Static scalar field condensation in regular asymptotically AdS reflecting star backgrounds

We study condensation behaviors of static scalar fields in the regular asymptotically AdS reflecting star spacetime. With analytical methods, we provide upper bounds for the radii of the scalar hairy reflecting stars. Above the bound, there is no scalar hair theorem for the star. Below the bound, we numerically obtain charged scalar hairy reflecting star solutions and in particular, the radii of the hairy stars are discrete, which is similar to known results in other reflecting object backgrounds. For every set of parameters, we search for the largest AdS hairy star radius, study effects of parameters on the largest hairy star radius and also find difference between properties in this AdS reflecting star background and those in the flat reflecting star spacetime. Moreover, we show that scalar fields cannot condense around regular AdS reflecting stars when the star charge is small or the cosmological constant is negative enough.


I. INTRODUCTION
The famous no-scalar-hair theorem announces that the static massive scalar fields cannot exist in asymptotically flat black holes and it was believed that this property is mainly due to the fact that a classical black hole horizon could irreversibly absorb matter and radiation fields, for references see [1]- [15] and a review see [16]. In fact, this no scalar hair behavior is not restricted to spacetime with a horizon. Hod firstly found that regular asymptotically flat neutral compact reflecting stars cannot support the static massive scalar fields [17].
This no hair theorem for regular reflecting stars leads to works searching for no scalar hair behavior in the horizonless spacetime.
As a further step, it was found that massless scalar fields nonminimally coupled to the gravity cannot exist around the asymptotically flat neutral compact reflecting stars [18]. In addition, Bhattacharjee and Sudipta extended the discussion of no hair behaviors to reflecting stars in the asymptotically dS gravity background [19]. It seems that no hair theorem may widely hold in regular reflecting objects gravities. Since there is no hair theorem for asymptotically flat charged black holes [20,21], it is interesting to further examine whether scalar fields can exist in the asymptotically flat charged reflecting objects background. Very different from cases in black holes, it was lately found that charged scalar fields can condense around a charged reflecting shell and it was found that the radius of the hairy shell is discrete in a range, where the spacetime outside the shell is supposed to be absolutely flat [22,23]. In fact, the existence of composed scalar fields and reflecting objects configurations doesn't depend on the choice of flat spacetime limit. In curved backgrounds of asymptotically flat reflecting stars, it was found that the reflecting star can support massive scalar fields and the scalar hairy reflecting star radius is also discrete [24,25].
A simple way to invade the no hair theorem of static scalar fields is enclosing the gravity system in a confined box [26][27][28][29][30]. And it is well known that the AdS boundary could also serve as an infinity potential to confine the scalar field and dynamical formations of scalar hair due to the confinement was studied in [31]. At present, AdS scalar hairy black holes have been widely studied with the interest of the application of holographic theories [32]- [45]. On the other side, the scalar field configurations were constructed in the regular AdS reflecting shell background with analytical methods [46]. For a AdS shell, we mean that the shell charge and mass is very small compared to the shell radius or the spacetime outside the shell is pure AdS.
Along this line, it is interesting to extend the discussion to the more general AdS reflecting star background by including the nonzero star charge and mass to examine whether there are still no scalar hair behaviors and also try to construct scalar field configurations supported by AdS charged reflecting stars. This paper is organized as follows. In section II, we introduce the gravity model constructed with a scalar field coupled to asymptotically AdS charged reflecting stars. In part A of section III, we obtain an upper bound for the scalar hairy reflecting star radius. And in part B of section III, we show that the scalar field could condense around a AdS reflecting star and study properties of the largest hairy star radius. The last section is a summary of our main results.

II. EQUATIONS OF MOTION AND BOUNDARY CONDITIONS
In this paper, we study the system of a scalar field coupled to a charged reflecting star in the four dimensional asymptotically AdS spacetime. And the corresponding Lagrange density reads As usual we define A µ as the ordinary Maxwell field with only nonzero tt component as A µ = − Q r dt and ψ = ψ(r) is the scalar field with only radial dependence. Here, m is the scalar field mass and q corresponds to the scalar field charge.
We set the ansatz of the AdS spherically symmetric star geometry as [47] where f (r) = r 2 L 2 − 2M r + Q 2 r 2 with M as the star mass, Q as the star charge and L corresponding to the AdS radius. We also define the radial coordinate r = r s as the radius of the reflecting star. In this work, we study the regular reflecting star without a horizon or there is g(r) > 0 for all r r s .
The scalar field equation can be obtained as We can set L = 1 in the calculation with the symmetry Near the AdS boundary, asymptotic behaviors of the scalar fields are and the scalar fields around the infinity behave as We also impose reflecting boundary conditions at the surface of the star as ψ(r s ) = 0.

III. SCALAR FIELD CONDENSATION BEHAVIORS AROUND ADS CHARGED REFLECTING STARS
A. Upper bounds for the radius of the scalar hairy AdS reflecting star With a new radial functionψ = √ rψ, the equation of the scalar field can be expressed as with g = r 2 From the boundary conditions (5) and (6), we havẽ ψ(r s ) = 0,ψ(∞) = 0.
The functionψ must have (at least) one extremum point r = r peak between the surface r s of the reflecting star and the AdS boundary r b = ∞. At this extremum point, the scalar field is characterized by According to the relations (7) and (9), we arrive at the inequality Then we have We divide radii of hairy reflecting stars into two types: r s < max{ For radii satisfying r s max{ and From the regular condition g(r) > 0 for all r > r s and relations (11)- (13), we arrive at According to (14), there is It also can be expressed with dimensionless quantities with the symmetry (4) as Then we have two types of hairy star radii So we obtain an upper bound for the regular AdS hairy reflecting star radii as B. Scalar field configurations supported by a regular AdS charged reflecting star In this part, we construct the regular AdS scalar hairy charged reflecting star solutions and also study properties of the hairy reflecting star radii. Since we impose reflecting boundary conditions for the scalar field at the star surface, the scalar field can be putted in the form ψ = ψ 0 (r − r s ) + · · · around the star radius. As the scalar field equation is linear with respect to ψ, we can fix ψ 0 = 1 in the calculation. We will numerically integrate the equation from various reflecting boundary to the infinity to search for hairy star radii r s satisfying the boundary conditions (5).
We show the case of qL = 2, m 2 L 2 = 1 2 , Q L = 4 and M L = 7.5 in Fig. 1. In the left panel, when we impose a reflecting boundary at r L = 1.900, the scalar field is zero around r L ≈ 3.061 and it decreases to be more negative far away. And with a little larger reflecting boundary at r L = 1.909 in the middle panel, the scalar field decreases to be zero at a larger coordinate r L ≈ 5.297 and becomes more negative far away from the reflecting boundary. For a reflecting boundary at r L = 1.918 in the right panel, there is no additional zero point of the scalar field and the scalar field increases to be more positive far from the star. In front cases, it is clearly that the reflecting boundaries cannot be fixed as the star radius since the scalar field should asymptotically approach zero far from the star surface according to (5). However, there may exist a critical reflecting boundary between r L = 1.909 and r L = 1.918 and for this critical reflecting boundary, the zero points of the scalar field is at the infinity or the scalar field asymptotically approaches zero according to (5). This possible critical reflecting boundary can be fixed as the radius of the AdS scalar hairy reflecting star. With more detailed calculations in cases of qL = 2, m 2 L 2 = 1 2 , Q L = 4 and M L = 7.5, we arrive at the critical discrete reflecting star radius rs L ≈ 1.91077 with the scalar field asymptotically approaches zero at the infinity. We have further checked that when the reflecting boundary r L → 1.91077, the corresponding scalar field converges to the nonzero limit with rs L = 1.91077. On the other side, the general solutions of equation (3) behave in the form ψ = A · 1 r λ − + B · 1 r λ + + · · · with λ ± = (3 ± √ 11)/2 as r → ∞. We have numerically found A < 0 for reflecting boundary r L below 1.91077 and there is A > 0 for reflecting boundary r L above 1.91077 in accordance with results in Fig. 1. So a critical radius rs L ≈ 1.91077 with A = 0 should exist and the corresponding scalar field behaves as ψ ∝  (17). Integrating the equation from rs L = 1.91077 to smaller radial coordinates in the right panel of Fig. 2  We show the largest hairy star radius mR s as a function of star mass and charge with dimensionless quantities according to the symmetry (4) in Fig. 3. With qL = 2, m 2 L 2 = 1 2 and Q L = 4 in the left panel, we see that the larger star mass qM corresponds to a larger mR s similar to known results in the asymptotically flat reflecting star background [25]. In contrast, we can see that larger qQ leads to a smaller mR s with qL = 2, m 2 L 2 = 1 2 and M L = 7.5 in the right panel, which is very different from cases of reflecting stars in the asymptotically flat background [25]. It is interesting to extend the discussion in the right panel of Fig. 3 to smaller star charge and also examine whether there are neutral AdS hairy reflecting stars. We show the largest star radius mR s and the largest black hole horizon mR h as a function of star charge with qL = 2, m 2 L 2 = 1 2 and M L = 7.5 in Fig. 4. Since the reflecting star is assumed to be regular, the star surface coordinate should be larger than the largest horizon of a black hole with the same charge and mass. We find that the largest radius is above the largest horizon in the range qQ 4 and they both increase as we decrease the star charge. When we choose smaller star charge, the largest star radius becomes nearer to the largest horizon and for small star charge, they almost coincide with each other. We also show values of the largest star radius mR s and the largest horizon mR h in Table I with qL = 2, m 2 L 2 = 1 2 , M L = 7.5 and various qQ. The data again shows that the star surface becomes nearer to the largest horizon as we decrease the star charge. And when the star charge is qQ ≈ 4, the star radius is almost equal to the horizon and our further numerical results suggest that there is no reflecting hairy star with the reflecting surface outside the horizon for qQ < 4. So we conclude that there is no regular asymptotically AdS hairy reflecting star solutions or no scalar hair theorem holds for small star charge.

IV. CONCLUSIONS
We studied static scalar field configurations supported by a reflecting star in the AdS spacetime.
With analytical methods, we provided upper bounds for the radius of the scalar hairy star as mr s max{m √ QL, m 3 √ 4M L 2 , 4 √ 2 √ qQmL}. Above the bound, the AdS reflecting star cannot support the scalar field or there is no scalar hair theorem. Below the bound, we numerically obtained the charged scalar hairy reflecting star solutions and the radius of the hairy star is discrete, which is similar to cases in other reflecting object backgrounds. For each fixed set of parameters, we searched for the largest radii mR s and also examined effects of parameters on the largest AdS hairy star radii. We found that larger star mass qM corresponds to a larger mR s similar to cases in the asymptotically flat reflecting star background. However, we saw that larger star charge qQ leads to a smaller mR s , which is very different from results in asymptotically flat reflecting star spacetime. And our further detailed calculation showed that scalar fields cannot condense around regular AdS reflecting stars or no scalar hair theorem also exists when the star charge is small.