Tolman-Bayin type static charged fluid spheres in general relativity

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by  Saibal Ray

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed2February2008 (MNLATEXstylefilev2.2)
Tolman-Bayin type static charged fluid spheres in general relativity
Saibal Ray1,2 & Basanti Das3
1DepartmentofPhysics,BarasatGovernmentCollege,Barasat700124,North24Parganas,WestBengal,India
2Inter-UniversityCentreforAstronomyandAstrophysics,POBox4,Pune411007,India;e-mail:[email protected]
3BeldaPrabhatiBalikaVidyapith,Belda,Midnapur721424,WestBengal,India.
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n
a ABSTRACT
J InastaticsphericallysymmetricEinstein-Maxwellspacetimetheclassofastrophysicalsolu-
tionfoundoutbyRayandDas(2002)andPantandSah(1979)arerevisitedhereinconnection
2
tothephenomenologicalrelationshipbetweenthegravitationalandelectromagneticfields.It

1 is qualitativelyshownthat the chargedrelativistic stars of Tolman (1939)and Bayin (1978)
v typeareofpurelyelectromagneticorigin.Theexistenceofthistypeofastrophysicalsolutions
5 is a probable extension of Lorentz’s conjecture that electron-like extended charged particle
0 possessesonly‘electromagneticmass’andno‘materialmass’.
0
1 Keywords: gravitation–relativity–stars:general–stars:interior.
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/
h 1 INTRODUCTION tutingmatterthree-quartersistobeascribedtotheelectromagnetic
p
field,andone-quartertothegravitationalfield”whereasLorentz’s
o- The study of the interior of stars is always fascinating to the as- (1904)conjectureofextendedelectronwasthat“thereisnoother,
r trophysicists, specially inconnection togeneral theory of relativ- no‘true’or‘material’mass,”andthusprovidesonly’electromag-
st ity.Thisisobviousbecauseofthefactthattowardsthelatestages netic masses of the electron’. Wheeler (1962) also believed that
a of stellar evolution, general relativistic effects become much im- electron has a ‘mass without mass’. Feynman (1964) termed this
v: portant.Oneoftheremarkableworksinthisdirectionwasthatof typeofmodelsas‘electromagneticmassmodels’inhisclassicvol-
i theTolman(1939) solutions.Tolmanextensivelystudiedthestel- ume. Starting from 60’s in the last century several authors (e.g.,
X lar interior and provided a class of explicit solution in terms of Florides, 1962; Cooperstock & De La Cruz, 1978; Tiwari et al.,
r knownanalyticfunctionsforthestatic,sphericallysymmetricequi- 1984;Gautreau,1985;Grøn,1986;PoncedeLeon,1987;andthe
a libriumfluiddistribution.SubsequentlyWyman(1949),Leibovitz referencestherein)tookuptheproblemagainandstudiedelectro-
(1969) and Whitman (1977) generalized some of Tolman’s solu- magneticmassmodelsforthestaticsphericallysymmetriccharged
tions.Bayin(1978)alsoobtainedsomemorenewanalyticsolutions perfect fluid distribution in the framework of general relativity.
relatedtostaticfluidspheresusingthemethodofquadratures. VeryrecentytheideaisextendedtotheEinstein-Cartantheoryand
Recently we (Ray & Das, 2002) have obtained the charged Kaluza-Klein theory by adding torsion and higher dimension re-
generalizationofBayin’swork(1978)motivatedbytheideathatin spectively (Tiwari & Ray, 1997; Ponce de Leon, 2003). Most of
stellar astrophysics the coupled Einstein-Maxwell field equations these workers exploit an equation of state ρ+p = 0 where, in
may have some physical implications. In connection to singular- general,thematterdensityρ>0andpressurep<0.Thistypeof
ityproblemitisobservedthatinthepresenceofcharge,thegrav- equationofstateimpliesthatthematterdistributionunderconsider-
itational collapse of a spherically symmetric distribution of mat- ationisintensionandhencethematterisknownintheliteratureas
tertoapointsingularitymaybeavoided.Themechanismissuch a‘false vacuum’ or ‘degenerate vacuum’ or ‘ρ-vacuum’ (Davies,
thatthegravitationalattractioniscounterbalancedbytherepulsive 1984; Blome & Priester, 1984; Hogan, 1984; Kaiser & Stebbins,
Coulombianforceinadditiontothethermalpressuregradientdue 1984).
to fluid. Also, it is seen that the presence of the charge function Itisinterestingtonotethatinthepresentstudy,eventhough
serves as a safety valve, which absorbs much of the fine-tuning, thesolutionsrelatedtopressureanddensityingeneralfollowthe
necessaryintheunchargedcase(Ivanov,2002).Thus,theproblem ordinaryequationofstate,viz.,ρ+p 6= 0butultimatelyincon-
ofcoupledcharge-matterdistributionsingeneralrelativityhasre- nectiontoelectromagnetic massmodels itturnsout tobetheex-
ceivedconsiderableattention. otic kind of equation of state (Davies, 1984; Blome & Priester,
Thepresentpaperisbasedonthesimpleinvestigationofthe 1984; Hogan, 1984; Kaiser & Stebbins, 1984) in both the cases
solutions already obtained by us (Ray & Das, 2002) and Pant & ofBayinandTolmansolutions.Wehaveinvestigatedherethatre-
Sah(1979)inconnectiontotheelectromagneticoriginofthegrav- lated to this type of vacuum- or imperfect-fluid equation of state
itationalmass.Itisworthwhiletomentionherethatthereisafairly thechargedanalogueofBayin(1978)andTolman(1939)typeas-
longhistoryofinvestigationsaboutthenatureofthemassofelec- trophysicalclassofsolutionshowtheelectromagneticfielddepen-
tron.Einstein(1919)himselfbelievedthat“...oftheenergyconsti- dency of gravitational mass. Therefore, theexistence of this type
2 Ray&Das
of solutions, in our opinion, is a probable extension of Lorentz’s Also,intermsofC(r)whenB(r)isgiven,thePfaffiandifferential
conjectureinconnectiontoastrophysicalmodels. equation(7)modifiesto
dC 1 dB B2−1 1 dB
= − + C
dr Br2 dr r3 B dr
(cid:18) (cid:19) (cid:16) (cid:17)
2 EINSTEIN-MAXWELLFIELDEQUATIONS 2 2B2q2
−C r + , (10)
r5
Wewritethelineelement forstaticsphericallysymmetric space-
whichisaRiccatiequationforC(r)withknownvalueofchargeq.
timesintheform
Bysolvingthesedifferentialequations(9)and(10),andalso
2 2 2 2 2 2 2 2 2
ds =A dt −B dr −r (dθ +sin θdφ ). (1) someothersimplecaseswe(Ray&Das,2002)obtainedthesolu-
tionsforEinstein-MaxwellfieldequationsrelatedtoBayin(1978)
inthestandardcoordinatesxi = (t,r,θ,φ),wherethequantities
typeastrophysicalclassofmodels.Thesolutionsthusobtainedfor
A(r)andB(r)arethemetricpotentials.
theparametersA,B,ρ,pandq respectivelythegravitationalpo-
TheEinstein-Maxwellfieldequations,forthemetric(1)intheco-
tentials,energy density, isotropicpressure and electriccharge are
movingcoordinatesreadas
involved with several integration constants. Some of these may,
1 2B′ 1 1 q2(r) inprinciple,bedeterminedbymatchingoftheinteriorsolutionto
B2 Br − r2 + r2 =8πρ+ r4 , (2) theexteriorReissner-Nordstro¨mmetricattheboundary r = aof
(cid:18) (cid:19) thesphericalmatterdistribution.TheexteriorReissner-Nordstro¨m
1 2A′ 1 1 q2(r) metricisgivenby
+ − =8πp− , (3)
B2 Ar r2 r2 r4
1 A′′ − A′B′(cid:18)+ 1 A′ −(cid:19)B′ =8πp+ q2(r), (4) ds2= 1− 2rm + qr22 dt2 − 1− 2rm + qr22 −1dr2
B2 A AB r A B r4 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:20) (cid:18) (cid:19)(cid:21) − r2(dθ2+sin2θdφ2). (11)
wheretheprimedenotes differentiationwithrespect toradial co-
ordinate r only. In the equation (2) - (4), the quantities ρ, p and Now,consideringthemetriccomponents g00,g11 and ∂∂g0r0 tobe
continuousacrosstheboundaryr =aofthesphereandassuming
qrepresenttheenergydensity,isotropicpressureandtotalelectric
forthetotalchargeonthesphere
chargerespectively.Thetotalchargewithinasphereofradiusris
definedas q(a)=Kan, (12)
r
q(r)=4π J0r2ABdr, (5) one can get the following cases of the gravitational mass (vide
Z0 equations(53),(56),(61),(65),(69)and(72)ofSection4inRay
&Das(2002))intheexplicitformswithelectriccharge.
Jibeingthe4-currenttakesheretheform,viatheelectromagnetic
fieldF01,as CaseI(i):Forn=1
2 q 2 2 q 3
01 q(r) m=q +a0a1 +a1 , (13)
F = . (6) K K
ABr2
(cid:16) (cid:17) (cid:16) (cid:17)
(ii):Forn=3
Now, eliminating p from equations (3) and (4) and assuming
A′/Ar=C(r)onecanget 2 q 2 q 5/3 q 2/3
m=a1 +4K +a0a1 , (14)
K K K
1 C 1
B3r2 + B3 dB− B2 dC CaseII(i)(cid:16):For(cid:17)n=1 (cid:16) (cid:17) (cid:16) (cid:17)
whichis−aP(cid:18)f(cid:16)aBfBfi2a2−nr3d1if+ferCeBn2t2(cid:17)iral−eq2urqa52ti(cid:16)(cid:19)ondirn(cid:17)t=hre0e.dimensionshavi(n7g) m= W102 (cid:20)W02(1−2K2)+(cid:16)Kq (cid:17)2(cid:18)C1− Kq22(cid:19)q(cid:21)1/32
× , (15)
thegeneralformas K
h i
(ii):Forn=3
f1(B,C,r)dB+f2(B,C,r)dC+f3(B,C,r)dr=0. (8)
1 2 q 2/3 2 2 q 4/3 1/2
m = W02 W0 +C1 K +(W0 K −1) K
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
3 ELECTROMAGNETICMASSMODELSFORSTATIC × q −2K1/3q5/3, (16)
K
CHARGEDFLUIDSPHERES
h i
CaseIII:Forn=1
3.1 Bayin’sclassofsolution 1+K2 2 q q
m= =(K −1) C3 −2 , (17)
The Pfaffian differential equation (7) can be solved in different C3 K K
ways as shown by us (Ray & Das, 2002) in details. It is shown CaseIV:Forn=1 h (cid:16) (cid:17) ih i
thatintermsofB(r)whenC(r)isknown,thePfaffiandifferential
equation(7)becomes m=3C62 3C5+ q 3 q 4, (18)
K K
dB = 1− 2rq22 B3+ C2r− r13 + ddCr B. (9) whereK,(cid:20)a0,a1,W(cid:16)0,C(cid:17)1,(cid:21)Ch3,Ci5 andC6 all areconstant quan-
dr " C+ r12 r3# (cid:20) C+ r12 (cid:21) tities.Now,mforthecasesI(i),I(ii)andIVareasusual positive
(cid:0) (cid:1)
Tolman-Bayintypestaticcharged... 3
whereasfortherestofthecasesII(i),II(ii)andIIItheconditions wherethegravitationalmassm(r)=M(r)+µ(r)beingdefined
for positivity are (1/2) > K2 > (q2/C1), K > (1/W0) and as
1<K <(C3q/2)respectively. r
2
Itisobservedfromtheexplicitformsoftheabovesetofex- M(r)=4π ρr dr (28)
pressions that the effective gravitational mass m, along with the Z0
centralpressuresanddensitiesatr =0(videequations(22),(27), and
(31),(40),(45)and(21),(28),(32),(39)respectivelyinRay&Das r
2 2
(2002)), is related tothe charge q of equation (12) of the spheri- µ(r)= (q /2r )dr, (29)
calsystem.Therefore,vanishingofthechargemakesallthephys- Z0
icalquantitiesincludingthegravitationalmassalsotovanish.This respectivelytheSchwarzschildmassandthemassequivalence of
meansthatthegravitationalmassoriginatesfromtheelectromag- electromagneticfield.Hence,thetotalgravitationalmass,m(r =
netic field alone. Thus, the gravitational mass is purely ‘electro- a),canbecalculatedas
magneticmass’(Lorentz,1904)andthistypeofmodelisknownas na2(2−n)+2q2
‘electromagneticmassmodel’intheliterature(Feynman,1964).It m= . (30)
2(1+2n−n2)a
isrelevanttonoteherethatthisparticularimportantfeatureofthe
solution set, viz., the electromagnetic nature of the gravitational If we now make the specific choice n = 0 for the parameter n
mass is obviously not available in the uncharged case of Bayin appearingintheabovesolutionsetthenonegetthefollowingex-
(1978) and thus indicates that the presence of charge allows for pressions.
awiderrangeofbehaviour. 2 2m q2
A = 1− + , (31)
a a2
(cid:20) (cid:21)
3.2 Tolman’ssolutionVI
Intheintroductionwehavementionedthatmotivatedbythework B−2= 1− 2m + q2 = 1− 2q2 , (32)
a a2 a2
ofTolman(1939),asimilarkindofnewclassofsolutionwasfound (cid:20) (cid:21) (cid:20) (cid:21)
byBayin(1978)andhenceinviewoftheresultsofsub-section3.1
relatedtoBayin’sworkitwillbeinterestingtoexaminethesolu- 1 q2
ρ(r)= , (33)
tionsofTolmanwhethertheyarealsoamemberofelectromagnetic 8πr2 a2
(cid:20) (cid:21)
massmodels.Asaready-madeexamplewewouldliketopresent
here the solution obtained by Pant & Sah (1979) to meet our, at 1 q2
leastpartial,requirement.Forastaticsphericallysymmetricdistri- p(r)=−8πr2 a2 , (34)
butionofchargedfluidthesolutionset(videequations(10a),(10b), (cid:20) (cid:21)
(10c)and(11)inPant&Sah(1979))isasfollows.
1 q 2q2 1/2
A2=eν =br2n, (19) σ(r)=±4πr2 a 1− a2 , (35)
h i(cid:20) (cid:21)
B−2 =e−λ =c, (20) q
E(r)= , (36)
ar
1 2
ρ= [1−c(n−1) ], (21) and
16πr2
q2
m= . (37)
1 2 a
p= [c(n+1) −1], (22)
16πr2 Thus, for vanishing electric charge all the physical quantities in-
cludinggravitationalmassvanishandthespacetimebecomesflat.
σ=± 1 [c{1−c(1+2n−n2)}]1/2, (23) Itisinterestingtonotethat,inthepresentsituation,theequations
4πr2 2
(33) and (34) related to the isotropic pressure and matter density
provide an equation of state ρ+p = 0, which is known as the
2 1 2
E = 2r2[1−c(1+2n−n )], (24) vacuum- or imperfect-fluid equation of state. As isevident, from
the equations (21) and (22), this is not true for the general case
where
whenn6=0andcanbereadas
b=a−2n 1− 2m + q2 , (25) 1 n(a2−2q2)
a a2 ρ+p= . (38)
(cid:20) (cid:21) 4πr2 a2(1+2n−n2)
(cid:20) (cid:21)
2m q2 2q2 2 −1 Hencestartingfromaperfectfluidtypeequationofstatevian=0
c= 1− a + a2 = 1− a2 (1+2n−n ) . (26) we are arriving at the imperfect-fluid type equation of state and
(cid:20) (cid:21) (cid:20) (cid:21) thusnhere istakingadefiniteand peculiar rolefor deciding the
Theabovesetofsolutions,inviewofc,withΛ = 0andB = 0 form of the equation of state. This particular aspect is also true
represents the charged analogue of Tolman’s (1939) solution VI viatheequations(2)and(3)fortheequations(19)and(20)which
andthusintheabsenceofthetotalchargeqreducestotheneutral reduce to the equations (31) and (32) respectively, with n = 0
one(thesub-case C of uncharged fluidsphere inthePant &Sah whenwegetν +λ = 0.Thisagain,fortheReissner-Nordstro¨m
(1979)).Now,theequation(2)canbeexpressedintheform metric (equation (11)) related to the spherically symmetric static
−2 2m(r) chargedfluiddistribution,canbeexpressedintheformg00g11 =
B =1− , (27) −1. Thus, in view of equations (2) and (3), we see that for the
r
4 Ray&Das
boundary conditionν +λ = 0onecan getρ+p = 0andvice givesenough scopetotheoreticalspeculationsandhencethecor-
versa,sothatλ=−ν ↔p=−ρ.1Thisresultmeansthatifρ>0 respondingmodelingandinvestigationsbecomeasmuchpertinent
then must be p < 0 for the inside of the fluid sphere though, in forthesecasesasfortheestablishedneutralsystems.
general, ρ and p arepositive for the condition 0 6 n 6 1. This
partially admits the comment by Ivanov that “... electromagnetic
massmodelsallseemtohavenegativepressure”.Partiallybecause, ACKNOWLEDGMENTS
inour opinion, therearesomeexamples ofelectromagnetic mass
One of the authors (SR) is thankful to the authority of Inter-
models wherepositive pressures arealsoavailable willbeshown
University Centre for Astronomy and Astrophysics, Pune, India,
elsewhere.
forprovidingAssociateshipprogrammeunderwhichapartofthis
workwascarriedout.
4 CONCLUSIONS
REFERENCES
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onPhysics,(Addison-Wesley,PaloAlto,Vol.II,Chap.28).
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FloridesP.S.,1962,Proc.Cam.Phil.Soc.,58110.
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We would also like to mention here that the works done by
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different investigatorson electromagnetic mass models so far are
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mainlyconcernedwiththestructureoftheclassicalelectron(spe- RayS.andDasB.,2002,Astrophys.SpaceSci.,282635.
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studiesdonot ruleout thepossibilityof theexistence of massive
astrophysical systems that are not electrically neutral (Treves &
Turella,1999).Themechanism,thoughnotcompletelyunderstood,
is mainly related to the acquiring a net charge by accretion from
thesurroundingmedium.Ontheotherhand,therearesomeother
views of acquiring charge by a compact star during its collapse
fromthesupernovastage.Inthisregarditwillbeworthmentioning
thattostudytheeffectofelectricchargeincompactstarsRayetal.
(2003)assumeanansatzsuchthatσ=αρwhereαisrelatedtothe
chargefractionf asα=869.24f andshowbynumericalcalcula-
tionthatinordertoseeanyappreciableeffectonthephenomenol-
ogyofthecompactstars,thetotalelectricchargeistobe∼ 1020
Coulomb.Therefore,inouropinion,evensucharemotepossibility
1 Acoordinate-independent statementoftherelationg00g11 = −1and
henceν+λ=0isgivenbyTiwarietal.(1984).