Radial stability analysis of the continuous pressure gravastar
aa r X i v : . [ g r- q c ] S e p Radial stability analysis of the continuous pressure gravastar
Dubravko Horvat ∗ , Saˇsa Iliji´c † , Anja Marunovi´c ‡ Department of Physics, Faculty of Electrical Engineering and Computing,University of Zagreb, Unska 3, HR-10 000 Zagreb, Croatia
Radial stability of the continuous pressure gravastar is studied using the conventionalChandrasekhar method. The equation of state for the static gravastar solutions is derived andEinstein equations for small perturbations around the equilibrium are solved as an eigenvalueproblem for radial pulsations. Within the model there exist a set of parameters leading to astable fundamental mode, thus proving radial stability of the continuous pressure gravastar.It is also shown that the central energy density possesses an extremum in ρ c ( R ) curve whichrepresents a splitting point between stable and unstable gravastar configurations. As suchthe ρ c ( R ) curve for the gravastar mimics the famous M ( R ) curve for a polytrope. Togetherwith the former axial stability calculations this work completes the stability problem of thecontinuous pressure gravastar. I. INTRODUCTION
Gravitational collapse as the stellar nuclear fuel is consumed could lead to black holes - objectswhich are accepted by scientific community but their undesired and even paradoxical features(singularities, horizon) have motivated research in a direction of finding massive objects (stars)without singularities and without horizon. One of these alternatives is a gravastar.Since the seminal work of Mazur and Mottola [1] the concept of the gra vitational va cuum star –the gravastar – as an alternative to a black hole has attracted a plethora of interest. In this versionof the gravastar a multilayered structure has been introduced: from the repulsive de Sitter core(where a negative pressure helps balance the collapsing matter) one crosses multiple layers (shells)and without encountering a horizon one eventually reaches the (pressureless) exterior Schwarzschildspacetime. Later some simplifications [2, 3] and modifications [4] have been introduced in theoriginal (multi)layer - onion-like picture.An important step was done when it was shown that due to anisotropy of matter comprising thegravastar [5] one can eliminate layer(s) and, by a continuous stress-energy tensor, the transitionfrom the interior de Sitter spacetime segment to the exterior Schwarzschild spacetime is possible ∗ [email protected] † [email protected] ‡ [email protected] [6] (see also [7]). The pressure anisotropy in the spherically symmetric geometry was perhapsfirst introduced by G. Lemaˆıtre [8] and suggested by Einstein (as quoted in [8]). The vanishingradial pressure with transversal pressure only was shown to be enough to support a stable object.Further development [9, 10] has brought different refinements to the original anisotropy notion.The pressure anisotropy, which is shown to be a necessary condition for the existence of a gravastar[5] is met also in boson star models [11] and wormholes [12]. The anisotropy (defined as a differencebetween the transversal and radial pressure) vanishes at the center ( r = 0) of the star as well as atits boundary ( r = R ). The gravastar has been confronted with its rivals - black holes [13, 14] andwormholes [12, 15, 16], and investigated with respect to energy conditions (violations) [17] and itscharged properties [18, 19]. An interesting question has been posed several times: is it possible todistinguish the gravastar from a black hole [20–22]? In Ref. [20] it was shown that gravitationalradiation could be used to tell a gravastar from a black hole. However, the definite answer to thisquestion has not been given at satisfactory level and the gravastar research is still a dynamic fieldwith recent papers like [23–26] etc .Almost every research mentioned above to some extent addresses the problem of the gravastarstability, since the stability problem is crucial for any object or situation to be considered asphysically viable. In Ref. [1] it was first shown that such an object is thermodynamically stable,while axial stability of thin-shells gravastars was tested in [3, 4]. Stability within the thin shellapproach based on the Darmois-Israel formalism was recently reviewed in [27]. In [20] stabilityanalysis of the thin shell gravastar problem is closely related to an attempt to distinguish thegravastar from a black hole by analysis of quasi-normal modes produced by axial perturbations.Problem of stability of a rotating the thin shell gravastar was addressed in [28]. Stability in the(multi)layer version of the gravastar was also considered in [14, 26, 30, 31].The axial stability of the continuous pressure gravastar was shown to be valid in [6]. Thisanalysis was based on the Ref. [32] where stability of objects with de Sitter centers was investigated.In this paper we analyze the radial stability of the continuous pressure gravastars [6] followingthe conventional Chandrasekhar method. Originally Chandrasekhar developed the method fortesting the radial stability of the isotropic spheres [33] in terms of the radial pulsations. In Ref.[36] Chandrasekhar’s method was generalized to anisotropic spheres. Stability of anisotropic starswas investigated before in [37, 38] and radial stability analysis for anisotropic stars using the quasi-local equation of state was given in [39]. The standard mathematical procedure is here applied toan object with a peculiar behavior of pressures (see below) and although the mathematical rigorwas never abandoned, the analysis due to the character of the object could be considered as a toymodel analysis of radial stability.This paper is organized as follows. In the next section the linearization of the Einstein equationsis given. Static solutions are described, an equation of state is derived and the pulsation equationis obtained. In Sec. III the eigenvalue problem for the radial displacement is presented. Resultsand discussion are given in the last section.Unless stated explicitly we shall work in units where G N = 1 = c . II. LINEARIZATION OF THE EINSTEIN EQUATION
In this paper the response of the continuous pressure gravastar model to small radial pertur-bations is considered. Assuming that the pulsating object retains its spherical symmetry, one canintroduce the Schwarzschild coordinates: ds = e ν ( r,t ) dt − e λ ( r,t ) dr − r dθ − r sin θdφ , (1)where λ and ν are, in this dynamical setting, time-dependent metric functions.The standard anisotropic energy-momentum tensor appropriate to describe continuous pressuregravastars is: T νµ = ( ρ + p r ) u µ u ν − g νµ p r + l µ l ν ( p t − p r ) + k µ k ν ( p t − p r ) , (2)where u µ is the fluid 4-velocity, u µ = dx µ /ds , l µ and k µ are the unit 4-vectors in the θ and φ directions, respectively, l µ = − r δ θµ , k µ = − r sin θ δ φµ .The velocity of the fluid element in the radial direction ˙ ξ is defined by:˙ ξ ≡ drdt = u r u t , (3)where ξ is the radial displacement of the fluid element, r → r + ξ ( r, t ). The components of the4-velocity are obtained by employing u µ u µ = 1 and Eq. (3): u µ = ( e − ν/ , ˙ ξe − ν/ , , . (4)The non-zero components of the energy-momentum tensor (2) linear in ˙ ξ are: T tt = ρ, T rr = − p r , T θθ = T φφ = − p t , T rt = ˙ ξ ( ρ + p r ) , T tr = e λ − ν ˙ ξ ( ρ + p r ) . (5)The components of the Einstein tensor for the metric (1) are: G tt = e − λ (cid:18) λ ′ r − r (cid:19) + 1 r , (6) G rr = − e − λ (cid:18) ν ′ r + 1 r (cid:19) + 1 r , (7) G rt = − e − λ ˙ λr , (8) G θθ = G φφ = − e − λ (cid:18) − ν ′ λ ′ − λ ′ r + ν ′ r + ν ′ ν ′′ (cid:19) + 12 e − ν ¨ λ + ˙ λ − ˙ λ ˙ ν ! . (9)Following the standard Chandrasekhar method, all matter and metric functions should only slightlydeviate from their equilibrium solutions, λ ( r, t ) = λ ( r ) + δλ ( r, t ) , ν ( r, t ) = ν ( r ) + δν ( r, t ) , (10) ρ ( r, t ) = ρ ( r ) + δρ ( r, t ) , p r ( r, t ) = p r ( r ) + δp r ( r, t ) , p t ( r, t ) = p t ( r ) + δp t ( r, t ) . (11)The subscript 0 denotes the equilibrium functions and δf ( r, t ) are the so-called Eulerian pertur-bations, where f ∈ { λ, ν, ρ, p r , p t , } . The Eulerian perturbations measure a local departure fromequilibrium in contrast to the Lagrangian perturbations, denoted as df ( r, t ), which measure adeparture from equilibrium in the co-moving system (fluid rest frame). The Lagrangian perturba-tions in the linear approximation play a role of a total differential and are linked to the Eulerianperturbations via the equation (see e.g. Ref. [40]): df ( r, t ) = δf ( r, t ) + f ′ ( r ) ξ. (12)A linearization of the Einstein equations G νµ = 8 πT νµ leads to the two sets of equations: one for theequilibrium (static) functions and the other for the perturbed functions. The equilibrium functionsobey the following set of equations:8 πρ = e − λ (cid:18) λ ′ r − r (cid:19) + 1 r , (13)8 πp r = e − λ (cid:18) ν ′ r + 1 r (cid:19) − r , (14)8 πp t = 12 e − λ (cid:18) − ν ′ λ ′ − λ ′ r + ν ′ r + ν ′ ν ′′ (cid:19) . (15)In practice, one usually combines these three equations into the Tolman-Oppenheimer-Volkoff(TOV) equation: p ′ r = −
12 ( ρ + p r ) ν ′ + 2 r Π , (16)where Π denotes the anisotropic term Π = p t − p r . The other set of equations emerging from thelinearization of the above Einstein equations yield the set of equations for the perturbed functions: (cid:16) re − λ δλ (cid:17) ′ = 8 πr δρ, (17) δν ′ = (cid:18) ν ′ + 1 r (cid:19) δλ + 8 πre λ δp r , (18)˙ δλ e − λ r = − π ˙ ξ ( ρ + p r ) , (19) e λ − ν ( ρ + p r ) ¨ ξ + 12 ( ρ + p r ) δν ′ + 12 ( δρ + δp r ) ν ′ + δp ′ r − r δ Π = 0 . (20)Equation (20) is known as the pulsation equation [36] and it serves to probe the radial stabilityof the system of interest. It is actually the TOV equation for the perturbed functions which isobtained – analogously as the non-perturbed TOV – by combining Einstein equations for perturbedfunctions.In order to solve the pulsation equation (20) for the gravastar all perturbed functions should beexpressed in terms of the radial displacement ξ (and its derivatives) and the equilibrium functions.In performing this, one first integrates Eq. (19) in time, yielding: δλ = − πre λ ξ ( ρ + p r ) . (21)Using this expression for δρ in Eq. (17) one obtains: δρ = − r (cid:2) r ( ρ + p r ) ξ (cid:3) ′ . (22)After inserting δλ in Eq. (18) a dependence on δp r remains, which should be expressed in terms ofthe displacement function (and its derivatives) and the equilibrium functions. To accomplish this,one ought to explore the system at hand in more detail.One of the possibilities, as suggested firstly by Chandrasekhar for isotropic structures [33]and more recently by Dev and Gleiser for anisotropic objects [36], is to make use of the baryondensity conservation to express the radial pressure perturbation in terms of the displacement func-tion and the static solutions. In this approach the adiabatic index appears as a free parameter.Chandrasekhar used this method to establish limiting values of the adiabatic index leading to an(un)stable isotropic object of a constant energy density. He showed that there were no stablestars of this kind if the adiabatic index was less than 4 / κM/R ( κ is a constant of order unitydepending on the structure of the star, M and R are the star’s mass and radius). In Ref. [36]the Chandrasekhar method was extended to various anisotropic star models and showed that thelimiting value of the adiabatic index is shifted to lower values, i.e. anisotropic stars can approachthe stability region with smaller adiabatic index than in the Chandrasekhar’s case.In this paper our primary concern is to probe the radial stability of one particular anisotropicobject – the gravastar. Due to the peculiar character of the gravastar (especially its radial pressure– see below) one cannot expect the adiabatic index to be constant along the whole object. In factthe adiabatic index is a function of the energy density and pressure(s). This is the main reason whyin this paper stability will not be tested by fixing the appropriate values of the adiabatic index thatguarantee stability. The required information will rather be extracted from a given static solutionby constructing the equation of state. A. Static solution
The procedure discussed so far is applicable to all spherically symmetric structures. To ap-ply it to gravastar configurations one has to recall the basic characteristics of gravastars in thecontinuous pressure picture [6]. The energy density ρ ( r ) is positive and monotonically decreasesfrom the center to the surface. Gravastars have a de Sitter-like interior, p r (0) = − ρ (0), and aSchwarzschild-like exterior. Furthermore, the atmosphere of the gravastar is defined as an outerregion, near to the surface, where ”normal” physics is valid [5], i.e. where both the energy-densityand the radial pressure are positive and monotonically decreasing functions of the radius. In thegravastar’s atmosphere the sound velocity v s , with v s = dp r dρ , (23)is real ( v s >
0) and subluminal ( v s < v s <
0. This is the main reason why, in probing the radial stability, we shall beprimarily concerned with the physical processes occurring in the gravastar’s atmosphere.To construct a static gravastar, the energy density profile and the anisotropic term are adoptedfrom the previous work [6, 18]: ρ ( r ) = ρ c (1 − ( r/R ) n ) , (24)Π ( r ) = βρ ( r ) m µ ( r ) . (25) Ρ (cid:144) Ρ c p r0 (cid:144) Ρ c p t0 (cid:144) Ρ c Μ max » (cid:144) R - - r0 d Ρ > r0 d Ρ < r (cid:144) FIG. 1: The energy density ρ /ρ c , radial pressure p r /ρ c , tangential pressure p t /ρ c and compactness µ as a function of radius r/R for { R, n, m } = { , , } . Three different values of the central energy density ρ c = { . , . , . } and their anisotropy strengths β = { . , . , . } correspond to thelower, middle and upper curve, respectively. r denotes the radius at which the sound velocity (23) vanishes(for the central curve). Here n , m are (free) parameters and ρ c = ρ (0) is the central energy density. β is theanisotropy strength measure and R is the radius of the gravastar for which p r ( R ) = 0. µ ( r )is the compactness function defined by µ ( r ) = 2 m ( r ) /r , where m ( r ) is the mass function m ( r ) = 4 π R ρ ( r ) r dr . The radial pressure p r is a solution of the TOV (16) and the tan-gential pressure is readily obtained from the anisotropy and the radial pressure by employing theidentity p t = p r + Π . The particular form of the anisotropic term is dictated by the behavior ofpressures at de Sitter core, since at r = 0 the anisotropy should vanish as seen from (16). Also,the above form of the anisotropy term ensures that the radial pressure vanishes at r = R . Thetransversal pressure vanishes as well although it is not necessary to be the case (see Ref. [6] forthe gravastar model with non vanishing transverse pressure.). The anisotropy strength measure iscontrolled as well by the energy conditions which have to be met.One such solution for fixed ( R, n, m ) = (1 , ,
3) is shown in Fig. 1 for three different valuesof the central energy density ρ c corresponding to three different values of the anisotropy strength β . Since the radius R is fixed there is an interplay between the central energy density ρ c andanisotropy strength β – a higher central energy density ρ c requires a smaller anisotropy strength β .We shall elaborate on this particular choice of parameters in Section IV, where the radial stabilityof these three gravastar configurations will be tested.In the inset of Fig. 1 the radial pressure close to the surface is extracted in order to show importantfeatures of the gravastar’s atmosphere. At the radius r the sound velocity of the fluid vanishes( dp r /dρ | r = r = 0) and hence r serves as a division point of the propagating (or physicallyreasonable) ( r > r , v s >
0) and non-propagating regions ( r < r , v s <
0) when probing radialpulsations of the gravastar.The dominant energy condition (DEC), i.e. p r , p t ≤ ρ , is obeyed by both radial and tangentialpressure throughout the gravastar. The compactness function has also been shown in Fig. 1. B. Equation of state
In this subsection we note that the equation of state (EoS) appropriate to describe thegravastar (inferred from the input functions (24) and (25)) is actually a functional of the energydensity (only), parameterized by the anisotropy strength β . Next this result is used to computethe Eulerian perturbation of the radial pressure δp r from the EoS, by perturbing the energydensity only. Ultimately this completes the task to express all perturbed functions in terms of thedisplacement (and its derivatives) and the static solutions.Generally, for isotropic structures, before solving the TOV, one assumes that the pressure p andthe energy density ρ are functions of the specific entropy s and the baryon density n . If a systemis described by the one-fluid model, then in static and dynamic settings it exhibits isentropicbehavior (constant s ), in which case one can set s = 0. Thus it is possible to eliminate the baryondensity n and express the pressure in terms of the energy density only, leading to a barotropicequation of state p = p ( ρ ). It is a rather simple task now to perturb this EoS and express theperturbed pressure in terms of the perturbed energy density.For anisotropic objects the EoS is highly dependent on the anisotropic term model (see e.g. the TOV (16)). The particular choice of the anisotropic term used here (25) is a functional (ora quasi-local variable ) of the energy density. This means that for a fixed anisotropy strength β there is a two parameter family of values { ρ c , R } belonging to the same EoS (see Fig. 2). Asa consequence, one can obtain perturbed (radial) pressure by perturbing the energy density only,and keeping the anisotropy strength β fixed.To illustrate this in more detail let us introduce an analytic form of the EoS which, to a good By the quasi-local variable we mean a function which is an integral in space of some local function – for example, themass function m ( r ) is a quasi-local variable of the energy density (which is a local function) as it is the volumeintegral of the energy density (the same holds for the compactness function). For a discussion of quasi-localvariables and quasi-local EoS see e.g. Refs. [34, 35] and Ref. [39]. Ρ (cid:144) Ρ c - - - - - p r0 (cid:144) Ρ c Ρ (cid:144) Ρ c - p r0 (cid:144) Ρ c FIG. 2: The radial pressure p r /ρ c is plotted against the energy density ρ /ρ c (EoS) for { R, n, m } = { , , } .Three different values of the central energy density ρ c = { . , . , . } and their anisotropy strengths β = { . , . , . } correspond to the lower, middle and upper curve, respectively. approximation, describes the gravastar configuration defined by (24) and (25): p r [ ρ ] = − ρ (cid:18) ρ c − α µ [ ρ ] (cid:19) . (26)Here α is closely related to the anisotropy strength β , µ [ ρ ] is the compactness function which isa functional of the energy density. Now it is clear that for a fixed α the (radial) pressure is fullydetermined by the energy density.Hence, following the reasoning outlined above and making use of Eq. (12), in the linear approx-imation the Eulerian perturbation for the radial pressure is: δp r = − p ′ r ξ + dp r [ ρ ] dρ ( δρ + ρ ′ ξ ) . (27)Here dp ro [ ρ ] /dρ denotes functional derivative of the radial pressure with respect to the energydensity. This is equal to dp ro /drdρ /dr as both the radial pressure and the energy density are functionsof radius r only.Similarly, the Eulerian perturbation of the anisotropy δ Π assumes the form: δ Π = − Π ′ ξ + d Π [ ρ ] dρ ( δρ + ρ ′ ξ ) . (28)With the above two expressions the pulsation equation (20) is fully determined. However, beforeproceeding to solve the pulsation equation it is useful to rewrite Eq. (27) in a slightly different form It is worth noting that the analytic form of the EoS (26) is not restricted to the chosen energy density (24). Forexample, it is also appropriate to describe a gravastar with the energy density of the form ρ ( r ) = ρ c e − η r . δρ = − ρ ′ ξ − ( ρ + p r ) e ν / r (cid:16) r e − ν / ξ (cid:17) ′ − r Π ξ. (29)Inserting this result in Eq. (27) the radial pressure perturbation becomes δp r = − p ′ r ξ − ( ρ + p r ) dp r [ ρ ] dρ e ν / r (cid:16) r e − ν / ξ (cid:17) ′ − r Π dp r [ ρ ] dρ ξ. (30)If one now identifies the adiabatic ”index” as γ = ρ + p r p r dp r [ ρ ] dρ , (31)the result derived in Eq. (30) reduces to that of Dev and Gleiser [36], Eq. (86). However, theexpressions for γ differ. Moreover, if one turns off anisotropy (Π = 0) Chandrasekhar’s result isobtained. III. THE PULSATION EQUATION AS AN EIGENVALUE PROBLEM
As in the Chandrasekhar method all matter and metric functions exhibit oscillatory behaviorin time, f ( r, t ) = e iωt f ( r ). Hence the pulsation equation assumes the form: P ξ ′′ + P ξ ′ + P ξ = − ω P ω ξ, (32)where P , P , P and P ω are polynomial functions of r , depending on the static solutions only (seeFig. 3). Eq. (32) represents an eigenvalue equation for the radial displacement ξ (with ω being aneigenvalue). Solutions of this differential equation are obtained by specifying boundary conditionsin the center and at the surface of the gravastar: ξ = 0 at r = 0 , (33)∆ p r = 0 at r = R. (34)The boundary condition in the center demands that there is no displacement of the fluid in thecenter of the gravastar. The boundary condition at the surface follows from the requirementthat the Lagrangian radial pressure perturbation has to vanish at the surface [40–42] . In themodel presented here where ∆ p r = ( dp r [ ρ ] /dρ )∆ ρ , the sound velocity vanishes at the surface,1 P P Ω » P » P (cid:144) R - - - FIG. 3: The polynomial functions P , P / , P /
100 and P ω from the pulsation equation (20) for { R, n, m } = { , , } and { ρ c , β } = { . , . } . dp r /dρ | r = R = 0. This means that, apart from being finite, there are no further restrictions on∆ ρ ( R ). This also implies that it is sufficient to demand that ξ ( R ) and ξ ′ ( R ) are bounded in orderto satisfy the boundary condition at the surface [41]. The choice ξ ′ ( R ) = 0 , (35)enables one to compare the results in the gravastar’s atmosphere with the radial oscillations of thepolytropes. This can be relevant as the EoS of the gravastar’s atmosphere close to the surface canbe approximated by the polytropic EoS p r ∝ ρ /n p , where n p is a polytropic index [6, 18].In order to study radial stability of the system described by Eq. (20) subject to the boundaryconditions (33) and (34), it is plausible to recast the pulsation equation into the standard Sturm-Liouville form (see e.g. Ref. [40]): ( P ξ ′ ) ′ + Q ξ = − ω W ξ, (36)where P = e R P / P dr and Q = P P P, W = P ω P P . (37)The leading coefficient in the pulsation equation P has three zeros - two at the ends { , R } andone in the interior region r ( dp r /dρ | r = r = 0), hence P / P has three singular points (see Fig.3), though all three are regular singular points or Fuchsian singularities [43] . A singular point r ∗ is regular (or Fuchsian) if the function P / P has a pole of at most first order, and the function P / P has a pole of at most second order at the singular point r = r ∗ . P the integral R P / P dr should be calculated, and since the interior singu-larity arises at r which is a division point between propagating and non-propagating domains, itis reasonable to divide the whole interval I = (0 , R ) in two parts: I = (0 , r ) and I = ( r , R ).In performing the integration numerically infinitesimally small regions around all three singularpoints { , r , R } are excluded, so that both integrals are rendered convergent and finite. As aconsequence, the leading coefficient in the Sturm-Liouville equation P is a positive function onthe (whole) interval I , whilst the weight function W is negative on the interval I and positiveon the interval I . As elucidated in the previous section, the interesting region is the gravastar’satmosphere, i.e. the second interval, I . In this region the standard Sturm-Liouville eigenvalueproblem formalism (see e.g. [41]) is applied, since P > W >
0. Therefore if ω is positive, ω itself is real and the solution is oscillatory. If on the other hand ω is negative, ω is imaginaryand the solution is exponentially growing or decaying in time, thus signalizing instabilities. Thenumber of nodes of the eigenvector ξ for a given eigenvalue ω is closely related to the stabilitycriteria. To be more precise, if for ω = 0 eigenvector ξ has no nodes, then all higher frequencyradial modes are stable. Otherwise, if for ω = 0 eigenvector ξ exhibits nodes, then all radial modesare unstable. Furthermore, if the system is stable, then the following relations hold ω < ω < · · · < ω n < . . . , (38)where n equals the number of nodes. IV. RESULTS AND DISCUSSION
In testing stability of certain configurations in general, it seems natural that one attemptsto find critical values of the parameters for which the system is marginally stable. Marginalstability means here that there exists a set of parameters for which the system exhibits the stable fundamental mode ( n = 0) for ω = 0. Then for the given set of parameters all higher frequencymodes are radially stable. For example, in the case of neutron stars (described by the polytropicEoS), there exists a critical value of the central energy density for which the stellar mass M as afunction of radius R is extremal. For such a critical value of the central energy density the starexhibits stable fundamental mode with ω = 0, which implies that all higher frequency modeswith the given central energy density are then radially stable. Furthermore, at the account of the M ( R ) curve one can then read off which configuration of the EoS will produce a stable star and3 Β cr Β < Β cr Β > Β cr STABLE UNSTABLE0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 R0.180.190.200.210.22 Ρ c FIG. 4: The central energy density ρ c plotted against the radius R . For { R, n, m } = { , , } the anisotropy-parameter β = { . , . , . } is constant on each curve and fixed by choosing the central energydensity to be ρ c = { . , . , . } from the lower to the upper curve, respectively. The minimum ofeach curve represents marginally stable configurations. which will not.The continuous pressure gravastar model described here displays a quite similar behavior. Foreach EoS (fixed β ) the extremum of the ρ c ( R ) curve represents critical values of the parameters { ρ c , R } for which the system exhibits a stable fundamental mode, ω = 0 (see Fig. 4). Thenfor such critical set of parameters all higher frequency modes are radially stable. Moreover, forsmaller radii the system exhibits stability, whereas for larger radii (than the critical one) it revealsinstability (see Fig. 4). In this sense the ρ c ( R ) curve for the gravastar mimics the well known M ( R )curve for a polytrope. To prove these statements, the behavior of the displacement function ξ isshown in Fig. 5 for three different cases. The radius R is, for simplicity, fixed (for all three cases)to be the critical radius of the central curve in Fig. 4. According to Fig. 4 one then expects thatthe radial displacements ξ derived from the lower, central and upper curve in Fig. 4 will generatestable, marginally stable and unstable fundamental mode respectively. This is exactly what isshown in Fig. 5. The central (solid) curve in Fig. 5 represents the marginally stable fundamentalmode – an eigenvector ξ is obtained for ω = 0. The upper (short-dashed) curve clearly showsstability of all radial modes as for ω = 0 there are no nodes, while the lower (long-dashed) curvereveals instabilities of all radial modes as there is a node in the fundamental mode. The lower,4 Ω = R = Ρ , Β cr Ρ > Ρ , Β < Β cr Ρ < Ρ , Β > Β cr (cid:144) R0246 Ξ FIG. 5: The displacement function ξ ( r ) for { R, n, m } = { , , } and ω = 0. Three different val-ues of the central energy density ρ c = { . , . , . } and their respective anisotropy strengths β = { . , . , . } correspond to the lower (unstable), middle (marginally stable) and upper(stable) curve respectively. middle and upper curves in Fig. 5 correspond to the upper, middle and lower curves in Fig. 1and Fig. 2, respectively. Here again one is able to relate this result to that of Ref. [36]: fromFig. 5, according to the values of the anisotropy strengths β , one can conclude that the anisotropyenhances stability.Albeit from the viewpoint of radial pulsations, the gravastar’s inner region does not seem to bephysically attractive as the sound velocity is imaginary there, it is important to add a couple ofcomments on the radial displacement’s behavior in that region. Pulsations are strongly attenuatedin the gravastar’s interior (see Fig. 5). This holds for all ω >
0. Therefore the radial pulsations ofthe gravastar as a whole can be seen as occurring prevalently in the gravastar’s atmosphere whereasentering the interior region they are exponentially (but smoothly) attenuated. This is actually whatone would intuitively expect from the repulsive gravitation caused by the de Sitter-like interior. In this paper the focus was set on one very specific star model - the gravastar. Thereforestandard stability analysis which has been applied here in every detail could be considered as a toymodel of the radial stability analysis. However it could be extended to a broader class of anisotropic A good example of such a space is an inflationary universe. The electric and magnetic fields of free photons in suchan inflationary (quasi-de Sitter) space get (exponentially) damped as ∝ /a , while the physical wavelength getsstretched as ∝ a . Here a denotes the scale factor of the Universe, which during inflation grows nearly exponentiallyin time. Acknowledgements
The authors would like to thank Andrew DeBenedictis and Tomislav Prokopec for useful com-ments on the manuscript. This work is partially supported by the Croatian Ministry of Scienceunder the project No. 036-0982930-3144.
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