 4 different horizons, which are: . Although the phantom energy is usually considered as a kind of dark energy, in this paper we will use the expression dark energy for the case where the condition    is satisfied and phantom energy otherwise. Hereinafter, we will use only some particular values of the parameter  which are analyzed in this work. See Table 1.

General Case
In the following have done a graphical analysis of several special cases. The influence of the cosmological constant was deeply discussed in a previous work [17], where In order to analyze the effect of the charge we have started from the cases with q = 0, considered in the previous work [17], and we have used the critical mass, when a stable gravastar was formed, changing the value of the charge. Recalling that the critical mass is defined as the mass for which   0    We call attention that in all the cases studied here, the formation of the apparent horizon can be avoided increasing the value of the charge, indicating that the shell can collapse to form naked singularity.

Conclusions
We constructed a gravastar model consisting by an interior de Sitter spacetime and an exterior spacetime with an external cosmological constant, described by a de Sitter-Reissner-Nordström metric. The charge is localized on the shell. Restricting the range of the initial radius, we obtain as the final structure bounded excursions, stable gravastars and also naked singularities.
We investigated the influence of the charge and we observed that increasing its value, and fixing a value for the mass, we can obtain a stable gravastar from a bounded excursion gravastar. For even higher values of  Table 1. The growth of the charge eliminates the stable structure. In addition, note that for , the shell collapses to a black hole, while for , we have the collapse of the shell in a naked singularity. q  10.0    Table 4. In this example, the shell is constituted by phantom energy and none stable structure is formed.  Table 5. Note that we have a gravastar enclosing a naked singularity.  the charge the apparent horizon can be avoided, which leads the formation of a naked singularity.
In the case of a shell formed by standard energy, with    q in its state equation (Figures 1, 6 and 7), for some fixed values of charge, above a lower limit  dependent on the mass and on cosmological constants, and small initial values of radius, the shell can collapse and form a naked singularity. There is also a possibly formation of black holes for some values of the charge, below , for fixed values of the mass. Increasing the value of the charge, we verify that initially bounded excursion configurations become more stable and there is another limit for the charge, c , for which the structure becomes a stable gravastar. For charges upper than c the shell collapses. Moreover, fixing the mass and varying the charge, we have a similar behavior, that is, a bounded excursion becomes more stable increasing the charge until reaches a stable gravastar and for other values of the charge, we have black holes and naked singularities. In the case of q  q q 1.7

 
3.0 (Figure 3), the shell is constituted by dark energy and for small values of the initial radius there is also a naked singularity formation. The same is found for a phantom dark energy shell with   0 (Figure 4). Finally, for a stiff fluid shell with   (Figures 2 and 5), we have bounded excursion formation, stable gravastar, black hole and naked singularity formation according to the values of the charge and for some values of initial radius. It is remarkable that the naked singularity formation appeared in this gravastar model is completely new. Then, beginning with a shell linking two spacetimes (de Sitter and de Sitter-Reissner-Nordström) in order to eliminate the horizons, as proposed by the gravastar's model, except for the cosmological horizon of the exterior spacetime, the shell can stay stable, forming a gravastar, or collapsing in a black hole or even a naked singularity, representing a new counterexample to the Cosmic Censorship. Then, this model definitively is not an alternative to the black hole, even naked singularities.