Born-Infeld Phantom Gravastars

arXiv:astro-ph/0503427v3 8 Feb 2006

Neven Bili´ c†§, Gary B Tupper‡ and Raoul D Viollier‡ † Rudjer Boˇskovi´c Institute, P.O. Box 180, 10002 Zagreb, Croatia ‡ Institute of Theoretical Physics and Astrophysics, Department of Physics, University of Cape Town, Private Bag, Rondebosch 7701, South Africa E-mail: [email protected], [email protected], [email protected] Abstract. We construct new gravitational vacuum star solutions with a Born-Infeld phantom replacing the de Sitter interior. The model allows for a wide range of masses and radii required by phenomenology, and can be motivated from low-energy string theory.

§ Author to whom correspondence should be addressed

Born-Infeld Phantom Gravastars

2

1. Introduction From its inception, the Schwarzschild solution of Einstein’s equations has been the subject of controversy over the possibility of compressing matter within the gravitational radius 2GM. Given the accumulated evidence for supermassive compact objects ranging from a few 106 M⊙ to a few 109 M⊙ , the existence of black-hole-like objects is beyond doubt [1, 2] What does remain an issue is whether the Schwarzschild metric correctly describes the physics of the interior. Alternatives to classical black holes have been proposed with no singularities in the interior [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The simplest model proposed for supermassive compact objects at the galactic centers is a self-gravitating degenerate fermion gas composed of, e.g., heavy sterile neutrinos [3, 4, 14, 15, 16] However, this scenario cannot cover the whole mass range of supermassive black-hole candidates with a single sterile neutrino species [17]. Recently, Chapline et al [5, 6] put forth an interesting proposal based on analogies to condensed matter systems where the effective general relativity was an emergent phenomenon. Specifically, they suggested that the sphere where the lapse function vanished marked a quantum phase transition, the lapse function increasing again at r < 2GM. As this required negative pressure, the authors of [6] assumed the interior vacuum condensate to be described by de Sitter space with the equation of state p = −ρ. Subsequently, the idea of gravitational vacuum condensate, or ‘gravastar’, was taken up by Mazur and Mottola [7, 9], replacing the horizon with a shell of stiff matter astride the surface at r = 2GM. Visser and Wiltshire [18] and recently Carter [19] also examined the stability of the gravastar using the Israel thin shell formalism [20]. Despite the fact that general relativity is an emergent phenomenon in string theory [21], the gravastar has met with a cool reception. Certainly, the assumption of a de Sitter interior presents a quandary: on the one hand, the quantum phase transition would suggest that the associated cosmological constant is a fundamental parameter; on the other hand, to accommodate the mass range of supermassive black hole candidates, it must vary over some six orders of magnitude. Thus, it seems prudent to explore other possibilities for the gravastar interior. One obvious extension of the de Sitter equation of state p + ρ = 0 would be an equation that satisfies p + ρ ≤ 0, thus violating the dominant energy condition. The fluid of which the equation of state violates the dominant energy condition has been dubbed phantom energy [22, 23] and has recently become a popular alternative to quintessence and to the cosmological constant [24]. The motivation for introducing the phantom energy is that the equation of state w ≡ p/ρ < −1 produces a superaccelerated cosmological expansion which seems to be favored by the combined analysis of the CMB and super nova type I data [25]. However, a superaccelerated expansion may also be obtained without violating the dominant energy condition in scalar-tensor theories of gravity [26] and in models with variable gravitational constant [27] or variable cosmological constant [28]. Some astrophysical aspects of the phantom have recently been discussed, such as phantom energy accretion [29], phantom energy wormholes [30],

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Born-Infeld Phantom Gravastars

and a possible relation of the phantom tachyon model to supermassive black holes [31]. In this paper we consider a gravastar interior consisting of a self-gravitating scalar field described by a Born-Infeld type Lagrangian which yields the Chaplygin gas equation of state [32, 33]. Hence, we look for static solutions of the self-gravitating Chaplygin gas. In particular, we consider static Chaplygin gas configurations in the phantom regime, i.e., when p + ρ < 0, and we show that these configurations could provide an alternative scenario for compact massive objects at galactic centers. The paper is organized as follows: In section 2 we introduce the basics of the model. In section 3 we investigate static solutions using Tolman-Oppenheimer-Volkoff equations. In section 4 we discuss a possible interpretation of galactic centers as BornInfeld phantom gravastars. A stability analysis is given in section 5 and we conclude the paper by section 6 2. The model Consider the equation of state A p=− ρ in the phantom regime, i.e., when √ ρ < A.

(1)

√

(2)

Equation (1) describes the Chaplygin gas which, for ρ ≥ A, has attracted some attention as a dark energy candidate [32, 33]. Astrophysical objects made of the socalled generalized Chaplygin gas [34] have recently been discussed [35]. The generalized Chaplygin gas has also been exploited in the phantom regime [36]. As we shall shortly demonstrate, static solutions to Einstein’s equations with matter described by (1) with (2) cover the range of masses and radii required to fit the phenomenology of supermassive dark compact objects at the galactic centers. Moreover, Eq. (1) yields the de Sitter gravastar solution in the limit when the central density of the static solution approaches √ the value A. √ The Chaplygin gas equation of state (1) with the condition ρ < A may be derived from the Dirac-Born-Infeld type Lagrangian √ √ LDBI = − A 1 + X , (3) where X = g µν ϑ,µ ϑ,ν .

(4)

Clearly, in the limit X → 0 this Lagrangian becomes a free scalar field Lagrangian with a “wrong sign” kinetic term, hence a phantomk. A phantom Lagrangian of the type (3) k It has recently been shown that quantum effects could lead to an effective dark-energy equation of state violating the dominant energy condition on cosmological scales even for a scalar field Lagrangian having the correct sign of the kinetic energy [37].

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Born-Infeld Phantom Gravastars

has been proposed in the context of the superaccelerated expansion [38]. Note that the Lagrangian (3) leads to a perfect fluid with the four-velocity ϑ,α uα = √ , X the pressure p = LDBI , and the density √ √ A ≤ A. ρ= √ 1+X

(5)

(6)

√ Clearly, the pressure and the density obey Eq. (1) and ρ ≤ A. It is worth√noting that the Chaplygin gas cosmological model [32], in which the condition ρ > A holds, is described by the Lagrangian (3) with a (−) sign in front of X. It is as though instead of the lapse function changing its sign, as it does in the Schwarzschild case for r < 2GM, the scalar kinetic energy changed its sign to become a phantom. Equivalent to LDBI of Eq. (3) is φ2 1 L=− X− 2 2

!

A φ2 + 2 , φ

(7)

as may be seen by eliminating φ2 through its equation of motion. Now recall that in four dimensions, a three-form field strength is dual to a scalar: H µνα = φ2 ǫµναβ ϑ,β . Then the content of the model is 1 1 L= Hµνα H µνα − 2 12φ 2

(8) A φ + 2 φ 2

!

.

(9)

Notably, this is also the content of low-energy string theory [21] in the Einstein frame where Hµνα is the Kalb-Ramond field, φ2 corresponds to the string coupling and the dilaton kinetic term is neglected in lieu of a potential assumed to arise from nonperturbative effects¶. A potential similar to that in (7) and (9) has been considered in the context of polymer scaling and black holes [39]. 3. Static solutions Next, we proceed to solve Einstein’s equations for a static, spherically symmetric configuration with an interior described by (1) and with a Schwarzschild exterior. Our approach is similar to that of Armendariz-Picon and Lim [40]. However, in contrast to us, they consider the classes of Lagrangian with strictly space-like gradient of φ, i.e. with X < 0, so that in their case the quantity (5) cannot be interpreted as a four velocity. As a consequence, their energy-momentum tensor does not have a standard perfect fluid form and the pressure is not isotropic. Here, we consider strictly time-like gradient of φ with X > 0, but we allow phantom-like Lagrangians which violate the dominant energy condition. ¶ Strictly, the exterior should have Hµνα H µνα = 6A and φ2 = 0 to be Schwarzschild.

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Born-Infeld Phantom Gravastars For the static, spherically symmetric line element ds2 = ξ 2 (r) dt2 −

dr 2 − r 2 dϑ2 + sin2 ϑ dϕ2 1 − 2GM(r)/r

(10)

with Tµν = diag (ρ, −p, −p, −p), Einstein’s equations become [41] M′ = 4πr 2 ρ, ξ′ = G ξ while Tµν ;ν = 0 gives

(11)

M + 4π r 3 p , r(r − 2G M)

(12)

ξ′ p = −(ρ + p) . ξ ′

(13)

We focus on the equation of state (1) to close the system and we require the solution to (11) and (12) to be regular at r = 0. Since by rescaling t one may set ξ(0) = 1, Eqs. (1) and (13) yield v u

2 ρ u t A − ρ0 . ξ(r) = ρ0 A − ρ2

(14)

Combining Eqs. (1), (12), and (13), one has ρ2 ρ =G 1− A ′

!

ρM − 4πAr 3 r(r − 2GM)

!

.

(15)

In Figs. 1, 2, and 3, respectively, √ we exhibit the resulting ρ(r)/ρ0 , ξ(r), and 2GM(r)/r for selected values √ of ρ0 / A. The solutions depend essentially on the magnitude of ρ0 relative to A. In the following we summarize the properties√of three classes of solutions √ corresponding to whether ρ0 is larger, smaller, or equal to A. i) For ρ0 > A, the density ρ increases and the lapse function ξ decreases with r starting from the origin up to the black-hole horizon radius Rbh , where 2GM(Rbh ) = Rbh . In the limit ρ0 → ∞, a limiting solution exists with a singular behavior 7A 1/3 ρ(r) ≃ ( ) (16) 18πGr 2 near the origin. √ ii) For ρ0 < A, both ρ and ξ decrease with r up to the radius R0 where they vanish. At that point the pressure p blows up to −∞ owing to (1). The enclosed mass M is always less than r/(2G), never reaching the black-hole horizon, i.e., the radius where 2GM(r) = √ r. √ iii) For ρ0 = A, the density ρ remains constant equal to A up to the de Sitter 3 radius RdS = 2GM = 8πGRdS ρ0 /3. Hence, the interior is de Sitter, precisely as in Chapline et al. [6]. The lapse function is given by r2 ξ = 1− 2 RdS

!1/2

(17)

Born-Infeld Phantom Gravastars

6

√ Figure 1. Density profile of the Chaplygin star for ρ0 / A = 1.2 (short dashed), 0.98 (dotted), 0.9 (long dashed). The limiting singular solution with r−2/3 behavior at small r is represented by the dot-dashed and the de Sitter gravastar by the solid line.

Figure 2. Lapse function ξ/ξ0 for various solutions as in Fig. 1.

Born-Infeld Phantom Gravastars

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Figure 3. Enclosed mass divided by the radius of the star for various solutions as in Fig. 1.

with RdS =

s

3 A−1/4 . 8πG

(18)

√ As ρ0 → A from above or from below, solutions i) or ii), respectively, converge to iii) except at the endpoint. The lapse function in iii) joins the Schwarzschild solution outside 2GM 1/2 , (19) r continuously, whereas in i) and ii) it happens discontinuously. As in the case of a de Sitter gravastar, in order to join our interior solutions to a Schwarzschild exterior at a spherical boundary of radius R, it is necessary to put a thin spherical shell+ at the boundary with a surface density and a surface tension satisfying Israel’s junction conditions [20]. In all the solutions discussed above the pressure is isotropic and does not vanish at the boundary. Hence, the pressure at the boundary must be compensated by a negative surface tension of the membrane. We postpone this issue for section 5 where we discuss the stability of the solution.

ξ(r) = 1 −

+

It has recently been demonstrated that the joining can be made continuous without the presence of a thin shell for a gravastar made of the fluid with an anisotropic pressure [42].

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Born-Infeld Phantom Gravastars 4. Black holes at galactic centers

Case ii), together with iii), is of particular interest as we would like to interpret the supermassive compact dark objects at the galactic centers in terms of phantom energy rather than in terms of a classical black hole. It is natural to assume that the most massive such object is described by the de Sitter gravastar, i.e., solution iii) (depicted by the solid line in Figs. 1, 2, and 3). If we identify the most massive black-hole candidate observed at the center of M87, with mass Mmax = 3 × 109 M⊙ , with the de Sitter gravastar, then A1/8 = 9.7keV4 , to be contrasted with the 10−3 eV values wanted for cosmology [33]. The radius of this object is RdS equal to the Schwarzschild radius 2GMmax . Clearly, solutions belonging to class ii), can fit all masses M < Mmax . However, for the phenomenology of supermassive galactic centers it is important to find, at least approximately, the mass-radius relationship for these √ solutions. This may be done in the low central density approximation, i.e., ρ0 ≪ A, which is similar to the Newtonian approximation but, in contrast to the Newtonian approximation, one cannot neglect the pressure term in Eq. (12). Moreover, as may be easily shown, in this approximation M ≪ r 3 p, so that the pressure term becomes dominant. Next, neglecting 2GM with respect to r, as in the usual Newtonian approximation, Eq. (15) simplifies to ρ′ = 4πGAr, with the solution ρ = ρ0

r2 1− 2 R0

!

;

R02 =

ρ0 , 2πGA

(20)

which gives a mass-radius relation M 16π 2 = GA = constant. R05 15

(21)

The mass-radius relationship M ∝ R05 which phantom gravastars obey, offers the prospect of unifying the description of all supermassive compact dark objects that have been observed at the galactic centers, as Born-Infeld phantom gravastars with masses ranging from Mmin = 106 M⊙ to Mmax = 3 × 109 M⊙ . Indeed, assuming that the most massive compact dark object, observed at the center of M87, is a Born-Infeld phantom gravastar near the black-hole limit, with Rmax = 2GMmax = 8.86 × 109 km = 8.21 lhr, the compact dark object at the center of our Galaxy, with mass MGC = 3 × 106 M⊙ , would have a radius RGC = 2.06 lhr if the scaling law (21) holds. This radius is well below the distances of the closest approach to Sgr A∗ which the stars SO-2 (Rmin = 17 lhr = 123 AU, [43]) and SO-16 (Rmin = 8.32 lhr = 60 AU [44]) recently had and beyond which the Keplerian nature of the gravitational potential of Sgr A∗ is well established. 5. Dynamical stability None of the solutions discussed in the preceding section will be stable unless there is a membrane, e.g. in the form of a thin shell, placed at the boundary with surface density σ and surface tension θ satisfying Israel’s junction conditions. Following Visser and Wiltshire [18] we consider a dynamical thin shell allowed to move radially at the

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Born-Infeld Phantom Gravastars

boundary of the phantom gravastar and discuss under which conditions will a gravastar configuration will be stable. A dynamical thin shell connecting two general static spherically symmetric spacetimes have also recently been considered [45] in a slightly different context. Israel’s junction conditions read [20] hh

Kab − δcb Kcc

ii

= 8πGSab ,

(22)

where Sab = diag (σ, θ, θ) is the surface stress energy, and [[f ]] denotes the discontinuity in f across the boundary, i.e., [[f (r)]] = lim (f (R + ǫ) − f (R − ǫ)) . ǫ→0

(23)

The tensor Kab is the extrinsic curvature defined by Kab = hca hdb nd;c

(24)

where na is a spacelike unit vector orthogonal to the timelike boundary and hab is the induced metric on the shell. The angular components of the extrinsic curvature may be easily calculated from (24) yielding 1 ˙ 1/2 , (25) Kϑϑ = Kϕϕ = (∆ + R) r precisely as in [18], where the dot denotes the derivative with respect to the proper time and where 2GM ∆=1− . (26) r The calculation of the time-time component Ktt is slightly more involved and is most easily done following Israel [20]. By making use of the Gauss normal coordinates and orthogonality from (24) it follows Kττ = ua ub Kab = −na ub ua ;b

(27)

where ua is the four-velocity of a point on the shell. Its non-vanishing components for the metric (10) are R˙ 1 1+ u = ξ ∆ t

!1/2

;

ur = R˙ .

(28)

Similarly, the non-vanishing components of na are ξ R˙ (∆ + R˙ 2 )1/2 nt = 1/2 ; nr = . (29) ∆ ∆ Using these expressions it may be shown that the four-acceleration ub ua ;b satisfies [20] 1 ¨ + Γr ua ub ). na ubua ;b = − (R (30) ab 2 1/2 ˙ (∆ + R ) A straightforward calculation yields " ′ ˙ 2 # ′ R ξ M 1 2 τ ¨ + (∆ + R˙ ) + . (31) R Kτ = ξ r ∆ (∆ + R˙ 2 )1/2

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Born-Infeld Phantom Gravastars Then, from (22) using (11) and (12) we find q

1 ∆ + R˙ 2 4πσ = − , G R

(32)

3 ¨ R R + GM − 4πGR ρ

4π(σ − 2θ) =

2

q

GR2 ∆ + R˙ 2

q

4πR ∆ + R˙ 2 (ρ + p) + . (33) ∆

To derive the stability condition a la Visser and Wiltshire [18], it is now sufficient to ¨ by −V ′ (R) in equations (32) and (33), where V (R) is a replace R˙ 2 by −2V (R) and R potential. The shell will be stable against small radial perturbation if there exists an ¯ such that equilibrium position R ¯ = 0; V (R)

¯ = 0; V ′ (R)

¯ > 0. V ′′ (R)

(34)

Then, by choosing a suitable potential V (R), equations (32) and (33) define a parametric equation of state θ = θ(σ) for the shell. For the static shell and the metric (10) we find 2GM 1− R

1/2

2GM(R) − 1− R

!1/2

= −4πGRσ,

(35)

and with the help of Eq. (12) we obtain 4πR3 A 2GM −1/2 − M(R) − M 1− R ρ(R) 2 = 4πR (σ − 2θ),

!

2GM(R) 1− R

!−1/2

(36)

where M is the total mass. Note that if the joining is affected at the point R0 where ξ = 0 and p is infinite (the point of naked singularity [42]), the surface density σ is finite, whereas the surface tension θ → −∞. Hence, to avoid the singularity the shell must be placed at some R < R0 . For the Newtonian gravastars discussed in the preceding section, equations (32) and (33) may be considerably simplified. Using (20), the approximation 2GM/R ≪ 1, and assuming that the boundary radius R is close to R0 , i.e., R0 − R ≪ 1. (37) y≡ R0 we find y2 3 √ σ = 2πGAR0 , (38) 1 − 2V √ y 2V ′ yV 1 1 − 2V + πGAR04 − 4πGAR03 √ . (39) θ=− 8πGR0 y 1 − 2V 1 − 2V In general, the potential V is a function of both R0 and y, depending on the chosen equation of state θ = θ(σ). The dynamical stability is achieved if for any R0 , V has a

Born-Infeld Phantom Gravastars

11

¯ minimum at some point y¯ = 1 − R/R which should not depend strongly on R0 and at which V (¯ y ) = 0. Instead of postulating θ = θ(σ) we can choose a desirable potential and determine the equation of state a posteriori. Static stability (indifferent balance) is obtained by setting V ≡ 0. Depending on the potential V (R0 , y), this may still be a complicated parametric equation of state. However, for the static shell, i.e. V ≡ 0, the equation of state simplifies to AR0 1/2 . (40) 32πGσ This equation describes a 2-dimensional generalized anti-Chaplygin gas [46] with the equation of state of the form p = −θ = A′ /σ α , with α = 1/2. θ=−

6. Conclusions In conclusion, we have shown that replacing the de Sitter interior of the gravastar by a Born-Infeld phantom allows a wide range of gravastar mass and radii related to the central density, or equivalently to the velocity of the phantom scalar. We have demonstrated that if the constant A, as the only free parameter of the model, is fixed by assuming that the most massive galactic center object is the maximal (de Sitter) gravastar, then the model is able to explain all supermassive compact dark objects at the center of the galaxies. Furthermore, as demonstrated above, the phantom gravastar model can lay claim to a connection with low-energy string theory. Acknowledgment The work of NB was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No. 0098002 and partially supported through the Agreement between the Astrophysical sector, SISSA, and the Particle Physics and Cosmology Group, RBI. Two of us (RDV and GBT) acknowledge grants from the South African National Research Foundation (NRF GUN-2053794), the Research Committee of the University of Cape Town, and the Foundation for Fundamental Research (FFR PHY-99-01241). References [1] F. Melia, Nature 437 (20 October 2005) 1105 [2] J. Kormendy, “The Stellar-Dynamical Search for Supermassive Black Holes in Galactic Nuclei”, in Coevolution of Black Holes and Galaxies, proceedings of Carnegie Observatories Centennial Symposium, Pasadena, California, 20-25 Oct 2002, Ed. L.C. Ho, (Cambridge Univ. Press, Cambridge 2004), pp 1-20, astro-ph/0306353. [3] R.D. Viollier, D. Trautmann and G.B. Tupper, Phys. Lett. B306 (1993) 79. [4] R.D. Viollier, Prog. Part. Nucl. Phys. 32 (1994) 51. [5] C. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, Phil. Mag. B 81 (2001) 235. [6] C. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, Int. J. Mod. Phys. A 18 (2003) 3587.

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[42] C. Cattoen, T. Faber and M. Visser, Class. Quant. Grav. 22 (2005) 4189, gr-qc/0505137 [43] A. Eckart et al., Mon. Not. R. Astron. Soc. 331 (2002) 917; R. Sch¨ odel et al., Nature 419 (2002) 694 [44] A. Ghez et al., Astrophys. J. 509 (1998) 678; A. Ghez et al, invited talk at the Galactic Center Conference, 4-8 October, 2002, Kalua-Kona (Hawaii). [45] F. S. N. Lobo, Phys. Rev. D 71 (2005) 124022, gr-qc/0506001; F. S. N. Lobo and P. Crawford, Class. Quant. Grav. 22 (2005) 4869, gr-qc/0507063 [46] V. Gorini, A. Kamenshchik, U. Moschella, and V. Pasquier, gr-qc/0403062.

arXiv:astro-ph/0503427v3 8 Feb 2006

Neven Bili´ c†§, Gary B Tupper‡ and Raoul D Viollier‡ † Rudjer Boˇskovi´c Institute, P.O. Box 180, 10002 Zagreb, Croatia ‡ Institute of Theoretical Physics and Astrophysics, Department of Physics, University of Cape Town, Private Bag, Rondebosch 7701, South Africa E-mail: [email protected], [email protected], [email protected] Abstract. We construct new gravitational vacuum star solutions with a Born-Infeld phantom replacing the de Sitter interior. The model allows for a wide range of masses and radii required by phenomenology, and can be motivated from low-energy string theory.

§ Author to whom correspondence should be addressed

Born-Infeld Phantom Gravastars

2

1. Introduction From its inception, the Schwarzschild solution of Einstein’s equations has been the subject of controversy over the possibility of compressing matter within the gravitational radius 2GM. Given the accumulated evidence for supermassive compact objects ranging from a few 106 M⊙ to a few 109 M⊙ , the existence of black-hole-like objects is beyond doubt [1, 2] What does remain an issue is whether the Schwarzschild metric correctly describes the physics of the interior. Alternatives to classical black holes have been proposed with no singularities in the interior [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The simplest model proposed for supermassive compact objects at the galactic centers is a self-gravitating degenerate fermion gas composed of, e.g., heavy sterile neutrinos [3, 4, 14, 15, 16] However, this scenario cannot cover the whole mass range of supermassive black-hole candidates with a single sterile neutrino species [17]. Recently, Chapline et al [5, 6] put forth an interesting proposal based on analogies to condensed matter systems where the effective general relativity was an emergent phenomenon. Specifically, they suggested that the sphere where the lapse function vanished marked a quantum phase transition, the lapse function increasing again at r < 2GM. As this required negative pressure, the authors of [6] assumed the interior vacuum condensate to be described by de Sitter space with the equation of state p = −ρ. Subsequently, the idea of gravitational vacuum condensate, or ‘gravastar’, was taken up by Mazur and Mottola [7, 9], replacing the horizon with a shell of stiff matter astride the surface at r = 2GM. Visser and Wiltshire [18] and recently Carter [19] also examined the stability of the gravastar using the Israel thin shell formalism [20]. Despite the fact that general relativity is an emergent phenomenon in string theory [21], the gravastar has met with a cool reception. Certainly, the assumption of a de Sitter interior presents a quandary: on the one hand, the quantum phase transition would suggest that the associated cosmological constant is a fundamental parameter; on the other hand, to accommodate the mass range of supermassive black hole candidates, it must vary over some six orders of magnitude. Thus, it seems prudent to explore other possibilities for the gravastar interior. One obvious extension of the de Sitter equation of state p + ρ = 0 would be an equation that satisfies p + ρ ≤ 0, thus violating the dominant energy condition. The fluid of which the equation of state violates the dominant energy condition has been dubbed phantom energy [22, 23] and has recently become a popular alternative to quintessence and to the cosmological constant [24]. The motivation for introducing the phantom energy is that the equation of state w ≡ p/ρ < −1 produces a superaccelerated cosmological expansion which seems to be favored by the combined analysis of the CMB and super nova type I data [25]. However, a superaccelerated expansion may also be obtained without violating the dominant energy condition in scalar-tensor theories of gravity [26] and in models with variable gravitational constant [27] or variable cosmological constant [28]. Some astrophysical aspects of the phantom have recently been discussed, such as phantom energy accretion [29], phantom energy wormholes [30],

3

Born-Infeld Phantom Gravastars

and a possible relation of the phantom tachyon model to supermassive black holes [31]. In this paper we consider a gravastar interior consisting of a self-gravitating scalar field described by a Born-Infeld type Lagrangian which yields the Chaplygin gas equation of state [32, 33]. Hence, we look for static solutions of the self-gravitating Chaplygin gas. In particular, we consider static Chaplygin gas configurations in the phantom regime, i.e., when p + ρ < 0, and we show that these configurations could provide an alternative scenario for compact massive objects at galactic centers. The paper is organized as follows: In section 2 we introduce the basics of the model. In section 3 we investigate static solutions using Tolman-Oppenheimer-Volkoff equations. In section 4 we discuss a possible interpretation of galactic centers as BornInfeld phantom gravastars. A stability analysis is given in section 5 and we conclude the paper by section 6 2. The model Consider the equation of state A p=− ρ in the phantom regime, i.e., when √ ρ < A.

(1)

√

(2)

Equation (1) describes the Chaplygin gas which, for ρ ≥ A, has attracted some attention as a dark energy candidate [32, 33]. Astrophysical objects made of the socalled generalized Chaplygin gas [34] have recently been discussed [35]. The generalized Chaplygin gas has also been exploited in the phantom regime [36]. As we shall shortly demonstrate, static solutions to Einstein’s equations with matter described by (1) with (2) cover the range of masses and radii required to fit the phenomenology of supermassive dark compact objects at the galactic centers. Moreover, Eq. (1) yields the de Sitter gravastar solution in the limit when the central density of the static solution approaches √ the value A. √ The Chaplygin gas equation of state (1) with the condition ρ < A may be derived from the Dirac-Born-Infeld type Lagrangian √ √ LDBI = − A 1 + X , (3) where X = g µν ϑ,µ ϑ,ν .

(4)

Clearly, in the limit X → 0 this Lagrangian becomes a free scalar field Lagrangian with a “wrong sign” kinetic term, hence a phantomk. A phantom Lagrangian of the type (3) k It has recently been shown that quantum effects could lead to an effective dark-energy equation of state violating the dominant energy condition on cosmological scales even for a scalar field Lagrangian having the correct sign of the kinetic energy [37].

4

Born-Infeld Phantom Gravastars

has been proposed in the context of the superaccelerated expansion [38]. Note that the Lagrangian (3) leads to a perfect fluid with the four-velocity ϑ,α uα = √ , X the pressure p = LDBI , and the density √ √ A ≤ A. ρ= √ 1+X

(5)

(6)

√ Clearly, the pressure and the density obey Eq. (1) and ρ ≤ A. It is worth√noting that the Chaplygin gas cosmological model [32], in which the condition ρ > A holds, is described by the Lagrangian (3) with a (−) sign in front of X. It is as though instead of the lapse function changing its sign, as it does in the Schwarzschild case for r < 2GM, the scalar kinetic energy changed its sign to become a phantom. Equivalent to LDBI of Eq. (3) is φ2 1 L=− X− 2 2

!

A φ2 + 2 , φ

(7)

as may be seen by eliminating φ2 through its equation of motion. Now recall that in four dimensions, a three-form field strength is dual to a scalar: H µνα = φ2 ǫµναβ ϑ,β . Then the content of the model is 1 1 L= Hµνα H µνα − 2 12φ 2

(8) A φ + 2 φ 2

!

.

(9)

Notably, this is also the content of low-energy string theory [21] in the Einstein frame where Hµνα is the Kalb-Ramond field, φ2 corresponds to the string coupling and the dilaton kinetic term is neglected in lieu of a potential assumed to arise from nonperturbative effects¶. A potential similar to that in (7) and (9) has been considered in the context of polymer scaling and black holes [39]. 3. Static solutions Next, we proceed to solve Einstein’s equations for a static, spherically symmetric configuration with an interior described by (1) and with a Schwarzschild exterior. Our approach is similar to that of Armendariz-Picon and Lim [40]. However, in contrast to us, they consider the classes of Lagrangian with strictly space-like gradient of φ, i.e. with X < 0, so that in their case the quantity (5) cannot be interpreted as a four velocity. As a consequence, their energy-momentum tensor does not have a standard perfect fluid form and the pressure is not isotropic. Here, we consider strictly time-like gradient of φ with X > 0, but we allow phantom-like Lagrangians which violate the dominant energy condition. ¶ Strictly, the exterior should have Hµνα H µνα = 6A and φ2 = 0 to be Schwarzschild.

5

Born-Infeld Phantom Gravastars For the static, spherically symmetric line element ds2 = ξ 2 (r) dt2 −

dr 2 − r 2 dϑ2 + sin2 ϑ dϕ2 1 − 2GM(r)/r

(10)

with Tµν = diag (ρ, −p, −p, −p), Einstein’s equations become [41] M′ = 4πr 2 ρ, ξ′ = G ξ while Tµν ;ν = 0 gives

(11)

M + 4π r 3 p , r(r − 2G M)

(12)

ξ′ p = −(ρ + p) . ξ ′

(13)

We focus on the equation of state (1) to close the system and we require the solution to (11) and (12) to be regular at r = 0. Since by rescaling t one may set ξ(0) = 1, Eqs. (1) and (13) yield v u

2 ρ u t A − ρ0 . ξ(r) = ρ0 A − ρ2

(14)

Combining Eqs. (1), (12), and (13), one has ρ2 ρ =G 1− A ′

!

ρM − 4πAr 3 r(r − 2GM)

!

.

(15)

In Figs. 1, 2, and 3, respectively, √ we exhibit the resulting ρ(r)/ρ0 , ξ(r), and 2GM(r)/r for selected values √ of ρ0 / A. The solutions depend essentially on the magnitude of ρ0 relative to A. In the following we summarize the properties√of three classes of solutions √ corresponding to whether ρ0 is larger, smaller, or equal to A. i) For ρ0 > A, the density ρ increases and the lapse function ξ decreases with r starting from the origin up to the black-hole horizon radius Rbh , where 2GM(Rbh ) = Rbh . In the limit ρ0 → ∞, a limiting solution exists with a singular behavior 7A 1/3 ρ(r) ≃ ( ) (16) 18πGr 2 near the origin. √ ii) For ρ0 < A, both ρ and ξ decrease with r up to the radius R0 where they vanish. At that point the pressure p blows up to −∞ owing to (1). The enclosed mass M is always less than r/(2G), never reaching the black-hole horizon, i.e., the radius where 2GM(r) = √ r. √ iii) For ρ0 = A, the density ρ remains constant equal to A up to the de Sitter 3 radius RdS = 2GM = 8πGRdS ρ0 /3. Hence, the interior is de Sitter, precisely as in Chapline et al. [6]. The lapse function is given by r2 ξ = 1− 2 RdS

!1/2

(17)

Born-Infeld Phantom Gravastars

6

√ Figure 1. Density profile of the Chaplygin star for ρ0 / A = 1.2 (short dashed), 0.98 (dotted), 0.9 (long dashed). The limiting singular solution with r−2/3 behavior at small r is represented by the dot-dashed and the de Sitter gravastar by the solid line.

Figure 2. Lapse function ξ/ξ0 for various solutions as in Fig. 1.

Born-Infeld Phantom Gravastars

7

Figure 3. Enclosed mass divided by the radius of the star for various solutions as in Fig. 1.

with RdS =

s

3 A−1/4 . 8πG

(18)

√ As ρ0 → A from above or from below, solutions i) or ii), respectively, converge to iii) except at the endpoint. The lapse function in iii) joins the Schwarzschild solution outside 2GM 1/2 , (19) r continuously, whereas in i) and ii) it happens discontinuously. As in the case of a de Sitter gravastar, in order to join our interior solutions to a Schwarzschild exterior at a spherical boundary of radius R, it is necessary to put a thin spherical shell+ at the boundary with a surface density and a surface tension satisfying Israel’s junction conditions [20]. In all the solutions discussed above the pressure is isotropic and does not vanish at the boundary. Hence, the pressure at the boundary must be compensated by a negative surface tension of the membrane. We postpone this issue for section 5 where we discuss the stability of the solution.

ξ(r) = 1 −

+

It has recently been demonstrated that the joining can be made continuous without the presence of a thin shell for a gravastar made of the fluid with an anisotropic pressure [42].

8

Born-Infeld Phantom Gravastars 4. Black holes at galactic centers

Case ii), together with iii), is of particular interest as we would like to interpret the supermassive compact dark objects at the galactic centers in terms of phantom energy rather than in terms of a classical black hole. It is natural to assume that the most massive such object is described by the de Sitter gravastar, i.e., solution iii) (depicted by the solid line in Figs. 1, 2, and 3). If we identify the most massive black-hole candidate observed at the center of M87, with mass Mmax = 3 × 109 M⊙ , with the de Sitter gravastar, then A1/8 = 9.7keV4 , to be contrasted with the 10−3 eV values wanted for cosmology [33]. The radius of this object is RdS equal to the Schwarzschild radius 2GMmax . Clearly, solutions belonging to class ii), can fit all masses M < Mmax . However, for the phenomenology of supermassive galactic centers it is important to find, at least approximately, the mass-radius relationship for these √ solutions. This may be done in the low central density approximation, i.e., ρ0 ≪ A, which is similar to the Newtonian approximation but, in contrast to the Newtonian approximation, one cannot neglect the pressure term in Eq. (12). Moreover, as may be easily shown, in this approximation M ≪ r 3 p, so that the pressure term becomes dominant. Next, neglecting 2GM with respect to r, as in the usual Newtonian approximation, Eq. (15) simplifies to ρ′ = 4πGAr, with the solution ρ = ρ0

r2 1− 2 R0

!

;

R02 =

ρ0 , 2πGA

(20)

which gives a mass-radius relation M 16π 2 = GA = constant. R05 15

(21)

The mass-radius relationship M ∝ R05 which phantom gravastars obey, offers the prospect of unifying the description of all supermassive compact dark objects that have been observed at the galactic centers, as Born-Infeld phantom gravastars with masses ranging from Mmin = 106 M⊙ to Mmax = 3 × 109 M⊙ . Indeed, assuming that the most massive compact dark object, observed at the center of M87, is a Born-Infeld phantom gravastar near the black-hole limit, with Rmax = 2GMmax = 8.86 × 109 km = 8.21 lhr, the compact dark object at the center of our Galaxy, with mass MGC = 3 × 106 M⊙ , would have a radius RGC = 2.06 lhr if the scaling law (21) holds. This radius is well below the distances of the closest approach to Sgr A∗ which the stars SO-2 (Rmin = 17 lhr = 123 AU, [43]) and SO-16 (Rmin = 8.32 lhr = 60 AU [44]) recently had and beyond which the Keplerian nature of the gravitational potential of Sgr A∗ is well established. 5. Dynamical stability None of the solutions discussed in the preceding section will be stable unless there is a membrane, e.g. in the form of a thin shell, placed at the boundary with surface density σ and surface tension θ satisfying Israel’s junction conditions. Following Visser and Wiltshire [18] we consider a dynamical thin shell allowed to move radially at the

9

Born-Infeld Phantom Gravastars

boundary of the phantom gravastar and discuss under which conditions will a gravastar configuration will be stable. A dynamical thin shell connecting two general static spherically symmetric spacetimes have also recently been considered [45] in a slightly different context. Israel’s junction conditions read [20] hh

Kab − δcb Kcc

ii

= 8πGSab ,

(22)

where Sab = diag (σ, θ, θ) is the surface stress energy, and [[f ]] denotes the discontinuity in f across the boundary, i.e., [[f (r)]] = lim (f (R + ǫ) − f (R − ǫ)) . ǫ→0

(23)

The tensor Kab is the extrinsic curvature defined by Kab = hca hdb nd;c

(24)

where na is a spacelike unit vector orthogonal to the timelike boundary and hab is the induced metric on the shell. The angular components of the extrinsic curvature may be easily calculated from (24) yielding 1 ˙ 1/2 , (25) Kϑϑ = Kϕϕ = (∆ + R) r precisely as in [18], where the dot denotes the derivative with respect to the proper time and where 2GM ∆=1− . (26) r The calculation of the time-time component Ktt is slightly more involved and is most easily done following Israel [20]. By making use of the Gauss normal coordinates and orthogonality from (24) it follows Kττ = ua ub Kab = −na ub ua ;b

(27)

where ua is the four-velocity of a point on the shell. Its non-vanishing components for the metric (10) are R˙ 1 1+ u = ξ ∆ t

!1/2

;

ur = R˙ .

(28)

Similarly, the non-vanishing components of na are ξ R˙ (∆ + R˙ 2 )1/2 nt = 1/2 ; nr = . (29) ∆ ∆ Using these expressions it may be shown that the four-acceleration ub ua ;b satisfies [20] 1 ¨ + Γr ua ub ). na ubua ;b = − (R (30) ab 2 1/2 ˙ (∆ + R ) A straightforward calculation yields " ′ ˙ 2 # ′ R ξ M 1 2 τ ¨ + (∆ + R˙ ) + . (31) R Kτ = ξ r ∆ (∆ + R˙ 2 )1/2

10

Born-Infeld Phantom Gravastars Then, from (22) using (11) and (12) we find q

1 ∆ + R˙ 2 4πσ = − , G R

(32)

3 ¨ R R + GM − 4πGR ρ

4π(σ − 2θ) =

2

q

GR2 ∆ + R˙ 2

q

4πR ∆ + R˙ 2 (ρ + p) + . (33) ∆

To derive the stability condition a la Visser and Wiltshire [18], it is now sufficient to ¨ by −V ′ (R) in equations (32) and (33), where V (R) is a replace R˙ 2 by −2V (R) and R potential. The shell will be stable against small radial perturbation if there exists an ¯ such that equilibrium position R ¯ = 0; V (R)

¯ = 0; V ′ (R)

¯ > 0. V ′′ (R)

(34)

Then, by choosing a suitable potential V (R), equations (32) and (33) define a parametric equation of state θ = θ(σ) for the shell. For the static shell and the metric (10) we find 2GM 1− R

1/2

2GM(R) − 1− R

!1/2

= −4πGRσ,

(35)

and with the help of Eq. (12) we obtain 4πR3 A 2GM −1/2 − M(R) − M 1− R ρ(R) 2 = 4πR (σ − 2θ),

!

2GM(R) 1− R

!−1/2

(36)

where M is the total mass. Note that if the joining is affected at the point R0 where ξ = 0 and p is infinite (the point of naked singularity [42]), the surface density σ is finite, whereas the surface tension θ → −∞. Hence, to avoid the singularity the shell must be placed at some R < R0 . For the Newtonian gravastars discussed in the preceding section, equations (32) and (33) may be considerably simplified. Using (20), the approximation 2GM/R ≪ 1, and assuming that the boundary radius R is close to R0 , i.e., R0 − R ≪ 1. (37) y≡ R0 we find y2 3 √ σ = 2πGAR0 , (38) 1 − 2V √ y 2V ′ yV 1 1 − 2V + πGAR04 − 4πGAR03 √ . (39) θ=− 8πGR0 y 1 − 2V 1 − 2V In general, the potential V is a function of both R0 and y, depending on the chosen equation of state θ = θ(σ). The dynamical stability is achieved if for any R0 , V has a

Born-Infeld Phantom Gravastars

11

¯ minimum at some point y¯ = 1 − R/R which should not depend strongly on R0 and at which V (¯ y ) = 0. Instead of postulating θ = θ(σ) we can choose a desirable potential and determine the equation of state a posteriori. Static stability (indifferent balance) is obtained by setting V ≡ 0. Depending on the potential V (R0 , y), this may still be a complicated parametric equation of state. However, for the static shell, i.e. V ≡ 0, the equation of state simplifies to AR0 1/2 . (40) 32πGσ This equation describes a 2-dimensional generalized anti-Chaplygin gas [46] with the equation of state of the form p = −θ = A′ /σ α , with α = 1/2. θ=−

6. Conclusions In conclusion, we have shown that replacing the de Sitter interior of the gravastar by a Born-Infeld phantom allows a wide range of gravastar mass and radii related to the central density, or equivalently to the velocity of the phantom scalar. We have demonstrated that if the constant A, as the only free parameter of the model, is fixed by assuming that the most massive galactic center object is the maximal (de Sitter) gravastar, then the model is able to explain all supermassive compact dark objects at the center of the galaxies. Furthermore, as demonstrated above, the phantom gravastar model can lay claim to a connection with low-energy string theory. Acknowledgment The work of NB was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No. 0098002 and partially supported through the Agreement between the Astrophysical sector, SISSA, and the Particle Physics and Cosmology Group, RBI. Two of us (RDV and GBT) acknowledge grants from the South African National Research Foundation (NRF GUN-2053794), the Research Committee of the University of Cape Town, and the Foundation for Fundamental Research (FFR PHY-99-01241). References [1] F. Melia, Nature 437 (20 October 2005) 1105 [2] J. Kormendy, “The Stellar-Dynamical Search for Supermassive Black Holes in Galactic Nuclei”, in Coevolution of Black Holes and Galaxies, proceedings of Carnegie Observatories Centennial Symposium, Pasadena, California, 20-25 Oct 2002, Ed. L.C. Ho, (Cambridge Univ. Press, Cambridge 2004), pp 1-20, astro-ph/0306353. [3] R.D. Viollier, D. Trautmann and G.B. Tupper, Phys. Lett. B306 (1993) 79. [4] R.D. Viollier, Prog. Part. Nucl. Phys. 32 (1994) 51. [5] C. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, Phil. Mag. B 81 (2001) 235. [6] C. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, Int. J. Mod. Phys. A 18 (2003) 3587.

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