Hair formation in the background of noncommutative reflecting stars

We investigate scalar condensations around noncommutative compact reflecting stars. We find that the neutral noncommutative reflecting star cannot support the existence of scalar field hairs. In the charged noncommutative reflecting star spacetime, we provide upper bounds for star radii. Above the bound, scalar fields cannot exist outside the star. In contrast, when the star radius is below the bound, we show that the scalar field can condense. We also obtain the largest radii of scalar hairy reflecting stars.


I. INTRODUCTION
The classical no scalar hair theorem provides insights into physical properties of black holes [1][2][3][4]. According to this theorem, asymptotically flat black holes cannot support static massive scalar fields, for references please refer to [5]- [17] and reviews can be found in [18,19]. It was usually believed that these no scalar hair behaviors are due to the existence of black hole horizons, which can inevitably absorb matter and radiation fields.
Whether the no scalar hair behavior exists in the horizonless spacetime is a question to be answered.
Recently, it was found that the no scalar hair behavior appears in the background of asymptotically flat horizonless neutral compact reflecting stars [20]. Moreover, it was shown that the massless scalar field nonminimal coupled to the gravity cannot condense around the asymptotically flat neutral compact reflecting star [21].
When extending the discussion to spacetimes with a positive cosmological constant, it was proved that massive scalar, vector and tensor hairs all die out outside asymptotically dS neutral reflecting stars [22].
However, the no hair theorem obtained in the regular neural gravity could be challenged in the charged spacetime. In the case of regular charged reflecting shell, it was analytically found that the scalar hair can form outside the shell when the shell radius is below an upper bound [23][24][25]. It was also shown that regular charged reflecting stars can support the existence of the scalar hair if the star radius is below an upper bound [26][27][28][29][30]. We mention that there are usually radii bounds, above which the scalar field cannot condense outside the objects [25][26][27][28][29][30].
Recently, the noncommutative gravity has attracted a lot of attentions based on the belief that the noncommutativity will appear at the Planck scale, where usual semiclassical approaches fail. The area law of noncommutative black holes has been studied [31][32][33][34][35][36][37]. Another important motivation to study noncommutative theories is it's natural emergence in string theory and some surprising consequences [38][39][40][41][42][43][44]. As we know, the discussions on no hair theorem in regular gravities were carried out in the commutative spacetimes. So it is interesting to extend the discussion to noncommutative backgrounds. And it is also meaningful to construct regular hairy configurations in the noncommutative gravity.
The rest of this work is organized as follows. In section II, we construct the gravity system composed of a scalar field and a charged reflecting star in the noncommutative geometry. In part A of section III, we analytically obtain an upper bound for the charged scalar hairy star radius. In part B of section III, we numerically study scalar hairy star solutions. The last section is devoted to the conclusion.

II. THE GRAVITY SYSTEM IN THE NONCOMMUTATIVE GEOMETRY
The idea of noncommutative spacetime was firstly introduced by Snyder [45] and such a noncommutative structure naturally emerges in string theory [46]. There are mainly two ways to obtain the noncommutative quantum field theory: the Weyl-Wigner-Moyal star product approach and the coordinate coherent state approach [47]. Most recently, inspired by the coordinate coherent state approach, P.Nicolini and other authors obtained noncommutative black hole metrics [35][36][37], where the noncommutativity is introduced by writing down the Gaussian distribution of mass and charge densities in the form ).
Here θ is the parameter used to describe the noncommutativity of the spacetime. And the mass M and charge Q diffuse throughout a region of linear size √ θ. When the object size is below the characteristic length √ θ, noncommutativity becomes non-negligible. And in the case of θ → 0, the model returns to the commutative one.
The solution is g(r) where γ is the incomplete gamma function in the form γ(n, z) = z 0 t n−1 e −t dt. Here M and Q are interpreted as the star mass and star charge respectively. We also define r s as the regular star radius and there is g(r) > 0 for r r s .
We study scalar condensations in the noncommutative charged reflecting star background. And the Lagrange density with scalar fields coupled to the Maxwell field reads Here ψ(r) is the scalar field and A µ corresponds to the Maxwell field. We also label q and m as scalar field charge and scalar field mass respectively.
We assume that the Maxwell field has only the nonzero t component in the form A t = φ(r)dt. Then the equation of the Maxwell field is [48][49][50][51][52][53] In this work, we neglect the scalar field's backreaction on the charged star. So the third term of equation (5) disappears and the electric potential is not affected by the non-commutativity. We take the electric potential in the form A t = − Q r [26,27] and the scalar field equation is We need to impose boundary conditions to solve the equation (6). At the star surface, we take the scalar reflecting condition that the scalar field vanishes. In the region far from the star, the general solutions behave in the form ψ ∼ A · 1 r e −mr + B · 1 r e mr , where A and B are integral constants. We set B = 0 in order to obtain the physical solution satisfying ψ(∞) = 0. So boundary conditions can be putted as With relations (6), (7) and the Hod's method in [20], it is easy to check that massive scalar fields cannot exist outside neutral noncommutative reflecting stars. In the next section, we turn to study scalar condensations in the background of charged noncommutative reflecting stars.

III. SCALAR FIELD CONDENSATIONS OUTSIDE NONCOMMUTATIVE CHARGED REFLECTING STARS
A. Upper bounds for radii of noncommutative hairy reflecting stars Introducing a new functionψ = √ rψ, we can express the equation (6) as The boundary conditions areψ Then the functionψ must possess at least one extremum point r = r peak above the star radius r s . At this extremum point, the following characteristic relations hold From (8) and (10), we deduce the inequality It can be transformed into With the relation γ(s + 1, z) = sγ(s, z) − x s e −z , the metric function g(r) can be expressed as Then we have And there is also the relation We mention that γ( 1 2 , r 2 4θ ) increases as a function of r with values of γ( 1 2 , r 2 4θ ) in the range [0, √ π]. According to (14), (15) and the fact that e − r 2 4θ decreases very quickly, g ′ asymptotically behaves as in the large r region.
In the case of r 2 Then there is g > 1 − 3M r + 1 2 Q 2 r 2 for r 2 4θ 6. We obtain an lower bound of the metric function as on conditions r > 3M + 9M 2 − Q 2 and r 2 4θ 6.
We assume that the star radius satisfies r 2 s 4θ 6, r s > 3Q 2 M and r s > 3M + 9M 2 − Q 2 , otherwise we will arrive at an upper bound According to (12), (30), (33) and r peak r s , there is Then we have In all, our analysis shows that the hairy star radius is below the bounds (34) or (36). With dimensionless quantities according to the symmetry (38), we obtain upper bounds for hairy star radii as Above this bound, the static scalar field cannot condense. Below the bound, we will numerically obtain hairy reflecting star solutions in the next part.
We mention that there are also upper bounds on the size of hairy black holes. According to the no short hair conjecture, the hairy black hole has an upper bound for the horizon r H < 2 3 (η) −1 , where η is the field mass [54]. In fact, the numerical results in [55] also support the idea that big black holes tend to have no massive hair. Similar to black hole theories, we find that big reflecting star cannot support the existence massive scalar hair. So it is natural that there is a maximal radius for the hairy star. In the following, we will numerically search for the maximal radius.

B. Scalar field configurations supported by noncommutative charged reflecting stars
We firstly show the metric solution g(r) with different values of the noncommutative parameter θ in Fig. 1.
It can be seen from the panels that the noncommutative spacetime is regular at r = 0 and the metric behaves like Schwarzschild geometry at r → ∞, in accordance with results in [35]. Our results also imply that in cases of very small θ, the metric almost coincides with the Schwarzschild geometry even at the points r ≈ 0, but the metric always keeps regular at the center with g(0) = 1. We point out that the regular star radius should be imposed above the outmost would-be horizon of the metric. In the following, we numerically obtain regular configurations composed of scalar fields and noncommutative charged reflecting stars. We take m = 1 according to the symmetry Around the star surface, the scalar field behaves in the form ψ = ψ 0 (r − r s )+ · · · . With the symmetry ψ → kψ of equation (6), we can set ψ 0 = 1 without loss of generality. Then we numerically search for the proper r s , where the corresponding scalar field satisfies the boundary condition (7) at the infinity.
In Fig. 2, we plot scalar fields in the background of compact stars with q = 2, Q = 4, M = 5 and θ = 50. With the star radius fixed at r s = 9.6719, the scalar field approaches zero at the infinity. If we impose the star radius a little larger or a little smaller than r s , the solutions asymptotically behave in the form ψ ∝ A · 1 r e −mr + B · 1 r e mr with nonzero B, which contradicts the infinity boundary condition (7). Our detailed calculations show that the hairy star radius is discrete. Integrating the equation from r s = 9.6719 to smaller radial coordinates, we find discrete points, which can be fixed as the hairy star radii. In Fig. 3 (37). Similar to cases in commutative spacetimes [23][24][25][26][27][28][29], we find many discrete hairy star radii. In the case of q = 2, Q = 4, M = 5 and θ = 50, we numerically find that mr s = 9.6719 is the largest hairy star radius below the upper bound mr s 2 √ 300 ≈ 34.641 of (37). For every given set of parameters, we can obtain the largest hairy star radius labeled as mR s . We also study effects of the noncommutative parameter on the largest hairy star radius. With dimensionless quantities according to the symmetry (38), we plot mR s as a function of m 2 θ with q = 2, Q = 4 and M = 5 in Fig. 4. It can be seen from the picture that larger m 2 θ corresponds to smaller mR s . The noncommutative physics is expected to be detected for a small size star r s ⋍ √ θ. However, the phenomenological impact of these results may be not visible since the presently accessible energy is √ θ < 10 −16 cm [35]. For large distance, the solution behaves very similar to the Schwarzschild metric. Recently, holographic superconductor models were constructed in the background of noncommutative AdS black holes [56,57], which provided a novel way to investigate the role of noncommutative geometry through the AdS/CFT duality.

IV. CONCLUSIONS
We studied static scalar field condensations outside noncommutative charged compact reflecting stars.
We found that the scalar field cannot condense outside neutral noncommutative reflecting stars. And in the background of charged noncommutative reflecting compact stars, we provided upper bounds for the star radius as mr s max 2m √ 6θ, 3mQ 2 M , 3mM + m 9M 2 − Q 2 , 2q 2 Q 2 − 1 8 , where m is the scalar field mass, q is the charge coupling parameter, M is the ADM mass, Q is the star charge and θ is the noncommutative parameter. Above the bound, the scalar field cannot condense and below the bound, we obtained numerical solutions of scalar hairy reflecting stars. With detailed calculations, we found that the scalar hairy reflecting star radius is discrete. We also examined effects of the noncommutative parameter on the largest radius of the scalar hairy reflecting star.