Stability Analysis of Lower Dimensional Gravastars in Noncommutative Geometry

Eur. Phys. J. C (2016) 76:641 DOI 10.1140/epjc/s10052-016-4500-3

Regular Article - Theoretical Physics

Stability analysis of lower dimensional gravastars in noncommutative geometry

Ayan Banerjee1,a, Sudan Hansraj2,b 1 Department of Mathematics, Jadavpur University, Kolkata 700032, India 2 Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

Received: 2 September 2016 / Accepted: 10 November 2016 / Published online: 23 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract The Bañados et al. (Phys. Rev. Lett 69:1849, the collapse of matter at or near where the event horizon is 1992), black hole solution is revamped from the Einstein expected to form i.e., alternative configurations of black holes field equations in (2 + 1)-dimensional anti-de Sitter space- could be formed by gravitational collapse of a massive star. time, in a context of noncommutative geometry (Phys. Rev. The quasi-normal modes of thin-shell non-rotating gravas- D 87:084014, 2013). In this article, we explore the exact tars were studied by Pani et al. [4] and they also considered gravastar solutions in three-dimensional anti-de Sitter space the gravitational wave signatures when no horizon is present given in the same geometry. As a first step we derive BTZ such as in the case of gravastars. To the best of the knowledge solution assuming the source of energy density as point-like of the authors no investigations into the wave signature from structures in favor of smeared objects, where the particle√ coalescing gravastars have been made to date. It has been mass M, is diffused throughout a region of linear size α speculated that the gravastars and black holes emit the same and is described by a Gaussian function of finite width rather gravitational wave signatures. than a Dirac delta function. We matched our interior solution In the gravastar model, the interior consists of a segment of to an exterior BTZ spacetime at a junction interface situated the de Sitter geometry, enclosed by a shell of Bose–Einstein outside the event horizon. Furthermore, a stability analysis condensate, all of which is surrounded by a Schwarzschild is carried out for the specific case when χ<0.214 under vacuum but without encountering a horizon. The de Sitter radial perturbations about the static equilibrium solutions. interior with negative pressure favoring expansion is neces- To give theoretical support we are also trying to explore their sary to provide a mechanism to counterbalance the gravita- physical properties and characteristics. tional collapse of the ultra-compact Bose–Einstein condensate (BEC), which itself is assumed to have the most extreme equation of state permissible by causality – that of stiff mat- 1 Introduction ter (p = ρ). Therefore the gravastar is a multilayered structure consisting of three different regions with three different The recent detection of gravitational waves ([3] and refer- equations of state (EOS): ences therein) carried implicit evidence for the existence of black holes since the cataclysmic event generating the wave (I) an internal core (de Sitter) with an EOS: p + ρ = 0, carried the expected signature of a coalescing binary black (II) a thin shell of ultra-stiff matter (BEC) with an EOS: hole system. Nevertheless, the question of what the final fate p =+ρ, of gravitational collapse is remains open. Black holes are (III) an outer vacuum Schwarzschild solution with EOS: p = one possibility but this does not preclude others. The gravas- ρ = 0, tar (gravitational vacuum star) model has been proposed by Mazur and Mottola [5,6], and it has attracted attention as In practice, the Mazur–Mottola model is a static spherically an alternative model to the black hole. The general idea is symmetric model with a five-layer solution of the Einstein preventing horizon (and singularity) formation, by stopping equations including two infinitesimally thin shells endowed with surface densities σ± and surface pressure p±. a e-mail: [email protected] Motivated by the work Visser and Wiltshire [7], one ana- b e-mail: [email protected] lyzed the dynamic stability against spherically symmetric 123 641 Page 2 of 7 Eur. Phys. J. C (2016) 76 :641 perturbations using the Israel thin-shell formalism while fore interesting to reduce the number of spatial dimensions Carter in [8] has extended gravastar stability with general- by 1 and to study general relativity in the simpler context ized exteriors (Reissner–Nordstrom). In Ref. [9]gravastar of (2 + 1) dimensions in the hope that it has the potential solutions have been studied within the context of nonlinear to generate non-trivial and valuable insight into some of the electrodynamics. Later some simplifications and important conceptual issues that arise in the (3 + 1) dimensional case, aspects of the gravastar have been studied in-depth in [10– especially in regard to the question of quantizing gravity. 14]. Moreover, the limits on gravastars and how an external Seminal contributions in this regard were made by Witten observer can distinguish it from a black hole have been stud- [41–43] who showed an equivalence between (2 + 1) gravity ied in [15,16]. theory and Chern–Simons theory. The usefulness of (2 + 1)- After the theoretical discovery of radiating black holes by dimensional gravity motivated us to show that neutral gravas- Hawking [17,18], based on quantum field theory, the ther- tars solutions do exist to avoid the event horizon formation, modynamical properties of black holes have been studied which may be considered as an alternative to BTZ in the extensively. Since this theoretical effort first disclosed the context of noncommutative geometry. mysteries of quantum gravity, considerable interest in this The motivation for this investigation is clear from the problem has developed in theoretical physics. It is generally above summary on the aspect of an exact gravastar solu- believed that spacetime as a manifold of points breaks down tion in the context of NC in (2 + 1)-dimension. Our paper is at very short distances of the order of the Planck length. organized as follows. In Sect. 2 we construct BTZ black hole In these circumstances noncommutative geometry [19,20] solution from an exact solution of the Einstein field equa- plays a key attribute in unraveling the properties of nature tions in the context of noncommutative geometry and spec- at the Planck scale. From the fundamental point of view ifying the mass function we present the structural equations of noncommutative geometry there is an interesting inter- of gravastar. In Sect. 3 we discuss the matching conditions play between mathematics, high energy physics as well as at the junction interface and determine the surface stresses. cosmology and astrophysics. In a noncommutative space- In Sects. 4 and 5 we investigate the linearized stability of time the coordinate operators on a D-brane [20,21] can be gravastars and determine the stability regions of the transi- encoded by the commutator [ˆxμ, xˆν] = iϑμν, where xˆ and tion layer. Finally, in Sect. 6 we draw the conclusions. iϑμν are the coordinate operators and an antisymmetric tensor of dimension (length)2, which determines the fundamental cell discretization of spacetime. As Smailagic et al. have 2 Interior geometry shown [22] that noncommutativity replaces point-like structures by smeared objects in flat spacetime. Thus it is reason- We will be concerned here the interior spacetime described able to believe that noncommutativity could eliminate the by the line element for a static spherically symmetry and time divergences that normally appear in general relativity that independent metric in (2 + 1) dimensions in the following appears in various form. As discussed in Ref. [23] the smear- form: ing effect is mathematically implemented as a substitution 2 rule: position Dirac delta function is substituted everywhere Φ( ) dr √ ds2 =−e2 r dt2 + + r 2dθ 2, (1) using a Gaussian distribution of minimal length α. 1 − 2m(r)/r In the same spirit, Nicolini et al. [23–25] have investigated the behavior of a noncommutative radiating Schwarzschild where Φ(r) and m(r) are arbitrary functions of the radial Φ( ) black hole. There is a lot of noncommutative effects have coordinate, r. Here the “gravity√ profile” factor r is related been performed to extend the solution for higher dimensional with the relationship A = 1 − m(r)/rΦ (r), which rep- black hole [26], charged black hole solutions [27,28] and resents the locally measured acceleration due to gravity charged rotating black hole solution [29,30]. A number of [44,45]. The convention used is that Φ (r) is positive or neg- studies have been performed in these directions where space- ative for an inwardly gravitational attraction or an outward time is commutative [31–34]. In the same context wormhole gravitational repulsion and m(r) can be interpreted as the solutions have been studied in [35–37]. Recently, Lobo and mass function. Garattini [38] showed that a noncommutative geometry back- We take the matter distribution to be anisotropic in nature ground is able to account for exact gravastar solutions and and therefore the stress-energy tensor for an anisotropic mat- studied the linearized stability. Gravastar solutions in lower ter distribution is provided by dimensional gravity have been studied in [39,40] in an anti- de Sitter background spacetime. Tij = (ρ + p⊥)ui u j + p⊥gij + (pr − p⊥)Xi X j , (2) It is well known that general relativity and its modified i counterparts are highly non-trivial systems to investigate in where u is the 3-velocity of the fluid and Xi is the unit space- the physically realistic four spacetime dimensions. It is there- like vector in the radial direction. ρ(r), pr (r), and p⊥(r) 123 Eur. Phys. J. C (2016) 76 :641 Page 3 of 7 641 represent the energy density, radial pressure and tangential having the minimal width, Gaussian, mass/energy distribu- pressure, respectively. tion The Einstein field equations Gμν + gμν =8πTμν,for M r 2 the spacetime given in Eq. (1) together with the energy- ρ = exp − , (10) momentum tensor given in Eq. (2), rendering G = c = 1, 4πα 4α provides the following relationships: where M is the total mass of the source. This is due to rm − m the coordinate coherent states approach to noncommuta- 8πρ + = , (3) r 3 tive geometry with the noncommutative parameter θ being a ∼ 2 Φ 2m small ( Planck length ) positive number. 8πp − = 1 − , (4) =−ρ r r r By solving the Einstein equations with an EOS pr , ( − ) as a matter source, we have the following relationship (see 2m 2 rm m 8πp⊥ − = 1 − Φ + Φ − Φ , Ref. [2]): r r 2 2 (5) 2Φ − r 2 e =−A + 2Me 4α − r , (11) In addition, we have the conservation equation in (2 + 1) √r →∞ dimensions: where A is an integration constant. In the limit α , Eq. (11) is reduced to the BTZ black hole where the constant 1 term A plays the role of the mass of the BTZ black hole, i.e., (ρ + p ) Φ + p + (p − p⊥) = 0, (6) r r r r A = M. In order to proceed with our investigation, we choose a where is the cosmological constant and Φ are arbitrary specific mass function m(r), for closing the system. For this functions of the radial coordinate r.Here denotes differen- purpose, we are now interested in the noncommutative geom- tiation with respect to the radial parameter r. Although we etry inspired mass function (see Refs. [46,47]) in the follow- shall not invoke isotropic particle pressure, it is interesting ing form: to note that the isotropy Eq. (4)=(5) M m˜ r 2 m = γ ,χ2 , (12) (m˜ −2)/2 Φ π 2 2M Φ + Φ 2 − r(m − 1) − m = 0, (7) r 2 χ 2 = 2/α γ a ; where M and b x is the Euler lower Gamma ostensibly nonlinear in Φ, may be reduced to the linear form function defined by a x du 1 γ ; ≡ a/b −u . y − r(m − 1) − m y = 0(8)x u e (13) r 2 b 0 u For a BTZ black hole, m˜ = 2, we obtain the following expres- by making the change of variables e2Φ(r) = y2(r). Equation sion for the mass function: (8) may be solved explicitly by r2/4θ 2 − r m(r) = M e t t = M − − . m − 1 m d 1 exp α (14) y = c exp − dr + c (9) 0 4 1 r r 2 2 At the origin, m(0) = 0, which is consistent with the solution where c1 and c2 are integration constants that may be settled of Eq. (12) and we notice that the parameter χ plays a critical by considering the boundary conditions. For stellar distribu- role in determining the horizons. tions is ignored and so (9) provides an algorithm to detect An interesting feature of the solution is the horizon. Corre- all static isotropic perfect fluid solutions in (2 + 1) dimen- sponding to A = M giveninEq.(11), and letting the function sions. Once a suitable form for m(r) is selected, Φ can be gtt(rh) = 0, gives the event horizon(s) which is depicted in determined (theoretically) and hence the density and pres- Fig. 1 for different values of χ. Thus we find three possible sure may be obtained to complete the model. But we shall cases [2]: not pursue these ideas here as we shall require anisotropic √ particle pressure for our model. (i) for M > M0 = 0.214 α there are two horizons i.e., χ> . We are going to solve the resulting Einstein’s equations, when 0 214; √ for static spherically symmetric perfect fluids in (2 + 1) (ii) for M = M0 = 0.214 α with one degenerate horizon dimensions, with a maximally localized source of energy i.e., when χ = 0.214; 123 641 Page 4 of 7 Eur. Phys. J. C (2016) 76 :641

3 of stress energy at the junction surface is given by ξ μ(τ, θ) ( (τ), (τ), θ) 2.5 = t a ⎛ ⎞ χ = 0.1 2 1 − 2m± +˙a2 μ dt da ⎝ a ⎠ χ = 0.214 U± = , , 0 = , a˙, 0 , (17) 1.5 dτ dτ − 2m± 1 a α

tt 1 g √ where the (±) correspond to the exterior and interior space-

0.5 χ = 0.8 times, with m± deﬁned as interior and exterior mass, respectively. 0 ± Also the normal unit vector (nμ ) to the boundary can be μ μ deﬁned as (with n nμ = 1 and U nμ = 0) −0.5 ⎛ ⎞ −1 1 − 2m± +˙a2 0 1 2 3 4 5 6 7 8 ± ⎝ a ⎠ nμ = −˙a, , 0 . (18) √r − 2m± α 1 a

Fig. 1 The function gtt cuts the r-axis gives the event horizons for At the junction surface the components of the extrinsic cur- different values of χ =0.8,χ = 0.214 and χ = 0.1, respectively vature tensor read 2 ν α β < < χ< . ± ∂ x ν± ∂x ∂x (iii) for 0 M M0 no horizon for 0 214, K =−nν + , (19) ij ∂ξi ∂ξ j αβ ∂ξi ∂ξ j where M0 is the existence of a lower bound for a black hole ξ i τ, θ mass and represents its ﬁnal state at the end of Hawking where =( ) represent the coordinates on the shell. Here, evaporation process. It is also clear from Fig. 1 that below in general Kij is discontinuous at the junction surface, the the minimal mass there is no black hole. discontinuity in the second fundamental forms is deﬁned as K = + − −. 3 Matching at junction interface and surface stresses ij Kij Kij (20)

For the specific gravastar model we match the interior gravas- Then using the Lanczos equation the Einstein equations lead tar geometry, given in Eq. (1), with an exterior geometry to the following form: associated with the BTZ solution, Si =− 1 Ki − δi Kk , −1 (21) ds2 =− −M − r 2 dt2+ −M − r 2 dr 2 + r 2dθ 2, j 8π j j k (15) Si where j is the surface stress-energy tensor on , with the Ki both interior and exterior matched at the junction surface , discontinuity of the extrinsic curvature defined by j . situated outside the event horizon, a > rh. As the gravastar Now, let us calculate the non-trivial components of the solution does not possess a singularity at the origin and has extrinsic curvature for the interior spacetime (1) and the exte- no event horizon, we are interested in the case χ<0.214 rior BTZ solution (15), given by and there is no event horizon yielding a solution. − +¨ Since the outer solutions have a zero stress-energy, while τ+ a a K τ = √ , (22) at the junction surface , both will have a non-zero stress- −M − a2 +˙a2 energy. The junction hypersurface is a timelike hypersurface μ defined by the parametric equation f (x (ξ i )) = 0, where ξ i m − m +ä τ− a2 a =(τ, θ) represents the intrinsic coordinates on the hypersur- K τ = , (23) − 2m(a) +˙2 face and τ is the proper time, respectively. 1 a a In order to proceed one can write the line element for and intrinsic metric to as 1 θ+ = − − 2 +˙2, 2 2 2 2 K θ M a a (24) ds =−dτ + a dθ . (16) a For the purpose of this paper we matched our interior geome- θ− 1 2m(a) K = 1 − +˙a2, (25) try by the exterior BTZ solution; the three velocity of a piece θ a a 123 Eur. Phys. J. C (2016) 76 :641 Page 5 of 7 641 where the prime denotes a derivative with respect to r and a 4 Stability analysis dot stands for d/dτ. Therefore, the stress-energy tensor (21) in the most general form of the surface energy density σ and In this section, we investigate the stability of the gravastar P Si (−σ, P) the surface pressure is j =diag . solution in a perturbative treatment of the shell dynamics, After some algebraic manipulation and using the Lanc- more precisely: by linearized stability of the solutions. For zos equation, we obtain the energy density and the surface that we rearrange Eq. (26) to obtain the thin-shell equation pressures given by of motion, 2 1 2m(a) a˙ + V (a) = 0, (31) σ =− −M − a2 +˙a2 − 1 − +˙a2 , 8πa a ⎡ where the potential V (a) is given by (26) − − 2 +˙2 +¨ 1 ⎣ M 2 a a a P = √ 2 8πa −M − a2 +˙a2 G1(a) + G2(a) G1(a) − G2(a) V (a) = − −[4πaσ(a)]2 , ⎤ 2 16πaσ(a) 1 − m − m +˙a2 +ä (32) − a ⎦ . (27) ( ) 1 − 2m a +˙a2 a 2m(a) 2 where we have used the notation G1(a) = 1 − +˙a σ √ a Note that by definition the surface tension has the oppo- G ( ) = − − 2 +˙2 P and 2 a M a a , respectively. site sign to the surface pressure .Now,weshallalsouse = π σ Si =[ μ ν]+ Now using the surface mass of the thin shell, ms 2 a , the conservation identity in the form j|i Tμνe j n −, + allows one to write the potential in the form where [X]− represents the discontinuity across the surface interface. The method is developed in Refs. [44,45]. To study T 2 2 the stability of the solutions under perturbations we encroach V (a) = S − − (2ms) , (33) ν 4ms on the momentum flux term Fμ = TμνU in the right hand side corresponding to the net discontinuity. With the defini- with tions of the conservation identity one can convert this into conserved energy and momentum of the surface stresses at G (a) + G (a) G (a) − G (a) S = 1 2 and T = 1 2 . (34) the junction interface. 2 2 It is useful to introduce the conservation equation in a form that relates the surface energy and surface pressure with the For the stability analysis of the static solutions at a0 under work done by the pressure and the energy flux on the shell the radial perturbations, we consider the Taylor expansion Si =− σ˙ + a˙ (σ + P) ( ) given by the equation τ|i a . Then the of the potential function V a around a0 up to second order, conservation identity provides the following relationship: given by 1 σ =− (σ + P) , (28) 1 a V (a) = V (a ) + V (a )(a − a ) + V (a )(a − a )2 0 0 0 2 0 0 σ˙ where σ = ˙ . Now taking into account Eqs. (26) and (27), 3 a + O (a − a0) , (35) Eq. (28) has⎡ the form ⎤ −M +˙a2 −ä 1 − 3m + m +˙a2 −ä where the prime denotes the derivative with respect to a. σ = 1 ⎣√ − a ⎦ , π 2 − − 2+˙2 The existence and stability of the static solutions depend 8 a M a a 1 − 2m +˙a2 a upon the inequalities that V (a0) has local minimum at a0 (29) and V (a0)>0. The first derivative of the potential is given by and at the static solution a it reduces to 0 T T ⎡ ⎤ ( ) = S − − , 3m V a 8msms (36) − 1 − + m (a0) 8ms ms 1 ⎣ M a0 ⎦ σ (a0) = − , π 2 2m 8 a0 −M − a2 1 − 0 a0 using the conditions V (a0) = 0, we can write the equilib- (30) rium relationship as which plays a crucial role in determining the stability regions T T ≡ = 1 S − . X ms (37) as we consider below. 8ms 8ms ms 123 641 Page 6 of 7 Eur. Phys. J. C (2016) 76 :641

Finally, the second derivative of the potential reads 5 Region of stability

( ) = S − − 2 V a 8msms 8ms We shall in this section consider the static solution for the T T T 2 stability analysis and deduce the stability region by consid- −1 + . (38) ering the inequalities Eqs. (43) and ( 44). For this purpose we 8 ms ms ms shall impose a positive surface energy density σ>0, which To evaluate a static equilibrium conﬁguration as regards indicates m(a)

2π m = (σ + P) η. (39) s a

Here the parameter η, which is interpreted as the subluminal speed of sound, has been used to present the stability regions without using the surface equation of state. Now, evaluated for a static equilibrium conﬁguration for the stability and taking into account Eq. (38), with V (a0)> 0, we have

dσ 2 η0 >, (40) da a=a0 by using Eq. (39), where η0 = η(a0) and , for notational simplicity, we define a simply behaving function of the form 1 2 2 ≡ X − Y , Fig. 2 The dimensionless parameter L = dσ /da|a corresponding to 2 (41) √ 0 2π a0 the value of χ =0.16with α = −0.02. The figure is shown for the specific case when m(a) < M/2 where S 1 T T T 2 Y = − + . (42) 8 64 ms ms ms

In order to analyze the stable equilibrium regions of the solution we adopt the following inequalities:

dσ 2 η0 > , if > 0, (43) da a=a0

dσ 2 η0 < , if < 0, (44) da a=a0 with the deﬁnition

−1 dσ 2 ≡ . √ (45) χ α − da a=a0 Fig. 3 The stability region for =0.16and = 0.02. The stability region is depicted below the surfaces, which is sufﬁcient close to the event horizon 123 Eur. Phys. J. C (2016) 76 :641 Page 7 of 7 641

(44). To justify our assumption we use a graphical represen- References tation due to the complexity of the expression , which is plotted in Fig. 2 for√ the specific case of m(a) = M/2 and when 1. M. Bañados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 χ = 0.16 with θ =−0.02. From Fig. 2 it is clear that the (1992) 2. F. Rahaman et al., Phys. Rev. D 87, 084014 (2013) solution does not correspond to any horizon. 3. B.P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016) In order to study the stability region we use the graphi- 4. P. Pani et al., Phys. Rev. D 81, 084011 (2010) cal representation (Fig. 3) for the case when χ = 0.16. We 5. P. O. Mazur, E. Mottola: arXiv:gr-qc/0109035 have examined the stability of the model based on the speed 6. P.O. Mazur, E. Mottola, Proc. Nat. Acad. Sci. 111, 9545 (2004) 7. M. Visser, D.L. Wiltshire, Class. Quantum Gravity 21, 1135–1152 of sound which should lie within the limit (0, 1]. 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