Gravastars with higher dimensional spacetimes
aa r X i v : . [ g r- q c ] M a y Gravastars with higher dimensional spacetimes
Shounak Ghosh a , Saibal Ray b , Farook Rahaman c , B.K. Guha a a Department of Physics, Indian Institute of Engineering Science and Technology,Shibpur, Howrah, West Bengal, 711103, India b Department of Physics, Government College of Engineering and Ceramic Technology,Kolkata 700010, West Bengal, India c Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India a Department of Physics, Indian Institute of Engineering Science and Technology,Shibpur, Howrah, West Bengal, 711103, India
Abstract
We present a new model of gravastar in the higher dimensional Einsteinianspacetime including Einstein’s cosmological constant Λ. Following Mazurand Mottola (2001, 2004) we design the star with three specific regions, asfollows: (I) Interior region, (II) Intermediate thin spherical shell and (III)Exterior region. The pressure within the interior region is equal to the nega-tive matter density which provides a repulsive force over the shell. This thinshell is formed by ultra relativistic plasma, where the pressure is directlyproportional to the matter-energy density which does counter balance therepulsive force from the interior whereas the exterior region is completelyvacuum assumed to be de Sitter spacetime which can be described by thegeneralized Schwarzschild solution. With this specification we find out aset of exact non-singular and stable solutions of the gravastar which seemsphysically very interesting and reasonable.
Keywords:
General relativity; Dark Energy; Gravastar
1. Introduction
In general relativity of Einstein there is an inherent feature of singularityat the end point of gravitationally collapsing system and has been remains ∗ Corresponding author.
E-mail addresses: [email protected] (SG), [email protected](SR), [email protected] (FR), [email protected] (BKG).
Preprint submitted to Annals of Physics May 22, 2018 n embarrassing situation to the astrophysical community. To overcome thisodd phase of a stellar body where all the physical laws break down, Mazurand Mottola [1, 2] proposed a new model considering the gravitationallyvacuum star which was termed in brevity as Gravastar, that brings up a newarena in the gravitational system. They generated a new type of solution tothis system of gravitational collapse by extending the idea of Bose-Einsteincondensation by constructing gravastar as a cold, dark and compact objectof interior de Sitter condensate phase surrounded by a thin shell of ultrarelativistic matter whereas the exterior region is completely vacuum, i.e. theSchwarzschild spacetime is at the outside. The shell is very thin but of finitewidth in the range R < r < R , where R = R and R = R + ǫ are theinterior and exterior radii of the gravastar, ǫ represents the thickness of theshell with ǫ ≪
1. With this unique specification we can divide the entiresystem of gravastar into three specific segments based on the equation ofstate (EOS) as follows: (I) Interior: 0 ≤ r < R , with EOS p = − ρ , (II)Shell: R ≤ r ≤ R , with EOS p = + ρ , and (III) Exterior: R < r , withEOS p = ρ = 0.The abovementioned model of gravastar has been studied by researcherswhich opened up a new challenges in the gravitational research to obtaina singularity free solution of the Einstein field equations. Therefore, it issupposed to be an alternative solution of black hole and has been studied byseveral authors in different context of astrophysical systems [3, 4, 5, 6, 7, 8,9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].The negative matter density in the interior region creates a repulsivepressure acting radially outward from the centre of the gravastar ( i.e. r =0) over the shell whereas the shell of positive matter density provides thenecessary gravitational pull to balance this repulsive force within the interior.It is assumed that the dark energy (or the vacuum energy) is responsible forthis repulsive pressure from the interior. In a general consideration, the EOS p = − ρ is suggesting that the repulsive pressure is an agent, responsiblefor accelerating phase of the present universe and is known as the Λ-darkenergy [21, 22, 23, 24, 25]. In literature this EOS is termed as a ‘false vacuum’,‘degenerate vacuum’, or ‘ ρ -vacuum’ [26, 27, 28, 29]. Therefore, in this contextone can note that gravastar may have some connection to the dark star [30,31, 32, 16].The EOS for the shell p = ρ represents essentially a stiff fluid modelas conceived by Zel’dovich [33] in connection to the cold baryonic universe.The idea has been considered by several scientists for various situations in2osmology [34, 35] as well as astrophysics [36, 37, 38].The so-called cosmological constant Λ was introduced by Einstein in hisfield equations to obtain a static and non-expanding solutions of the universe.In his model this constant with the right sign could produce a repulsivepressure to exactly counter balance the gravitational attraction and thuscould provide a statical model of the universe.After the experimental verification of expanding universe by Edwin Hub-ble between 1922 to 1924 [39] and the success of FLRW cosmology madeEinstein realize that the universe has been expanding with an acceleration.That is why later on Einstein discarded the cosmological constant from hisfield equation. However, though it is abandoned by Einstein but for thephysical requirement to describe one-loop quantum vacuum fluctuations, theCasimir effect [40], cosmological constant had to appear once again in thetheory with a form as T ij = Λ g ij / πG , where T ij and g ij are the stress en-ergy tensor and the metric tensor respectively and G is the usual Newtonianconstant.Recent observations conducted by WMAP suggests that 73% of the to-tal mass-energy of the universe is dark energy [41, 42]. It is believed thatthis dark energy plays an important role for the evolution of the universeand in order to describe the dark energy scientists have recall the erstwhilecosmological constant. Therefore, in the modern cosmology this cosmolog-ical constant is treated as a strong candidate for the dark energy which isresponsible for the accelerating phase of the present universe.It is observed that in superstring theory spacetime is considered to beof dimensions higher than four. To be consistent with the physically ac-ceptable models the 4-dimensional present spacetime fabric is assumed to beself-compactified form of manifold with multidimensions or extra spatial di-mensions [43, 44, 45, 46, 47, 48]. Following this aspect very recently Bhar [49]has proposed a charged gravastar in higher dimension by admitting confor-mal motion and later on worked out a problem by Ghosh et al. [20] withoutadmitting the conformal motion in the framework of Mazur and Mottolamodel. Some authors [50, 51, 52] have studied higher dimensional works ad-mitting one parameter Group of Conformal motion. Usmani et al. [16] havealso found solution of neutral gravastar in higher dimension without admit-ting the conformal motion. These works provide an alternative solution tothe static black holes. The present study on gravastar basically is an exten-sion of the work of Usmani et al. [16] and its generalization to the higherdimensional spacetime in presence of the cosmological constant (Λ).3herefore, the main motivation of this work is to study the effects of thecosmological constant for construction of gravastars and also to study thehigher dimensional effects, if any. The present investigations are based onthe plan as follows: The background of the model has been implementedthrough the Einstein geometry in Sec. 2, whereas the solution of interiorspacetime, the thin shell and exterior spacetime of the gravastar have beendiscussed in Sec. 3. We have shown the matching condition in Sec. 4 anddiscussed the junction conditions for the different regions of the gravastar inSec. 5. Some physical features of the model, viz. the proper length, Energy,Entropy are explored in the Sec. 6 which followed by the discussions andconcluding remarks in Sec. 7.
2. The Einsteinian relativistic spacetime geometry
The Einstein-Hilbert action coupled to matter is given by I = Z d D x √− g (cid:18) R D πG D + L m (cid:19) , (1)where the curvature scalar in D -dimensional spacetime is represented by R D , with G D is the D -dimensional Newtonian constant and L m denotes theLagrangian for the matter distribution. We obtain the following Einsteinequation by varying the above action with respect to the metric G Dij = − πG D T ij , (2)where G Dij denotes the Einstein’s tensor in D -dimensional spacetime.The interior of the star is assumed to be perfect fluid type and can begiven by T ij = ( ρ + p ) u i u j + pg ij , (3)where ρ represents the energy density, p is the isotropic pressure, and u i isthe D -velocity of the fluid.Here in the present study it is assumed that the gravastars in higher di-mensions have the D -dimensional spacetime with the structure R XS XS d ( d = D − S and the time axis is rep-resented by R . For this purpose, we consider a static spherically symmetricmetric in D = d + 2 dimension as ds = − e ν dt + e λ dr + r d Ω d , (4)4here d Ω d is the linear element of a d -dimensional unit sphere, parameterizedby the angles φ , φ , ......, φ d as follows: d Ω d = dφ d + sin φ d [ dφ d − + sin φ d − { dφ d − + ......... + sin φ ( dφ +sin φ dφ ) ....... } ].Now the Einstein field equations for the metric (4), together with theenergy-momentum tensor in presence of the non-zero cosmological constantΛ, yield − e − λ " d ( d − r − dλ ′ r + d ( d − r = 8 πG D ρ + Λ , (5) e − λ " d ( d − r + dν ′ r − d ( d − r = 8 πG D p − Λ , (6) e − λ " ν ′′ − λ ′ ν ′ ν ′ − ( d − λ ′ − ν ′ ) r + ( d − d − r − ( d − d − r = 8 πG D p − Λ , (7)where ‘ ′ ’ denotes differentiation with respect to the radial parameter r . Fol-lowing geometrical unit system we have assumed c = 1 throughout the paper.The general relativistic conservation of energy-momentum, T ij ; j = 0, canbe expressed in its general form with D -dimension as12 ( ρ + p ) ν ′ + p ′ = 0 . (8)In the next Sec. 3 we shall formulate special explicit forms of the energyconservation equations for all the three regions, viz. interior, intermediatethin shell and exterior spacetimes.
3. Modelling a gravastar under general relativity
The interior region of the gravastar is so designed that the negative pres-sure which is acting radially outward from the central part of the star couldbalance the inward gravitational pulling from the shell of the spherical sys-tem. To fulfil this criterion, we choose the EOS for the interior region in thefollowing form [1], p = − ρ. (9)5sing Eq. (8) and the above EOS (9), we obtain p = − ρ = − ρ c , (10)where ρ c is the constant density of the interior region.Using Eq. (10) in the field equation (5), one obtains the solution of λ as e − λ = 1 − πG D ρ c d ( d + 1) r − r d ( d + 1) + C r − d , (11)where C is an integration constant. We observe from the fourth factor ofthe above expression that there involves a constraint d = 1 which means wemust consider d ≥
2. Hence the solution to be regular at r = 0, one candemand for C = 0. Thus essentially we get e − λ = 1 − πG D ρ c d ( d + 1) r − r d ( d + 1) . (12)The variation of e − λ with respect to the radial coordinate r is plotted inFig. 1. Figure 1: Variation of the e − λ with the radial coordinate r (km) for different dimensionsin the interior region where the specific legends used are shown in the respective plots Now by employing Eq. (9) in Eqs. (5) and (6), we obtainln k = λ + ν ⇒ e ν = ke − λ , (13)6here k is a constant of integration.Thus we have the following interior solutions for the metric potentials λ and ν as follows ke − λ = e ν = k " − πG D ρ c d ( d + 1) r − d ( d + 1) r , (14)From Eq. (10) it is observed that the matter density remains constantover the entire interior spacetime. Thus we can calculate the active gravita-tional mass M ( r ) in higher dimensions as M ( r ) = Z R = R π d +12 Γ (cid:16) d +12 (cid:17) r d ρ c dr = π d +12 ρ c ( d + 1)Γ (cid:16) d +12 (cid:17) R d +1 , (15)where R is the internal radius of the gravastar. So the usual gravitationalmass for a d -dimensional sphere can be expressed by Eq. (15). This explicitlyshows that the active gravitational mass M ( r ) is directly dependent on thefollowing two factors - the radius R as well as the matter density ρ . Here we assume that the thin shell contains ultra-relativistic fluid of softquanta and obeys the following EOS p = ρ. (16)By inspection we note that it is very difficult to obtain a general solu-tion of the Einstein field equations within the shell of non-vacuum region.Therefore, we try to find an analytic solution within the thin shell limit,0 < e − λ ≡ h <<
1. To do so we set h to be zero to the leading order. Underthis approximation, the field Eqs. (5) - (7) along with the above EOS, canbe reformatted in the following form − h ′ r + ( d − r = 8 πG D ρd + Λ d , (17) − ( d − r = 8 πG D pd − Λ d , (18) h ′ ν ′ d − h ′ r − ( d − d − r = 8 πG D p − Λ . (19)7ow using Eqs. (17) and (18), we find out an expression for h as h = e − λ = k + 2( d −
1) ln r − r d , (20)where k is an integration constant. It is to note that the range of r lieswithin the thickness of the shell, i.e. R = R and R = R + ǫ , where ǫ thethickness of the shell ( ǫ << ν ) can be found as e − ν = k " r d r − d + d , (21)where k is the integration constant.Again using Eqs. (8) and (21) we get the pressure and matter densitywithin the shell of the gravastar as p = ρ = ρ e − ν = ρ k " r d r − d + d . (22)The variation of p = ρ with respect to the radial coordinate r is plottedin Fig. 2. Figure 2: Variation of the pressure p = ρ (km − ) of the ultra relativistic matter in theshell with the radial coordinate r (km) for different dimensions
8n the intermediate thin shell, therefore, by virtue of Eq. (22) the en-ergy conservation equation (8) takes the special explicit form in the pressuregradient as follows: p ′ = 2 ρ k " r d r − d + d r r − d + d − dr . (23) The EOS for the exterior region is defined as p = ρ = 0. In higher dimen-sions one can expect that the exterior solution is nothing but a generalizationof the Schwarzschild solution. Now, following the work of Tangherlini [53]this can be obtained as ds = − − µr d − − r ( d + 1) d ! dt + − µr d − − r ( d + 1) d ! − dr + r d Ω d , (24)where µ is a constant and is given by µ = 16 πG D M / Ω d in higher dimension,with M as the mass of the gravastar and Ω d as the area of a unit d -spherewhich is defined by Ω d = 2 π ( d +12 ) / Γ( d +12 ).However, Eq. (24) appears to have a cosmological constant p = − ρ andhence due to the EOS p = ρ = 0, the above metric eventually reduces to ds = − (cid:18) − µr d − (cid:19) dt + (cid:18) − µr d − (cid:19) − dr + r d Ω d . (25)
4. Matching condition for finding out expressions of constants
For the construction of the gravastar the metric potential g rr must becontinuous at the interface between the core and the shell at r = R (interiorradius) and also at junction of the shell and the exterior region at r = R (exterior radius). Using this matching condition one can obtain the valuesof the integration constants k and k as shown below.To find the value of k we gave matched the metric potential at r = R .Using Eq. (12) and (20)we have the value of k as k = 1 − πG D ρ c d ( d + 1) R + 2Λ R d + 1 − d −
1) ln R . (26)Again from Eqs. (21) and (24) at r = R we get the value of k as k = 2Λ R − d + dR d (cid:18) − µR d − − R ( d +1) d (cid:19) , (27)9e choose the numerical values of µ = 3 . M ⊙ , R =10 km and R =10.009 km.
5. Junction condition
There are three regions in the gravastar configuration, viz. the interiorregion, thin shell and exterior region. The shell joins the interior and exteriorregions at the junction interface. The metric coefficients are continuous atthe shell, however we do not have any confirmation of the continuity of theirderivatives.Following the condition of Darmois-Israel [54, 55] now we provide theintrinsic surface stresses at the junction interface in the form S ij = − π ( κ ij − δ ij κ kk ) , (28)where k ij = K + ij − K − ij , that shows the discontinuity of the extrinsic curvaturesor second fundamental forms. Here the signatures − and + describes theinterior and exterior boundaries respectively of the gravastar.Now this extrinsic curvature connect the two sides of the thin shell as K ± ij = − n ± ν " ∂ X ν ∂ξ ∂ξ j + Γ ναβ ∂X α ∂ξ i ∂X β ∂ξ j | S , (29)where n ± ν is the unit normals to the surface S can be defined as n ± ν = ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g αβ ∂f∂X α ∂f∂X β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∂f∂X ν , (30)with n ν n ν = 1.Following the methodology of Lanczos [56] one can obtain the surfaceenergy-momentum tensor on the thin shell as S ij = diag [ − Σ , p θ , p θ ..., p θ d ],where Σ is the surface energy density and p θ = p θ = ... = p θ d = p t are thesurface pressures which respectively can be determined byΣ = − d πR q f (31) p t = − d − d Σ + f ′ π √ f . (32)10sing the above equations one can obtainΣ = − d πR vuut − µR d − − R Λ( d + 1) d − vuut − πG D ρ c d ( d + 1) R − R d ( d + 1) (33)and p t = 18 πR ( d − − ( d − µ R d − − R d +1 q − µR d − − R Λ( d +1) d − ( d − − πG D ρ c d +1 R − R d +1 q − πG D ρ c d ( d +1) R − R d ( d +1) . (34)Now, it is easy to find out the mass m s of the thin shell as m s = 2 π d +12 Γ( d +12 ) R d Σ = Ω ′ d R d vuut − πG D ρ c R d ( d + 1) − R d ( d + 1) − vuut − µR d − − R Λ d ( d + 1) , (35)where Ω ′ d = π d +12 Γ( d +12 ) d πR .Using Eq. (35) we can determine the total mass of the gravastar in termsof the mass of the thin shell in the form µ = m s Ω ′ d R vuut − πG D ρ c d ( d + 1) R − R d ( d + 1) − m s Ω ′ d R d + 16 πG D ρ c d ( d + 1) R d +1 . (36)
6. Physical features of the models
As we consider the thickness of the intermediate shell is very small (0 <ǫ ≪ r = R and r = R + ǫ joining the region I and region III respectively, where R describes the phase boundary of the region I.Using the Eq. (22) we calculate the energy within the shell as E = Z R + ǫR π d +12 Γ (cid:16) d +12 (cid:17) r d ρdr = π d +12 Γ (cid:16) d +12 (cid:17) ρ k Z R + ǫR r d r − d + d ! . (37)11o solve the above integration of Eq. (37) over the limit R to R + ǫ , letus form a special differential equation in terms of the parameter F ( r ) whichcan be represented as dF ( r ) dr = r d r − d + d . (38)Now using the above consideration we can solve the the integration asfollows Z R + ǫR dF ( r ) dr dr = [ F ( r )] R + ǫR = F ( R + ǫ ) − F ( R ) ≈ ǫ dFdr ! r = R . (39)As ǫ ≪
1, so we consider up to the first order term of the Taylor seriesexpansion for the expression in Eq. (39).Therefore, combining Eqs. (37) and (39) we get E ≈ π d +12 Γ (cid:16) d +12 (cid:17) ǫρ k R d R − d + d ! , (40)where in the square bracket of Eqs. (37) and (40) we have used the factorΩ d as the surface area of a unit d -sphere which has already been defined byΩ d = 2 π d +12 / Γ( d +12 ).From Eq. (40) we observe that the energy within the shell is directlyproportional to the thickness of the shell ( ǫ ).The variation of E with r for different dimensions is shown in Fig. 3. Onecan note that the energy is increasing from the interior boundary to the exte-rior boundary. This is clearly indicating that the shell is getting harder fromthe interior to exterior boundary, which suggests that the exterior boundaryis more dense than the interior boundary as obtained in Fig. 2. The plotalso indicates that energy is decreasing as the dimension increases. The entropy within the shell in higher dimensions can be obtained usingthe following equation S = Z R + ǫR π d +12 Γ( d +12 ) r d s ( r ) √ e λ dr, (41)12 igure 3: Variation of the energy E (km) of the shell with r (km) for different dimensions where s ( r ) is the entropy density, following the prescription Mazur and Mot-tola [1] we can write it as follows s ( r ) = ξ k B T ( r )4 π ¯ h = ξ k B ¯ h r p π (42)where ξ is a dimensionless constant and T is the local temperature whichdepends on the radial coordinate r .By using the above equation Eq. (42) in Eq. (41), we have S = 2 π d +12 Γ( d +12 ) ξk B ¯ h s ρ k π Z R + ǫR r d q (2Λ r − d + d )[ k + 2( d −
1) ln r − r d ] . (43)The above integration can be solved for small thickness limit ( ǫ ≪
1) byusing the Taylor series expansion and we obtain S ≈ π d +12 Γ( d +12 ) ξk B ¯ h s ρ k π ǫR d q (2Λ R − d + d )[ k + 2( d −
1) ln R − R d ] . (44)It is observed from Eq. (44) that the entropy within the shell is directlyproportional with the thickness of the shell. The variation of the entropy ( S )with r for different dimensions shown in Fig. 4 and it shows almost similarin nature as the variation of energy obtained in Fig. 3.13 igure 4: Variation of the entropy of the shell with r (km) for different dimensions Now, the proper length between two interfaces of the shell can be calcu-lated by using the following equation ℓ = Z R + ǫR √ e λ dr = Z R + ǫR dr q k + 2( d −
1) ln r − r d ≈ ǫ q k + 2( d −
1) ln R − R d . (45)Variation of the proper length has been shown in Fig. 5. Let us assume that p θ = p θ = p θ = ... = p t = − T , where T is thesurface tension. Therefore, Eqs. (33) and (34) yield T = ω ( R )Σ . (46)Thus, the EOS parameter can be found as ω ( R ) = − d − d − ( d − µ dRd − − R d ( d +1) q − µRd − − R d +1) d − d − d − πGDρcd ( d +1) R − R d ( d +1) q − πGDρcd ( d +1) R − R d ( d +1) . hq − µR d − − R Λ( d +1) d − q − πG D ρ c d ( d +1) R − R d ( d +1) i . (47)14 igure 5: Variation of the proper length of the shell with the radial coordinate r (km) fordifferent dimensions To check the stability of the gravastar in higher dimensions we are fol-lowing the prescription suggested by Mazur and Mottola [1, 2]. For doingthis we have maximized the entropy functional with respect to all variationsof the mass function µ ( r ) and investigate the sign of second variation of theentropy functional. As the end points of the boundary are fixed at R and R respectively, so the first variation of S must vanish at R and R , i.e., δS = 0by Einstein’s field equations of Eq. (5) to Eq. (7 for a static, sphericallysymmetric star.Now, the higher dimensional generalization of the entropy functional canbe obtained as S = vuut π d +12 Γ( d +12 ) ξk B h √ π Z R R r d dr dµdr ! h − / , (48)where h = 1 − µr d − − r d ( d +1) .Now from Eqs. (20), (24) and (26) we get the form of µ ( r ) as µ ( r ) = 16 πG D ρ c r d − R d ( d + 1) + 2Λ r d +1 d + 1 − R r d − d + 1 + 2( d − r d − ln R r . (49)15ence we can have second variation of the entropy function as δ S = vuut π d +12 Γ( d +12 ) ξk B h √ π Z R R r d dr dµdr ! − h − / − d ( δµ ) dr ! +( δµ ) h r d − dµdr ! dµdr ! . (50)As a next step we consider the following linear combination δµ = χψ ,where ψ does vanish at the endpoints. Now insert this into the second vari-ation (Eq. 50), after integrating by parts with δµ = 0 at the extreme pointsof the shell, we are left with δ S = − vuut π d +12 Γ( d +12 ) ξk B h √ π Z R R r d dr dµdr ! − h − / χ " dψdr < . (51)It clearly indicates that the entropy function with higher dimensions hasthe maximum value for all radial variations which are vanished at extremepoints of the boundary of the shell. Therefore, following Mazur and Mot-tola [2] we can conclude that “perturbations of the fluid in region II that arenot radially symmetric decrease the entropy..., which demonstrates that thesolution is stable to all small perturbations keeping the endpoints fixed”. Inessence, generalization to the higher dimension does not affect the stabilityof the gravastar. To discuss the size of gravastars we follow its phenomenology and startfrom the interior condensate phase which obey the EOS p = − ρ . This isexactly that of the cosmological dark energy which is thought to be respon-sible for the accelerating phase of the present universe [21, 22]. It is arguedby Mottola [57] that the gravastar solution automatically fixes the vacuumcondensate energy density ( ρ cond ) in terms of its size. This raised an inter-esting possibility that the observable universe itself could be the interior ofa gravastar. In that case, following Mottola [57] the observed cosmologicaldark energy of our universe can be calculated as ρ cond ≃
72 % of ρ crit = (0 .
72) 3 c H πG ≃ . × − erg/cm , (52)16here it is assumed that some 72% of the critical energy density ρ crit definedby the present value of the Hubble parameter H would be identified withthe condensate energy density.Let us now set the energy scale of the interior de Sitter energy density to TeV , so that from above Eq. (52) we have ρ cond ≃ . × T ev/km . (53)This is a huge energy which is required to get a gravastar that is com-parable to the size of the observable universe. One can even find a link ofthis energy to the scale of inflation or GUT (10 GeV ) as the above energyturns out to be ρ cond ≃ . × Gev/km .
7. Discussions and Conclusions
In the present study of gravastars with higher dimensional manifold andin the presence of cosmological constant, we have considered several aspectsof the system. To examine a new model of gravastar in comparison withthe type proposed by Mazur-Mottola [1, 2], we are especially searching forits generalization to: (i) the extended D dimensional spacetime from the 4dimensions and (ii) the cosmological constant. It is worth to mention thatusing the consideration (i) along with the inclusion of Maxwell’s spacetime wehave already obtained an interesting class of solutions [20]. However, in thepresent work we have used the consideration (i) combined with the cosmolog-ical constant to explore possibility of a physically viable astrophysical systemwhich can be considered as an alternative to D -dimensional Schwarzschild-Tangherlini category black hole [53].In this section we are discussing some key physical features of the model(observed as well as speculated) as follows:(i) We have obtained several physical parameters, e.g. metric potentials,proper length of the shell, energy, entropy etc. and our results match withthe results of Usmani et al. [16] without the cosmological constant in higherdimensional spacetime. The variations of the parameters as shown in theplots (Figs. 1-5) indicate in favour of the physical acceptability and for theexistence of gravastars.(ii) We have calculated different parameters for three specific regions ofthe gravastar. All the features of the solutions suggest that the cosmologicalconstant plays an important role for the construction of gravastars.17iii) From Eqs. (10), (12), (14) and Fig. 1 we can observe that thepressure, matter density and the metric potentials are finite at the centre(i.e. r = 0) of the gravastar. So the solutions that we have obtained arecompletely singularity free and also maintain regularity conditions inside thestar.(iv) The variation of the energy and the entropy is shown in Fig. 3 andFig. 4 respectively. From these figures it can be observed that both energyand entropy are decreasing with the dimensions. This in turn indicates thatthe shell becomes less compact and the matter density must decrease withdimensions. Exactly the same nature of variation for the matter densitywithin the shell has been observed in Fig. 2.(v) From Fig. 5 we note that the proper length is decreasing with theincrease in dimensions, which suggests that the shell become thinner in higherdimensions. This is again well justification of the Fig. 2, i.e. as the matterdensity decreases the energy as well as the entropy will also decrease.(vi) In connection to Figs. 1 and 2 let us now look at the issue: howlarge the star will be as a function of the energy density and number ofdimensions. For the interior region, the equation of state ρ = ρ c (Eq. (10))does not contribute anything to the size of the gravastar. However, via Eq.(13) we note from Fig. 1 that the size of the gravastar depends entirelyon the increasing order of dimensions. In the case of intermediate region,the thin shell follows the expression (23) which is characterized by Fig. 2and indicates that the density decreases with increasing mode of dimension.Hence, the total mass of the gravastar being constant the volume of the starbecomes larger with decreasing density. Thus, for both the cases we observethat the size of the gravastar increases with higher order of dimensions.(vii) In comparison with the previous work by Ghosh et al. [20] it canbe seen that the presence of cosmological constant for the construction ofgravastar in higher dimensions shows almost the similar effects as observed inthe presence of charge in the gravastar. This observation therefore indicatesthat the repulsive nature of the Colombian charge and cosmological constanthave the similar role as far as gravastar is concerned.(viii) It is observed that all the obtained results of the present model ongravastar are overall very much indicative that higher dimensional approachto construct a gravastar seems theoretically sound and solutions are physi-cally acceptable. However, a close observation of different profiles and plotsshow that the higher dimensional modelling of a gravastar does not indicateany significant difference in nature of the physical parameters from that of the18rdinary four dimensional spacetime with or without cosmological constant.As a final comment we would like to mention of a methodology whichis instructive to prove the existence of a gravastar in comparison to blackhole. The scheme has been first developed by Kubo and Sakai [58] for thecase of spherical thin-shell model of a gravastar provided by Visser and Wilt-shire [59] and later on have been exhibited by Das et al. [60]. The methodis involved in finding out microlensing effects for the gravastar, where themaximal luminosity could be considerably larger than the black hole withthe same mass.In this context, one may raise the following issue: Could gravastar bea substantial fraction of the dark matter while evading microlensing con-straints? As gravastar is an alternative to black hole and at present we donot have conclusive evidences of gravastar, so the explanation may be suit-able in terms of rather black hole. The primordial black holes are consideredto be one of the plausible candidates of dark matter which may be detectedthrough the gravitational micro-lensing effect. The EROS and MACHO sur-veys have put a limit on the abundance of primordial black holes in the range10 − kg which indicate that primordial black holes within this rangecannot constitute an important fraction of the dark matter [61, 62]. How-ever, the micro-lensing constraints could naturally be evaded in the case ofregrouping of primordial black holes in dense halos [63]. As these surveysand information related to primordial black holes may be put as input in thecase study of gravastars. Acknowledgements
The authors FR and SR are grateful to the authority of Inter-UniversityCenter for Astronomy and Astrophysics, Pune, India for providing Associate-ship under which a part of this work was carried out. SR is also grateful tothe authority of The Institute of Mathematical Sciences, Chennai, India forthe assistance and facilities. The authors all are grateful to the anonymousreferee for the pertinent comments which have helped them to upgrade themanuscript substantially.
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