Radiating stars with generalised Vaidya atmospheres

0.14 MB·English

by  S. D. Maharaj

General Relativity and Gravitation manuscript No.
(will be inserted by the editor)
Radiating stars with generalised Vaidya atmospheres
S. D. Maharaj · G. Govender · M.
Govender
Received: date/Accepted: date
3
1 Abstract Wemodelthegravitationalbehaviourofaradiatingstarwhentheexte-
0
riorgeometryisthegeneralisedVaidyaspacetime.Theinteriormatterdistribution
2
isshear-freeandundergoingradialheatflow.Theexteriorenergymomentumten-

n sor isa superpositionof a nullfluid and a stringfluid. An analysisofthejunction
a
conditionsat thestellarsurfaceshows thatthepressureat theboundarydepends
J
ontheinteriorheatfluxandtheexteriorstringdensity.Theresultsforarelativis-
8
tic radiatingstarundergoingnonadiabaticcollapseareobtainedas a special case.

For a particular model we demonstrate that the radiating fluid sphere collapses
]
c without the appearance of the horizon at the boundary.
q
- Keywords heat conducting fluids · radiatingstars · relativisticastrophysics
r
g
[
1 Introduction

1
v
The study of radiating stars in the context of general relativity has generated
7
much interest in researchers because of the variety of applications in relativistic
1
astrophysics.Thesestudiesareimportantastheyenableustoinvestigatephysical
4
1 featuressuchassurfaceluminosity,dynamicalstability,particleproductionatthe
. stellarsurface,relaxationeffects, causal temperaturegradientsand otherthermo-
1
dynamicalprocesses. Some relevantreferences investigatingthese issues aregiven
0
3 byDiPriscoetal.[1],Govenderetal.[2],Herreraetal.[3]andPinheiroandChan
1 [4]. Relativistic radiating stars are also important in the process of gravitational
: collapse, describing the final state of stars, formation of singularities and black
v
hole physics, in four and higher dimensions. Recent investigations in this regard
i
X are contained in the works of Goswamiand Joshi [5], Joshi [6] and Madhav et al.
r [7].In particular,thevalidityofthecosmiccensorshipconjecturecan betestedin
a
this physical scenario.
S.D.Maharaj·G.Govender ·M.Govender
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of
KwaZulu-Natal,PrivateBagX54001,Durban4000,SouthAfrica
E-mail:[email protected]
2 S.D.Maharajetal.
Themodelofarelativisticradiatingstarundergoingdissipationwascompleted
bySantos[8]byanalysingthejunctionconditionsatthestellarsurface.Bymatch-
ing a shear-free interior spacetime to the radiating Vaidya exterior spacetime, he
showed that at the surface the pressure is nonvanishing and proportional to the
heat flux. Subsequently several explicit relativistic radiating stellar models have
been found by investigatingthe appropriate boundary condition. Kramer [9] and
Maharaj and Govender [10] generated nonstatic radiating spheres from a static
model by allowingcertain parametersto becomefunctions of time.Kolassis et al.
[11]andThirukkaneshandMaharaj[12]assumedgeodesicfluidtrajectoriestopro-
ducenewradiatingmodels.IntheapproachofDeOlivieraetal.[13]andNogueira
and Chan [14] the model has an initial static configuration before the radiating
sphere starts gradually to collapse. Exact solutions for shear-free interiors which
are conformally flat generate radiating stellar models as shown by Herrera et al.
[15],Herreraetal.[16],MaharajandGovender[17]andMisthryetal.[18].Stellar
models which are radiating with nonzero shear are difficult to analyse because of
the complexity of the boundary condition. However even in this case there have
been advances in obtaining exact solutions. Particular exact models have been
found by Naidu et al. [19], Rajah and Maharaj [20] and Pinheiro and Chan [21].
InthispaperweseektogeneralisetheSantosjunctionconditionsbymatching
a shear-free interior spacetime to the generalised Vaidya exterior spacetime. The
energy momentumtensorof thegeneralisedVaidyaspacetimemaybe interpreted
as a superposition of two fluids, a null dust and a null string fluid. The physical
propertiesofthegeneralisedVaidyaspacetimehavebeendiscussedbyHusain[22]
andWangand Wu [23].GlassandKrisch[24] haveinterpretedtheexteriorspace-
timeasasuperpositionoftwofluidsoutsidearelativisticstar,theoriginalVaidya
null fluid and a new null fluid composed of strings. By assuming diffusive trans-
port for the string fluid Glass and Krisch [25] found new solutions to Einstein’s
equations with transverse stresses. Physically reasonable energy transport mech-
anisms have been generated by Krisch and Glass [26] in the stellar interior with
thegeneralisedVaidyametricastheexteriorspacetime.Theseinvestigations,and
othertreatments,havelargelyfocussedonphysicalprocessesintheexteriorofthe
stellar model with a generalised Vaidya atmosphere.To fully describe a radiating
stellar model requires generationof the junction conditionsat the stellar surface.
We follow the convention that the coupling constant 8πG and the speed of
c4
light c are unity; the metric has signature (− + ++). In Sect. 2 we present the
field equations for the interior and exterior spacetimes. In Sect. 3 the matching
of the interior and exterior spacetimes across the stellar surface is outlined. The
new set of junction conditions are derived for the generalised Vaidya spacetime.
We indicate how the new junction conditions generalise the junction conditions
previously derived by Santos [8]. The physical significance of our new result is
highlightedintermsofastringfluid.We alsoconsiderthenew junctioncondition
in the context of conservation of momentum flux across the stellar boundary. In
Sect. 4 we generatea particularmodel and demonstratethat the new generalised
junctionconditionhasasolution.Inthismodelitispossiblefortheradiatingfluid
spheretocollapsewithouttheappearanceofaboundary.Theresultsofthispaper
are briefly summarised in Sect. 5.
RadiatingstarswithgeneralisedVaidyaatmospheres 3
2 Field Equations
Spacetime needs to be divided into two distinct regions, the interior spacetime
M− andtheexteriorspacetimeM+ forastellarmodel.Theboundaryofthestar
Σ serves as the matching surface for M− and M+. The boundary or stellar sur-
faceis atimelikethree-dimensionalhypersurface.We requireforthefirstjunction
condition that
(ds2+)Σ =(ds2−)Σ =ds2Σ (1)
sothatthelineelementsmatchontheboundaryΣ.Thesecondjunctioncondition
isgeneratedbythecontinuityoftheextrinsiccurvatureofΣ acrosstheboundary
given by
(K+ ) =(K− ) (2)
αβ Σ αβ Σ
Note that the junction conditions (1) and (2) are equivalent to the Lichnerowicz
[27] and O’ Brien and Synge [28] junction conditions.
The line element for the interiormanifoldM− is given by
ds2 =−A2(t,r)dt2+B2(t,r)[dr2+r2(dθ2+sin2θdφ2)] (3)
in comoving and isotropic coordinates. The interior spacetime is expanding and
accelerating but is shear-free. A physically relevant interior matter distribution
that is consistent with (3) is given by
−
T =(µ+p)u u +pg +q u +q u (4)
ab a b ab a b b a
where µ is the energy density, p is the isotropic pressure, q is the radial heat
a
flux vector and ua = 1δa is the comoving fluid four-velocity. The Einstein field
A 0
equations G− =T− for the interiormanifold M− are given by
ab ab
B˙2 1 B′′ B′2 4B′
µ=3 − 2 − + (5a)
A2B2 B2 B B2 r B
!
1 B¨ B˙2 A˙ B˙
p= −2 − +2
A2 B B2 AB
!
1 B′2 A′B′ 2A′ 2B′
+ +2 + + (5b)
B2 B2 A B r A r B
!
B¨ A˙ B˙ B˙2 1 A′
p=−2 +2 − +
A2B A3B A2B2 rAB2
1 B′ A′′ B′2 B′′
+ + − + (5c)
rB3 AB2 B4 B3
2 B˙′ B′B˙ A′B˙
q =− − + + (5d)
AB2 B B2 A B
!
where dots and primes denote differentiationwith respect to t and r respectively.
The results (1)-(5) were first obtained by Santos [8].
The line element for the exteriormanifold M+ is taken to be
ds2 =− 1−2m(v,r) dv2−2dvdr+r2(dθ2+sin2θdφ2) (6)
r
(cid:18) (cid:19)
4 S.D.Maharajetal.
wherem(v,r)isthemassfunction,andisrelatedtothegravitationalenergywithin
agivenradiusr(LakeandZannias[29],PoissonandIsrael[30]).Thismetricisoften
called the generalised Vaidya spacetime since it reduces to the Vaidya spacetime
when m = m(v) which is the mass of the star as measured by an observer at
infinity. It has been demonstrated by Husain [22] and Wang and Wu [23] that an
energy momentum tensor consistent with (6) is
T+ =T(n)+T(m)
ab ab ab
=εl l +(ρ+P)(l n +l n )+Pg (7)
a b a b b a ab
which represents a superposition of a null dust and a null string fluid. In general
T+ representsaTypeIIfluidasdefinedbyHawkingandEllis[31].Thenullvector
ab
la isadoublenulleigenvectoroftheenergymomentumtensorT+.Theweakand
ab
strong energy conditions, and the dominant energy conditions are satisfied for
properchoicesofthemassfunctionm(v,r).In(7)wehaveintroducedthetwonull
vectors
1 m(v,r)
l =δ0, n = 1−2 δ0+δ1 (8)
a a a 2 r a a
(cid:20) (cid:21)
where l la = n na = 0 and l na = −1. The Einstein field equations G+ = T+
a a a ab ab
for the exterior manifoldM+ are then given by
m
ε =−2 v (9a)
r2
mr
ρ =2 (9b)
r2
mrr
P =− (9c)
r
where we have used the notation
∂m ∂m
mv = ∂v , mr = ∂r
Weinterpretεasthedensityofthenulldustradiation;ρandP arethenullstring
density and null string pressure, respectively. Note that (7)-(9c) were derived by
Wang and Wu [23].
3 Generalised Santos conditions
It is possible to match the spacetimes (3) and (6) across the boundary Σ. Since
the derivationis similarto the Santos [8] treatmentwe provideonly an outlineof
the argument for our more general case with m=m(v,r). The intrinsic metric to
the hypersurfaceΣ is defined by
ds2 =−dτ2+Y2 dθ2+sin2θdφ2 (10)
Σ
(cid:16) (cid:17)
For the interiorspacetime M− we obtain
A(r ,t)dt=dτ (11a)
Σ
r B(r ,t) =Y(τ) (11b)
Σ Σ
RadiatingstarswithgeneralisedVaidyaatmospheres 5
For the exterior region M+ we generate the results
r (v)=Y(τ) (12a)
Σ
2m dr dv −2
1− +2 = (12b)
r dv dτ
(cid:18) (cid:19)Σ (cid:18) (cid:19)Σ
Equations (11) and (12) correspond to the first junction condition (1). Observe
that the quantityτ was defined on the surface Σ as an intermediatevariable. On
eliminatingτ we have
2m drΣ 1/2
A(r ,t)dt= 1− +2 dv (13a)
Σ rΣ dv
(cid:18) (cid:19)
r (v)=rB(r ,t) (13b)
Σ Σ
Equations (13) are the necessary and sufficient conditions for the first junction
condition (1) to be valid.
The intrinsic curvature for the interior spacetimeM− has the form
− 1 A′
K = − (14a)
11 B A
(cid:18) (cid:19)Σ
− ′
K = r(rB) (14b)
22 Σ
K− =(cid:2)sin2θK(cid:3)− (14c)
33 22
The extrinsic curvature for the exterior spacetime M+ has the form
K+ = ˜v˜−v˜m +v˜mr (15a)
11 "v˜ r2 r #
Σ
2m
K+ = v˜ 1− r+r˜r (15b)
22 r
(cid:20) (cid:18) (cid:19) (cid:21)Σ
K+ =sin2θK+ (15c)
33 22
where ˜r = dr and v˜ = dv. Observe the appearance of the term containing mr
dτ dτ
in K+ which does not exist in the treatment of Santos [8]. As we shall see later
11
this has a profound effect on the physics of the model. Equations (14) and (15)
correspond to the second junction condition (2). The mass profile in terms of the
metric functions can be generated by eliminatingr,˜r and v˜. We observethat
m(v,r)= rB 1+r2B˙2 − 1 (B+rB′)2 (16)
" 2 A2 B2 !#
Σ
which is the totalgravitationalenergy contained withinthe stellarsurface Σ. We
also establish the relationship
1 A′ B′ B˙ −1
− = 1+r +r
B A B A
(cid:18) (cid:19) !
× mr + B′ + rB′2 − r B˙2 − 1 rB˙′ +rB¨ −rB′B˙ −rB˙ A˙ (17)
" r B2 2 B3 2BA2 A B A B2 A2!#
Σ
6 S.D.Maharajetal.
Thisexpressionmaybe simplifiedfurther:multiplywith1+rB′ +rB˙ and utilise
B A
(5b) and (5d). We then arriveat the result
mr
p= qB−2 (18)
r2B2 Σ
(cid:16) (cid:17)
whichgeneralisesthejunctionconditionofSantos[8].Hencewehavedemonstrated
that the junction conditions (2) are equivalent to
m(v,r)= rB 1+r2B˙2 − 1 (B+rB′)2 (19a)
" 2 A2 B2 !#
Σ
mr
p = qB−2 (19b)
r2
Σ
(cid:16) (cid:17)
Equations (19) are the necessary and sufficient conditions for the second junc-
tion condition(2) to be valid. We point out that the mathematicalapproach and
procedure that we have followed is similar to Santos [8]. However in our case the
external stellar atmosphere is the generalised Vaidya spacetime. The form of the
equations(10)-(19b)canberelatedtotheequationsofSantos[8]sincetheyhavea
similarstructure.Thefactthatm=m(v,r)fundamentallyaffectsthefinalresult.
The equations for the extrinsic curvature (15a), the matching condition (17) and
junction condition (18) are fundamentallydifferent.
We have generated the relationships (13) and (19) so that the junction con-
ditions (1) and (2) are satisfied for the shear-free interior spacetime (3) and the
generalised Vaidya exterior spacetime (6) across the hypersurface Σ. This gener-
alisestheSantos[8]resultforarelativisticradiatingstarwhenm=m(v).Observe
that when m depends on the coordinatev only then (19b) becomes
p=qB (20)
at the boundary Σ, which is the earlier Santos junction condition. When (20) is
valid then the pressure p on the boundary depends only on the heat flux q. We
haveshown herethatif m=m(v,r)then (19b) is valid,and thepressurep on the
boundary depends on the heat flux q and the gradient mr(v,r).
ThegeneralisedVaidyaspacetimehasphysicalsignificanceandcontainsmany
known solutions of the Einstein field equations with spherical symmetry. It con-
tainsthemonopolesolution,thedeSitterandAnti-deSittersolutions,thecharged
Vaidyasolutionandtheradiatingdyonsolution.Thephysicalfeaturesandtheen-
ergy momentum complexes, that provide acceptable energy momentum distribu-
tionsforthesesystems,havebeenstudiedbyBarriolaandVilenkin[32],Chamorro
and Virbhadra [33], Virbhadra [34]-[36] and Yang [37]. Glass and Krisch [24]-[25]
and Krisch and Glass [26] have interpreted the generalised Vaidya spacetime to
represent a superposition of an atmosphere composed of two fluids: a string fluid
and a null dust fluid. This atmosphere may model several physical situations at
different distance scales, eg. the exterior regions of black holes (distance scale of
multiples of the Schwarzschild radius) and globular clusters containing a com-
ponent of dark matter (distance scale of the order of parsecs). The additional
term2mr intheboundarycondition(19b)arisesfromthematchingatthesurface
r2
Σ. This quantity has physical significance and can be interpreted as a particular
RadiatingstarswithgeneralisedVaidyaatmospheres 7
contributionfromtheenergymomentumtensor.Weobservethattheterm2mr in
r2
(19b)isthesamequantityasthatin(9b).Thereforewemayinterpretthequantity
2mr as the string density ρ.
r2
We can thereforewrite (19b) in the more transparentform
p=[qB−ρ] (21)
Σ
attheboundaryΣ.Consequentlyforaradiatingstarwithoutgoingdissipationin
theformofradialheatflow,withthegeneralisedVaidyaspacetimeastheexterior,
the pressure on the surface depends on the interior heat flux q and the exterior
stringdensityρ.Theappearanceof thequantityρ in (21)allowsformoregeneral
behaviour that was the case in the Santos [8] treatment. From (20) we observe
that q = 0 implies that p = 0 on Σ and the exterior manifold M+ must be the
Schwarzschild exterior metric with m being constant. In (21) we note that we
obtain the Schwarzschild exterior geometry when q = 0 = ρ. However it is clear
from (21) that when qB =ρ then p=0 on Σ and the exterior spacetimeremains
the generalisedVaidya spacetimewith m=m(v,r). In addition,when q=0 then
p=−ρ on Σ and the interioris not radiating.
Itispossibletoprovideaphysicalinterpretationofourresultbyconsideration
ofthemomentumfluxacrosstheboundaryΣ.Sincethequantity(19a)represents
the total gravitational energy for a sphere of radius r within Σ we can write
m(v,r)=m(t,r). Partiallydifferentiating(19a) eventuallyleads to the result
2 2
pΣ =−r2v˜2mv − r2mr (22)
which reduces to the corresponding Santos [8] equation when m = m(v). The
radial flux of momentum across the hypersurface Σ is defined by
F± =e±an±bT± (23)
0 ab
where e±a and n±b are vectors which are respectively tangent and normal to Σ.
0
For conservation of momentum flux across Σ we must have
F+ =F− (24)
In the interiormanifold M− we can generate the quantity
−
F =−qB (25)
In the exterior manifoldM+ we can produce the quantity
2
F+ = v˜2m (26)
r2 v
Then equations (22)-(26) yield the result
2 mr
p = qB−
Σ r2B2
(cid:18) (cid:19)Σ
which is the same as (19b). Therefore the junction condition (19b) corresponds
to the conservation of the radial flux of momentum across the hypersurface Σ. It
represents the local conservation of momentum.
8 S.D.Maharajetal.
4 A particular model
To illustratethe utility of the generalised Santos condition we consider a specific
example relating to horizons. For simplicity we take the mass function to be of
the form m=p˜r2 where p˜is an arbitrary constant. For this example we consider
a particularformof themetriccoefficientsgivenin(3).We choosethecoefficients
to be separable in r and t so that
A=a(r), B =b(r)R(t) (27)
The field equations (5) then yield the quantities
1 3 1 2b′′ b′2 4b′
µ = R˙2− − + (28a)
R2 a2 b2 b b2 rb
" !#
1 1 1 b′2 2a′b′ 2 a′ b′
p = − 2RR¨+R˙2 + + + + (28b)
R2 a2 b2 b2 ab r a b
" (cid:16) (cid:17) (cid:18) (cid:19)!#
2a′R˙
q =− (28c)
R3a2b2
and the condition for the isotropyof pressure
a′′ b′′ b′2 a′b′ a′ b′
+ −2 −2 − − =0 (29)
a b b2 ab ra rb
The boundary condition(19b) now yields at r=r the equation
Σ
2RR¨+R˙2+MR˙ =N +P (30)
where we have used the particular choice of the mass function m = p˜r2, the
potentials(27) and the system (28). In the above we have set
′
2a
M =− (31a)
b
a2 b′2 a′b′ 2 a′ b′
N = +2 + + (31b)
b2 b2 ab r a b
" (cid:18) (cid:19)#
a2p˜
P =6 (31c)
b2
which are constants in the integrationprocess as (30) holds on the boundary Σ.
Equation(30)does nothavea generalsolutionin closedform;forthepurpose
ofourinvestigationweobservethatitadmitsanelementaryparticularsolutionof
the form
R(t)=−Ct (32)
where C is given by
1 1
C = M± M2+4(N +P) 2
2
(cid:20) (cid:16) (cid:17) (cid:21)
RadiatingstarswithgeneralisedVaidyaatmospheres 9
This solutionis useful in that at thesurfaceof thecollapsingstar theratio mΣ is
rΣ
independent of time. The mass profile given by (19a) and the solution (32) yield
the followingexpression
2m r2b2C2 rb′ r2b′2
Σ =2 − − (33)
rΣ " 2a2 b 2b2 #
Σ
which is valid on the stellar surface. It is clear that the quantities in the above
equationareevaluatedattheboundaryΣ; thereissufficientfreedomprovidedby
r,a,b andC sothattherighthandsidecanbeconstrainedinsuchamannerthat
the ratio 2mΣ is strictly less than unity. Consequently no horizon will form. We
rΣ
can illustratethis explicitlyby settingb(r)=1 and a(r)= 1ξr2+β where β and
2
ξ are arbitraryconstants. Then we get
2m C2r2
1− Σ = 1− (34)
rΣ " β+ 1ξr2 2#
2 Σ
WechooseC2tobelessthan β2+βξ+1ξ2r(cid:0)2onΣso(cid:1)that1−2mΣ isnotzeroforall
r2 4 rΣ
time. Consequently the boundary surface never reaches the horizon. This simple
example demonstrates the absence of the horizon and is similar to the result of
Banerjee et al. [38] which holds in the conventional Vaidya spacetime. We regain
theirresults when P =0 in (31). We interpretour resultto mean thatthereis no
accumulationofenergyintheinterioroftheradiatingstarasitreleasesenergyat
the rateof generationduring the collapse.
5 Conclusion
In this work we have produced a general model of a relativistic radiating star by
performing the smooth matching of a shear-free interior spacetime to the gener-
alised Vaidya exterior spacetime, across a timelike spatial hypersurface. We have
demonstratedthatwiththegeneralisedVaidyaradiatingmetric,thejunctioncon-
ditions on the stellar surface change substantially, and consequently represents a
moregeneralatmospheresurroundingthestar.Theatmosphereisasuperposition
of null dust and a string fluid. We find that the density of the string fluid affects
the pressure at the stellar boundary. We have shown explicitlythat
p=qB−ρ
string
at the stellar surface. If the weak and strong energy conditions or the dominant
energy conditions are satisfied then ρ ≥0 (µ6=0) and ρ ≥P ≥
string string string
0 (µ6=0)respectively.Thisindicatesthatforoutgoingheatflux ingravitational
collapse,theexteriorstringfluiddensityreducesthepressureonthestellarbound-
ary. It is interestingto note that we have shown using a geometric approach that
the derivative of the mass function with respect to the exterior radial coordinate
is relatedto the stringdensity.By means of a simpleexamplewehaveconsidered
theformationof horizonsfor a heat conductingsphere whichradiatesenergy into
thegeneralisedVaidyaexternalatmosphereduringcollapse.Thehorizondoesnot
appear in this exampleat any stageof the collapse.
10 S.D.Maharajetal.
Acknowledgements MG and GG thank the National Research Foundation and the Uni-
versity of KwaZulu-Natal for financial support. SDM acknowledges that this work is based
uponresearchsupportedbytheSouthAfricanResearchChairInitiativeoftheDepartmentof
Science and Technology and the National Research Foundation. The authors would also like
tothankSanjayWaghandSubharthiRayforconstructivecriticismsandusefuldiscussions.
References
1. DiPrisco,A.,Herrera,L.,LeDenmat, G.,MacCallum,M.A.H.,Santos, N.O.:Phys. Rev.
D76,064017(2007)
2. Govender, M.,Maharaj,S.D.,Maartens,R.:Class.Quantum Grav.15,323(1998)
3. Herrera,L.,LeDenmat,G.,Santos, N.O.:Phys.Rev.D79,087505(2009)
4. Pinheiro,G.,Chan,R.:Gen.Relativ.Gravit.40,2149(2008)
5. Goswami,R.,Joshi,P.S.:Phys.Rev.D76,084026 (2007)
6. Joshi,P.S.:Pramana-J.Phys.69,119(2007)
7. Madhav,T.A.,Goswami,R.,Joshi,J.S.:Phys.Rev.D72,084029(2005)
8. Santos, N.O.:Mon.Not.R.Astron.Soc.216,403(1985)
9. Kramer,D.:J.Math.Phys.33,1458(1992)
10. Maharaj,S.D.,Govender, M.:Aust.J.Phys.50,959(1997)
11. Kolassis,C.A.,Santos, N.O.,Tsoubelis,D.:Astrophys.J.327,755(1988)
12. Thirukkanesh,S.,Maharaj,S.D.:J.Math.Phys.50,022502(2009)
13. de Oliviera, A.K.G., Santos, N.O, Kolassis, C.A.: Mon. Not. R. Astron. Soc. 216, 1001
(1985)
14. Nogueira,P.C.,Chan,R.:Int.J.Mod.Phys.D13,1727(2004)
15. Herrera,L.,LeDenmat,G.,Santos,N.O.,Wang,A.:Int.J.Mod.Phys.D13,583(2004)
16. Herrera,L.,DiPrisco,A.,Ospino,J.:Phys.Rev.D74,044001 (2006)
17. Maharaj,S.D.,Govender, M.:Int.J.Mod.Phys.D14,667(2005)
18. Misthry,S.S.,Maharaj,S.D.,Leach, P.G.L.:Math.Meth.Appl.Sci.31,363(2008)
19. Naidu,N.F.,Govender, M.,Govinder,K.S.:Int.J.Mod.Phys.D15,1053(2006)
20. Rajah,S.S.,Maharaj,S.D.:J.Math.Phys.49,012501(2008)
21. Pinheiro,G.,Chan,R.:Gen.Relativ.Gravit.43,1451(2011)
22. Husain,V.:Phys.Rev.D53,1759(1996)
23. Wang,A.,Wu,Y.:Gen.Relativ.Gravit.31,107(1999)
24. Glass,E.N.,Krisch,J.P.:Phys.Rev.D57,5945(1998)
25. Glass,E.N.,Krisch,J.P.:Class.QuantumGrav.16,1175(1999)
26. Krisch,J.P.,Glass,E.N.:J.MathPhys.46,062501 (2005)
27. Lichnerowicz,A.:TheoriesRelativistesdelaGravitationetdel’Electromagnetisme.Mas-
son,Paris(1955)
28. O’Brien,S.,Synge, J.L.:DublinInst.Adv.Stud.A9,1(1952)
29. Lake,K.,Zannias,T.:Phys.Rev.D43,1798(1991)
30. Poisson,E.,Israel,W.:Phys.Rev.D41,1796(1990)
31. Hawking,S.W.,Ellis,G.F.R.:Thelargescalestructureofspacetime.CambridgeUniversity
Press,Cambridge(1973)
32. Barriola,M.,Vilenkin,A.:Phys.Rev.Lett.63,341(1989)
33. Chamorro,A.,Virbhadra,K.S.:Pramana-J.Phys.45,181(1995)
34. Virbhadra,K.S.:Phys.Rev.D41,1086(1990)
35. Virbhadra,K.S.:Phys.Rev.D42,2919(1990)
36. Virbhadra,K.S.:Phys.Rev.D60,104041(1999)
37. Yang,I.C.:Chin.J.Phys.45,497(2007)
38. Banerjee,A.,Chatterjee, S.,Dadhich,N.K.:Mod.Phys.Lett. A17,2335(2002).