On instabilities of stationary scalar field configurations supported by reflecting compact stars
aa r X i v : . [ g r- q c ] S e p On instabilities of stationary scalar field configurations supported by reflectingcompact stars
Yan Peng ∗ School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
Abstract
We study instabilities of the system composed of stationary scalar fields and asymptotically flathorizonless reflecting compact stars. In the probe limit, we obtain bounds on the scalar field fre-quency. Below this bound, stationary hairy stars are expected to suffer from nonlinear instabilitiesunder massless field perturbations. In other words, we prove that stationary scalar hairy stars areunstable for scalar fields with small frequency.
PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z ∗ [email protected] I. INTRODUCTION
There is accumulating evidence that fundamental scalar fields may exist in nature [1]. Theoretically, thescalar field can be either static or stationary. The famous black hole no scalar hair theorems state an interestingproperty that static scalar hair usually cannot exist in the asymptotically flat black hole backgrounds, seereferences [2]-[10] and reviews [11, 12]. In contrast, it has recently been shown that rotating black holes allowthe existence of stationary massive scalar field hairs [13]-[22]. Moreover, with scalar fields confined in a box,static hairy black hole solutions were constructed in [23–25] and their dynamical formation was studied in[26]. We should mention that the instability properties of hairy black holes were investigated [27, 28].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 withreflective surface boundary conditions [29]. In fact, being filled with matter, it is natural to assume that thecompact star surface would have reflective properties [30]. When considering nonminimal couplings betweenscalar fields and curvature, static scalar hairs also cannot exist outside asymptotically flat horizonless neutralreflecting compact stars [31]. In the asymptotically dS background, no static scalar hair theorems still holdfor the horizonless neutral reflecting compact star [32]. So the no static scalar hair behavior is a very generalproperty in the spacetime of horizonless neutral reflecting compact stars.On the other side, null circular geodesics may exist in the compact object spacetime [33–37]. For horizonlesscompact objects, if null circular geodesics exist, in general, there will be pairs of the null circular geodesicsand the innermost null circular geodesic is stable [38, 39]. So horizonless compact objects with null circulargeodesics is expected to be unstable since massless fields can pile up on the innermost stable null geodesic[40, 41]. One known way to evade the no hair theorem for horizonless reflecting compact star is to consider acharged background [42]-[52]. The null circular geodesic was used to study instabilities of charged horizonlessstatic scalar hairy compact stars [53, 54]. Recently, Hod provided another interesting way to evade the nohair theorem for horizonless reflecting compact stars, which is considering stationary scalar field hairs [55].So it is interesting to disclose the (in)stability of such horizonless stationary scalar hairy reflecting compactstars through properties of null circular geodesics.In the following, we study the (in)stability of horizonless stationary scalar hairy reflecting compact starsin the asymptotically flat gravity. We analytically obtain bounds for the scalar field frequency. Below thisbound, 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 L = −|∇ µ Ψ | − m Ψ , (1)where Ψ is the scalar field with mass m.The line element of the spherically symmetric compact star reads [33] ds = − f e − χ dt + dr f + r ( dθ + sin θdφ ) . (2)Here f and χ are metric functions depending on the radial coordinate r. We define the star radius as r s . Sincethe spacetime is horizonless, there is f ( r ) > r > r s . θ and φ are angular coordinates.The equations of metrics and scalar fields are [53–59] f ′ = − πrρ + (1 − f ) /r, (3) χ ′ = − πr ( ρ + p ) /f, (4)( ∇ ν ∇ ν − m )Ψ = 0 (5)with ρ = − T tt , p = T rr 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 typesolution χ ( r ) = 0 and f ( r ) = 1 − Mr with M as the star mass. We take stationary scalar fields in the formΨ( t, r ) = e − iωt ψ ( r ) , (6)where ω is the frequency.And the scalar field equation is ψ ′′ + ( 2 r + f ′ f ) ψ ′ + ( ω f − m f ) ψ = 0 (7)with f = 1 − Mr [60–64].At the star surface r s , we impose the scalar reflecting condition. At the infinity, the general asymptoticbehavior is ψ ∼ A · r e −√ m − ω r + B · r e √ m − ω r , where A and B are integral constants. Boundness of thescalar field at infinity requires B = 0 [65]. So boundary conditions are ψ ( r s ) = 0 , ψ ( ∞ ) = 0 . (8)It was shown that horizonless compact objects with null circular geodesics usually have pairs of null circulargeodesics and the innermost null circular geodesic is stable [38]. As massless fields tend to pile up on thestable null circular geodesic, horizonless compact stars with null circular geodesics are unstable to masslessfield perturbations [40, 41]. So we can study the instability of horizonless stationary hairy configurationsby examining whether there is null circular geodesic in the spacetime. In the probe limit, there is exteriornull circular geodesic when the would-be null circular geodesic radius is above compact star surface. In thefollowing analysis, we will obtain the instability condition by imposing exterior null circular geodesic radiiabove upper bounds of hairy star radii.We introduce a new function ˜ ψ = √ rψ . According to the scalar field equation (7), the equation of thefunction ˜ ψ can be expressed as r ˜ ψ ′′ + ( r + r f ′ f ) ˜ ψ ′ + ( − − rf ′ f + ω r f − m r f ) ˜ ψ = 0 (9)with f = 1 − Mr .Boundary conditions of the function ˜ ψ are˜ ψ ( r s ) = 0 , ˜ ψ ( ∞ ) = 0 . (10)According to (10), at least one extremum point r = r peak of the function ˜ ψ exists between the surface r s and the infinity. At this extremum point, there are the following relations { ˜ ψ ′ = 0 and ˜ ψ ˜ ψ ′′ } f or r = r peak . (11)Relations (9) and (11) yield the inequality − − rf ′ f + ω r f − m r f > f or r = r peak . (12)It can be transformed into m r f ( r ) ω r − rf f ′ − f f or r = r peak . (13)The regular condition of the spacetime requires r s > M , otherwise there is a horizon at r = 2 M above thestar surface. With r s > M , there are relations r > r s > M, (14) f = 1 − Mr = 1 r ( r − M ) > , (15) rf ′ = r ( 2 Mr ) = 2 Mr > . (16)According to (13-16), the following inequality holds m r f ( r ) ω r f or r = r peak . (17)The relation (17) yields the inequality m r (1 − Mr ) ω r f or r = r peak . (18)We can transform (18) into m − ω m Mr peak . (19)From (19), we obtain bounds on the extremum point r = r peak in the form r peak m Mm − ω . (20)It is also the bound on hairy star radii as r s r peak m Mm − ω . (21)Following approaches in [66, 67], we derive the characteristic relation for null circular geodesics. TheLagrangian describing geodesics is 2 L = − e − χ f ˙ t + 1 f ˙ r + r ˙ φ , (22)where a dot represents a derivative with respect to the affine parameter along the geodesic.The Lagrangian is independent of t and φ . This implies that the existence of two constants of motionlabeled as E and L. According to (22), the generalized momenta can be expressed as p t = − e − χ f ˙ t = − E = const, (23) p φ = r ˙ φ = L = const, (24) p r = 1 f ˙ r . (25)The Hamiltonian of the system is H = p t ˙ t + p r ˙ r + p φ ˙ φ − L , which implies2 H = − E ˙ t + L ˙ φ + 1 f ˙ r = δ = const. (26)For timelike geodesics, there is δ = 1 and in this paper with null geodesics, we take δ = 0.From (26), we obtain the expression ˙ r = f [ E ˙ t − L ˙ φ ] . (27)With relations (23) and (24), we get expressions for ˙ t and ˙ φ in the form˙ t = e χ Ef , ˙ φ = Lr . (28)Putting (28) into (27), we arrive at the relation˙ r = f [ e χ E f − L r ] = e χ E − L fr . (29)At the null circular geodesic, there are relations ˙ r = 0 and ( ˙ r ) ′ = 0 [67]. The equation ˙ r = 0 yields E = L fr e χ . (30)The requirement ( ˙ r ) ′ = 0 yields( ˙ r ) ′ = χ ′ e χ E + 2 L fr − L f ′ r = 0 . (31)According to (30) and (31), the characteristic equation of null circular geodesics is2 f ( r γ ) + r γ [ χ ′ ( r γ ) f ( r γ ) − f ′ ( r γ )] = 0 . (32)Without backreaction of scalar fields, the exterior spacetime of the compact star can be described by χ = 0and f = 1 − Mr . The exterior null circular geodesic radius is r γ = 3 M. (33)In order to obtain the instability condition, we impose that the radius of the would-be outer null circulargeodesics is above the hairy star radius bound (21), which leads to the existence of the exterior null circulargeodesic r γ = 3 M [53, 54]. So the horizonless hairy star is unstable on condition that r γ = 3 M > m Mm − ω . (34)From (34), we obtain bounds for the frequency of stationary scalar fields as ω m . (35)It is known that the static ( ω = 0) scalar field usually cannot exist outside horizonless neutral reflectingcompact stars. In contrast, stationary ( ω = 0) scalar fields can condense outside the horizonless neutralreflecting compact star. In this work, we further show that horizonless neutral stationary hairy reflectingcompact stars are unstable for frequency below the bound (35). III. CONCLUSIONS
We investigated the model of stationary scalar fields linearly coupled to asymptotically flat horizonlessneutral compact stars with reflecting boundary conditions. We studied instabilities of scalar hairy reflectingstars through properties of null circular geodesics. We analytically obtained bounds on the frequency ofstationary scalar field as ω m , where ω 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 areexpected to be dynamically unstable under perturbations of massless fields. That is to say the horizonlessstationary scalar hairy compact reflecting star is unstable for scalar fields with small frequency. Acknowledgments
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