Gravastars or Black Holes as Consequence of the Einstein's Theory ...

Gravastars or Black Holesas Consequence of theEinstein’s Theory of GravityM. de Fátima A. da Silvamfasnic@gmail.comP. Rocha - Universidade Federal Fluminense,R. Chan - Observatório Nacional,M. F. A. da Silva - Universidade do Estado do Rio de Janeiro,A. Wang - Baylor University

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum◮ an infinitely thin shell

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum◮ an infinitely thin shell◮ a finite thin layer (envelope)

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum◮ an infinitely thin shell◮ a finite thin layer (envelope)◮ another infinitely thin shell

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum◮ an infinitely thin shell◮ a finite thin layer (envelope)◮ another infinitely thin shell◮ the exterior Schwarzschild vacuum

IntroductionThe pioneer model of gravastar was proposedby Mazur and Mottola (2004), with five layers,◮ an interior de Sitter vacuum◮ an infinitely thin shell◮ a finite thin layer (envelope)◮ another infinitely thin shell◮ the exterior Schwarzschild vacuum◮

Following the pioneers, Visser and Wiltshire(2004) adopted a simplifier model with 3layers,◮ an interior de Sitter vacuum

◮ Following the pioneers, Visser and Wiltshire(2004) adopted a simplifier model with 3layers,◮ an interior de Sitter vacuum◮ only one infinitely thin shell

◮ Following the pioneers, Visser and Wiltshire(2004) adopted a simplifier model with 3layers,◮ an interior de Sitter vacuum◮ only one infinitely thin shell◮ the exterior Schwarzschild vacuum

◮ Following the pioneers, Visser and Wiltshire(2004) adopted a simplifier model with 3layers,◮ an interior de Sitter vacuum◮ only one infinitely thin shell◮ the exterior Schwarzschild vacuum

◮ The equation governing the shell’s motion is12ȧ2 + V (a) = 0,

◮ The equation governing the shell’s motion is12ȧ2 + V (a) = 0,◮Stable gravastars:V (a 0 ) = 0, V ′ (a 0 ) = 0, V ′′ (a 0 ) > 0. If and only if thereexists such an a 0 , the model is said to be stable.

◮ The equation governing the shell’s motion is12ȧ2 + V (a) = 0,◮◮Stable gravastars:V (a 0 ) = 0, V ′ (a 0 ) = 0, V ′′ (a 0 ) > 0. If and only if thereexists such an a 0 , the model is said to be stable.”Bounded excursion” gravastars:As VW noticed, there is a less stringent notion of stability, theso-called ”bounded excursion” models, in which there existtwo radii a 1 and a 2 such thatV (a 1 ) = 0, V ′ (a 1 ) ≤ 0, V (a 2 ) = 0, V ′ (a 2 ) ≥ 0, withV (a) < 0 for a ∈ (a 1 , a 2 ), where a 2 > a 1 .

◮ Based on the discussions about the gravastarpicture, many contributions of other authors canbe found in the literature:

◮ Based on the discussions about the gravastarpicture, many contributions of other authors canbe found in the literature:◮ C. Cattoen, T. Faber, I. Dymmikova, B. M. N. Carter, O.Bertolami, J. Páramos, De benedictis, C. B. M. H.Chirenti, L. Rezzolla, F. S. N. Lobo and our group,between others.

◮ Based on the discussions about the gravastarpicture, many contributions of other authors canbe found in the literature:◮ C. Cattoen, T. Faber, I. Dymmikova, B. M. N. Carter, O.Bertolami, J. Páramos, De benedictis, C. B. M. H.Chirenti, L. Rezzolla, F. S. N. Lobo and our group,between others.◮ In previous works we studied the gravastarsmodels proposed by VW (JCAP 6, 25 (2008),JCAP 11, 10 (2008)), and found that, amongother things, such configurations canindeed be constructed, although the regionfor the formation of these types ofgravastars is very small in comparison tothat of black holes.

◮ Here, we generalize theses results to the casewhere the equation of state of the infinitely thinshell is given by p = (1 − γ)σ with γ being aconstant, with an interior phantom/standardenergy fluid (JCAP 03,10 (2009)).

◮ Here, we generalize theses results to the casewhere the equation of state of the infinitely thinshell is given by p = (1 − γ)σ with γ being aconstant, with an interior phantom/standardenergy fluid (JCAP 03,10 (2009)).◮ We constructed three-layer dynamical models, inanalogy to the VW models, and then, we showthat both types of stable gravastars and blackholes exist for several situations. However, we donot find a configuration with a pure dark energyin the interior. In the phase space, the region ofgravastars and of black holes are non-zero.

Dynamical Three-layer Prototype GravastarsThe interior: anisotropic dark energy fluid (Lobo’smodel CQG 23, 1525 (2006)) with a metric given byds 2 − = −f 1 dt 2 + f 2 dr 2 + r 2 dΩ 2 ,where dΩ 2 ≡ dθ 2 + sin 2 (θ)dφ 2 , andf 1 = (1 + br 2 ) 1−ω2 (1 + 2br 2 ) ω , f 2 = 1 + 2br 21 + br 2 ,with ¯m(r) = br 32(1+2br 2 )

The corresponding energy density ρ, radial andtangential pressures p r and p t are given,respectively, by( ) ( ωb 3 + 2br2p r = ωρ =8πp t = −ωρ + b 2 r { 2 (1 + ω)(3 + 2br 2 ) [ (1 + 3ω) + 2br 2 (1 + ω) ](1 + 2br 2 ) 2 ),−8ω(5 + 2br 2 )(1 + br 2 ) } { 32π [ (1 + 2br 2 ) 3 (1 + br 2 ) ]} .

◮The exterior: spacetime is given by the Schwarzschild metricds+ 2 = −fdv 2 + f −1 dr 2 + r 2 dΩ 2 ,where f = 1 − 2m/r.

◮◮The exterior: spacetime is given by the Schwarzschild metricwhere f = 1 − 2m/r.ds 2 + = −fdv 2 + f −1 dr 2 + r 2 dΩ 2 ,The thin shell: the hypersurface divided these twospacetimes is given by the metricdsΣ 2 = −dτ 2 + R 2 (τ)dΩ 2 .Since ( )ds−2 = ( )ds 2 Σ + = Σ ds2 Σ , we find that r Σ = r Σ = R, andf 1 ṫ 2 − f 2 Ṙ 2 = 1, f ˙v 2 − Ṙ 2= 1.f

The interior and exterior extrinsic curvature are given byKττ − = 1 {[2 (1 + bR 2 ) −ω/2 ṫ 4(1 + bR 2 ) ω/2 bR 2 Ṙ 2+2(1 + bR 2 ) ω/2 Ṙ 2 − (1 + 2bR 2 ) ω√ 1 + bR 2 bR 2 ṫ 2−(1 + 2bR 2 ) ω√ 1 + bR 2 ṫ 2 ](2bR 2 ω + 2bR 2 + 3ω + 1)−2(1 + bR 2 ) ω/2 (1 + 2bR 2 )Ṙ 2 }(1 + 2bR 2 ) −2 (1 + bR 2 ) −1 bR + Ṙẗ − ¨Rṫ,K − θθ= ṫ(1 + bR 2 )R1 + 2bR 2 , K − φφ = K − θθ sin2 (θ),K + ττ = ˙v m(4m2 ˙v 2 − 4mR ˙v 2 − 3R 2 Ṙ 2 + R 2 ˙v 2 )(2m − R)R 3 + Ṙ¨v − ¨R ˙v,K + θθ= − ˙v(2m − R), K + φφ = K + θθ sin2 (θ).

◮ From the Lake’s work (PRD 19, 2847 (1979)) we have⇒ [K θθ ] = K + θθ − K − θθ = −M.which furnishes[]√1/2M = −R 1 − 2m 1 + bR 2 + Ṙ 2 (1 + 2bR 2 )R + Ṙ 2 +R(1 + bR 2 ) −(ω+1)/4 (1 + 2bR 2 ) ω+22 ,

◮ From the Lake’s work (PRD 19, 2847 (1979)) we have⇒ [K θθ ] = K + θθ − K − θθ = −M.which furnishes[]√1/2M = −R 1 − 2m 1 + bR 2 + Ṙ 2 (1 + 2bR 2 )R + Ṙ 2 +R(1 + bR 2 ) −(ω+1)/4 (1 + 2bR 2 ) ω+22 ,◮ andṀ+8πRṘϑ = 4πR 2 [T αβ u α n β ] = πR 2 (T + αβ uα +n β + − T − αβ uα −n β −which reduces toṀ + 8πRṘ(1 − γ)σ = 0 ⇒ M = kR 2(γ−1) , if we assumethe equation of state given by ϑ = (1 − γ)σ.),

Recall that 1 2Ṙ 2 + V (R) = 0 ⇒ V (R) = − 1 2Ṙ 2 , we find{V (R, m, ω, b, γ) = −b (ω+2)−2b (3ω+4)/22 R 2(γ−1) b (ω+1)/412 R 4(γ−1) b (ω+1)/21[b (−ω)2 b (ω+1)/21 R 2−b −(ω+1)2 b (ω+3)/21 R 2 − 2b (−ω)2 b (ω+1)/21 mR + b 1 R 2 + b 2 R 2+2b 2 mR + b 2 R 4(γ−1)] 1/2 (ω+2) + b 2 R 2 b (ω+1)/21−b (2ω+3)2 R 2 − 2b (ω+2)2 mRb (ω+1)/21+2b (2ω+3)2 mR + b (2ω+3)2 R 4(γ−1) − b (ω+2)1 R 2} [] } 2+b (ω+1)2 b (ω+3)/21 R 2 /{2R 2 b 2 b (ω+1)2 − b (ω+1)/21 .where b 1 ≡ 1 + bR 2 and b 2 ≡ 1 + 2bR 2 , and we have rescalledm, b, R as m → mk − 122γ−3 , b → bk2γ−3 , R → Rk− 12γ−3 .

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Clearly, for any given constants m, ω, b and γ, the potentialV (R) uniquely determines the collapse of the prototypegravastar. Depending on the initial value R 0 , the collapse canform either a black hole, a gravastar, a Minkowski, or aspacetime filled with phantom fluid.To avoid any kind of initial horizons, cosmological or event, weimpose R 0 > 2m.Since the potential V (R) is so complicated, it is too difficultto study it analytically. Instead, in the following we shall studyit numerically. Before doing so, we shall present theclassifications of matter as dark energy or phantom energy foranisotropic interior fluid. Taking several values of ω in theintervals −1 < ω < −1/3 and ω > 0, we could not found anycase where the interior dark energy exist.

Fig 4: In this figure we show the intervals of ω for which the weakand strong energy conditions are independent of the coordinate Rand the parameter b. The condition ρ + p r > 0 is violated forω < −1, while the conditions ρ + p t > 0 and ρ + p r + 2p t > 0 aresatisfied for ω < −1 and −1/3 < ω < 0, for any values of R andb. For the others intervals of ω the analysis of the energyconditions depends on a complex relation of R and b.

In order to fulfill the energy condition σ + 2p ≥ 0 of the shelland assuming that p = (1 − γ)σ we must have γ ≤ 3/2. Onthe other hand, in order to satisfy the condition σ + p ≥ 0, weobtain γ ≤ 2. Hereinafter, we will use only some particularvalues of the parameter γ which are analyzed in this work.Table: This table summarizes the classification of matter on thethin shell, based on the energy conditions. The last columnindicates the particular values of the parameter γ, where weassume that σ ≥ 0.Matter Condition 1 Condition 2 γNormal Matter σ + 2p ≥ 0 σ + p ≥ 0 -1 or 0Dark Energy σ + 2p < 0 σ + p ≥ 0 7/4R. Ph. Energy σ + 2p < 0 σ + p < 0 3A. Ph. Energy σ + 2p ≥ 0 σ + p < 0 Not possible

In the following we will discuss three physical possibilities forthe type of system that can be formed from the study of thepotential V (R, m, ω, b, γ):◮ (a) Black hole or dispersion of the matter,

In the following we will discuss three physical possibilities forthe type of system that can be formed from the study of thepotential V (R, m, ω, b, γ):◮ (a) Black hole or dispersion of the matter,◮ (b) Gravastar or normal star,

In the following we will discuss three physical possibilities forthe type of system that can be formed from the study of thepotential V (R, m, ω, b, γ):◮ (a) Black hole or dispersion of the matter,◮ (b) Gravastar or normal star,◮ (c) Black hole or phantom gravastar.

Black Hole or Dispersion of the MatterFor m > m c the potential V (R) is strictly negative. Then, thecollapse always forms black holes. For m = m c , there are twodifferent possibilities, depending on the choice of the initialradius R 0 . In particular, if the star begins to collapse withR 0 > R c , it will approach to the minimal radius R c . Once itreaches this point, the shell will stop collapsing. However, thispoint is unstable and any small perturbations will lead the stareither to expand for ever and leave behind a flat spacetime, orto collapse until R = 0, whereby a Schwarzschild black hole isfinally formed. On the other hand, if the star begins tocollapse with 2m c < R 0 < R c as shown in the followingfigures, the star will collapse until a black hole is formed. Form < m c , the potentials V (R) for each case have a positivemaximal, and the equation V (R, m < m c ) = 0 has twopositive roots R 1,2 with R 2 > R 1 > 0.

For these cases there are also two possibilities, depending onthe choice of the initial radius R 0 :◮ If R 0 > R 2 , the star will first contract to its minimal radiusR = R 2 and then expand to infinity, whereby a spacetimefulfilled by phantom energy fluid is finally formed.

For these cases there are also two possibilities, depending onthe choice of the initial radius R 0 :◮ If R 0 > R 2 , the star will first contract to its minimal radiusR = R 2 and then expand to infinity, whereby a spacetimefulfilled by phantom energy fluid is finally formed.◮ If 2m < R 0 < R 1 , the star will collapse continuously untilR = 0, and a black hole will be finally formed.

Case B:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case C:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case H:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case I:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case K:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case L:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Gravastar or Normal StarIn this case we can see that V (R) = 0 now can have one, twoor three real roots, depending on the mass of the shell. Form > m c we have, say, R i , where R i+1 > R i . If we chooseR 0 > R 3 (for m = m c we have R 2 = R 3 ), then the star will notbe allowed in this region because the potential is greater thanthe zero. However, if we choose R 1 < R 0 < R 2 , the collapsewill bounce back and forth between R = R 1 and R = R 2 . Sucha possibility is shown in the following figures. This is exactlythe so-called ”bounded excursion” model mentioned in VW,and studied in some details in two previous works of ours. Ofcourse, in a realistic situation, the star will emit bothgravitational waves and particles, and the potential shall beself-adjusted to produce a minimum at R = R static whereV (R = R static ) = 0 = V ′ (R = R static ) whereby a gravastar ora normal star is finally formed, although we have found, inthat previous works, that for the gravastar with De Sitter’sinterior, the potential tends to −∞ when R tends to ∞.

Here it is completely different since the potentialnow tends to +∞ when R tends to ∞. Thus, inthe cases studied here we do not have situationswhere the star expands leaving behind a flatspacetime, as in (JCAP 6, 25 (2008), JCAP 11, 10 (2008)).

Case A:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case G:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Black Hole or Phantom GravastarIn this case the potential takes the shape given by figurebellow, from which we can see that V (R) = 0 now has fourreal roots, say, R i , where R i+1 > R i . If we choose R 0 > R 4 ,then again the star will not be allowed in this region becausethe potential is greater than zero. However, if we chooseR 3 < R 0 < R 4 , the collapse will bounce back and forthbetween R = R 3 and R = R 4 , as in the previous case. But, ifwe choose R 2 < R 0 < R 3 , we can note that this region isforbidden because either the potential is imaginary or greaterthan zero. However, if we choose R 1 < R 0 < R 2 , the collapsewill bounce back and forth between R = R 1 and R = R 2 . IfR 0 < R 1 the system will collapse until R = 0, whereby aSchwarzschild black hole is finally formed.

Case J:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Case J:The potential V (R) and the energy conditionsEC1≡ ρ + p r + 2p t , EC2≡ ρ + p r and EC3≡ ρ + p t

Table: Summary of all possible kind of energy of the interior and ofthe shell.Case Interior Energy Shell Energy Figures StructuresA Standard Standard 10 Normal StarB Standard Dark 4 BH/DispersionC Standard R. Phantom 5 BH/DispersionD Dark Standard Interior not foundE Dark Dark Interior not foundF Dark R. Phantom Interior not foundG R. Phantom Standard 11 GravastarH R. Phantom Dark 6 BH/DispersionI R. Phantom R. Phantom 7 BH/DispersionJ A. Phantom Standard 12 Gravastar or BHK A. Phantom Dark 8 BH/DispersionL A. Phantom R. Phantom 9 BH/Dispersion

Conclusions◮ We have studied the problem of the stability of gravastarsby constructing dynamical three-layer models of VW,which consists of an internal phantom fluid, a dynamicalinfinitely thin shell of perfect fluid with the equation ofstate p = (1 − γ)σ, and an external Schwarzschild space.

Conclusions◮ We have studied the problem of the stability of gravastarsby constructing dynamical three-layer models of VW,which consists of an internal phantom fluid, a dynamicalinfinitely thin shell of perfect fluid with the equation ofstate p = (1 − γ)σ, and an external Schwarzschild space.◮ We have shown explicitly that the final output can be ablack hole, a ”bounded excursion” stable gravastar, aMinkowski, or a phantom spacetime, depending on thetotal mass m of the system, the parameter ω, theconstant b, the parameter γ and the initial position R 0 ofthe dynamical shell.

Conclusions◮ We have studied the problem of the stability of gravastarsby constructing dynamical three-layer models of VW,which consists of an internal phantom fluid, a dynamicalinfinitely thin shell of perfect fluid with the equation ofstate p = (1 − γ)σ, and an external Schwarzschild space.◮ We have shown explicitly that the final output can be ablack hole, a ”bounded excursion” stable gravastar, aMinkowski, or a phantom spacetime, depending on thetotal mass m of the system, the parameter ω, theconstant b, the parameter γ and the initial position R 0 ofthe dynamical shell.◮ All these possibilities have non-zero measurements in thephase space of m, b, ω, γ and R 0 , although the region ofgravastars is very small in comparison with that of blackholes.

It is interesting to note that we can have black hole formationeven with an interior phantom energy for any given γ.The results obtained here further confirm our previous conclusion:Even though the existence of gravastars cannot becompletely excluded in these dynamical models, ourresults do indicate that, even if gravastars indeedexist, they do not exclude the existence of blackholes.