Hair distributions in noncommutative Einstein-Born-Infeld black holes

We study hair mass distributions in noncommutative Einstein-Born-Infeld hairy black holes with non-zero cosmological constants. We find that the larger noncommutative parameter makes the hair easier to condense in the near horizon area. We also show that Hod's lower bound can be evaded in the noncommutative gravity. However, for large black holes with a non-negative cosmological constant, Hod's lower hair mass bound almost holds in the sense that nearly half of the hair lays above the photonsphere.


I. INTRODUCTION
The famous black hole no hair theorem introduced by Wheeler [1][2][3] was motivated by the uniqueness theorem that a Einstein-Maxwell black hole can be determined only by the three conserved global charges associated with Gauss laws as ADM mass M, electric charge Q and angular momentum J [4][5][6][7][8][9][10]. In accordance with this no hair theorem, stationary black holes indeed cannot support the existence of scalar fields, massive vector fields and spinor fields in the exterior spacetime, for references see [11][12][13][14][15][16][17][18].
However, nowadays we are faced with the surprising discovery of various types of hairy black holes in theories like Einstein-Yang-Mills, Einstein-Skyrme, Einstein-non-Abelian-Proca, Einstein-Yang-Mills-Higgs, Einstein-Yang-Mills-Dilaton and non-static spin gravies, which cannot be unique described by the three conserved charges M, Q and J, for references please refer to [19]- [39] and reviews can be found in [40,41]. Recently, a no short hair theorem was proposed as an alternative to the classical no hair theorem based on the fact that the hair satisfying the weak energy condition and the energy-momentum tensor dominant condition must extend above the photonsphere [42]. And it was found that no short scalar hair behaviors also exist in non-spherically symmetric non-static kerr black holes [43]. It also provided a nice heuristic picture that the formation of hair is due to the self-interaction which can bind together the hair below the photonsphere and hair above the photonsphere relatively distant from the horizon [42,43]. It should be emphasized that for various types of black hole hair, these two conditions are indeed satisfied.
Along this line, it is interesting to study the hair distribution outside the black hole horizon. For the limit case of the linear Maxwell field, Hod showed that the region above the photonsphere contains at least half of the total mass of Maxwell fields and also found that this lower bound holds for various genuine hairy black holes in Einstein-Yang-Mills, Einstein-Skyrme, Einstein-non Abelian-Proca, Einstein-Yang-Mills-Higgs and Einstein-Yang-Mills-Dilaton systems [44]. And Hod further conjectured that the hair mass lower bound exists in all hairy black holes. In fact, it was found that the non-linear Einstein-Born-Infeld black holes also satisfy this lower bound that half of the Born-Infeld hair is above the photonsphere [45]. As a further step, we showed that the Hod's lower bound holds in asymptotically dS Einstein-Born-Infeld hairy black holes [46].
The known results imply that Hod's lower bound of hair mass ratio may be a general property in the hairy black hole background. However, all of these calculations were based on a commutative spacetime. Recently, noncommutative black holes have been studied on the motivation that noncommutativity is expected to be relevant at the Planck scale where it is known that usual semiclassical considerations break down. For example, modifications to the semiclassical area law in the noncommutative (NC) spacetime have been obtained [47][48][49][50][51][52]. Another important motivation to study noncommutative theories is due to its natural emergence in string theory and some surprising consequences [53][54][55][56][57][58]. In this work, we plan to extend the discussion of hair distributions to noncommutative spacetimes and also examine whether the Hod's lower hair mass bound holds in noncommutative hairy black holes.
In the following, we introduce noncommutative Einstein-Born-Infeld black holes and disclose effects of parameters on hair distributions. We also examine whether the Hod's lower bound holds in this noncommutative model. And we will summarize our main results at the last section.

II. HAIR MASS BOUNDS IN NONCOMMUTATIVE HAIRY BLACK HOLES
In this paper, we choose the background of noncommutative Einstein-Born-Infeld hairy black holes and the corresponding Lagrangian density with non-zero cosmological constant Λ is [59][60][61][62] Here R is the scalar curvature, b is the Born-Infeld factor parameter and the limit of b → ∞ corresponds to the Maxwell field case. Now we introduce the line element of Einstein-Born-Infeld black holes with noncommutative mass deformation as follows [51,63] The metric function is and γ is the incomplete gamma function defined as γ(n, z) = z 0 t n−1 e −t dt. We also label θ as the noncommutative parameter and the model goes back to the commutative case in the limit of θ → 0.
The mass in a sphere of radius r is It was found that the black hole horizon r H and the photonsphere r δ can be conveniently used to describe spatial distribution of the matter field outside the horizon [42,44]. And the spatial distribution of the hair is characterized by the dimensionless hair mass ratio is the hair mass above the photonsphere and is the hair mass between the horizon and the photonsphere. Here, the black hole horizon r H is defined by According to the approach in [44], the radius r δ of the null circular geodesic (photonsphere) is determined by the relation And the hair mass ratio can be expressed as  [44,45]. In fact, this lower bound also exists in asymptotically dS static Einstein-Born-Infeld hairy black holes [46]. In the following, we extend the discussion to the case of noncommutative static Einstein-Born-Infeld hairy black holes.
Case I: Λ = 0 We calculate the hair mass ratio in the noncommutative Einstein-Born-Infeld asymptotically flat black holes. We In Fig. 2, we plot the minimum ratio can be invaded in the noncommutative static asymptotically flat Einstein-Born-Infeld black hole model. We also mention that the Hod's lower bound is more likely to be invaded in the small charge region. In contrast, it should be emphasized that the Hod's lower bound always holds in the commutative static EBI black holes with non-negative cosmological constants [45,46].
With the relations (15), (16) and (17), we see that large cosmological constants lead to a mass ratio above the Hod's lower mass bound. With detailed calculation, we find that It means that the Hod's bound can be invaded by imposing an AdS boundary. Due to the confinement of the AdS boundary, this result is natural and effects of negative cosmological constants on hair distribution should be qualitatively the same for other types of black hole hairs.
In the front analysis, we find that the Hod's lower bound can be widely invaded in the noncommutative gravity and the Hod's bound is not such a general property as cases in the commutative case. However, we will show in the following that the Hod's bound almost holds in noncommutative black holes of large size with non-negative cosmological constants. Our numerical data shows that the following relation exactly holds with non-negative cosmological constants and various other parameters In the large black hole limit or r δ r H ≫ θ, the model goes back to the commutative case and the relation (20) is equivalent to Hod's lower bound in the commutative black hole. Here, we further find that (20) holds beyond the large black hole limit. According to (20) and the fact that 1, the ratio (7) can be expressed as Considering that γ( 3 2 , r 2 4θ ) increases as a function of r with values of γ( 3 2 , r 2 4θ ) in the range [0, , we obtain lower bound of the ratio in the noncommutative gravity as According to results in [46], the ratio of (20) is equal to 2 in the limit of large b and small Q. So the lower bound (22) should be also the approximate formula of the hair mass ratio in the case of large b and small Q. That is to say in the nearly neutral black hole with large b, which is also well supported by our numerical data. For example, in the case of M = 1.0, Q = 0.
In summary, we show that the Hod's bound can be invaded in the noncommutative Einstein-Born-Infeld black holes. We obtain a lower bound (22) expressed with black hole horizon and noncommutative parameters.
And (22) shows that the Hod's bound almost holds for large black holes in flat or dS backgrounds. Since there is also no scalar hair theorem in regular neutral reflecting stars [64] and static scalar fields can condense around charged reflecting stars [65], it is also very interesting to extend the discussion to the horizonless reflecting star background.

III. CONCLUSIONS
We studied hair distributions of the static spherically symmetric Einstein-Born-Infeld black hole in the noncommutative geometry. We used the photonsphere to divide the matter into two parts and obtained lower bounds of the mass ratio. We found that the noncommutative parameter makes the hair easier to condense in the near horizon area. We further showed that the Hod's bound can be invaded in the noncommutative hairy black holes and the Hod's bound is not such a general property as cases in the commutative case. We also mentioned that the Hod's lower bound is more likely to be invaded in the small charge region. However, for large black holes with a non-negative cosmological constant, the Hod's lower hair mass bound almost holds in a sense that nearly half of the hair lays above the photonsphere.