aa r X i v : . [ g r- q c ] O c t Radiating Gravastars
R. Chan , ∗ M.F.A. da Silva , † Jaime F. Villas da Rocha , ‡ and Anzhong Wang § Coordena¸c˜ao de Astronomia e Astrof´ısica,Observat´orio Nacional, Rua General Jos´e Cristino, 77,S˜ao Crist´ov˜ao 20921-400, Rio de Janeiro, RJ, Brazil Departamento de F´ısica Te´orica, Instituto de F´ısica,Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524,Maracan˜a 20550-900, Rio de Janeiro - RJ, Brasil Universidade Federal do Estado do Rio de Janeiro,Instituto de Biociˆencias, Departamento de Ciˆencias Naturais,Av. Pasteur 458, Urca, CEP 22290-240, Rio de Janeiro, RJ, Brazil GCAP-CASPER, Department of Physics,Baylor University, Waco, TX 76798, USA (Dated: October 2, 2018)
Abstract
Considering a Vaidya exterior spacetime, we study dynamical models of prototype gravastars,made of an infinitely thin spherical shell of a perfect fluid with the equation of state p = σ , enclosingan interior de Sitter spacetime. We show explicitly that the final output can be a black hole, anunstable gravastar, a stable gravastar or a ”bounded excursion” gravastar, depending on how themass of the shell evolves in time, the cosmological constant and the initial position of the dynamicalshell. This work presents, for the first time in the literature, a gravastar that emits radiation. PACS numbers: 98.80.-k,04.20.Cv,04.70.Dy ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: anzhong˙[email protected] . INTRODUCTION Gravastar was proposed as an alternative to black holes. The initial model of Mazur andMottola (MM) [1], consists of five layers: an internal core 0 < r < r , described by thede Sitter universe, an intermediate thin layer of stiff fluid r < r < r , an external region r > r , described by the Schwarzschild solution, and two infinitely thin shells, appearing,respectively, on the hypersurfaces r = r and r = r . The intermediate layer is constructed insuch way that r is inner than the de Sitter horizon, while r is outer than the Schwarzschildhorizon, eliminating the existence of any horizon. Configurations with a de Sitter interiorhave long history which we can find, for example, in the work of Dymnikova and Galaktionov[2]. After this work, Visser and Wiltshire [3] (VW) pointed out that there are two differenttypes of stable gravastars which are stable gravastars and ”bounded excursion” gravastars.In the spherically symmetric case, the motion of the surface of the gravastar can be writtenin the form [3], 12 ˙ R + V ( R ) = 0 , (1)where R denotes the radius of the star, and ˙ R ≡ dR/dτ , with τ being the proper time ofthe surface. Depending on the properties of the potential V ( R ), the two kinds of gravastarsare defined as follows. Stable gravastars : In this case, there must exist a radius a such that V ( R ) = 0 , V ′ ( R ) = 0 , V ” ( R ) > , (2)where a prime denotes the ordinary differentiation with respect to the indicated argument.If and only if there exists such a radius R for which the above conditions are satisfied, themodel is said to be stable. Among other things, VW found that there are many equationsof state for which the gravastar configurations are stable, while others are not [3]. Carterstudied the same problem and found new equations of state for which the gravastars arestable [4], while De Benedictis et al [5] and Chirenti and Rezzolla [6] investigated the stabilityof the original model of Mazur and Mottola against axial-perturbations, and found thatgravastars are stable to these perturbations, too. Chirenti and Rezzolla also showed thattheir quasi-normal modes differ from those of black holes with the same mass, and thus canbe used to discern a gravastar from a black hole.2 Bounded excursion” gravastars : As VW noticed, there is a less stringent notion ofstability, the so-called ”bounded excursion” models, in which there exist two radii a and a such that V ( R ) = 0 , V ′ ( R ) ≤ , V ( R ) = 0 , V ′ ( R ) ≥ , (3)with V ( R ) < R ∈ ( R , R ), where R > R .Lately, we studied both types of gravastars [7]-[12], and found that, such configurationscan indeed be constructed, although the region for the formation of them in the phase spaceis very small in comparison to that of black holes.Based on the discussions about the gravastar picture some authors have proposed alter-native models [13]-[18]. In addition, since in the study of the evolution of gravastar there isa possibility of black hole formation, we can find some works considering the hypothesis ofdark energy black hole [19][13][20][21][22].In the last four years we have adopted a different approach (from VW [3]), which meansthat we started from an equation of state and found the potential of the shell [7][8]. Gen-eralizing the exterior spacetime in order to include a cosmological constant, we study thegravastar model in a de Sitter-Schwarzschild spacetime, which allowed to investigate the roleof cosmological constant in its evolution [10]. Following this direction, we also considereda de Sitter-Reissner-Nordstr¨om exterior spacetime [11]. On the other hand, we studied theeffects of changing the interior of the gravastar, filling it with anisotropic fluids, which canbe characterized by different kinds of dark energy [9][12].Here we are interested in the study of a gravastar model whose interior consists of a deSitter spacetime and an exterior radiative Vaidya’s spacetime. The paper is organized asfollows: In Sec. II we present the metrics of the interior and exterior spacetimes, and writedown the motion of the thin shell in the form of equation (1). In Sec. III we study the modelby using the small radiating source approximation. In Sec. IV we discuss the formation ofblack holes and gravastars, when the mass of the thin shell increases, while in Sec. V westudy the case where the mass of the thin shell decreases. Finally, in Sec. VI we present ourmain conclusions. 3 I. DYNAMICAL THREE-LAYER PROTOTYPE GRAVASTARS
The interior spacetime is described by the de Sitter’s metric given by ds i = − f dt + f − dr + r d Ω , (4)where f = 1 − ( r/L ) , L = p / Λ and d Ω = dθ + sin ( θ ) dφ .The exterior spacetime is given by the Vaidya’s metric ds e = − F dv − d r dv + r d Ω , (5)where F = 1 − m ( v ) r . The metric of the hypersurface on the shell is given by ds = − dτ + R ( τ ) d Ω , (6)where τ is the proper time.Since ds i = ds e = ds , we find that r Σ = r Σ = R , and f ˙ t − f − ˙ R = 1 , (7) " F + 2 ˙ R ˙ v ˙ v = 1 , (8)where the dot denotes the ordinary differentiation with respect to the proper time. On theother hand, the interior and exterior normal vectors to the thin shell are given by n iα = ( − ˙ R, ˙ t, , ,n eα = ( − ˙ R, ˙ v, , . (9)Then, the interior and exterior extrinsic curvatures are given by K iττ = − [(3 L ˙ R − L ˙ t + 2 L R ˙ t − R ˙ t ) R ˙ t − ( L + R )( L − R )( ˙ R ¨ t − ¨ R ˙ t ) L ] × ( L + R ) − ( L − R ) − L − (10) K iθθ = ˙ t ( L + R )( L − R ) L − R (11) K iφφ = K iθθ sin ( θ ) , (12) K eττ = ˙ v (2 m ˙ v − mR ˙ R − mR ˙ v + ˙ mR ˙ v ) R − , (13) K eθθ = − ˙ v (2 m − R ) + R ˙ R, (14)4 eφφ = K eθθ sin ( θ ) . (15)Since [23] [ K θθ ] = K eθθ − K iθθ = − M, (16)where M is the mass of the shell, we find that M = ˙ v (2 m − R ) + ˙ t (1 − aR ) R. (17)Then, substituting equations (7) and (8) into (17) we get M = − R ˙ R + (2 m − R ) ˙ v + R " ˙ R + 1 − (cid:18) RL (cid:19) / . (18)In order to keep the ideas of MM as much as possible, we consider the thin shell as consistingof a fluid with the equation of state, p = σ , where σ and p denote, respectively, the surfaceenergy density and pressure of the shell. Then, the equation of motion of the shell is givenby [23] ˙ M + 8 πR ˙ Rp = 4 πR [ T αβ u α n β ] = 4 πR (cid:16) T eαβ u αe n βe − T iαβ u αi n βi (cid:17) , (19)where u α is the four-velocity.The exterior energy-momentum tensor is given by T eαβ = ǫl α l β (20)where l α = δ vα , l α l α = 0 , (21) ǫ = − ˙ v πR dmdv . (22)Since u eα = ( ± ( ˙ R + F ˙ v ) , ± ˙ v, , , (23)(in this paper we shall choose the plus signal), we find T eαβ u αe n βe = − ˙ m ˙ v πR . (24)Since the interior spacetime is de Sitter, we get˙ M + 8 πR ˙ Rσ = − ˙ m ˙ v . (25)5n order to solve this equation, let us assume that˙ m ˙ v = k = k ˙ M , (26)where k and k are constant. Note that k < ǫ >
0. Thus, recalling that σ = M/ (4 πR ), we find that the solution of equation (25) is given by, M = kR − k , (27)where k is a positive integration constant. Note also that k = − k ≪ k < M >
0, meaning that the mass of the shell is increasing. In this case,the interior de Sitter energy transfers to the thin shell, feeding the outgoing radiation. If k > M <
0, meaning that the mass of the shellis decreasing. In this case, the thin shell looses mass in order to maintain the outgoingradiation. Thus, for three particular values of k , which cover all the possibilities describedabove, we have M = kR − , if taking the limit k → kR +2 , if k = − kR − , if k = +1, (28)which will be studied in more details in the next three sections.Substituting equation (27) into equation (18), we find that the potential in the form ofequation (1) can be written as V ( R, m, L, k ) = − L R k R − k (cid:16) − R mL − L R k R − k +2 R k R − k L + 4 R m L + 4 RmL k R − k + R + k R − k L (cid:17) . (29)Rescaling m, L and R as, m → mk k k ,L → Lk k k ,R → Rk k k , (30)6e find that equation (29) can be written as V ( R, m, L ) = − L R × (cid:16) − R k k mL − L R + 2 R L + 4 R k k m L + 4 RmL + R k k + R − k L (cid:17) . (31)Clearly, for any given constants m and L , equation (31) uniquely determines the collapse ofthe prototype gravastar. Depending on the initial value R , the collapse can form either ablack hole, a gravastar or a de Sitter spacetime. In the last case, the thin shell first collapsesto a finite non-zero minimal radius and then expands to infinity. To guarantee that initiallythe spacetime does not have any kind of horizons, cosmological or event, we must restrict R to the range, 2 m < R < L, (32)where R is the initial collapse radius. III. GRAVASTARS/BLACK HOLES WITH SMALL EMISSION OF RADIATION
Here we will consider the limit k →
0. Then, we find that V ( R, m, L ) = − L R (cid:0) − L R + 2 R L + R − R mL + 4 R m L + 4 R mL + L (cid:1) , (33)fro which we find that lim R → V ( R, m, L ) = lim R →∞ V ( R, m, L ) = −∞ .The first derivative of the potential, equation (33), is given by dVdR = − L R (cid:0) R L + 5 R − R mL + 8 R m L − R mL − L (cid:1) . (34)Thus the solutions for dVdR = 0 are m = 18 R L (cid:16) R + L + √ R − R L + 25 L (cid:17) , (35)and m = 18 R L (cid:16) R + L − √ R − R L + 25 L (cid:17) . (36)Note that m and m are always positive, as can be seen from Fig. 1.7 FIG. 1: Case k ≃
0. The masses m (left) and m (right) where the first derivative of the potential V ( R ) is zero. We note that both masses are positive. FIG. 2: Case k ≃
0. The second derivative of the potential d VdR ( R, m, L ) calculated at m = m (left) and m = m (right). We note that d VdR ( R, m = m , L ) can be positive or negative (thefrontier between the two regions is given by equation (39)) and that d VdR ( R, m = m , L ) is alwaysnegative. R V R (b)(a)
FIG. 3: The possible type of potentials.
The second derivative of the potential is given by d VdR ( R, m, L ) = − R L (cid:0) R L + 45 R − R mL + 24 R m L + 4 R mL + 21 L (cid:1) . (37)Substituting equation (35) into equation (37) we have d VdR ( R, m = m , L ) = − L R × (cid:16) − R L − R − R √ R − R L + 25 L + 125 L + 5 L √ R − R L + 25 L (cid:17) . (38)Solving d VdR ( R, m = m , L ) = 0 we get L f ≈ . R . (39)Substituting equation (36) into equation (37) we have d VdR ( R, m = m , L ) = − L R × (cid:16) − R L − R + 21 R √ R − R L + 25 L + 125 L − L √ R − R L + 25 L (cid:17) . (40)Thus, we can see from Fig. 2 that the second derivative of the potential is always positiveat m = m and negative at m = m . This means that the form of the potential is given byFigs. 3a and 3b. 9 FIG. 4: Case k ≃
0. The potential V ( R, m, L ) calculated at m = m (left) and at m = m (right).We note that both potentials are negative. Substituting equation (35) into equation (33) we have V ( R, m = m , L ) = − L R × (cid:16) R L + 25 L + 5 L √ R − R L + 25 L − R L + 9 R + 3 R √ R − R L + 25 L (cid:17) , (41)and substituting equation (36) into equation (33) we get V ( R, m = m , L ) = − L R × (cid:16) R L + 25 L − L √ R − R L + 25 L − R L + 9 R − R √ R − R L + 25 L (cid:17) . (42)Thus, from Fig. 4 we can see that the potential is always negative at both m = m and m = m . This means that the form of the potential is given by the lowest curves of Figs.3a and 3b, respectively. Hence, the collapse can either form gravastars or black holes. IV. GRAVASTARS/BLACK HOLES WHEN THE THIN SHELL MASS IN-CREASES
Now, let us assume that k = −
2. Then, we find that10
FIG. 5: Case k ≃
0. The potential V ( R, m, L ) calculated at m = m , R c = 2 and L c = 5. Thisrepresents the formation of a ”bounded excursion” gravastar. V ( R, m, L ) = − L R (cid:0) − R L + 2 L R + 4 m L − mL R + 4 R L m + R + R L (cid:1) . (43)Note again that lim R → V ( R, m, L ) = lim R →∞ V ( R, m, L ) = −∞ .The first derivative of the potential, equation (43), is given by dVdR = − L R (cid:0) L R + 2 mL R − R L m + R + R L − m L (cid:1) . (44)Thus, the solutions for dVdR = 0 are m = R L (cid:16) − L + √ L + 9 L (cid:17) (45)and m = R L (cid:16) − L − √ L + 9 L (cid:17) . (46)Note from Fig. 8 that m is always negative, while m is always positive. Since the massmust be always positive, thus the unique reasonable solution for dVdR = 0 is given by m .11 FIG. 6: Case k ≃
0. The potential V ( R, m, L ) calculated at m = m , R c = 2 and L c =5 . The second derivative of the potential is given by d VdR ( R, m, L ) = − R L (cid:0) L R − mL R + 4 R L m + R + R L + 40 m L (cid:1) . (47)Thus, from Fig. 9 we can see that the second derivative of the potential is always negativeat m = m . This means that the form of the potential is given by Fig. 3b.Substituting equation (45) into equation (47) we have d VdR ( R, m = m , L ) = − L × (cid:16) L + 9 + √ L + 9 L + 9 L − L √ L + 9 L (cid:17) . (48)Substituting equation (45) into equation (43) we have V ( R, m = m , L ) = − L × (cid:16) − L + 30 R L + 9 R − R √ L + 9 L + 9 L R + 3 R L √ L + 9 L (cid:17) . (49)12 FIG. 7: Case k ≃
0. The potential V ( R, m, L ) calculated at m = m , R c = 2 and L c = 10. Thisrepresents the formation of a black hole. We notice that V ( R, m = m , L ) can be positive or negative, depending on the radius R and the cosmological constant L (see figure 10). This means that there exist possibilities offormation of both gravastars and black holes. A. Total Gravitational Mass
In order to study the gravitational effect generated by the two components of the gravas-tar, i.e., the interior de Sitter and the thin shell in the exterior region, we need to calculatethe total gravitational mass of a spherical symmetric system. Some alternative definitionsare given by [24],[25] and [26]. Here we consider the Tolman’s formula for the mass, whichis given by M G = Z R Z π − π Z π √− g T αα drdθdφ, (50)13 IG. 8: Case k = −
2. The masses m (left) and m (right) where the first derivative of thepotential V ( R ) is zero. We can note that m is always positive and m is always negative. Thus,we have only m as solution of dVdR = 0FIG. 9: Case k = −
2. The second derivative of the potential d VdR ( R, m, L ) calculated at m = m .We note that d VdR ( R, m = m , L ) is always negative. FIG. 10: Case k = −
2. The potential V ( R, m, L ) calculated at m = m . We note that V ( R, m = m , L ) can be positive or negative. R 43.532.5-1.5 21.5-2.5-1 1-2
FIG. 11: Case k = −
2. The potential V ( R, m, L ) calculated at m = m , R c = 2 and L c = 10.This represents the formation of a black hole. √− g is the determinant of the metric. For the special case of a thin shell we have M G = Z R Z π − π Z π √− g T αα δ ( r − R ) d r dθdφ. (51)Thus, the Tolman’s gravitational mass of the thin shell is given by M shellG = 3 M, (52)and for the interior de Sitter (dS) spacetime we have M dSG = −
23 Λ i R . (53)Thus, the de Sitter interior presents a negative gravitational mass, since Λ i >
0, in agreementwith its repulsive effect.Now we can write the total Tolman’s gravitational mass of the gravastar as M totalG = M shellG + M dSG = 3 M −
23 Λ i R . (54)This mass should also represent the Vaidya exterior mass ( m = M totalG ) of the gravastar.This last equation can explain how the mass of the shell can increase with the time. Since m must decrease with the time because of the emission of radiation, the unique way that M may increase with the time is that the radius R is increasing with the time. V. GRAVASTARS/BLACK HOLES WHEN THE THIN SHELL MASS DE-CREASES
Now, let us assume that k = +1. Then, we find V ( R, m, L ) = − L R (cid:0) − L R + 2 R L + R − R mL + 4 R m L + 4 L mR + L (cid:1) . (55)Note again that lim R → V ( R, m, L ) = lim R →∞ V ( R, m, L ) = −∞ .The first derivative of the potential, equation (55), is given by dVdR = − L R (cid:0) R L + 2 R − R mL + 2 R m L − L mR − L (cid:1) . (56)Thus the solutions for dVdR = 0 are m = 14 R L (cid:16) R + L + √ R + 2 R L + 9 L (cid:17) , (57)16 IG. 12: Case k = +1. The mass m , where the first derivative of the potential V ( R ) is zero. Wenote from equation (57) that m is always positive. However, m can be positive or negative. and m = 14 R L (cid:16) R + L − √ R + 2 R L + 9 L (cid:17) . (58)We note from equation (57) that m is always positive and from Fig. 12 that m may bepositive or negative, depending on the radius R and the cosmological constant L .The second derivative of the potential is given by d VdR ( R, m, L ) = − R L (cid:0) R L + 14 R − R mL + 2 R m L + 2 L mR + 5 L (cid:1) . (59)Thus, we can see from Fig. 13 that the second derivative of the potential can be positiveor negative at m = m , depending on the radius R and the cosmological constant L . Thismeans that the form of the potential is given by Figs. 3a and 3b, respectively. Besides, thesecond derivative of the potential is always negative at m = m . This means that the formof the potential is given by Fig. 3b.Substituting equation (57) into equation (59) we have d VdR ( R, m = m , L ) = − L R × (cid:16) R + 5 R √ R + 2 R L + 9 L − L − L √ R + 2 R L + 9 L (cid:17) , (60)17 IG. 13: Case k = +1. The second derivative of the potential d VdR ( R, m, L ) calculated at m = m (left) and at m = m (right). We note that d VdR ( R, m = m , L ) can be positive or negative (thefrontier between the two regions is given by equation (62)) and that d VdR ( R, m = m , L ) is alwaysnegative. and substituting equation (58) into equation (59) we get d VdR ( R, m = m , L ) = − L R × (cid:16) − R + 5 R √ R + 2 R L + 9 L + 9 L − L √ R + 2 R L + 9 L (cid:17) . (61)Solving d VdR ( R, m = m , L ) = 0, we get L f ≈ . R . (62)Substituting equation (57) into equation (55) we have V ( R, m = m , L ) = − L R × (cid:16) − L R + 18 R L + 9 R + 3 R √ R + 2 R L + 9 L + 9 L + 3 L √ R + 2 R L + 9 L (cid:17) , (63)18 FIG. 14: Case k = +1. The potential V ( R, m, L ) calculated at m = m (left) and at m = m (right). We note that V ( R, m = m , L ) is always negative and that V ( R, m = m , L ) can bepositive or negative. and substituting equation (58) into equation (55) we get V ( R, m = m , L ) = − L R × (cid:16) − L R + 18 R L + 9 R − R √ R + 2 R L + 9 L + 9 L − L √ R + 2 R L + 9 L (cid:17) . (64)We notice that V ( R, m = m , L ) is always negative (see figure 14). Since V ( R, m = m , L )can be positive or negative, depending on the radius R and the cosmological constant L , wemay have again formation of gravastars or black holes. VI. CONCLUSIONS
In this paper, we have studied the problem of the stability of gravastars by constructingdynamical three-layer models of VW [3], which consists of an internal de Sitter spacetime,a dynamical infinitely thin shell of perfect fluid with the equation of state p = σ , and anexternal Vaidya’s spacetime.We have shown explicitly that the final output can be a black hole, an unstable gravastar,19 FIG. 15: Case k = +1. The potential V ( R, m, L ) calculated at m = m , R c = 2 and L c = 5 . a stable gravastar or a ”bounded excursion” gravastar, depending on the time evolution ofthe shell mass, the parameter L and the initial position R of the dynamical shell. Acknowledgments
The financial assistance from FAPERJ/UERJ (MFAdaS) are gratefully acknowledged.The authors (RC, MFAdaS, JFVR) acknowledges the financial support from FAPERJ(no. E-26/171.754/2000, E-26/171.533/2002, E-26/170.951/2006, E-26/110.432/2009 andE26/111.714/2010). The authors (RC, MFAdaS and JFVdR) also acknowledge the finan-cial support from Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq- Brazil (no. 450572/2009-9, 301973/2009-1 and 477268/2010-2). The author (MFAdaS)also acknowledges the financial support from Financiadora de Estudos e Projetos - FINEP- Brazil (Ref. 2399/03). The work of AW was supported in part by DOE Grant, DE-FG02-20
R 3.50.3 30.20.1 2.50-0.1 21.5
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