arXiv:0910.3230v1 [gr-qc] 16 Oct 2009NewPhysicallyRealisticSolutionsforChargedFluidSpheresJBurkeandDHobillDepartmentofPhysicsandAstronomy,UniversityofCalgary,Calgary,Alberta,Canada,T2N1N4E-mail: hobill@crag.ucalgary.caAbstract. Anewclass of solutions to the coupled, spherically symmetricEinstein-Maxwell equationsforacompactmaterial sourceisconstructed. Someof these solutions canbe made tosatisfyanumber of requirements for beingphysicallyrelevant, includinghaving acausal speedof sound. Inthe case ofvanishingcharge these solutions reduce tothose foundbyBayinandTolman.Onlythelattercanbeconsideredashavingphysicallyrealisticproperties.PACSnumbers: 04.20.Jb,04.40.Nr1. IntroductionOneof the most fundamental problems ingeneral relativityis theconstructionofexactsolutionstotheEinsteinequations(ortheEinsteinequationscoupledtootherequationsassociatedwithfundamentalphysicaltheories). Anevenmorechallengingproblemis is to ndsolutions that might represent physicallyrealizable systemsthat couldpossiblyoccur innature. It is well knownthat of the manysolutionstothesystemsof equationsderivedfromtheEinsteinequationsleadtounphysicalbehaviour. Inmanycases, thespacetimes constructedmaybetrulysingular eveninthe presence of matter or other elds. Alternativelynegative energydensitiesand/or negative pressures are oftenrequired inorder tomeet regularity conditionsorboundaryconditionsthatareimposeduponthemetricorothergeometricobjects.Inothercaseswhereenergydensitiesandpressuresarepositive,oftenonendsthatthespeedof soundviolatescausalitybybeinggreaterthanthespeedof light. Forexampleconstantdensitycongurations,wherethepressurehasaradialdependencetoensurethattheinteriorsolutionremainsstaticwill leadtoinnitesoundspeeds.Chargeddustsolutionsontheotherhandhavezerosoundspeed. Theyrequiresuchanebalance between themass and charge distributionsthat theyare almost alwaysunstable to any perturbations. They either undergo gravitational collapse or y apartduetoelectrostaticrepulsion.This paper will present a newsolution to the spherically symmetric timeindependentEinstein-Maxwellsystemofequationsthatgovernthebehaviourofthespace-time in the interior of a charged uid sphere. At the outer boundary the solutionswill be matchedto the external vacuumsolutionfor the eldequations i.e. theNewPhysicallyRealisticSolutionsforChargedFluidSpheres 2Reissner-Nordstr omsolution. It will bedemonstratedthat someof thesesolutionshaveparametervaluesthatleadtophysicallyrealisticproperties.While the total number of known exact interior solutions to the Einstein-Maxwellequations is much smaller than their uncharged counterparts, the number of solutionsthat canbesaidtobephysicallyrelevantissmall. Areviewof suchsolutionshasbeenprovidedbyIvanov[1] andthismakesagoodstartingplaceforunderstandingthedierent approachesthat havebeentakenandtheresults obtainedfromthoseapproaches. In some instances corrections to some of the original papers are provided.Themost commonlyusedmethods employeither isotropiccoordinates (wheretheproblemwasrstdiscussedbyPapapetrou[2] andMajumdar[3])orcurvature(i.e.Schwarzschild-like) coordinates.Asmentionedintheintroductoryparagraphabove,thispaperwillbeprimarilyconcerned withtheconstruction of physicallyinteresting solutionsthat mightarisefromregular initial conditions for charge and mass densities along with realisticpressuredistributions. Themotivations for this aretondcompact chargeduidcongurations that might represent realistic sources for the exterior vacuum Reissner-Nordstr omsolutions. Inparticularonecanaskwhetheritispossibletondrealisticsolutions for the extreme case where the total charge is equal to the total mass.Asecondmotivationis todeterminethepossibleendstates forsystems that havecollapsed gravitationally but due to the repulsive nature of the electrostatic interactionareunabletoformblackholesandresultinstablecongurations.The approach to be taken here is to write the Einstein-Maxwell systemincurvature coordinates. The Einstein equations will yield three equations for two metricfunctionsintermsofthemassdensity,thechargeaspectandtheuidpressure. TheMaxwell equations simply state the the electrostatic Maxwell eld is just the Coulombeldinasphericallysymmetricspacetime. Thesolutionwill begivenintermsofexplicitclosed-formfunctionsoftheradialcoordinateforthetwometriccoecients,amassdensity, achargeaspect(theintegratedchargedensity)andthepressureoftheuid.ThereviewbyIvanovoutlines that variousans atsethat havebeenutilizedtondexactsolutions. EssentiallyonegivestwooutofthevefunctionsandusestheEinstein-Maxwellequationstoconstructtheremainder. Ofcoursesomemethodsofsolutionare easier thanothers, particularlyif one is not concernedwithensuringthatthesolutionhaveaphysicalinterpretation. Thesimplestmethodistogivethefunctional dependenceofthemetricfunctionsandconstructtheremainingvariablesfrom the derivatives on the left-hand-side of the dierential equations. On the otherhandgivingthemassand/orchargedistributionsalongwithanequationofstateoftheformP=P()requiressolvingasetofnonlinearcoupleddierentialequations.Butthemethodhas theadvantage thatthereissome control over thephysics. Othermethodsuseacombinationofthetwoapproaches.The requirements that make a solution physically relevant has had muchdiscussioninthe literature. (See e.g. the reviews byDelgatyandLake [4] andFinchandSkea[5] that analyzeover onehundredknownsolutions for uncharged,sphericallysymmetric,uidsolutionsingeneralrelativity.) Thosethatweexpecttomeetarethefollowing:(i) Themetriccoecientsareregulareverywhereincludingattheorigin r = 0.(ii) Themetricfunctions matchtotheReissner-Nordstr omfunctions at theuid-vacuuminterface.NewPhysicallyRealisticSolutionsforChargedFluidSpheres 3(iii) Themassdensityispositiveanddecreasingoutward toward theboundary.(iv) TheintegratedchargeandmassincreaseoutwardtogivetheRNparametersattheboundary.(v) ThepressurePispositiveandniteeverywhereinsidetheuid(vi) Thepressurevanishesattheuidboundarywiththevacuum.(vii) Thespeedofsoundvs= (dP/d)1/2iscausal(0 v 1).(viii) Thespeedofsoundvs= (dP/d)1/2monotonicallydecreaseswithincreasingr.(ix) Both the pressure and mass density are decreasing functions of r: dP/dr < 0 andd/dr < 0.This list isnot necessarilyexhaustivebut doesincludeall of therequirementspresented by Delgaty and Lake in their extensive review of exact uid sphere solutions.Supplementary conditions that dene what Finch and Skea call interesting solutionshavebeenreviewedinthatmanuscript. Theseextraconditionsinclude: anexplicitequation of state in the form P= P(), and non-numerical solutions to the dierentialequations that governing the behaviour of some of the dependent functions. While thelatterconditionwillbemetinwhatfollows,thewehavenotsucceededinprovidinganequationofstateoutsideofgivingboththepressureandmassdensityintermsoffunctionsofaradial coordinate.In comparing the behaviour of 127 known solutions against the requirements listedabove, DelgatyandLakefoundthat only9of thosesatisedall of theconditions.Clearlysuchsolutions are rare. Manysolutions have pathologies associatedwithnegative pressures, and/or non causal sound speeds. As will be shown in what followsmanyofthenewsolutionsalsodemonstratesuchpathological behaviour.The outlinefor this manuscript is as follows: the next sectionintroduces thecoordinates, thelineelement, andtheformoftheEinstein-Maxwellequationstobesolved. Astrategyforsolvingfortheunknownfunctionsisthenpresented. Section3 thenapplies the methodto obtaina general formof the solutions interms ofanalyticfunctions. Special casesareanalyzed todeterminewhichsolutionsandwhatparameters lead to physically relevant behaviour by meeting all of the conditions listedabovesimultaneously. Finally,thelastsectionconcludeswithsomediscussions.2. TheEinstein-Maxwell EquationsIn curvature coordinates, the line element of a spherically symmetric, static spacetimecanbewrittenintheform:ds2= edt2edr2r2d2(1)where d2= d2+sin2d2is the metric on the unit two-sphere in terms of sphericalpolaranglesand,andandarer-dependentfunctions.ThecoupledEinsteinMaxwellequationscanbewritteninthefollowingform:rre+1r2(1 e) = +q2r4(2)rre1r2(1 e) = P q2r4(3)e_rr2rr4+(r)24+rr2r_= P+q2r4(4)NewPhysicallyRealisticSolutionsforChargedFluidSpheres 4wherethesubscript(r)representsaderivativewithrespecttorand = 8G/c4. Inwhatfollows, geometricunitsareusedwhereG = c = 1.Thechargefunctionq(r)isobtainedbyintegratingthechargedensity(r)overasphericalpropervolume:q(r) =2_r0(r)r2e/2dr.Inwhatfollows, itwill beassumedthatthechargefunctionisknownandthiswillbeusedtoobtainthechargedensityonceasolutionforthemetricfunction(r)isdetermined.Thespherical symmetryof theproblemallowsonlyaradial componenttotheelectriceldwhichcanbecomputedfromanelectrostaticpotential,(r). Thisleadsdirectlytotheonenon-zerocomponentoftheMaxwelleldtensor:F01=r= qr2e(+)/2.Equation (2) provides a single equation for and since the the left-hand-side is atotalderivativeofthefunctionr(1 e)multipliedbyr2oneintroducesthemassaspect:M(r) =2_r0r2dr +12_r0q2r2drwhichleadsdirectlytoanexpression for(r):z e= 1 2Mr(5)Thereforeif themassdensityandthechargeaspectaregivenasfunctionsof ronecanndanexpression forthemassaspectandthereforeforthefunctionz(r).Assumingthatthemetricfunctionhasbeendeterminedonecanimmediatelyobtainanexpressionforthepressurebytakingasumof theequations(2)and(3).Thenoneobtains:(P+ ) =er(r + r) =zrrzrr(6)Finally using equations (2) - (4) one can obtain the following equations byintroducingthefunctiony= e/2:2r2zyrr + (r2zr2rz)yr +_rzr2z + 2 4q2r2_y = 0 (7)2r2zyrr + (r2zr2rz)yr + (5rzr + 2z 2 + 4r2)y= 0 (8)2r2zyrr + (r2zr + 6rz)yr + (rzr + 2z 2 4Pr2)y = 0 (9)Whilethethirdequation(9)couplestoequation(6)thersttwoequationsareequivalentandleadtosecondorderlineardierentialequationsforthefunctiony.Theprocedureforconstructingasolution isasfollows: (i)givethemassdensityandthechargeaspectfunctionsasexplicitfunctionsintheradial coordinater; (ii)compute the mass aspect which leads to an explicit solution for the function z(r); (iii)obtainasolutionforthefunctiony(r)usingthedierentialequation(7)or(8); (iv)nallysolveforthepressure, p(r)usingequation(6).The advantages to this procedure are that one can begin with a radial dependencethat guarantees that some of the requirements that the solution be physicallyNewPhysicallyRealisticSolutionsforChargedFluidSpheres 5signicantaremetfromtheverybeginning. Hereitassumedthatsolutionsfortheconditions onthemetricfunctions will followif the input is physicallymotivated.Giventhattherewillbetwoconstantsofintegrationassociated withthesolutionforyandsomefreeparametersassociatedwiththechoicesforand, thereshouldbeenoughfreedomtoensurethattheremainingequationsaresatised.Ofcoursethedicultyofsolvingtheequationsfory(r)dependsonthechoicesthat are made for (r) and q(r). In the remainder of this paper we present a choice thatleads to some easily solvable equations after making the appropriate set of coordinatetransformations. Wethenexplorethesesolutionstodeterminewhetherornottheycanmeettheconditionsforphysical acceptability.3. ASetofSolutionsOnemustbeginwithdeningthemassdensityasafunctionoftheradial coordinateandwechoose: = 01r22r4(10)Thecharge functionmust alsobegiven andin ordertoensurethatitisconcentratedattheouteredgeofthesphereitisgivenasasimplepowerfunction:q= q2r4. (11)With these given, the rst metric function we can solve for is . Using the substitutionz= e,wehavez= 1 2M(r)r,wherethemassfunctionM(r)isgivenbyM(r) =12r_0_ +q2r4_r2dr. (12)Weintegrateandobtainz= 1 ar2br4cr6,wherea,b,andcareconstants:a =130; (13)b =151; (14)andc =17_q22 2_.TosolveforthesecondmetricfunctionweusetheEinsteinequationswrittenaslinear second-order dierential equations for y= e/2. In particular, where subscriptsdenotederivativeswithrespecttothegivenvariable,2r2zyrr +_r2zr2rz_yr +_rzr2z + 2 4q2r2_y= 0.NewPhysicallyRealisticSolutionsforChargedFluidSpheres 6Achangeof variabletox=r2gives amass densityfrom(10) andachargefunctionfrom(11)of,respectively, = 01x 2x2andq= q2x2.Thedierentialequationisnowzyxx +12zxyx_110+17_8q22 2_x_y = 0,wherez= 1 ax bx2cx3. (15)Wechoose2= 8q22sothat2> 0andtheconstantcbecomesc = q22= 182< 0 (16)andthedierentialequation,withanotherchangeofvariable=x0_0dxz=x0_0dx1 ax bx2cx3,can now be written as the linear second-order constant coecient dierential equationofasimpleharmonicoscillator:y + dy= 0. (17)Theconstantddependson1:d =1201= b4. (18)Equation(17)hasthefollowingthreetypesofsolutions:y= C1 + C2 (19)y= C1ed+ C2ed(20)andy= C1 sin_d_+ C2 cos_d_(21)ford = 0,d < 0,andd > 0,respectively.ThenalfunctiontobefoundisthepressureP. Wehavetheequation(P+ ) =er(r + r) =zrrzrr(22)whichcanalsobewrittenintermsofthenewcoordinatesxand:P= 4zyy2zx. (23)Theconstants C1and C2are determined byconditions on yand thepressure. At theboundary of theuid sphere, which is determined by P (x0) = 0, the metric functionsmustmatchtheexternalReissner-Nordstr om metric,givenbye= e= 1 2m(x0)rq2(x0)r2. (24)NewPhysicallyRealisticSolutionsforChargedFluidSpheres 7Thatis,attheboundarywehavey(0) =_z(x0). (25)Apossiblesecondconditioncomesfromsettingtheboundarywhere (x0)=0.Inthiscase, equation(22)impliesthatthederivativesofandwithrespecttormustbeequal. Thiscanbewrittenas2y(x0) = zx(x0) (26)This second condition, however, is not necessarily required. The mass density doesnothavetovanishattheboundary. Inthiscasetheboundaryconditionsmustbemodied. We still impose the junction condition with the external Reissner-Nordstr ommetric, equation(25), however, sincethe mass densitydoes not gotozeroat theboundary, equation (26) no longer applies. Instead we solve for thepressure, given byequation(23),goingtozero. Weobtainy0y(0)=_2zx(x0) + (x0)4_z(x0)_andsincey(0) =_z(x0)fromthejunctioncondition,wehavey0=14 (2zx(x0) + (x0)) . (27)Thecharge densitycan befoundwithqand zdetermined. Thecharge densityis =2qrr2e/2(28)= 8cxz.Inaddition,thetotalmassoftheuidspherecanbefoundusingtherelationm(r) = M(r) +q22r2.whereM(r)isgiveninequation(12). Wesolveandndthetotal massouttotheboundaryx0tobem(x0) =a2x3/20+b2x5/20(29)whilethetotalchargetotheboundaryisq(x0) =cx20. (30)Yet another conditionthat couldbeusedis tomatchthe metricfunctions totheextremeReissner-Nordstr om solution,whereweequateequations(29)and(30):a2x3/20+b2x5/20=cx20(31)Wenowconsiderthefollowingcasesforthesolutionsofequation(17).NewPhysicallyRealisticSolutionsforChargedFluidSpheres 83.1. Cased = 0(b = 1= 0)When1=0theequationforyislinearin, equation(19). Wenotethatif themassdensitydoesnotgotozeroattheboundary, itispossibletohaveapositivepressure solution. That is, thecontribution from themassmust begreater than fromthecharge. In this case, we use the junction condition (25) and equation (27) to solvefortheconstantsC1andC2. WendC1=_z(x0) C20,asbefore,andC2=a4+c2x20.Figure(1)showsasolutionwherethemassdensityisalwaysgreaterthanthechargedensityandthepressurestartspositiveandmonotonicallygoestozeroattheboundary. Thecaseshownhasthefollowingchoicesfora,c,andx0:a = 1c = 16x0= 1Thespeedofsoundintheuidsphereisalsoofinterest. Thisspeedshouldbepositiveandcausal, thatis0 dPd 1. (32)Figure(1)isaplotof thespeedof soundinthismediumwhichindicatesthatthespeediscertainlynotcausal inthiscase.If this solution is matched to the extreme Reissner-Nordstr om solution, we imposetheconditiongiven inequation(31). Whereb = 0,theboundaryx0issolvedfor:x0=12a2(c).Unfortunately,thisradial boundaryalsoproducesnegativepressuresover partoftheuidsphere. Figure(2)showstheexamplewitha = 1c = 16x0= 1Thespeedof soundissingularatthecenterof thesphereanddecreaseswhenboththepressureandmassdensityaredecreasingThespeedisnegativewhenthepressureincreases. ThespeedisplottedinFigure(2).Ifweimposethecondition(x0) = 0,theboundaryoftheuidsphereisx0=_02=3a8(c)NewPhysicallyRealisticSolutionsforChargedFluidSpheres 90 0.2 0.4 0.6 0.8 1radial coordinate, x00.20.40.60.81metric coefficientg001/g11(a)0 0.2 0.4 0.6 0.8 1radial coordinate, x01234density, pressurechargemasspressure(b)0 0.2 0.4 0.6 0.8 1radial coordinate, x00.10.20.30.40.5aspect function, m, Qchargemass(c)0 0.2 0.4 0.6 0.8 1radial coordinate, x0246810(sound speed)^2(d)Figure1. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb = 0.andequation(26)impliesC2=a16.Thejunctioncondition(25)producesC1=_z(x0) a160.Inthiscase,thepressure, fromequation(23),readsP= 4z_C2C1 + C2_a 2cx2. (33)The pressure will be positive at the center (x = =0) so long as C1 0,1< 0)Thiscaseinvolvesthereal exponential solutionforygivenbyequation(20). If wedonotimposetheconditionthatthemassdensitygoestozeroattheboundary,wecansolve fortheconstantsC1andC2usingthejunctioncondition (25)and equation(27). ThesetwoequationsresultinC1=12ed0__z(x0) +1d_a4+b4x0 +c2x20__C2=12ed0__z(x0) 1d_a4+b4x0 +c2x20__These constants do give positive pressures as shown in the following example. Choosethefollowingvaluesfortheconstantsa,b,c,andx0:a = 1b = 1c = 2x0= 0.09NewPhysicallyRealisticSolutionsforChargedFluidSpheres 110 0.5 1 1.5radial coordinate, x00.20.40.60.81metric coefficientg001/g11(a)0.50.51.52.53.54.5density, pressure0 0.5 1 1.5radial coordinate, xchargemasspressure(b)0 0.5 1 1.5radial coordinate, x00.20.40.60.81apect function, m, Qchargemass(c)0246810(sound speed)^20 0.5 1 1.5radial coordinate, x(d)Figure3. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb = 0and(x0) = 0.OurresultsareshowninFigure(4).Onceagain, thespeedof soundinthismediumdPdisevaluatedand, asshowninFigure(4), isnegativeforthiscase. Thisisduetothefactthatthemassdensityincreasesovertherangeofinterestwhilethepressuredecreases.AmatchtotheextremeReissner-Nordstr om solutionleadstoaboundarygivenbythesolutiontoequation(31):x0=_cb1bc ab_2wheretheminussigncaseischosentoensurethatthemassdensityremainspositiveovertheregionoftheuidsphere. Usingthesamevaluesfortheotherconstantsasthepreviousexample,Figure(5)isobtained.Thespeedofsoundbecomessingularwhend = 0.If themassdensityisforcedtogotozeroattheboundaryof theuidsphere((x0) = 0),wesolvefortheboundary:x0= 122+122_21 + 402(34)=5b16(c)+116(c)_25b2+ 96a(c)NewPhysicallyRealisticSolutionsforChargedFluidSpheres 120 0.02 0.04 0.06 0.08 0.1 0.12radial coordinate, x0.850.90.951metric coefficientg001/g11(a)0 0.02 0.04 0.06 0.08 0.1 0.12radial coordinate, x01234density, pressurechargemasspressure(b)0 0.02 0.04 0.06 0.08 0.1 0.12radial coordinate, x00.0050.010.015aspect function, m, Qchargemass(c)0.160.120.080.040(sound speed)^20 0.02 0.04 0.06 0.08 0.1 0.12radial coordinate, x(d)Figure4. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb > 0.The two conditions given by equations (25) and (26) allow us to solve for the constantsC1andC2,C1=12ed0__z(x0) +zx(x0)2d_C2=12ed0__z(x0) zx(x0)2d_Thepressure, fromequation(23),isnowP= 4dz_C1edC2edC1ed+ C2ed_a bx 2cx2(35)TheconditionforthepressuretobepositiveatthecenteroftheuidsphereisC1C2C1 + C2>a2b.ThisclearlyrequiresC1>C2butthissetsnoclearconditiononzx(x0)duetothepresenceoftherealexponentialsinC1andC2. Thepressuresappeartobenegativewhenthecontributionfromthechargeoverwhelmsthecontributionfromthemass,especiallyneartheboundarywherethemassdensityisvirtuallynil. Theconstantsa, b, and c are thesame as in theprevious example whilex0isnow given by equationNewPhysicallyRealisticSolutionsforChargedFluidSpheres 130 0.05 0.1 0.15 0.2radial coordinate, x0.70.80.91metric coefficientg001/g11(a)0.50.51.52.53.54.5density, preessure0 0.05 0.1 0.15 0.2radial coordinate, xchargemasspressure(b)0 0.05 0.1 0.15 0.2radial coordinate, x00.010.020.030.040.05aspect function, m, Qchargemass(c)1086420246810(sound speed)^20 0.05 0.1 0.15 0.2radial coordinate, x(d)Figure5. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforextremeReissner-Nordstr omsolutionandb > 0.(34). Figure(6)showsthisnegativepressurecase. Thespeedof soundinthiscasebecomessingularatthepointofmaximummassdensity(whered = 0).Wecanalsoinvestigatethesub-caseof 0=0withbothconditionson(x0).Thus, boththemassandchargedensitiesarezeroatthecenterof theuidsphere.Thisistheonlyrelevantcaseinwhichtoexaminethissub-caseasthemassdensitywill remainpositiveforalargeenoughchoiceof1. When(x0)>0wedesirethemassdensitytobelargerelative tothecharge density,sothefollowingvaluesfortheconstantsa,b,c,andx0arechosen:a = 0b = 10c = 2x0= 0.3WedoinfactobtainpositivepressuresasshowninFigure(7). Thespeedofsoundinthis caseremainsnegativeduetotheincreasingmassdensitywithadecreasingpressure. ThespeedisshowninFigure(7).TheextremeReissner-Nordstr omsolutionhasaboundarygivenfromequation(31)of:x0=4(c)b2NewPhysicallyRealisticSolutionsforChargedFluidSpheres 140 0.2 0.4 0.6 0.8radial coordinate, x0.40.60.81metric coefficientsg001/g11(a)20246density, pressure0 0.2 0.4 0.6 0.8radial coordinate, xchargemasspressure(b)0 0.2 0.4 0.6 0.8radial coordinate, x00.20.40.6aspect function, m, Qchargemass(c)1086420246810(sound speed)^20 0.2 0.4 0.6 0.8radial coordinate, x(d)Figure6. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb > 0and(x0) = 0.Thisleadstonegativepressuresoverpartof thesphereasintheexamplegiveninFigure(8)wherethefollowingvaluesfortheconstantsareuseda = 0b= 7c= 2Thespeedof soundinthiscasebeginspositiveyetbecomesnegative asthepressureincreasesandthendecreases. ThisspeedisgiveninFigure(8).However, when(x0) = 0and0= 0themassdensityremainsbelowthechargedensityand, hence, negativepressuresresult. Forexample, wechoosethefollowingvaluesforbandc:b = 2c = 2.Theresultingpressureisshown inFigure(9).Thespeedof soundinthiscasebecomessingularwhenthemassdensitypeaks(i.e. d = 0).NewPhysicallyRealisticSolutionsforChargedFluidSpheres 150 0.1 0.2 0.3 0.4radial coordinate, x0.20.40.60.81metric coefficientg001/g11(a)0 0.1 0.2 0.3 0.4radial coordinate, x0246810density, pressurechargemasspressure(b)0 0.1 0.2 0.3 0.4radial coordinate, x00.050.10.150.2aspectfunctions, m, Qchargemass(c)0.40.30.20.10(sound speed)^20 0.1 0.2 0.3 0.4radial coordinate, x(d)Figure7. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dfora = 0andb > 0.3.3. Cased > 0(b < 0,1> 0)This case involves the sine and cosine solution for y given by equation (21). Now, if weusethejunctioncondition(25)andequation(27),theconstantsC1andC2becomeC1=_z(x0) sin_b40 +1b_a2+b2x0 + cx20_cos_b40C2=_z(x0) cos_b401b_a2+b2x0 + cx20_sin_b40.Weobtainpositivepressuresinthiscaseaswell. Thefollowingexample,Figure(10),usesthefollowingvaluesfortheconstantsa,b,c,andx0:a = 1b = 1c = 2x0= 0.05The speed of sound in this case, as shown in Figure (10), does follow the conditionsgivenbyequation(32).NewPhysicallyRealisticSolutionsforChargedFluidSpheres 160 0.05 0.1 0.15 0.2radial coordinate, x0.70.80.91metric coefficientg001/g11(a)0.50.51.52.53.54.55.5density, pressure0 0.05 0.1 0.15 0.2radial coordinate, xchargemasspressure(b)0 0.05 0.1 0.15 0.2radial coordinate, x00.010.020.030.04aspect function, m, Qchargemass(c)0.020.0100.01(sound speed)^20 0.05 0.1 0.15 0.2radial coordinate, x(d)Figure8. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforextremeReissner-Nordstr omsolution, a = 0,andb > 0.Thus,thiscasesatisestheconditionsforaphysicallyrealisticsolution, that: atthecenterof thesphere, themassmvanishesande=1; themetricfunctionsy2and zmatch the external Reissner-Nordstr om metric at the boundary; the pressure ispositive and monotonically goes to zero at the boundary; the mass density is positive;andthespeedofsoundinthismediumisbothpositiveandcausal.Matching to the extreme Reissner-Nordstr om solution results in another situationwherethepressuredoesnotmonotonicallyvanishattheboundary, giveninFigure(11) forthesamevalues asin thelast example. Theboundary isgiven from equation(31):x0=_cb1bc ab_2.Thespeedof soundisfractional butdoesnotremainpositiveasillustratedinFigure(11).Theconditionthatthemassdensitygoestozeroyieldstheboundaryx0= 122+122_21 + 402=5b16(c)+116(c)_25b2+ 96a(c).NewPhysicallyRealisticSolutionsforChargedFluidSpheres 170 0.2 0.4 0.6 0.8radial coordinate, x0.70.80.91metric coefficientg001/g11(a)202468density, pressure0 0.2 0.4 0.6 0.8radial coordinate, xchargemasspressure(b)0 0.2 0.4 0.6 0.8radial coordinate, x00.20.40.6aspect function, m, Q chargemass(c)1086420246810(sound speed)^20 0.2 0.4 0.6 0.8radial coordinate, x(d)Figure9. Plotsof: (a)themetricfunctionsy2andz; (b)thepressureP, themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dfora = 0,b > 0,and(x0) = 0.Thejunctioncondition(25)andequation(26)areusedtosolvefortheconstantsC1andC2:C1=_z(x0) sind0 +zx(x0)2dcosd0C2=_z(x0) cosd0zx(x0)2dsind0.Thepressure, fromequation(23),isgivenbyP= 4dz_C1 cosd C2 sindC1 sind + C2 cosd_a bx 2cx2. (36)TheconditionthatthepressurebepositiveatthecenterleadstotherelationC1C2>a4d,whichisnotsatised, ingeneral. WeshowanexampleinFigure(12)withnegativepressureswherethefollowingvalueswereusedThe speed of sound in this case, begins at a positive fractional value but decreasesthroughzerotonegativevaluesasthepressurerstdecreasesandthenincreasestozeroattheboundary.FinallyOnemightaskwhichsolutions(ofthe127NewPhysicallyRealisticSolutionsforChargedFluidSpheres 180 0.02 0.04 0.06radial coordinate, x0.920.940.960.981metric coefficientg001/g11(a)0 0.02 0.04 0.06radial coordinate, x0123density, pressurechargemasspressure(b)0 0.02 0.04 0.06radial coordinate, x00.0020.0040.006aspect function, m, Qchargemass(c)0 0.02 0.04 0.06radial coordinate, x00.050.10.150.2(sound speed)^2(d)Figure10. Plotsof: (a)themetricfunctionsy2andz;(b)thepressureP,themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb < 0.4. ConclusionsIt has been shownthat by giving the mass density and charge aspect functionsexplicitlywasfourth-orderpolynomialsinthecurvatureradialcoordinateleadstoasimplelinearsecond-order dierentialequationwithconstantcoecientsthatcanbeeasily solved. With an appropriate choice of the constants appearing in the polynomial,someof thesolutionscanbemadetosatisfytherequirementsforbeginphysicallyacceptable. Giventhesmallnumberofknown solutionsforbothcharged andneutraluidspheres that obeysuchcriteria, the congurations of chargeandmass foundinthesesolutionsmaybeexpectedtoformfromarealisticcollapsetoacondensedsphericalobject.Whilematchingtotheexternal Reissner-Nordstrom solution causesnoproblemssince this onlydetermines the values for the constants of integration, it has beenshown that with this family of solutions it is impossible to nd realistic congurationsthatleadtoextremeReissner-Nordstromsolutions. Inall caseswherethesolutionwasmatchedtotheextremeReissner-Nordstrom vacuumsolution,problemsaroseinmaintainingaspeedofsoundthatisboundedbythespeedoflight.It remainstobedeterminedhowstablethesecongurationsare, andthis willbestudiedinthefuture. Giventhattheequationofstatefortheuidhasnotbeenwritten in the form p = p() does make thestability analysis a bit more dicult sinceNewPhysicallyRealisticSolutionsforChargedFluidSpheres 190 0.025 0.05 0.075 0.1radial coordinate, x0.850.90.951metric coefficientsg001/g11(a)0.50.51.52.53.5density, pressure0 0.025 0.05 0.075 0.1radial coordinate, xchargemasspressure(b)0 0.025 0.05 0.075 0.1radial coordinate, x00.0050.010.015aspect function, m, Qchargemass(c)0.10.0500.050.10.150.2(sound speed)^20 0.025 0.05 0.075 0.1radial coordinate, x(d)Figure11. Plotsof: (a)themetricfunctionsy2andz;(b)thepressureP,themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforextremeReissner-Nordstr omsolutionandb < 0.itisnotknown ifonecanuseadiabaticityargumentswithoutdirectlycheckingthemindependently. Howeverageneral stabilityanalysisof this andother knownexactsolutionsshouldbeinterestingfromthepointof viewof tryingtounderstandwhatchargeduidcongurationscouldpossiblybeformedfromthegravitationalcollapseofmaterialthathasanexcesschargedensity.Finally,oneshouldbeabletoextendthismethodtootherexplicitfunctionsforboththe mass densityandchargeaspect. Higher order polynomials areexpectedtoproducesimilar results andone canexpect that these wouldforcemoreof thechargetobedistributedclosertothesurfaceofthesphere. Furtherstudiesonwhatthemostgeneral polynomialsmightbeandtherelationshipsbetweenthechargeandmassdistributionswillbestudiedinthefuture.4.1. AcknowledgmentsDHwouldliketothankD.GiangfordiscussionsconcerningmethodsforsolvingtheEinstein-Maxwellsystemofequations. Thisresearch wasfundedthroughanNSERC(Canada)DiscoveryGrant.NewPhysicallyRealisticSolutionsforChargedFluidSpheres 200 0.1 0.2 0.3 0.4radial coordinate, x0.70.80.91metric coefficientg001/g11(a)0.50.51.52.53.54.55.5density, pressure0 0.1 0.2 0.3 0.4radial coordinate, xchargemasspressure(b)0 0.1 0.2 0.3 0.4radial coordinate, x00.050.10.15aspect function m, Qchargemass(c)0.20.100.1(sound speed)^20 0.1 0.2 0.3 0.4radial coordinate, x(d)Figure12. Plotsof: (a)themetricfunctionsy2andz;(b)thepressureP,themass and charge densities and ;(c) the mass andcharge functionsm(r) andq(r)and(d)dP/dforb < 0and(x0) = 0.5. References1 S.D.Majumdar, Phys.Rev.,72,390(1947).2 A.Papapetrou, Proc.Roy.IrishAcad.,A51,191(1947).3 B.V.Ivanov,Phys. Rev.D65,104001, (2002).4 M.S.R. Delgaty and K. Lake, Comput. Phys. Commun. 115, 395, (1998)[arXiv:gr-qc/9809013].5 M.R.FinchandJ.E.F.Skea,unpublishedpreprint,www.dft.if.euerj.br/users/JimSkea/papers/pfrev.ps.5 R.C.Tolman,Phys.Rev. 55,363, (1939).7 S.Bayin,Phys.Rev.D18,2745, (1975).