On instabilities of stationary scalar field configurations supported by reflecting compact stars

TL;DR

This paper investigates the stability of scalar fields around horizonless reflecting stars, showing that stationary scalar hairy stars are unstable for low-frequency massless scalar fields, highlighting potential nonlinear instabilities.

Contribution

It provides theoretical bounds on scalar field frequencies and demonstrates the instability of stationary scalar hairy stars under certain conditions.

Findings

Stationary scalar hairy stars are unstable for low-frequency massless scalar fields.

Bounds on scalar field frequencies are derived in the probe limit.

Nonlinear instabilities are expected below certain frequency thresholds.

Abstract

We study instabilities of the system composed of stationary scalar fields and asymptotically flat horizonless reflecting compact stars. In the probe limit, we obtain bounds on the scalar field frequency. Below this bound, stationary hairy stars are expected to suffer from nonlinear instabilities under massless field perturbations. In other words, we prove that stationary scalar hairy stars are unstable for scalar fields with small frequency.

Equations

$$\mathcal{L}=-|{\nabla }_{\mu }\Psi {|}^2-m^2{\Psi }^2,$$

$$d{s}^2$$

$$f^{\prime }=-8\pi r\rho +(1-f)/r,$$

$${\chi }^{\prime }=-8\pi r(\rho +p)/f,$$

$$({\nabla }^{\nu }{\nabla }_{\nu }-m^2)\Psi =0$$

$$\Psi (t,r)=e^{-i\omega t}\psi (r),$$

$${\psi }^{\prime \prime }+(\frac{2}{r}+\frac{f^{\prime }}{f}){\psi }^{\prime }+(\frac{{\omega }^2}{f^2}-\frac{m^2}{f})\psi =0$$

$$\psi (r_s)=0,\psi (\infty )=0.$$

$$r^2{\tilde{\psi }}^{\prime \prime }+(r+\frac{r^2{f}^{\prime }}{f}){\tilde{\psi }}^{\prime }+(-\frac{1}{4}-\frac{r{f}^{\prime }}{2f}+\frac{{\omega }^2{r}^2}{f^2}-\frac{m^2{r}^2}{f})\tilde{\psi }=0$$

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

$$\{{\tilde{\psi }}^{\prime }=0and\tilde{\psi }{\tilde{\psi }}^{\prime \prime }⩽0\}forr=r_{peak}.$$

$$-\frac{1}{4}-\frac{r{f}^{\prime }}{2f}+\frac{{\omega }^2{r}^2}{f^2}-\frac{m^2{r}^2}{f}⩾0forr=r_{peak}.$$

$$m^2{r}^{2}f(r)⩽{\omega }^2{r}^2-\frac{rf{f}^{\prime }}{2}-\frac{1}{4}{f}^{2}forr=r_{peak}.$$

$$r⩾r_s⩾2M,$$

$$f=1-\frac{2M}{r}=\frac{1}{r}(r-2M)⩾0,$$

$$r{f}^{\prime }=r(\frac{2M}{r^2})=\frac{2M}{r}⩾0.$$

$$m^2{r}^{2}f(r)⩽{\omega }^2{r}^{2}forr=r_{peak}.$$

$$m^2{r}^2(1-\frac{2M}{r})⩽{\omega }^2{r}^{2}forr=r_{peak}.$$

$$m^2-{\omega }^2⩽\frac{2m^{2}M}{r_{peak}}.$$

$$r_{peak}⩽\frac{2m^{2}M}{m^2-{\omega }^2}.$$

$$r_s⩽r_{peak}⩽\frac{2m^{2}M}{m^2-{\omega }^2}.$$

$$2\mathcal{L}=-e^{-\chi }f{\dot{t}}^2+\frac{1}{f}{\dot{r}}^2+r^2{\dot{\varphi }}^2,$$

$$p_t=-e^{-\chi }f\dot{t}=-E=const,$$

$$p_{\varphi }=r^2\dot{\varphi }=L=const,$$

$$p_r=\frac{1}{f}\dot{r}.$$

$$2\mathcal{H}=-E\dot{t}+L\dot{\varphi }+\frac{1}{f}{\dot{r}}^2=\delta =const.$$

$${\dot{r}}^2=f[E\dot{t}-L\dot{\varphi }].$$

$$\dot{t}=\frac{e^{\chi }E}{f},\dot{\varphi }=\frac{L}{r^2}.$$

$${\dot{r}}^2=f[\frac{e^{\chi }{E}^2}{f}-\frac{L^2}{r^2}]=e^{\chi }{E}^2-\frac{L^{2}f}{r^2}.$$

$$E^2=\frac{L^{2}f}{r^2{e}^{\chi }}.$$

Full text

We study instabilities of the system composed of stationary scalar fields and asymptotically flat horizonless reflecting compact stars. In the probe limit, we obtain bounds on the scalar field frequency. Below this bound, stationary hairy stars are expected to suffer from nonlinear instabilities under massless field perturbations. In other words, we prove that stationary scalar hairy stars are unstable for scalar fields with small frequency.

I Introduction

There is accumulating evidence that fundamental scalar fields may exist in nature Franz . Theoretically, the scalar field can be either static or stationary. The famous black hole no scalar hair theorems state an interesting property that static scalar hair usually cannot exist in the asymptotically flat black hole backgrounds, see references Bekenstein -Bar and reviews Bekenstein-1 ; CAR . In contrast, it has recently been shown that rotating black holes allow the existence of stationary massive scalar field hairs Hod-1 -st11 . Moreover, with scalar fields confined in a box, static hairy black hole solutions were constructed in Dolan ; Basu ; Oscar and their dynamical formation was studied in Sanchis . We should mention that the instability properties of hairy black holes were investigated sta1 ; sta2 .

Interestingly, no static scalar hair behavior also appears in horizonless neutral compact object spacetimes. Hod firstly proved no static scalar hair theorems for asymptotically flat horizonless neutral compact stars with reflective surface boundary conditions Hod-6 . In fact, being filled with matter, it is natural to assume that the compact star surface would have reflective properties bg . When considering nonminimal couplings between scalar fields and curvature, static scalar hairs also cannot exist outside asymptotically flat horizonless neutral reflecting compact stars Hod-7 . In the asymptotically dS background, no static scalar hair theorems still hold for the horizonless neutral reflecting compact star Bhattacharjee . So the no static scalar hair behavior is a very general property in the spacetime of horizonless neutral reflecting compact stars.

On the other side, null circular geodesics may exist in the compact object spacetime P1 ; P2 ; P3 ; P4 ; P5 . For horizonless compact objects, if null circular geodesics exist, in general, there will be pairs of the null circular geodesics and the innermost null circular geodesic is stable IN1 ; IN2 . So horizonless compact objects with null circular geodesics is expected to be unstable since massless fields can pile up on the innermost stable null geodesic IS1 ; IS2 . One known way to evade the no hair theorem for horizonless reflecting compact star is to consider a charged background Hod-8 -YP-5 . The null circular geodesic was used to study instabilities of charged horizonless static scalar hairy compact stars ISPY ; ISP . Recently, Hod provided another interesting way to evade the no hair theorem for horizonless reflecting compact stars, which is considering stationary scalar field hairs SH . So it is interesting to disclose the (in)stability of such horizonless stationary scalar hairy reflecting compact stars through properties of null circular geodesics.

In the following, we study the (in)stability of horizonless stationary scalar hairy reflecting compact stars in the asymptotically flat gravity. We analytically obtain bounds for the scalar field frequency. Below this bound, stationary hairy stars are unstable. We give conclusions at the last section.

II Bounds for the frequency of stationary scalar fields

We study a gravity model of stationary scalar fields linearly coupled to horizonless reflecting compact stars. And the matter field Lagrange density is

[TABLE]

where $\Psi$ is the scalar field with mass m.

The line element of the spherically symmetric compact star reads P1

[TABLE]

Here $f$ and $\chi$ are metric functions depending on the radial coordinate r. We define the star radius as $r_{s}$. Since the spacetime is horizonless, there is $f(r)>0$ for all $r\geqslant r_{s}$. $\theta$ and $\phi$ are angular coordinates.

The equations of metrics and scalar fields are ISP ; ISPY ; SH ; dyp ; metric1 ; metric2 ; metric3

[TABLE]

[TABLE]

[TABLE]

with $\rho=-T^{t}_{t}$, $p=T^{r}_{r}$ as the matter field energy density and the radial pressure respectively.

In this work, we neglect scalar fields’ backreaction on the background. So there is Schwarzschild type solution $\chi(r)=0$ and $f(r)=1-\frac{2M}{r}$ with $M$ as the star mass. We take stationary scalar fields in the form

[TABLE]

where $\omega$ is the frequency.

And the scalar field equation is

[TABLE]

with $f=1-\frac{2M}{r}$ blc1 ; blc2 ; blc3 ; blc4 ; blc5 .

At the star surface $r_{s}$, we impose the scalar reflecting condition. At the infinity, the general asymptotic behavior is $\psi\sim A\cdot\frac{1}{r}e^{-\sqrt{m^{2}-\omega^{2}}r}+B\cdot\frac{1}{r}e^{\sqrt{m^{2}-\omega^{2}}r}$, where A and B are integral constants. Boundness of the scalar field at infinity requires $B=0$ BC . So boundary conditions are

[TABLE]

It was shown that horizonless compact objects with null circular geodesics usually have pairs of null circular geodesics and the innermost null circular geodesic is stable IN1 . As massless fields tend to pile up on the stable null circular geodesic, horizonless compact stars with null circular geodesics are unstable to massless field perturbations IS1 ; IS2 . So we can study the instability of horizonless stationary hairy configurations by examining whether there is null circular geodesic in the spacetime. In the probe limit, there is exterior null circular geodesic when the would-be null circular geodesic radius is above compact star surface. In the following analysis, we will obtain the instability condition by imposing exterior null circular geodesic radii above upper bounds of hairy star radii.

We introduce a new function $\tilde{\psi}=\sqrt{r}\psi$. According to the scalar field equation (7), the equation of the function $\tilde{\psi}$ can be expressed as

[TABLE]

with $f=1-\frac{2M}{r}$.

Boundary conditions of the function $\tilde{\psi}$ are

[TABLE]

According to (10), at least one extremum point $r=r_{peak}$ of the function $\tilde{\psi}$ exists between the surface $r_{s}$ and the infinity. At this extremum point, there are the following relations

[TABLE]

Relations (9) and (11) yield the inequality

[TABLE]

It can be transformed into

[TABLE]

The regular condition of the spacetime requires $r_{s}\geqslant 2M$, otherwise there is a horizon at $r=2M$ above the star surface. With $r_{s}\geqslant 2M$, there are relations

[TABLE]

[TABLE]

[TABLE]

According to (13-16), the following inequality holds

[TABLE]

The relation (17) yields the inequality

[TABLE]

We can transform (18) into

[TABLE]

From (19), we obtain bounds on the extremum point $r=r_{peak}$ in the form

[TABLE]

It is also the bound on hairy star radii as

[TABLE]

Following approaches in td1 ; td2 , we derive the characteristic relation for null circular geodesics. The Lagrangian describing geodesics is

[TABLE]

where a dot represents a derivative with respect to the affine parameter along the geodesic.

The Lagrangian is independent of t and $\phi$. This implies that the existence of two constants of motion labeled as E and L. According to (22), the generalized momenta can be expressed as

[TABLE]

[TABLE]

[TABLE]

The Hamiltonian of the system is $\mathcal{H}=p_{t}\dot{t}+p_{r}\dot{r}+p_{\phi}\dot{\phi}-\mathcal{L}$, which implies

[TABLE]

For timelike geodesics, there is $\delta=1$ and in this paper with null geodesics, we take $\delta=0$.

From (26), we obtain the expression

[TABLE]

With relations (23) and (24), we get expressions for $\dot{t}$ and $\dot{\phi}$ in the form

[TABLE]

Putting (28) into (27), we arrive at the relation

[TABLE]

At the null circular geodesic, there are relations $\dot{r}^{2}=0$ and $(\dot{r}^{2})^{\prime}=0$ td2 . The equation $\dot{r}^{2}=0$ yields

[TABLE]

The requirement $(\dot{r}^{2})^{\prime}=0$ yields

[TABLE]

According to (30) and (31), the characteristic equation of null circular geodesics is

[TABLE]

Without backreaction of scalar fields, the exterior spacetime of the compact star can be described by $\chi=0$ and $f=1-\frac{2M}{r}$. The exterior null circular geodesic radius is

[TABLE]

In order to obtain the instability condition, we impose that the radius of the would-be outer null circular geodesics is above the hairy star radius bound (21), which leads to the existence of the exterior null circular geodesic $r_{\gamma}=3M$ ISP ; ISPY . So the horizonless hairy star is unstable on condition that

[TABLE]

From (34), we obtain bounds for the frequency of stationary scalar fields as

[TABLE]

It is known that the static ($\omega=0$) scalar field usually cannot exist outside horizonless neutral reflecting compact stars. In contrast, stationary ($\omega\neq 0$) scalar fields can condense outside the horizonless neutral reflecting compact star. In this work, we further show that horizonless neutral stationary hairy reflecting compact stars are unstable for frequency below the bound (35).

III Conclusions

We investigated the model of stationary scalar fields linearly coupled to asymptotically flat horizonless neutral compact stars with reflecting boundary conditions. We studied instabilities of scalar hairy reflecting stars through properties of null circular geodesics. We analytically obtained bounds on the frequency of stationary scalar field as $\frac{\omega^{2}}{m^{2}}\leqslant\frac{1}{3}$, where $\omega$ and m are the frequency and mass of the scalar field respectively. Below this bound, stationary scalar hairy configurations supported by horizonless reflecting compact stars are expected to be dynamically unstable under perturbations of massless fields. That is to say the horizonless stationary scalar hairy compact reflecting star is unstable for scalar fields with small frequency.

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.