Table Of ContentEur. Phys. J. C manuscript No.
(will be inserted by the editor)
Gravastars with higher dimensional spacetimes
S. Ghosha,1, B.K. Guhab,2, Saibal Rayc,3
1DepartmentofPhysics,IndianInstitute ofEngineeringScienceandTechnology, B.Garden,
Howrah711103,WestBengal,India
2Department of Physics, Indian Institute of Engineering Science and Technology, B.
Garden,Howrah711103, WestBengal,India
3Department ofPhysics, Government CollegeofEngineeringandCeramicTechnology,
Kolkata700010,WestBengal,India
Received: date/Accepted: date
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1
Abstract WepresentanewmodelofgravastarinthehigherdimensionalEinstein-
0
2 MaxwellspacetimeincludingEinstein’scosmologicalconstantΛ.FollowingMazur
and Mottola [1,2] we obtain a set of solutions for gravastar. This gravastar is
n
described by three different regions namely, (I) Interior region, (II) Intermediate
a
J thin spherical shell and (III) Exterior region. The pressure within the interior
regionisequaltothenegativematterdensitywhichprovidesarepulsiveforceover
2
theshell.Thisthinshellisformedbyultrarelativisticplasma,wherethepressureis
] directlyproportionaltothematter-energydensitywhichdoescounterbalancethe
c
repulsiveforcefromtheinteriorwhereastheexteriorregionis completelyvacuum
q
assumed to be de Sitter spacetime which can be described by the generalized
-
r Schwarzschildsolution.Withthisspecificationwefindoutasetofexactandnon-
g
singular solutions of the gravastar which seems physically very interesting and
[
reasonable.
1
v Keywords General relativity;Gravastar;Dark Energy
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4
0
1 Introduction
1
0
. IngeneralrelativityofEinsteinthereisaninherentfeatureofsingularityattheend
1
point of gravitationally collapsing system and has been remains an embarassing
0
situationto theastrophysicalcommunity.To overcomethis odd phaseof a stellar
7
1 body where all the physical laws break down, Mazur and Mottola [1,2] proposed
: a new model considering the gravitationally vacuum star which was termed in
v
brevity as Gravastar, that brings up a new arena in the gravitational system.
i
X They generated a new type of solution to this system of gravitationalcollapseby
r extending the idea of Bose-Einstein condensation by constructing gravastar as a
a cold, dark and compact object of interior de Sitter condensate phase surrounded
byathinshellofultrarelativisticmatterwhereastheexteriorregioniscompletely
vacuum, i.e. the Schwarzschild spacetime is at the outside. The shell is very thin
ae-mail:[email protected]
be-mail:[email protected]
ce-mail:[email protected]
2
but of finite width in the range r1 < r < r2, where r1 and r2 are the interior
and exterior radii of the gravastar. With this unique specification we can divide
the entire system of gravastar into three specific segments based on the equation
of state (EOS) as follows: (I) Interior: 0 r < r1, with EOS p = ρ, (II) Shell:
≤ −
r1 r r2, with EOS p=+ρ, and (III) Exterior: r2 <r, with EOS p=ρ=0.
≤ ≤
Theabovementionedmodelofgravastarhasbeenstudiedbyresearcherswhich
opened up a new challenges in the gravitational research to obtain a singular-
ity free solution of the Einstein field equations. Therefore, it is supposed to be
an alternative solution of black hole and has been studied by several authors in
differentcontextofastrophysicalsystems[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,
18,19,20].
The negativematterdensity in the interiorregion creates a repulsivepressure
acting radiallyoutwardfrom the centre of the gravastar(i.e. r=0) over the shell
whereas the shell of positive matter density provides the necessary gravitational
pulltobalancethisrepulsiveforcewithintheinterior.Itisassumedthatthedark
energy (or the vacuum energy) is responsible for this repulsive pressure from the
interior.Inageneralconsideration,theEOSp= ρissuggestingthattherepulsive
−
pressureisanagent,responsibleforacceleratingphaseofthepresentuniverseand
is known as the Λ-dark energy [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,
inthiscontextonecannotethatgravastarmayhavesomeconnectiontothedark
star [30,31,32,16].
The EOS for the shell p = ρ represents essentially a stiff fluid model as con-
ceivedbyZel’dovich[33]inconnectiontothecoldbaryonicuniverse.Theideahas
been consideredbyseveralscientistsforvarioussituationsincosmology[34,35]as
well as astrophysics[36,37,38].
Einstein introduced cosmological constant Λ in his field equations to make it
consistentwiththeMachprincipletoobtainastaticandnon-expandingsolutions
of the universe without having any valid physical interpretation of the proposed
model. In his model the constant Λ with the right sign could produce a repulsive
pressure to exactly counter balance the gravitational attraction and hence could
keep the model of the universe static.
ButaftertheexperimentalverificationofexpandinguniversebyEdwinHubble
between 1922 to 1924[39] and the success of FLRW cosmologymadeEinstein re-
alizethattheuniversehasbeen expandingwithanacceleration.Thatiswhylater
on Einstein discardedthecosmologicalconstantfromhis field equation.However,
though it is abandoned by Einstein but for the physical requirement to describe
one-loop quantum vacuum fluctuations,the Casimireffect [40], cosmologicalcon-
stanthadtoappearonceagaininthetheorywithaformasTij =Λgij/8πG,where
Tij and gij are the stress energy tensor and the metric tensor respectively and G
is the usual Newtonian constant.
RecentobservationsconductedbyWMAPsuggeststhat73%ofthetotalmass-
energy of the universe is dark energy [41,42]. It is believed that this dark energy
playsanimportantrolefortheevolutionoftheuniverseandinordertodescribethe
darkenergyscientistshaverecalltheerstwhilecosmologicalconstant.Therefore,in
themodern cosmologythis cosmologicalconstantis treatedas a strongcandidate
for the dark energy which is responsible for the accelerating phase of the present
universe.
3
Very recently a charged gravastar in higher dimension has proposed by Bhar
[43] admittingconformal motion and also by Ghosh et al. [20] without admitting
theconformalmotionintheframeworkofMazurandMottolamodel.Theseworks
provide an alternative solution to the static black holes. Usmani et al. [16] have
alsofoundsolutionofneutralgravastarinhigherdimensionwithoutadmittingthe
conformalmotion.Thepresent study on gravastarbasicallyis an extensionof the
workofUsmanietal.[16]asmentionedaboveanditsgeneralizationtothehigher
dimensionalspacetimeinpresenceofthecosmologicalconstant(Λ).Therefore,the
main motivation of this work is to study the effects of the cosmological constant
for construction of gravastars and also to study the higher dimensional effects, if
any.
Thepresentinvestigationsarebasedontheplansasfollows:Thebackgroundof
themodelhasbeenimplementedbyEinstein-MaxwellgeometrydiscussinSect.2,
whereasthehesolutionofinteriorspacetime,thethinshellandexteriorspace-time
ofthegravastarhasbeendiscussedinSect.3.Thenwehavediscussedthejunction
conditions for the different regions of the gravatar in Sect. 4. In the Sect. 5 we
explore some physical features of the model, viz. proper length, Energy, Entropy
and also study their variation with the radial parameter which followed by the
discussion and concluding remarks at the end in Sect. 6.
2 The Einstein-Maxwell spacetime geometry
The Einstein-Hilbertaction coupled to matter is given by
I = dDx√ g RD +Lm , (1)
− 16πGD
Z (cid:18) (cid:19)
wherethecurvaturescalarinD-dimensionalspacetimeisrepresentedbyRD,with
GD is theD-dimensionalNewtonianconstantand Lm denotesthe Lagrangianfor
thematterdistribution.We obtainthefollowingEinsteinequationbyvaryingthe
above action with respect to the metric
GDij = 8πGDTij, (2)
−
where GD denotes the Einstein’s tensor in D-dimensional spacetime.
ij
The interior of the star is assumed to be perfect fluid type and can be given
by
Tij =(ρ+p)uiuj +pgij, (3)
where ρ represents the energy density, p is the isotropic pressure, and ui is the
D-velocity of the fluid.
Hereinthepresentstudyitisassumedthatthegravastarsinhigherdimensions
have the D-dimensional spacetime with the structure R1XS1XSd(d = D 2),
wherethe rangeof theradialcoordinateis S1 and the timeaxis is represente−d by
R1.Forthispurpose,weconsiderastaticsphericallysymmetricmetricinD=d+2
dimension as
ds2 = eνdt2+eλdr2+r2dΩ2, (4)
− d
where dΩ2 is the linear element of a d-dimensionalunit sphere, parameterizedby
d
the angles φ1,φ2,......,φd as follows:
4
dΩd2=dφ2d+sin2φd[dφ2d−1+sin2φd−1{dφ2d−2+.........+sin2φ3(dφ22+sin2φ2dφ21).......}].
Now the Einstein field equations for the metric (4), together with the energy-
momentum tensor given in Eq. (3) in presence of the non-zero cosmological con-
stant Λ, yield
′
−e−λ d(d2r−21) − d2λr + d(d2r−21) =8πGD ρ+Λ, (5)
(cid:20) (cid:21)
′
e−λ d(d2r−21) + d2νr − d(d2r−21) =8πGD p−Λ, (6)
(cid:20) (cid:21)
e−λ ′′ λ′ν′ ν′2 (d 1)(λ′ ν′) (d 1)(d 2) (d 1)(d 2)
ν + − − + − − − −
2 " − 2 2 − r r2 #− 2r2
=8πGD p Λ, (7)
−
where ‘’ denotes differentiation with respect to the radial parameter r. Here we
′
have assumed c=1 in geometricalunit.
In general relativity the conservation of energy-momentum is expressed with
the aid of a stress-energy-momentum pseudotensor, i.e. Tij;j = 0 and can be
expressed in its general form with D-dimension as
1 ′ ′
(ρ+p)ν +p =0. (8)
2
In the next Sect. 3 we shall formulate special explicit forms of the energy
conservation equations for all the three regions, viz. interior, intermediate thin
shell and exterior spacetimes.
3 The gravastar models
3.1 Interior spacetime
In the interior region of the gravastar it is assumed that the negative pressure is
acting radially outward from the center of the spherically symmetric system to
balancethe inwardpull from the shell. FollowingMazur-Mottola[1], the EOS for
the interiorregion can be provided in the form
p= ρ. (9)
−
Using Eq. (8) and the aboveEOS (9), we obtain
p= ρ=ρc, (10)
−
where ρc is the critical density of the interior region. Using Eq. (9) in the field
equation (5), one obtains the solutionof λ as
e−λ=1− 1d6(πdG+D1ρ)cr2− d(2dΛ+r21) +C1r1−d, (11)
where C1 is an integration constant. Since d 2 for dimension higher than three
≥
andthesolutionisregularatr=0,sowedemandforC1 =0.Thusessentiallywe
get
e−λ =1 16πGDρcr2 2Λr2 . (12)
− d(d+1) − d(d+1)
5
Using Eq. (9) one may obtain from Eqs. (5) and (6), the followingrelation
lnk=λ+ν eν =ke−λ, (13)
⇒
where k is an integrationconstant.
Thuswehavethefollowinginteriorsolutionsforthemetricpotentialsλand ν
as follows
ke−λ =eν =k 1 (C2 C3)r2 , (14)
− −
h i
where C2 = 16dπ(dG+D1ρ)c and C3 = d(d2+Λ1).
From Eq. (10) it is observed that the matter density remains constant over
the entire interiorspacetime.Thus we can calculatethe activegravitationalmass
M(r) in higher dimensions as
M(r)= r1=R 2πd+21 rdρcdr= 2πd+21ρc Rd+1, (15)
Z0 Γ d+21 (d+1)Γ d+21
(cid:16) (cid:17) (cid:16) (cid:17)
where R is the internal radius of the gravastar. So the usual gravitational mass
in for a d-dimensional sphere dimension can be represented by Eq. (15), which is
directly proportionalto the radius R and the matter density ρ.
Intheinteriorregion,therefore,theenergyconservationequation(8)takesthe
special explicit form as follows:
(p+ρ) 16πGDρc 2Λ r
′ d(d+1) − d(d+1)
p = . (16)
1 1h6πGDρc 2Λ ri2
− d(d+1) − d(d+1)
h i
3.2 Intermediatethin shell
Here we assume that the thin shell contains ultra-relativisticfluid of soft quanta
and obeys the EOS
p=ρ. (17)
It is difficult to obtain a general solution of the field equations in the non-
vacuum region,i.e. withinthe shell.Therefore,wetry tofind an analyticsolution
within the thin shell limit, 0<e−λ h<<1. To do so we set h to be zero to the
≡
leading order. Under this approximation, the field Eqs. (5) - (7) along with the
above EOS, can be reformated in the followingform
′
h + (d−1) = 8πGDρ + Λ, (18)
− 2r 2r2 d d
(d−1) = 8πGDp Λ, (19)
− 2r2 d − d
′ ′ ′
hν (d 1)h (d 1)(d 2)
4 + −2r − − 2r2 − =8πGDp−Λ. (20)
Now using Eqs. (18) and (19), we find out an expression for h as
h=e−λ=k1+2(d 1)lnr 2Λr2, (21)
− − d
6
where k1 is an integration constant. Here the range of r lies within the thickness
of the shell, i.e. r1 = R,r2 = R+ǫ, where ǫ << 1 which is the thickness of the
shell.
The other metric potential (ν) can be found as
e−ν =k2 2Λr2r2dd2+d , (22)
| − |
where k2 is the integrationconstant.
Again using Eqs. (8) and (22) we get the pressure and matter density within
the shell of the gravastaras
p=ρ=ρ0e−ν =ρ0k2 2Λr2r2dd2+d . (23)
| − |
In the intermideatethin shell, therefore, the energy conservation equation (8)
takes the special explicit form as follows:
′ 2Λ
p =r(p+ρ) d . (24)
2Λr2 d2+d −
(cid:20) − (cid:21)
Fig.1 Variationofthepressureanddensityoftheultrarelativisticmatterintheshellagainst
r fordifferentdimensionswherethespecificlegendsusedareshownintherespectiveplots
3.3 Exterior spacetime
The EOS for the exterior region is defined as p = ρ = 0. In higher dimensions
one can expect that the exterior solution is nothing but a generalization of the
Schwarzschild solution. Now, following the work of Tangherlini [44] this can be
obtained as
µ 2Λr2 µ 2Λr2 −1
ds2 = 1 dt2+ 1 dr2+r2dΩ2, (25)
− − rd−1 − (d+1)d − rd−1 − (d+1)d d
(cid:18) (cid:19) (cid:18) (cid:19)
7
whereµ is a constantand is given by µ=16πGDM/Ωd in higher dimension,with
M asthemassofblackholeandΩ astheareaofaunitd-spherewhichisdefined
d
by Ωd =2π(d+21)/Γ(d+21).
In the exterior region, therefore, the energy conservation equation (8) takes
the special explicit form as follows:
′ (p+ρ)[d(d2 1)µ 4Λrd+1]
p = − − . (26)
−2[d(d+1)rd d(d+1)µr 2Λrd+2]
− −
4 Junction Condition
Therearethreeregionsinthegravastarconfiguration,viz.theinteriorregion,thin
shell and exterior region. The shell joins the iterior and exterior regions at the
junction interface.The metric coefficients arecontinuousat theshell, however we
donothaveanyconfirmationofthecontinuityoftheirderivatives.Atthisjuncture
weusetheconditionofDarmois-Israel[45,46]forcalculationofthesurfacestresses
at the junction interface. The intrinsic surface stress-energy tensor Sij takes the
form
1
Sij =−8π(κij−δijκkk), (27)
where kij =Ki+j −Ki−j, that shows the discontinuityof the extrinsiccurvatures or
secondfundamentalforms.Herethesignatures and+describestheinteriorand
−
exterior boundaries respectively of the gravastar.
Now this extrinsic curvature connect the two sides of the thin shell as
Ki±j =−n±ν ∂∂ξ21X∂ξνj +Γανβ∂∂Xξiα∂∂Xξjβ |S, (28)
(cid:20) (cid:21)
±
where nν is the unit normals to the surface S can be defined as
−1
n±ν =± gαβ∂∂Xfα∂∂Xfβ 2 ∂∂Xfν, (29)
(cid:12) (cid:12)
(cid:12) (cid:12)
with nνnν =1. (cid:12)(cid:12) (cid:12)(cid:12)
Followingthe methodologyof Lanczos [47] one can obtain the surface energy-
momentum tensor on the thin shell as Sij =diag[−Σ,pθ1,pθ2..., pθd], where Σ is
the surface energy density and pθ1 =pθ2 =...=pθd =pt are the surface pressures
which respectively can be determined by
d
Σ = f (30)
−4πR
d 1 p f′
pt = − Σ+ . (31)
− d 8π√f
Using the above equations one can obtain
Σ = d 1 µ 2R2Λ 1 16πGDρcR2 2ΛR2 (32)
−4πR"s − Rd−1 − (d+1)d −s − d(d+1) − d+1#
8
and
pt= 1 (d−1)− (2dR−d1−)1µ − 2dΛ+R12 (d−1)− 16πdG+D1ρcR2− 2ddΛ+R12 . (33)
4πR 1 µ 2R2Λ − 1 16πGDρcR2 2ΛR2
− Rd−1 − (d+1)d − d(d+1) − d+1
q q
Now, it is easy to find out the mass ms of the thin shell as
ms= Γ2(πdd++211)RdΣ =ΩdRd s1− 16dπ(GdD+ρ1c)R2 − 2dΛ+R12 −s1− Rdµ−1 − d(2dR+2Λ1)!.
2
(34)
Using Eq. (34) we can determine the total mass of the gravastar in terms of
the mass of the thin shell in the form
µ= d(Rdd++11)[16πGDρc+2Λ(d−1)]+ΩmdsR"2s1− 1d6(πdG+D1ρ)cR2− 2dΛ+R12 − ΩmdRsd#.
(35)
5 Physical features of the models
5.1 Energy content
As we consider the thickness of the intermediate shell is very small (0 < ǫ 1),
≪
so the phase boundary can be described by the interfaces at r = R and r =
R+ǫ joiningtheregion Iand regionIII respectively,whereR describes thephase
boundary of the region I.
Using the Eq. (23) we calculatethe energy within the shell as
E =ZRR+ǫΓ2πdd+2+211 rdρdr =Γ2πdd+2+211 ρ0k2ZRR+ǫ |2Λr2r−3dd2+d|
(cid:16) (cid:17) (cid:16) (cid:17) (36)
Tosolvetheaboveintegration(36)overthelimitRtoR+ǫ,letusformaspecial
differential equation in terms of the parameter F(r) which can be represented as
dF(r) r3d
= . (37)
dr 2Λr2 d2+d
| − |
Now using the above considerationwe can solve the the integrationas follows
R+ǫ dF(r)dr =[F(r)]R+ǫ=F(R+ǫ) F(R)=ǫ dF . (38)
dr R − dr
ZR (cid:18) (cid:19)r=R
Asǫ 1,soweconsideruptothefirstordertermoftheTaylorseriesexpansion
≪
for the expression in Eq. (38).
Therefore, combiningEqs. (36) and (38) we get
2πd+21 R3d
E =Γ d+21 ǫρ0k2|2ΛR2−d2+d|, (39)
(cid:16) (cid:17)
9
wherethesquarebracketoftheintegralwehaveused thefactorΩ as theareaof
d
a unit d-sphere which has already been defined by Ωd =2π(d+21)/Γ(d+21).
From Eq. (39) we observe that the energy within the shell is directly propor-
tional to the thickness of the shell (ǫ).
Fig. 2 VariationoftheEnergyoftheshellwithr fordifferentdimensions
The variation of E with r for different dimensions is shown in Fig. 2. One
can note that the energy is increasing from the interior boundary to the exterior
boundary.Thisisclearlyindicatingthattheshellisgettingharderfromtheinterior
toexteriorboundary,whichsuggeststhattheexteriorboundaryismoredensethan
theinteriorboundaryas obtainedin Fig. 1. Theplotalso indicatesthatenergy is
decreasing as the dimension increases.
5.2 Entropy
FollowingtheprescriptionbyMazur-Mottola[1]wecanusethefollowingequation
to calculatethe entropy within the shell as,
S = 2πd+21 ξkB ρ0k2 R+ǫ ǫr2d . (40)
Γ(d+21) ~ r 2π ZR (d2−d−2Λr2)(k1+2(d−1)lnr− 2Λdr2)
q
Theaboveintegrationcan be solved for smallthickness limit(ǫ 1) by using
≪
the Taylorseries expansion and we obtain
S = 2πd+21 ξkB ρ0k2 ǫR2d . (41)
Γ(d+21) ~ r 2π (d2−d−2ΛR2)(k1+2(d−1)lnR− 2ΛdR2)
q
It is observed from Eq. (41) that the entropy within the shell is directly pro-
portional with the thickness of the shell. The variation of the entropy (S) with r
for different dimensions shown in Fig. 3. and it shows almost similar in nature as
the variationof energy obtained in Fig. 2.
10
Fig. 3 VariationoftheEntropyoftheshellwithr fordifferentdimensions
5.3 Proper length
Now, the proper length between two interfaces of the shell can be calculated by
using the followingequation
R+ǫ R+ǫ dr
ℓ= eλdr =
ZR p ZR k1+2(d−1)lnr− 2Λdr2
q ǫ
= .
k1+2(d−1)lnR− 2ΛdR2
q (42)
Fig. 4 Variationoftheproperlengthoftheshellwiththeradialcoordinater
