# No hair theorem for massless scalar fields outside asymptotically flat horizonless reflecting compact stars

## Abstract

In a recent paper, Hod started a study on no scalar hair theorem for asymptotically flat spherically symmetric neutral horizonless reflecting compact stars. In fact, Hod’s approach only rules out massive scalar fields. In the present paper, for massless scalar fields outside neutral horizonless reflecting compact stars, we provide a rigorous mathematical proof on no hair theorem. We show that asymptotically flat spherically symmetric neutral horizonless reflecting compact stars cannot support exterior massless scalar field hairs.

## 1 Introduction

Recently, the first ever image of a black hole has been captured by a network of eight radio telescopes around the world [1]. These discoveries open up hope to directly test various black hole theories from astronomical aspects. One remarkable property of classical black holes is the famous no hair theorem [2, 3, 4, 5, 6, 7, 8, 9]. If generically true, such no hair theorem would signify that asymptotically flat black holes cannot support scalars, massive vectors and Abelian Higgs hairs in exterior regions, for recent references see [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and reviews see [26, 27]. It was believed that no hair behaviors are due to the existence of one-way absorbing horizons.

However, it was recently found that no hair behavior also appears in the background of horizonless reflecting compact stars. In the asymptotically flat gravity, Hod firstly proved no static scalar hair theorem for neutral horizonless reflecting compact stars [28]. In the asymptotically dS gravity, it was found that neutral horizonless reflecting compact stars cannot support the existence of massive scalar, vector and tensor field hairs [29]. When considering a charged background, large reflecting shells can exclude static scalar field hairs [30, 31, 32]. Similarly, static scalar field hairs cannot exist outside charged reflecting compact stars of large size [33, 34, 35, 36, 37]. With field-curvature couplings, such no hair theorem could also hold in the horizonless gravity [38, 39, 40]. Moreover, we proved no hair theorem for horizonless compact stars with other boundary conditions [41, 42, 43].

As is well known, scalar field mass usually plays an important role in the scalar hair formation. For massless scalar fields \(\psi (r)\), no scalar hair theorem was investigated in the background of horizonless compact stars [28], where the relation \(\psi (r_{{ peak}})\psi ''(r_{{ peak}})<0\) at the extremum point \(r=r_{{ peak}}\) is essential in Hod’s present proof. However, the general characteristic relation at the extremum point should be \(\psi (r_{{ peak}})\psi ''(r_{{ peak}})\leqslant 0\) and in fact, \(\psi ''(r_{{ peak}})=0\) holds for some solutions. So Hod’s approach only ruled out massless scalar fields with \(\psi ''(r_{{ peak}})\ne 0\) and nontrivial solutions with \(\psi ''(r_{{ peak}})=0\) cannot be excluded. Then it is of some importance to search for a mathematical proof on no hair theorem for massless scalar field hairs.

In the following, we consider static massless scalar fields in the background of asymptotically flat spherical neutral horizonless reflecting compact stars. We provide a rigorous mathematical proof on no hair theorem for massless scalar fields. We summarize main results in the last section.

## 2 No massless scalar hair for horizonless reflecting compact stars

- (I)
Firstly, there is \(\psi '(r_{{ peak}})=0\). Otherwise, \(\psi (r)=\psi (r_{{ peak}})+\psi '(r_{{ peak}})(r-r_{{ peak}})+\cdots \) and \(\psi (r)\) cannot has extremum value at the point \(r_{{ peak}}\).

- (II)
In the case of \(\psi ''(r_{{ peak}})\ne 0\), we will have \(\psi ''(r_{{ peak}})< 0\). Otherwise, \(\psi \) cannot has local maximum extremum value at the point \(r_{{ peak}}\) since \(\psi (r)=\psi (r_{{ peak}})+\frac{\psi ''(r_{{ peak}})}{2}(r-r_{{ peak}})^2+\cdots \). In this work, we only consider the case of positive local maximum value according to the symmetry \(\psi \rightarrow -\psi \) of Eq. (3).

- (III)
In the case of \(\psi ''(r_{{ peak}})= 0\), we will have \(\psi ^{(3)}(r_{{ peak}})= 0\). Otherwise, \(\psi \) cannot have local maximum extremum value at the point \(r_{{ peak}}\) since \(\psi (r)=\psi (r_{{ peak}})+\frac{\psi '''(r_{{ peak}})}{3}(r-r_{{ peak}})^3+\cdots \).

- (IV)
In the case of \(\psi ''(r_{{ peak}})= 0\) and \(\psi ^{(4)}(r_{{ peak}})\ne 0\), we will have to impose \(\psi ^{(4)}(r_{{ peak}})< 0\) to obtain a local maximum extremum value for \(\psi \) at the point \(r_{{ peak}}\). In this case, there is the relation \(\psi (r)=\psi (r_{{ peak}})+\frac{\psi ^{(4)}(r_{{ peak}})}{24}(r-r_{{ peak}})^4+\cdots \).

## 3 Conclusions

We studied no hair theorem for static massless scalar fields outside the asymptotically flat spherically symmetric horizonless reflecting compact stars. We obtained the characteristic relations (12) at extremum points, which are in contradiction with the Eq. (18). That is to say there is no nontrivial scalar field solution of Eq. (3). We concluded that asymptotically flat spherically symmetric horizonless reflecting compact stars cannot support the existence of exterior massless scalar fields. In this work, we provided a rigorous mathematical proof on no hair theorem for massless scalar fields.

## Notes

### Acknowledgements

This work was supported by the Shandong Provincial Natural Science Foundation of China under Grant no. ZR2018QA008. This work was also supported by a grant from Qufu Normal University of China under Grant no. xkjjc201906.

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