Charged Perfect Fluid Sphere in Higher Dimensional Spacetime

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by  Piyali Bhar

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Charged Perfect Fluid Sphere in Higher Dimensional
Spacetime
Piyali Bhara,1, Tuhina Mannab,2, Farook
Rahamanc,3, Saibal Rayd,4, G.S. Khadekare,5 .
1Gopinathpur HighSchool (H.S),Haripal,Hooghly712403, WestBengal,India
2DepartmentofMathematics,Jadavpur University,Kolkata700032, WestBengal,India
6 3Department of Commerce (Evening), St. Xaviers Colege, 30 Mother Teresa Sarani,
1 Kolkata700016,WestBengal,India
4Department ofPhysics,Government CollegeofEngineering&CeramicTechnology, 73
0
A.C.B.Lane,Kolkata700010, WestBengal,India
2
5Department of Mathematics, R.T.M.Nagpur University, Mahatma Jyotiba Phule

n Educational Campus,AmravatiRoad,Nagpur440033, Maharastra,India
a Received: date/Accepted: date
J

7
Abstract Present paper provides a new model for perfect fluid sphere filled with

charge in higher dimensional spacetime admitting conformal symmetry. We con-
]
h sider a linear equation of state with coefficients fixed by the matching conditions
p at the boundary of the source corresponding to the exterior Reissner-Nordstr¨om
- higher dimensional spacetime. Several physical features for different dimensions,
n
e starting from four up to eleven, are briefly discussed. It is shown that all the fea-
g tures as obtained from the present model are physically desirable and valid as
s. far as the observed data set for the compact star SAX J1808.4−3658 (SS2) is
c concerned.
i
s
Keywords General Relativity; linear equation of state; higher dimension;
y
h compact star
p
[

1 Introduction
1
v
7 Withtherecentadvancementinsuperstringtheoryinwhichthespacetimeiscon-
4 sidered to be of dimensions higher than four, the studies in higher dimensional
6 spacetime has attained new importance. Throughout the last decade a number
2 of articles have been published in this subject both in localized and cosmological
0
domains. It is a common trend to believe that the 4-dimensional present space-
.
1 time structure is the self-compactified form of manifold with multidimensions.
0 Therefore, it is argued that theories of unification tend to require extra spatial
6
dimensions to be consistent with the physically acceptable models [1,2,3,4,5,6].
1
It has been shown that some features of higher dimensionalblack holes differ sig-
:
v nificantly from 4-dimensional black holes as higher dimensions allow for a much
i
X
ae-mail:[email protected]
r be-mail:[email protected]
a
ce-mail:[email protected]
de-mail:[email protected]
ee-mail:gkhadekar@rediffmail.com
2
richer landscape of black hole solutions that do not have 4-dimensional counter-
parts [7]. Some recent higher dimensional works admitting one parameter Group
of Conformalmotion can be seen in the Refs. [8,9].
The study of charged fluid sphere has attained considerable interest among
researchersinlastfewdecades.Itisobservedthatafluidsphereofuniformdensity
withanetsurfacechargebecomesmorestablethanwithoutcharge[10].According
toKrasinski[11]inthepresenceofcharge,thegravitationalcollapseofaspherically
symmetricdistributionofmattertoapointsingularitymaybeavoided.Sharmaet
al.[12] arguethat in thissituationthe repulsiveColombianforcecounterbalances
the gravitational attraction in addition to the pressure gradient. To study the
cosmic censorship hypothesis and the formation of naked singularities Einstein-
Maxwellsolutionsarealsoimportant[13].Thepresenceofchargeaffectsthevalues
for redshift, luminosity and maximum mass for stars. For a charged fluid spheres
the gravitational field in the exterior region is described by Reissner-Nordstr¨om
spacetime. Charged perfect fluid sphere satisfying a linear equation of state was
discussed by Ivanov [14]. In this paper the author reduced the system to a linear
differential equation for one metric component. Regular models with quadratic
equation of state was discussed by Maharaj and Takisa [15].
The obtained solutionsof the Einstein-Maxwellsystem of equations are exact
and physically reasonable. A physical analysis of the matter and electromagnetic
variablesindicatesthatthemodelis wellbehavedand regular.In particularthere
is no singularity in the proper charge density at the stellar center. A Charged
anisotropic matter with linear equation of state has discussed by Thirukkanesh
and Maharaj [16]. In connection with this we want to mention a recent work of
Varela et al. [17]. In this paper the author considered a self-gravitating, charged
and isotropicfluid sphere.
To solve Einstein-Maxwell field equation they have assumed both linear and
nonlinear equation of state and discussed their result analytically. Rahaman et
al. [18] have obtained a singularity free solutions for anisotropic charged fluid
spherewithChaplyginequationofstate.TheauthorsusedKrori-Baruaansatz[19]
to solve the system.
Thewellknowninheritancesymmetryisthesymmetryunderconformalkilling
vectors (CKV) i.e.
L g =ψg , (1)
ξ ik ik
where L is the Lie derivative of the metric tensor which describes the interior
gravitationalfield ofacompactstarwithrespecttothevectorfieldξ andψ isthe
conformalfactor.Inadeepersensethisinheritancesymmetryprovidesthenatural
relationshipbetweengeometryandmatterthroughtheEinsteinfieldequations.It
is supposed that the vector ξ generates the conformal symmetry and the metric
g is conformallymapped onto itself along ξ. Harko et al. [20,21] have shown that
neither ξ nor ψ need to be static even through one consider a static metric.
Therearemanyearlierworksonconformalmotioninliterature.Theexistence
of one parameter group of conformal motion in Einstein-Maxwell spacetime have
been studied in [22,23,24]. Anisotropic sphere admitting one-parameter group of
conformal motion has been discussed by Herrera and Leo´n [25]. A class of solu-
tions for anisotropic stars admitting conformal motion has been studied in [26].
Charged gravastar admitting conformal motion has been studied by Usmani et
al. [27]. Bhar [28] has generalized this result in higher dimensional spacetime.
3
Relativistic stars admitting conformal motion has been analyzed by Rahaman et
al.[29].Isotropicandanisotropicchargedspheresadmittingaoneparametergroup
of conformal motions was analyzed in [30]. Anisotropic spheres admitting a one
parametergroupofconformalmotionshasbeendiscussedbyHerrera&Leo´n[31].
Chargedfluidspherewithlinearequationofstateadmittingconformalmotionhas
been studied in Ref. [32]. The authors have also discussed about the dynamical
stability analysis of the system. Ray et al. have given an electromagnetic mass
model admitting conformal killing vector [33,34]. By assuming the existence of
a one parameter group of conformal motion Mak & Harko [35] have described
an charged strange quark star model. The above author have also discussed con-
formally symmetric vacuum solutions of the gravitational field equations in the
brane-world models [36]. Bhar [37] has described one parameter group of confor-
mal motion in the presence of quintessence field where the Vaidya-Titekar [38]
ansatz was used to develop the model.
Theobtainedresultsareanalyzedphysicallyaswellaswiththehelpofgraph-
ical representation. In a very recent work Bhar et al. [44] provide a new class of
interior solutions for anisotropic stars admitting conformal motion in higher di-
mensional noncommutativespacetime. The Einstein field equations are solved by
choosinga particulardensitydistributionfunctionof Lorentziantypeas provided
by Nazari and Mehdipour [42,43] under a noncommutativegeometry.
Inspired by these early works in the present paper we have used the Einstein-
Maxwellspacetimegeometrytodescribeaself-gravitatingchargedanisotropicfluid
sphere satisfying a linear equation of state admitting conformal motion in higher
dimensions. Once we specify the equation of state (EOS) we have integrated the
Tolman-Oppenheimer-Volkoff(TOV)equationstoderivethegrossfeaturesof the
stellarconfiguration.We propose to apply this model to describe charged strange
quark stars. The paper has been divided into the following parts : In Sect. 2 we
haveobtainedthe Einstein-Maxwellfield equationsfor staticsphericallysymmet-
ric distribution of charged matter. In Sect. 3 the conformal killing equations are
solvedandusedtheinheritancesymmetrywhichisthesymmetryunderconformal
killing vectors (CKV). The exterior spacetime using RN metric and investigation
of the matching condition are also done here along with the matching of the ex-
terior higher dimensional spacetime and interior spacetime at the boundary. In
Sect. 4 various physical properties are analyzed such as (i) stabilitycondition via
the TOV equations are integrated to obtain the gravitational (Fg) and hydro-
static (F ) forces, (ii) Energy conditions, namely, Null energy condition (NEC),
h
Weak energy condition (WEC) and Strong energy conditions are discussed and
the corresponding graphs for different dimensions plotted against r and (iii) the
compactnessfactorandredshiftareinvestigated.Finallysomeconcludingremarks
are passed in Sect. 5.
2 The interior spacetime and Einstein-Maxwell Field equations
To describe the static spherically symmetry spacetime in higher dimension we
consider the line element in the standard form as
ds2 =−eν(r)dt2+eλ(r)dr2+r2dΩ2, (2)
n
4
where
dΩn2 =dθ12+sin2θ1dθ22+sin2θ1sin2θ2dθ32+...+Πin=−11sin2θidθn2 (3)
and λ and ν arefunctionsof radialcoordinater. Heredimensionof thespacetime
is assumed as D = n+2 so that for n = 2 it reduces to ordinary 4-dimensional
spacetimegeometry.
Now,theEinstein-Maxwellfieldequationsintheirfundamentalformsaregiven
by
1
R − g R=−κ(T matter+T charge), (4)
ij 2 ij ij ij
where the energy-momentumtensor of perfect fluid distributionis
T matter =(ρ+p)u u −pg , (5)
ij i j ij
and the energy-momentumof the electromagneticfield is
1 1
T charge= [−gklF F + g F Fkl]. (6)
ij 4π ik jl 4π ij kl
The electromagneticfield equations are given by
[(−g)1/2Fij], =4πJi(−g)1/2, (7)
j
and
F =0, (8)
[ij,k]
wheretheelectromagneticfieldtensorF isrelatedtotheelectromagneticpoten-
ij
tials as F =A −A which is equivalent to the equation (6), viz., F =0.
ij i,j j,i [i,j,k]
Also, ui is the 4-velocity of a fluid element, Ji is the 4-current and κ = −8π (in
relativisticunitG=c=1).Hereandinwhatfollowsacommadenotesthepartial
differentiationwith respect to the coordinateindices involvingthe index.
The Einstein-Maxwellfield equations in higher dimension can be written as
e−λ nλ′ − n(n−1) + n(n−1) =8πρ+E2, (9)
2r 2r2 2r2
(cid:18) (cid:19)
e−λ n(n−1) + nν′ − n(n−1) =8πp−E2, (10)
2r2 2r 2r2
(cid:18) (cid:19)
1e−λ 1(ν′)2+ν′′− 1λ′ν′+ (n−1)(ν′−λ′)+ (n−1)(n−2)
2 2 2 r r2
(cid:20) (cid:21)
(n−1)(n−2)
− =8πp+E2. (11)
2r2
n+1
(rnE)′= 2π 2 rnσ(r)eλ2, (12)
Γ n+1
2
whereσ(r)isthechargedensityonthe(cid:0)n-sph(cid:1)erewithn=D−2,Dbeingdimension
of the spacetime.
5
The above equation equivalentlygives
n+1
E(r)= 1 2π 2 rnσeλ2dr, (13)
rn Γ n+1
Z 2
where ρ, p, E are respectively the matte(cid:0)r den(cid:1)sity, isotropic pressure and electric
field of the charged fluid sphere. Here ‘prime’ denotes the differentiation with
respect to the radial coordinater.
3 The solution under conformal Killing vector
The conformalKilling equation (1) becomes
L g =ξ +ξ =ψg . (14)
ξ ik i;k k;i ik
Now the conformal Killing equation for the above line element (2) gives the
followingequations
ξ1ν′=ψ, (15)
ξn+2 =C1, (16)
ψr
ξ1 = , (17)
2
ξ1λ′+2ξ1,1=ψ, (18)
where C1 is a constant.
The above equations consequently gives
eν =C2r2, (19)
2
2
eλ = C3 , (20)
ψ
(cid:18) (cid:19)
ξi =C1δni+2+ ψ2r δ1i, (21)
(cid:16) (cid:17)
Where C2 and C3 are constantsof integrations.
Equations (15)-(17) help us to write Einstein-Maxwell field equations (5)-(9)
in the followingform
n(n−1) ψ2 nψψ′
1− − =8πρ+E2, (22)
2r2 C2 rC2
(cid:18) 3(cid:19) 3
n ψ2
(n+1) −(n−1) =8πp−E2, (23)
2r2 C2
(cid:20) 3 (cid:21)
nψψ′ ψ2 (n−1)(n−2)
+n(n−1) − =8πp+E2. (24)
rC2 2r2C2 2r2
3 3
6
One can note that in Eqs. (18)-(20) we have four unknowns (ρ, p, ψ, E) with
three equations. So to solve the above equations let us assume that the pressure
is linearlydependent on the density, i.e.
p=αρ+β, (25)
where 0 < α < 1 and β is some arbitrary constant. It is to note that α here is
a constant which has a relation with the sound speed as dpr/dρ = α, and β is
arbitraryconstant that is related to the dimension of the spacetime.
SolvingEqs. (18)-(20)with the help of equation (21),one can obtain
ψ2 = (n−n1)2n((1α++α1))−C322α + (1+16απ)β(nC3+2 1)r2+C4r−2(n+1n+α3α−2α), (26)
where C4 is a constant of integration. Let us assume C4 =0 to avoid the infinite
mass at the origin.Now e−λ can be obtained as
e−λ = ψ2 = (n−1)2 (α+1) + 16πβ r2. (27)
C2 n n(1+α)−2α (1+α)(n+1)
3
The matter density and isotropic pressure can be obtained as
1 (n−1)2 βn
ρ= − , (28)
n(1+α)−2α 8πr2 1+α
α (n−1)2 αn
p= +β 1− . (29)
n(1+α)−2α 8πr2 1+α
(cid:16) (cid:17)
The expression of electric field becomes
(n−1)(1−α) 1 q
E2 = = . (30)
n(1+α)−2α 2r2 rn−2
3.1 The exterior spacetimeand matchingcondition
The solution of the Einstein-Maxwell equation in higher dimensional spacetime
for r >R is given by the followingReissner-Nordstr¨om (RN) spacetime in higher
dimension as
µ Q2 µ Q2 −1
ds2 =− 1− + dt2+ 1− + dr2+r2dΩ2, (31)
rn−1 r2(n−1) rn−1 r2(n−1)
(cid:18) (cid:19) (cid:18) (cid:19)
whereµisrelatedtothemassM asµ= 16πGM andQisitscharge.Soourinterior
nΩn
solution should match with Eq. (31) at the boundary r=R.
The continuityof the metric eν gives the constant C2 as
1 µ Q2
C2 =sR2 1− Rn−1 + R2(n−1) (32)
(cid:20) (cid:21)
7
Fig.1 Variationofdensityρagainstrintheinteriorofthecompactstarfordifferentdimen-
sions:4D and5D intheleftpanelwhereas6D,9D and11D intherightpanel
and the continuityof eλ gives
Q2 =R2(n−1) (n−1)2 α+1 + 16πβR2 −1+ µ . (33)
n n+nα−2α (α+1)(n+1) Rn−1
(cid:20) (cid:21)
The intensity of the electric field at the boundary can be obtained as
Q(R) (n−1)2 α+1 16πβR2 µ
E(R)= =R + −1+ . (34)
Rn−2 n n+nα−2α (α+1)(n+1) Rn−1
r
The profile of the electric field is shown in Fig. 3. The figure shows that E2 is
a monotonic decreasing function of r which attains maximum value for 11D and
minimum value for 4D.
3.2 The Junction Condition
Herethemetriccoefficientsarecontinuousatr=R,butthatdoesnotensurethat
their derivatives are also continuous at the junction surface. In other words the
affine connections may be discontinuous at the junction surface r = R. To take
care of this weuse the Darmois-Israelformationto determinethe surface stresses
at the junction boundary. The intrinsic surface stress energy tensor as given by
Lancozs equations is as follows:
Obviouslythemetriccoefficientsarecontinuousatbut itdoesnotensurethat
their derivatives are also continuous at the junction surface. In other words the
affine connections may be discontinuous there. To take care of this let us use the
Darmois-Israelformationtodeterminethesurfacestressesatthejunctionbound-
ary.TheintrinsicsurfacestressenergytensorS isgivenbyLancozsequationsin
ij
the followingform
1
Si =− (κi −δiκk). (35)
j 8π j j k
8
The discontinuityin the second fundamental form is given by
κ =K+−K−, (36)
ij ij ij
where the second fundamentalform is given by
K± =−n± ∂2Xν +Γν ∂Xα∂Xβ | . (37)
ij ν ∂ξi∂ξj αβ ∂ξi ∂ξj S
(cid:20) (cid:21)
Here n± are the unit normal vector defined by
ν
−1
n± =± gαβ ∂f ∂f 2 ∂f , (38)
ν ∂Xα∂Xβ ∂Xν
(cid:12) (cid:12)
(cid:12) (cid:12)
withnνnν =1.Hereξiistheintri(cid:12)nsiccoordinate(cid:12)ontheshell.+and−corresponds
to exterior i.e, RN spacetime in higher dimension and interior (our) spacetime
respectively.
Considering the spherical symmetry of the spacetime surface stress energy
tensorcan bewrittenas Si =diag(−Σ,P,P,...,P) whereΣ and P arethesurface
j
energy density and surface pressure respectively and can be provided by
n µ q2 n (n−1)2 α+1 16πβ
Σ =− 1− + + + r2,
4πR Rn−1 R2(n−1) 4πR n n+nα−2α (1+α)(n+1)
r r
(39)
n−1 µ q2 µ q2
P = 1+ − 1+ −
4πR 2Rn−1 R2n−1 2Rn−1 R2n−2
(cid:18) (cid:19)r
n (n−1)2 α+1 16πβR2
− + . (40)
4πR n n+nα−2α (α+1)(n+1)
r
The mass of the thin shell can be obtained as
n+1
ms = Γ2πn+21 RnΣ. (41)
2
(cid:0) (cid:1)
Now using Eqs. (40) and (41) one can obtain the mass of the charged fluid
sphere in terms of the mass of the thin shell as
µ=Rn−1 1+ q2 −G2+2BG−B2 , (42)
R2(n−1)
(cid:20) (cid:21)
where
G= 2msΓ n+21
nRn−1(cid:0)πn−21(cid:1)
and
(n−1)2 α+1 16πβR2
B= + .
n n+nα−2α (α+1)(n+1)
r
9
4 Physical Analysis of the solutions
For a physically meaningful solution one must have pressure and density are de-
creasing function of r. For our model
dp (n−1)2 α
=− <0, (43)
dr n+α(n−2)4πr3
dρ (n−1)2 1
=− <0. (44)
dr n+α(n−2)4πr3
The above expression indicates that both ρ and p are monotonic decreasing
functionofr,i.e.thehavemaximumvalueatthecenterofthestaranditdecreases
radiallyoutwards.Theconstantof integrationβ can be obtainedby imposingthe
condition p(r=R)=0 as
α(α+1)(n−1)2
β= . (45)
8πR2(n+nα−2α)(nα−α−1)
The above equations consequently gives
α(α+1)(n−1)2
Rn= ,
8πβ(n+nα−2α)(nα−α−1)
r
where n= D−2, D being the dimension of the spacetime. One can easily verify
that
dp
=α.
dρ
To satisfy the causality conditions one must have 0 < dp < 1 which implies
dρ
0 < α < 1. To find the radius of different dimensional charged star let us fix
α=0.4. From the expression of Rn one can note that for α=0.4, n+(n−2)α is
always positive and nα−α−1 is negative for n = 2 and n = 3, i.e. for four and
five dimensional spacetime. So we must have β < 0 for n =2 and n = 3. On the
otherhandnα−α−1 ispositiveforsixdimensionalonwardswhenα=0.4.So we
havetotakepositivebetaforsixdimensionalonwards.Sotofindtheradiusofthe
chargedstarindifferentdimensionwehavechooseα=0.4,β =−0.001for4Dand
5D spacetime and α = 0.4,β = 0.001 for the spacetime onwards six dimension.
The radius of the charged star in different dimension are shown in Table 1. From
Fig.2,weseethattheradiusof thestaris foundwherethegraphsof p(r)cutthe
r-axis and one can note that the radius of the charged star in 5D is greater than
4D for fixed values of α and β mentioned in the figure. On the other hand the
radiusofthechargedstardecreaseswhenthedimensionincreases,i.e.forforfixed
values of α and β mentioned in the figure the radius is maximum for 6D charged
star and is minimumfor 11D charged star.
The gravitationalmass inside the charged sphere can be obtained as
m(r)= r 2πn+21 rn ρ+ E2 dr
Γ n+1 8π
Z0 2 (cid:20) (cid:21)
= πn−2(cid:0)1 (cid:1) 2n−(1+α) rn−1− 8nπβ rn+1 .
4Γ n+1 2(n+nα−2α) (1+α)(n+1)
2 (cid:20) (cid:21)
(46)
(cid:0) (cid:1)
10
Fig. 2 Radiiofthestarsarefoundwhereradialpressurescutraxisfordifferentdimensions:
4D and5D intheleftpanelwhereas6D,9D and11D intherightpanel
The profile of the mass function for different dimensional compact stars are
showninFig.7.Thefigureindicatesthatm(r)is amonotonicincreasingfunction
of r and m(r)> 0 inside the charged fluid sphere. Moreover as r → 0, m(r)→ 0,
i.e. the mass function is regular at the center of the charged star.
The charged density can be obtained as
Γ n+1 n−1 (n−1)(1−α) (n−1)2 α+1 16πβ
σ= 2 + r2.
2(cid:0)πn+21(cid:1) r2 r2(n+nα−2α)r n n+nα−2α (1+α)(n+1)
(47)
Theprofileof thechargeddensityis shownin Fig.4. Thefigureindicatesthat
it is a monotonic decreasing function of r and its values increases as dimensions
increases, i.e. the value of σ is maximumfor 11D and minimum for 4D.
Fig.3 TheElectricfieldareplottedagainstrfordifferentdimensions:4Dand5Dintheleft
panelwhereas6D,9D and11D intherightpanel