No scalar hair theorem for neutral Neumann stars: static massive scalar fields nonminimally coupled to gravity

TL;DR

This paper demonstrates that Neumann boundary condition compact stars cannot support static massive scalar fields with nonminimal coupling, extending no hair theorems beyond Dirichlet conditions.

Contribution

It proves the nonexistence of scalar hair for Neumann boundary condition compact stars with nonminimal coupling, generalizing previous no hair theorems.

Findings

Neumann stars cannot support static massive scalar fields.

Scalar hair is excluded for nonminimal coupling under Neumann conditions.

Results extend no hair theorems to new boundary conditions.

Abstract

In a recent paper, Hod proved that spherically symmetric Dirichlet reflecting compact stars cannot support static nonminimally coupled scalar fields. In the present paper, we study the validity of no hair theorems for compact stars with Neumann surface boundary conditions. We find that Neumann compact stars cannot support static massive scalar field hairs with a generic dimensionless nonminimal coupling parameter.

Equations

$$d{s}^2$$

$$\nu (r\to \infty )\backsim O(\frac{1}{r}),\lambda (r\to \infty )\backsim O(\frac{1}{r}).$$

$$\mathcal{L}=R-\xi R{\psi }^2-|{\nabla }_{\alpha }\psi {|}^2-{\mu }^2{\psi }^2,$$

$$R(r\to \infty )\to 0.$$

$$0⩽\xi ⩽\frac{1}{4}.$$

$${\psi }^{\prime \prime }+(\frac{2}{r}+\frac{{\nu }^{\prime }}{2}-\frac{{\lambda }^{\prime }}{2}){\psi }^{\prime }-({\mu }^2+\xi R)e^{\lambda }\psi =0.$$

$$\psi \sim A\cdot \frac{1}{r}{e}^{-\mu r}+B\cdot \frac{1}{r}{e}^{\mu r},$$

$${\psi }^{\prime }(r_s)=0,\psi (\infty )=0.$$

$$R=-\frac{8\pi }{1-8\pi {\xi }^2{\psi }^2}\{e^{-\lambda }[\xi (\frac{12}{r}+3{\nu }^{\prime }-3{\lambda }^{\prime })\psi {\psi }^{\prime }+6\xi \psi {\psi }^{\prime \prime }+(6\xi -1)({\psi }^{\prime }{)}^2]-2{\mu }^2{\psi }^2\}.$$

$$F\cdot {\psi }^{\prime \prime }+[F\cdot (\frac{2}{r}+\frac{{\nu }^{\prime }}{2}-\frac{{\lambda }^{\prime }}{2})+8\pi \xi (6\xi -1)\psi {\psi }^{\prime }]{\psi }^{\prime }-{\mu }^2{e}^{\lambda }(1+8\pi \xi {\psi }^2)\psi =0,$$

$$\psi (r_s)=0,\psi (r_s)>0and\psi (r_s)<0.$$

$$\psi (r_s)>0.$$

$$8\pi \xi (6\xi -1){\psi }^{\prime 2}={\mu }^2{e}^{\lambda }(1+8\pi \xi {\psi }^2).$$

$$F(r\to \infty ,\xi )\to 1+8\pi \xi (6\xi -1)\psi (\infty {)}^2=1>0.$$

$$F(r,\xi )⩾0.$$

$${\psi }^{\prime \prime }(r_s)⩽0and{\psi }^{\prime \prime }(r_s)>0.$$

$$\{{\psi }^2>0,{\psi }^{\prime }=0and\psi {\psi }^{\prime \prime }⩽0\}forr=r_s.$$

$$F\cdot \psi {\psi }^{\prime \prime }+[F\cdot (\frac{2}{r}+\frac{{\nu }^{\prime }}{2}-\frac{{\lambda }^{\prime }}{2})+8\pi \xi (6\xi -1)\psi {\psi }^{\prime }]\psi {\psi }^{\prime }-{\mu }^2{e}^{\lambda }(1+8\pi \xi {\psi }^2){\psi }^2<0,$$

$$\{{\psi }^2>0,{\psi }^{\prime }=0and\psi {\psi }^{\prime \prime }⩽0\}forr=r_{peak}.$$

$$F\cdot \psi {\psi }^{\prime \prime }+[F\cdot (\frac{2}{r}+\frac{{\nu }^{\prime }}{2}-\frac{{\lambda }^{\prime }}{2})+8\pi \xi (6\xi -1)\psi {\psi }^{\prime }]\psi {\psi }^{\prime }-{\mu }^2{e}^{\lambda }(1+8\pi \xi {\psi }^2){\psi }^2<0.$$

Full text

In a recent paper, Hod proved that spherically symmetric Dirichlet reflecting compact stars cannot support static nonminimally coupled scalar fields. In the present paper, we study the validity of no hair theorems for compact stars with Neumann surface boundary conditions. We find that Neumann compact stars cannot support static massive scalar field hairs with a generic dimensionless nonminimal coupling parameter.

I Introduction

The classical no hair theorem Bekenstein -JBN plays an important role in the development of black hole theories. It states an intriguing property that asymptotically flat black holes cannot support static scalar fields, for recent progress see references mr1 -CAHE and reviews Bekenstein-1 ; CAR . It was usually believed that this no hair property is due to the existence of black hole absorbing horizons.

However, no scalar hair behavior also appears in the horizonless spacetime. It was firstly proved that asymptotically flat neutral Dirichlet reflecting horizonless compact stars cannot support massive scalar field hairs Hod-6 . In the asymptotically dS gravity, massive scalar, vector and tensor fields also cannot exist outside neutral horizonless Dirichlet reflecting compact stars Bhattacharjee . Then whether no hair theorem exists in the charged horizonless gravity is still an question to be answered. In fact, it was shown that static scalar fields cannot condense outside charged reflecting shells of large radii Hod-8 ; Hod-9 ; Yan Peng-1 . Large charged reflecting stars also cannot support static scalar field hairs Hod-10 ; Yan Peng-2 ; Yan Peng-3 ; Yan Peng-4 ; Yan Peng-5 ; YP1 ; YP2 ; YP3 . Moreover, it was found that scalar fields cannot exist outside compact stars with Neumann surface boundary conditions Yan Peng-6 ; Yan Peng-7 .

All front no scalar hair theorems only consider minimally coupled scalar field hairs. Interestingly, no scalar hair theorem also holds with nonminimal field-curvature couplings ADB ; AS ; Hod-11 ; Hod-12 . Considering a nonminimal coupling parameter $\xi$, compact stars with generic boundary conditions can rule out the existence of scalar hairs in ranges $\xi<0$ and $\xi>\frac{1}{4}$ and compact stars with Dirichlet reflecting boundary conditions can also rule out exterior scalar hair for $0\leqslant\xi\leqslant\frac{1}{4}$ Hod-13 ; Hod-14 ; Hod-15 . So it is interesting to study no scalar hair properties with other boundary conditions in the range $0\leqslant\xi\leqslant\frac{1}{4}$. In this work, we plan to investigate the no nonminimal scalar hair behavior in the background of horizonless compact stars with Neumann boundary conditions.

The rest of this work is as follows. We construct the model of static scalar fields nonminimally coupled to gravity in the background of Neumann compact stars. We find that no scalar hair theorem holds for generic coupling parameters. At last, we summarize the main results.

II No hair theorem for scalar fields outside Neumann stars

We study the model of massive scalar fields nonminimally coupled to the compact star gravity. And the asymptotically flat spherically symmetric spacetime reads ADB ; AS ; Hod-11 ; Hod-12 ; Hod-13

[TABLE]

The functions $\nu$ and $\lambda$ only depend on the radial coordinate r. We define the radial coordinate $r=r_{s}$ as the star radius. Asymptotic flatness of the spacetime requires the behaviors ADB

[TABLE]

The Lagrange density describing scalar fields nonminimally coupled to gravity is Hod-12 ; Hod-13

[TABLE]

where $\psi(r)$ is the scalar field with mass $\mu$. We label R as scalar Ricci curvature of the spacetime. Asymptotic flatness of the spacetime yields

[TABLE]

The dimensionless parameter $\xi$ describes the nonminimal coupling strength between scalar fields and curvature. Hod proved that compact stars with generic boundary conditions can rule out the existence of scalar hairs in ranges $\xi<0$ and $\xi>\frac{1}{4}$ Hod-13 . So it remains to study no hair theorem for parameters satisfying

[TABLE]

The scalar field equation is

[TABLE]

Around the infinity, the scalar field asymptotically behaves in the form

[TABLE]

where A and B are integral constants. The physical solution requires $B=0$ Hod-12 . At the star surface, we take the Neumann boundary condition. It yields boundary conditions

[TABLE]

The Ricci scalar curvature is

[TABLE]

Substituting (9) into the scalar field equation (6), we arrive at

[TABLE]

where $F(r,\xi)=1+8\pi\xi(6\xi-1)\psi^{2}$.

We divide the proof of no hair theorem into three cases

[TABLE]

It is known that the nontrivial scalar field cannot exist for $\psi(r_{s})=0$ Hod-13 . Considering the symmetry $\psi\rightarrow-\psi$ of equation (10), it remains to prove no hair theorem for

[TABLE]

In the range $0\leqslant\xi\leqslant\frac{1}{4}$, there is $F(r,\xi)\geqslant 0$, which is important in the following analysis Hod-13 . The proof of $F(r,\xi)\geqslant 0$ is as follows. In the range $\frac{1}{6}\leqslant\xi\leqslant\frac{1}{4}$, there is $F(r,\xi)\geqslant 0$. For $0\leqslant\xi<\frac{1}{6}$, $F(r,\xi)$ cannot switch signs. Otherwise, $F(r,\xi)$ vanishes at some point $r_{0}$. And at this point $r=r_{0}$, there is the relation

[TABLE]

In the regime $0\leqslant\xi<\frac{1}{6}$, the functional expression on the left side of (13) is non-positive whereas the functional expression on the right side of (13) is positive. One therefore deduces that the radial function $F(r,\xi)$ cannot switch signs. At the infinity, $F(r,\xi)$ behaves as

[TABLE]

So we find $F(r,\xi)>0$ in the case of $0\leqslant\xi<\frac{1}{6}$. As a summary, for $0\leqslant\xi\leqslant\frac{1}{4}$, there is the relation

[TABLE]

We divide the analysis into two cases

[TABLE]

In the case of $\psi^{\prime\prime}(r_{s})\leqslant 0$, we obtain following relations at the star surface $r=r_{s}$ as

[TABLE]

At the star radius $r_{s}$, relations (15), (17) and $0\leqslant\xi\leqslant\frac{1}{4}$ give the characteristic inequality

[TABLE]

which is in contradiction with equation (10).

In another case of $\psi^{\prime\prime}(r_{s})>0$, with the condition $\psi^{\prime}(r_{s})=0$, there is $\psi^{\prime}(r)>0$ around $r_{s}$. With increase of the radial coordinate, the scalar field firstly becomes more positive and finally approaches zero at the infinity. In this case, there is at least one positive maximum extremum point $r=r_{peak}$ between the star surface $r_{s}$ and the infinity boundary. At this extremum point, the scalar field is characterized by following relations

[TABLE]

At this extremum point $r=r_{peak}$, relations (15), (19) and $0\leqslant\xi\leqslant\frac{1}{4}$ lead to the inequality

[TABLE]

It can be easily seen that relations (18) and (20) are in contradiction with equation (10). So nontrivial scalar field solution of equation (10) cannot exist. Here we prove no nonminimally coupled scalar hair theorem for $0\leqslant\xi\leqslant\frac{1}{4}$. Also considering known results that compact stars with generic boundary conditions cannot support scalar hairs in ranges $\xi<0$ and $\xi>\frac{1}{4}$ Hod-13 , we conclude that scalar hair cannot form outside regular neutral Neumann stars for any coupling parameter $\xi$.

III Conclusions

In the background of Neumann stars, we studied no hair theorem for static massive scalar fields nonminimally coupled to the spherically symmetric asymptotically flat horizonless gravity. We considered the field-curvature coupling and scalar fields’ backreaction on the background. We obtained the characteristic inequalities (18) at the star surface and (20) at extremum points, which are in contradiction with the scalar field equation (10). It means that there is no nontrivial scalar field solution. So we concluded that asymptotically flat spherically symmetric regular Neumann stars cannot support the existence of exterior massive scalar field hairs for generic nonminimal coupling parameters.

Acknowledgements.

We would like to thank the anonymous referee for the constructive suggestions to improve the manuscript. 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.