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DraftversionJanuary13,2015 PreprinttypesetusingLATEXstyleemulateapjv.11/10/09 RADIALSTABILITYINSTRATIFIEDSTARS JonasP.Pereira1,2andJorgeA.Rueda2,3,4 DraftversionJanuary13,2015 ABSTRACT Weformulatewithinageneralizeddistributionalapproachthetreatmentofthestabilityagainstradialpertur- bationsforbothneutralandchargedstratifiedstarsinNewtonianandEinstein’sgravity. We obtainfromthis approachtheboundaryconditionsconnectingtwoanyphaseswithinastarandunderlineitsrelevanceforreal- isticmodelsofcompactstarswithphasetransitions,owingtothemodificationofthestar’ssetofeigenmodes 5 withrespecttothecontinuouscase. 1 Subjectheadings:Thin-ShellFormalism–NeutronStars–RadialPerturbations—PhaseTransitions 0 2 n 1. INTRODUCTION tum Chromo-Dynamics (QCD) could occur within the core a ofthestaritself(see,e.g.,Glendenning1996,andreferences There is theoretical evidence that compact stars, such J therein). Fromalltheabovewecanconcludethatultradense as neutron stars, are made up of several matter phases 2 starssuchasneutronstarsshowsnecessarilyanontrivialstrat- (Shapiro&Teukolsky1986). Thisisaconsequenceoftheas- 1 ification. Betweenanytwophases, whichcanbeverydiffer- tonishinglyhighdensityexcursiontheycouldattainintheirin- ent, it is reasonable to investigate the situation where some nerregions. Forinstance,aneutronstarisingeneralthought ] quantities are discontinuous, e.g. the energy-densityand the c ascomposedbyatleasttodifferentregions:thecrustandthe pressure. Suchdiscontinuitiescan beharnessedbyappropri- q core. Startingfromverylowdensitiesofafewgramspercu- ate surface tensions. These surface quantities influence the - biccentimeterclosetotheirsurfaces,anduptodensitiesofthe gr orderofthenuclearsaturationvalue,ρnuc ≈2.7×1014gcm−3, sptraobbilleitmywofhaichsy,satsewmeasdhdailnlgshnoewwhbeoreu,nmdaordyifcyotnhdeistieotnosfetoigtehne- [ the crust of a neutron star is thought to be in a solid-like state. The core, with densities that might overcome by or- frequenciesandeigenmodesofastar. 1 ders of magnitude ρ , is instead thought to be in a liquid- In this work we analyze the problem of perturbations in v likestate. Thedetailsnuocfthetreatmentofthethermodynamic systems constituted of various phases that are split by sur- 1 faces that host nontrivial degrees of freedom. This analysis transition(e.g.MaxwellorGibbsphaseconstruction),aswell 2 is thought to be a generalization of the treatment for con- as the conditions on density and pressure at which such a 6 tinuous systems (see Herrera&Santos 1997, and references transition occurs, are still a matter of debate. The applica- 2 therein,foracomprehensiveanalysisofproperties,typesand tionoftheGibbsconstructionwithmorethanoneconserved 0 stability of continuous anisotropic fluids also in presence of charge (e.g. baryon and electric) leads to the appearance of 1. mixed phases, in between of the pure phases, with an equi- radiation and heat flux). By investigating the dynamics of 0 librium pressure that varies with the density, leading to a perturbations, we are automatically probing the stability of 5 spatially extended phase-transition region of non-negligible systems. We shall trammel ourselves to the simplest possi- 1 thicknesswithrespecttothestar’sradius(Glendenning1992; blecase: sphericallysymmetricextendedbodieswhereradial : perturbationstake place. In orderto modelthe problem, we v Glendenning&Pei1995;Christiansen&Glendenning1997; shallassumethesesurfacesofdiscontinuityseparatingtwoar- i Glendenning&Schaffner-Bielich 1999; Christiansenetal. X bitraryphasesareverythinandageneralizeddistributionalap- 2000;Glendenning2001). Onthecontrary,inthetraditional proach(Poisson2004;Raju1982b)shallbeadopted.Westart r Maxwellconstructionthe phasesare in “contact”each other. a ouranalysisin the Newtoniancase in orderto gainintuition Itisworthmentioningthatinthesetreatmentsthepurephases oftherelevantaspectsoftheproblem,andfinallywegeneral- are subjected to the conditionof local chargeneutrality, and ize it to generalrelativity. Our purposeis solely to expound sotheydonotaccountforthepossibleinteriorCoulombfields. theproblemandseektosolveitasgenericallyaswecan.Our Indeed,thecompleteequilibriumofthemulticomponentfluid analysisisfarfrombeingcompleteandhastobeconsidered inthecoresofcompactstarsneedsthepresenceofaCoulomb asa firststeptowardsdeeperinvestigationsandscrutiniesof potentialformedbyelectricchargeseparationduetogravito- specificcasesshallbetheobjectofposteriorstudies. polarization effects (Rotondoetal. 2011; Ruedaetal. 2011), Weshowinthisworkthatphasetransitionsinthepresence favoringa sharpcore-crusttransitionthatensuresthe global, ofsurfacedegreesoffreedomcanbeenclosedintoadditional butnotthelocal,chargeneutrality(seeBelvedereetal.2012, boundaryconditionstotheproblem. Besides, ourformalism 2014, and references therein). Besides the core-crust transi- tellsusthatsuchboundaryconditionsareonlyself-consistent tion,additionalphase-transitionsastheonesallowedbyQuan- when the set of eigenfrequencies of the perturbation modes is related to the global system, not with individual phases. 1Universite´ deNiceSophiaAntipolis,CEDEX2,GrandChaˆteauParc This is consistent with the well-known results from coupled Valrose,BP2135,06103Nice,France springs, where there are only global frequencies. The pres- 2DipartimentodiFisicaandICRA,SapienzaUniversita` diRoma,P.le AldoMoro5,I–00185Rome,Italy enceoffurtherboundaryconditionsnaturallymodifythepos- 3ICRANet,P.zzadellaRepubblica10,I–65122Pescara,Italy sible set of eigenfrequencies, since we are inserting further 4ICRANet-Rio,CentroBrasileirodePesquisasF´ısicas,RuaDr.Xavier restrictive aspects to the physical oscillation modes. There- Sigaud150,RiodeJaneiro,RJ,22290–180,Brazil fore,measurementsonthepulsationmodesinastarcouldtell [email protected];[email protected] 2 PereiraandRueda us very precisely about its internalstructure, being a sort of of freedom have themselves a dynamics, described generi- fingerprint,which couldhelp to understandbetter the nature callybythethin-shellformalismorDarmois-Israelformalism ofthesesystems. (Israel 1966, 1967; Lobo&Crawford 2005; Poisson 2004). Thearticleisorganizedasfollows. Insection2weformu- Weareherehoweverparticularlyinterestedinanotheraspect latetheproblemofradialperturbationsofstarswithinterfaces oftheproblem,namelytounderstandtherolesuchdegreesof intheclassicalNewtoniancase. Forthesakeofexemplifica- freedomplayonthestabilityofthesystemwhenitsparts(de- tion, we compute the specific solution of the stability equa- finednaturallybythehypersurfacethathoststheaforesaidde- tion in section 3 for the simple case of incompressible stars, greesoffreedom)areperturbed. Therefore,beforeanything, scrutinizingwhichcircumstancesdeliverusenougharbitrary itisassumedthatforbeingmeaningfultotalkaboutthissce- constantsto fix additionalboundaryconditions. Insection 4 nario, one has that the hypersurfaces of discontinuity them- we give the formulationin the case of systems with electric selves are stable (see Pereiraetal. 2014, for further details). chargebutstillwithinNewtoniangravity. Theformulationin If this is not the case, any displacement of the hypersurface thecaseofgeneralrelativityintheneutralcaseispresentedin ofdiscontinuitywouldtriggeracataclysmicsetofeventsthat section5,whileinsection6weextendthegeneralrelativistic wouldhaveasaresultthedisruptionofthesystem. treatment to systems endowed with electric charge. Finally, Oneexpectsthatthestratifiedproblemcouldbeaccounted in section 7 we summarize and discuss the main results of foradditionalboundaryconditionstothesystem. Thereason thiswork. We use unitssuchthatc = G = 1andthe metric forthisisthattheperturbationsintheupperandlowerregions signature 2throughoutthearticle,unlessotherwisestated. with respectto a givensurface ofdiscontinuitywouldbe de- − scribed by the same physics[e.g. Eq. (2)], as well as the to- 2. STABILITYOFCLASSICALSYSTEMSWITHPHASE talityofthematches. Theonlymissingpointswouldbetheir TRANSITIONS connection(allowingcombinationsofsolutionstobealsoso- Assume a continuous classical astrophysical system with lutionstothephysicalequationsinvolved)andgeneralization sphericalsymmetry.Whenitsvolumeelementsareperturbed by means of surface quantities. For example, the existence radially, it is well-known (see, e.g., Shapiro&Teukolsky of a surface tension would accountto an extra surface force 1986)thattheevolutionofperturbationsoftheform term. The same ensues with the presence of a surface mass (enclosedbyasurfacemassdensity)andtheassociatedpres- ξ˜(r,t)=ξ(r)eiωt, (1) ence of a surface gravitational force. Therefore, in order to properly describe stratified systems, one must make use of withωanarbitraryconstant,isdescribedby distributions(Poisson2004;Raju1982b). d 1 d(r2ξ) 4dP We proceed now with the distributional generalization of ΓP ξ+ω2ρξ =0, (2) theequationsdescribingcontinuousfluidsundergravitational dr " r2 dr #− r dr fields. Assume that a surface harboring surface degrees of where P(r) is the pressure of the background system under freedom of a system in equilibrium is at r = R. The first hydrostaticequilibriumand equationtobegeneralizedintermsofdistributionsinthiscase is the equation of hydrostatic equilibrium. This should now Γ(cid:17) ρ ∂P, (3) read P ∂ρ dP 2 +ρg(r) Pδ(r R)=0, (5) with ρ(r)the massdensityofthe system. Formulatedin this dr − R − way, we have at hands an eigenvalue problem. For the case whereg(r) has beendefined as the normof the gravitational ofcontinuoussystems,theboundaryconditionstobesupple- field,solutionto(thedistributional)Poisson’sequation ~g= mentedtoEq.(2)arequitesimple. Theyaredirectlyrelated 4πGρ,~g= g(r)rˆ,thatcanalwaysbewrittenas ∇· tothesphericalsymmetryofthesystem,aswellastothevan- − − ishingofitspressureonitsbordereveninthepresenceofper- GM(r) r turbations(othersituationswhereasurfacetensionbepresent g(r)= , M(r)(cid:17)4π ρ(r¯)r¯2dr¯. (6) couldalsobeenvisagedandweshallattempttoelaborateon r2 Z0 theminthesequel).Inotherwords,weimposethat Besides, in Eq. (5) stands for the surface tension (Peters ξ(0)=0, and ξ(R )=finite, (4) 2013) onPthe surface of discontinuity at r = R. The expres- s sionontheaforesaidequationistheresultofrestoringsurface where Rs was defined as the radius of the star, such that forces(thisisthereasontheyhaveanoppositedirectiontothe P(Rs) = 0. For further details about these boundary con- pressuregradient)atasmallsurfacearea(thefactor2comes ditions (Shapiro&Teukolsky 1986). Eq. (2) supplemented from the principalcurvaturesin a surface element, which in withEq.(4)constitutesaSturm-Liouvilleproblem,wherethe the spherically symmetric case are equal). The gravitational aspects of its solutions are already known. Concerning the forcefromthesheetofmassatr=Risnaturallyincorporated eigenfrequencies,ω2,theyareallrealandformadiscretehi- onthedistributionaldefinitionsofρandg(r)asweshallshow erarchicalset. WhenoneseeksforstablesolutionstoEq.(2), in the sequel. The mass density, on the other hand, mustbe oneseeksforsolutionswithpositiveω2,speciallyforitsfun- expressedas damentalmode. AsitcanbeseenfromEq.(1),negativeval- uesofω2indicateinstabilitiesintheassumedbackgroundsys- ρ(r)=ρ (r)θ(R r)+ρ+(r)θ(r R)+σδ(r R), (7) − tem,whichleadstotheconclusionthattheydonotlingeron − − − intime. Theywouldeitherimplodeorexplode. withθ(r R)theHeavisidefunction,whosederivativeisthe − We now turn to the more involved problem of permitting Diracdeltafunctionδ(r R)inthesenseofdistributions.From − the system to be stratified and harboring surface degrees of theexistenceofasurfacemassdensity,itcanbecheckedthat freedomon the interface of two given phases. Such degrees the gravitational field is discontinuous at a given surface of Radialstabilityinstratifiedstars 3 discontinuityofthesystem(atR),i.e. since ∆δ(r R) = 0 and ∆R = ξ˜. Now we assume that = (σ).−This means that we are endowing the fluid at [g(R)]+ =4πGσ, (8) P P thesurfaceofdiscontinuitywithadiabaticpropertiesandthe − wherewehaveintroducedtheconvention[A]+ (cid:17) A+ A ,the underlying microphysics is not contemplated in this proce- − jump of A(r) across r = R. Therefore, g(r)−could−be repre- dure. Forcontinuousmedia,thetotalmassintheinterfaceof senteddistributionallyas twophasesisgenerallynotaconstant. Thismeansthatmass fluxesare allowedto takeplace. Thisgenericallywouldren- g=g+(r)θ(r R)+g−(r)θ(R r). (9) derthemass ofeachphasenotconstant,an aspectnottaken − − intoaccountinEq.(2). Nevertheless,ifthedisplacementsof The above equation means that the associated distributional thesurfaceofdiscontinuityaresmallandoscillatory,wehave gravitationalpotentialφ(r) (~g = φ,~g = grˆ) is always a continuousfunction,thoughnotd−iff∇erentiabl−eatasurfaceof thatonaveragethemassesoneachphaseareconserved(here it becomes clear why the surface of discontinuity should be discontinuity. stable).Foradiabaticprocesses,wehave Besides, we will assume that also the pressure can be dis- continuousatr=R,beinghencewrittenas ∂ ∆ =η2∆σ, η2 (cid:17) P, (14) P(r)=P (r)θ(R r)+P+(r)θ(r R). (10) P ∂σ − − − withη2thesquareofthespeedofthesoundinthefluidatthe The reason of why this is so will be clarified when we deal surfaceofdiscontinuity. Themissing term ∆σ canbe found with our original problem in the scope of general relativity. viathe thin-shellformalismwhenthe classicallimitis taken TheheuristicargumentcorroboratingthevalidityofEq.(10) there. Generically, in the static and spherically symmetric isthatitismeaninglesstocolligateasurfacetermtotheradial case,σcanbewrittenas(Lobo&Crawford2005) pressure,since itsassociated forcewouldnecessarilybenor- maltoitandthereforewouldnotlieonthesurface. Onlytan- c2 gentialpressuresshouldhave tied within their surfaceterms. σ= [e−β(R)]+, (15) FromEqs.(10),(7),(6)(8)and(9),wehave −4πGR − withtheclassicallimitβ(r) GM(r)/(rc2) 1. Itiseasyto R G ≈ ≪ = [P(R)]++ [M2(R)]+ (11) checkthatEq.(15) reducesto Eq.(8)in the aforementioned P 2 − 16πR3 − limit. Whenperturbed,itcanbeshownthatβ β+δβ,with and δβ = 4πGrρ0e2β0ξ˜/c2 (Misneretal. 1973),→ρ0 here mean- − dP ing the mass density in the hydrostatic backgroundsolution. ± +ρ±(r)g±(r)=0. (12) Hence, dr ∆σ=δσ+σ ξ˜ = [ρ ξ˜]++σ ξ˜, (16) The arithmetic average present in Eq. (11) [see the def- ′0 − 0 − ′0 inition of g(r) and the value of σ] is a general conse- where we also considered σ0 the background solution quence of the product of delta functions with Heaviside [Eq.(8)]. onesinthegeneralizedsenseofdistributions(Raju1982b,a). Another simpler way of obtaining Eq. (16) would be Notice from Eq. (11) that its first term is the known through the dynamics of δg [δ~g= -(δg)rˆ]. In the spherically Young-Laplace equation for spherical surfaces at equilib- symmetriccasewehaveδg= 4πGρξ˜(Shapiro&Teukolsky rium(Rodr´ıguez-Valverdeetal.2003;Peters2013,see,e.g.,), 1986),andsinceδg(cid:17)g g0,w−ithg0thenormofthegravita- wherejustgeometricaspectsaretakenintoaccountforthesur- tionalfieldwithoutthep−erturbationξ˜,fromEq.(8),wefinally facetension. Itssecondterm,though,isthegravitationalsur- obtain facetension,uniquelyduetothenon-zerosurfacemass. Ifthe [δg(R)]+ struarrfya,cbeuttepnrsoiopnorwtieorneanlutollt,htheesuprrfeascseurmeajsusmdpencsoiutyldannodtmbeusatrbbei- ∆σ= 4πG − +σ′0(R)ξ˜=σ′0(R)ξ˜−[ρ0ξ˜]+−. (17) a monotonicallydecreasingfunctionofthe radialcoordinate Besides,fromEqs.(6)and(8),forthecasewhenthejumpof [see Eqs.(8)and (11)]. FromEq.(11), oneseesfurtherthat ξ˜isnullatasurfaceofdiscontinuity(thatshallbejustifiedin the force per unit area associated with the surface tension is thesequel),oneshowsthattheaboveequationcanbefurther exactly the one necessary to counterbalance both the forces simplifiedto comingfromthepressuregradientatRandthesurfacegravi- ∆σ= 2σξ˜. (18) tationalforce,asitshouldbe. −r Oneseesthattheaboveproceduregeneralizesournotionof Nowweshowthegeneralequationgoverningthepropagation hydrostaticequilibriumineachphasethestratifiedsystemhas ofradialperturbations.ForPdefinedasinEq.(10),ρinterms [seeEq.(12)]andautomaticallygivesthesurfacetensionatR ofEq.(7),gasgivenbyEq.(9),andfinally thatguaranteesthehydrostaticequilibriumforarbitrarypres- surejumpsandsurfacemasses. Wewillkeepthesamephilos- ξ˜(r,t)=ξ˜+(r,t)θ(r R)+ξ˜ θ(R r), (19) − ophynowconcerningthegeneralizationofEq.(2). Fromour − − thustheequationgoverningtheevolutionofperturbationson generalizedhydrostaticequilibriumequation,wehavethatan agivenvolumeelementofthefluidis importanttermforthedeductionoftheequationgoverningra- dialperturbationswouldbetheapplicationoftheLagrangian dv ∂P 2 σ pohpyersaictoarl q∆ua[∆ntAity(cid:17)inAth(te,req+uiξ˜li)b−riuAm0(at,nrd),pAer0tuarnbdedAcabseeisn,grea- ∆(cid:26)ρ dtr+∂r+ρg(r)−" RP + 2(v˙+r +v˙−r)#δ(r−R)(cid:27)=0. (20) spectively]onthesurfaceforceinEq.(5),i.e.[seeEq.(20)] Noticethatweassumedthatv˙ isadistributionlikeξ˜. Physi- r ∆ ξ˜ callythismustbetakenbecausethephasesarealways“local- ∆ Pδ(r R) (cid:17) Pδ(r R) P δ(r R), (13) izable”anddonotmix.Inotherwords,thisconstraintreflects "R − # R − − R2 − 4 PereiraandRueda the intuitive fact that the surface of discontinuity should be andtheboundarycondition well-defined.Themathematicalreasonofwhythisissoshall be givenbelow and it is related to the well-posednessof the [ξ(R)]+ =0, or ξ+(R)=ξ−(R)(cid:17)ξ(R), (28) − problem. andthereforetheonlymeaningfulΓaregivenby When developed,takinginto accountthe hydrostaticequi- libriumequation[seeEq.(5)],itcanbesimplifiedto Γ(r)=Γ (r)θ(R r)+Γ+(r)θ(r R). (29) − − − ∂ ΓP ∂ (r2ξ˜) 4∂Pξ˜ d2ξ˜ρ= 2ξ˜ 2η2σ 3 GatheringtheaboveequationsonEq.(21),weobtain ∂r " r2 ∂r #− r ∂r − dt2 "R2 − P (cid:16) (cid:17) d ΓP ∂ 4dP σ d2ξ˜+ d2ξ˜ (r2ξ) ξ+ω2ρξ = + − δ(r R). (21) dr " r2 ∂r #− r dr −2 dt2 dt2 !# − 2 ξ(R) 3 2η2σ ω2σ δ(r R). (30) Firstnotethatcoefficientsmultipliedbyδ2(r−R)orδ′(r−R) − "R2 (cid:16) P− (cid:17)− # − inEq. (21)mustbeallnull. TakingintoaccountEq. (19),it meansthat Onessees from Eq. (30) that in the case where and σ are P [ξ˜]+ =0. (22) null,theclassicalexpression,Eq.(2),isrecovered. − Summing up, substituting Eqs. (27) and (7) into Eq. (30), Thisautomaticallywarrantsξ¨˜ asadistributionwithoutDirac one sees that the only way to satisfy such an equation is by deltas, as we have advanced previously. In order to obtain imposingthat Eq. (21), we used the results that for a distribution A(r) = A+(r)θ(r−R)+A−(r)θ(R−r), ddr"Γ±P±r12ddr(r2ξ±)#− 4r ddPr±ξ±+ω2ρ±ξ± =0, (31) ∂A ∆A=δA+ ξ˜ [A]+ξ˜δ(r R), (23) ∆ ∂A = ∂ (∆A∂)r −∂ξ˜∂A− − "ΓPddr(r2ξ)#++2(3P−2η2σ−2R[P(R)]+−)ξ(R)=0, (32) ∂r! ∂r − ∂r ∂r − and for completeness, condition (28). Eq. (31) is obtained +1 ∂ξ˜+ + ∂ξ˜− [A(R)]+δ(r R). (24) here as a consequence of our distributional search for solu- 2" ∂r ∂r # − − tionstothe radialLagrangiandisplacements. Thisisexactly what one expects under physical arguments. Eqs. (32) and Besides the above mathematical properties, we have also (28) are our desired boundaryconditionsto be further taken madeuseof into account [besides Eq. (4)] at the interface of any two ρ ∂ σ ∂ξ˜+ ∂ξ˜ phases. ∆ρ= (r2ξ˜)+ + − δ(r R), (25) For the case where [P(R)]+ = [Γ(R)]+ = σ = = 0, we −r2∂r 2 ∂r ∂r ! − havethatalsothederivativeo−fξiscontin−uousandPthereforeξ whichisadirectconsequenceofassumingthatthetotalmass isadifferentiablefunctionanywhere,asitshouldbesincewe in each phase is constant, even in the presence of perturba- are defining here a continuous system. Nevertheless, when- tions. This is just guaranteed if the surface of discontinuity ever the aforementionedconditionsdo not take place, richer is stable, a prime hypothesis for having a well-posed stabil- scenariosrise. Even in the case of a phase transition at con- ity problem. We also have assumed that P = P(ρ), which stant pressure and negligible surface mass, the discontinuity implies that ∆P = Γ P ∆ρ /ρ . In deriving Eq. (21), we ofΓandtheexistenceof generallyrenderthederivativeof ± ± ± ± ± P further took into account Eq. (11). We finally stress that a ξdiscontinuous. simpler way to obtain Eq. (21) is to recall that ∆M = 0, which guaranteesthat ∆g = 2gξ˜/r. The fact that ∆M = 0 3. ASPECIFICEXAMPLE:UNIFORMDENSITYSTARS − meansthatcomovingobserverswiththefluiddonotnoticea Wewouldliketostressthattheboundaryconditionwehave masschange. Theaforementionedresultcanalsobedirectly derivedpreviouslyisactuallyveryrestrictive. Thereasonfor showedbyEqs.(6),(24)and(25). this is that the physically acceptable cases [solutions to Eqs. It can be seen that only solutions of the type ξ˜±(r,t) = (31) associated with a surface of discontinuityat r = R] are eiω±tξ±(r)forEq.(21)arejustmeaningfulif onlythe ones thatdeliver enougharbitraryconstantsof inte- grationforsatisfyingEqs.(27)and(32). Thisshouldbetaken ω+ =ω (cid:17)ω. (26) − into account together with the physical requirement of only admitting finite ξ everywhere and that are null at the origin Thisistheonlywaytoeliminatethetimedependenceabove inEq.(21),andalsotoguaranteethatthejumpofξ˜benullfor [seeEq.(4)]. Inthefollowingweshallseeaparticularexam- plewherealloftheseaspectsareevidenced. anysurfaceofdiscontinuityatanytime. Therefore,wearrive Letusnowinvestigateastarmadeoutoftwophases,each attheimportantconclusionthatevenastratifiedsystemwhere withauniformmass-density.Letusassumealso,forthesake oscillatoryperturbationstakeplaceshouldbedescribedbya of simplicity and exemplification, that the associated Γ for solesetoffrequencies.Eachmemberofthissetdescribesthe eachregionisanarbitraryconstant. Aswewillsee,although eigenfrequency of whole system, instead of one or another thiscanbeconsideredonlyasafirstacademicexample,ital- phase. Nevertheless,werecallthatatthesurfaceofdisconti- readyevidencessomeaspectsstratified systemsshouldhave. nuitythefrequenciesareinprinciplenotdefined. Bearingin For this case it is straightforward to solve Poisson’s equa- mindtheaboveconclusions,wehavethat,usingEq.(1), tion and the equation of hydrostatic equilibrium (see, e.g., ξ(r)=ξ (r)θ(R r)+ξ+(r)θ(r R) (27) Shapiro&Teukolsky 1986, for further details) and we have − − − Radialstabilityinstratifiedstars 5 forr <R ξ inthisregion,theoutercounterpartofEq.(39)mustagain takeplace.Nevertheless,fors= 2,oneshouldalsoimpose P−(r)= 2πGρ2−( 2 r2), (cid:17) 3p−0 , (33) A+ =(m −2)(m+1). (40) 3 R − R 2πGρ2 m,s=−2 − − FromEq.(36),oneseesfromthiscasethatitsassociatedfun- where p isanarbitraryconstantthatcorrespondstothepres- −0 damentalmode(m = 0)isunstable. Thismeansinprinciple sureofsystemattheorigin.Forr>R,instead thatthissolutiontotheouterregionshouldbeexcluded,leav- 2πGρ2 ing out just the one from the case s = 1, where we should P+(r)= +(R2 r2). (34) consider 3 s − A+ =(m+1)(m+4). (41) m,s=1 Theconstantmass-densityintheinnerandouterregionshas been definedas ρ and ρ+, respectively. The pressure at the Notwithstanding,ourpreviousanalysisexhibitsclearprob- − lems:therearenotenougharbitraryconstantstofixEqs.(28) originp couldalwaysbechosensuchthatitmatchesthepres- −0 and (32) and the eigenfrequencies in each region are differ- sureatthebaseoftheouterphase,andasitcanbeseenfrom ent. However, we shall show that the condition of having a Eq. (34), we have also introduced the condition of having a sameeigenfrequencyforthewholesystem,asrequiredbyour nullpressureatthestar’ssurface. SubstitutingEqs.(33)and formalism, addresses all the problems. Obviouslythe stable (34)intoEq.(31),weareledto eigenfrequencies of the star are only related to the solution d2ξ 2 dξ 2 s = 1. However, they couldrise here either from aspects of (1 x2) ± + 4x ± + A ξ± =0, (35) theinneror the outerphasesof the star. Letussee howthis − ± dx2± x± − ±!dx± ±− x2±! conclusionensues.Assumeinitiallythattheonlypossibleω’s whereweassumedthatx+ (cid:17)r/Rs, x (cid:17)r/ ,and aregivenbyEq.(39),associatedwiththemodesm−s=1. So,for − R havingfiniteξ+ relatedto s = 1, onemustimposethatthere 3ω2 8 existsam+ totheouterphasesuchthatthenumeratorofthe A± (cid:17) 2πGρ±±Γ± + Γ± −2. (36) associateds=r1ecurrence relation be null. It can be shown that thisisjustthecaseif We now solve Eq. (35) by the method of Frobenius. For a sakeofsimplicity,weshalldropthe notation.Wetherefore 5+ 9+4A+ (m ) assumesolutionsoftheform ± m+ = − q s=1 −s=1 . (42) s=1 2 ξ = ∞ a xn+s, (37) Therefore,Eq.(42)demandsthat n Xn=0 9+4A+ =(2p+1)2, p 2, p N. (43) s=1 ≥ ∈ wherea andsarearbitraryconstantstobefixedbyprimarily n Forthecase s = 2toξ+,itcanbeshownthatthecondition wdeemllaansdiξn(gx)thisatatlhweayfisrsfitnciotne.ditBioynsoufbsEtqit.ut(i4n)gisEsqa.ti(s3fi7e)di,natos gfoivretnhebyexEiqst.e(n4c3e)−.oTfhaemm+so=d−2eirteslealtfeidsto am+s=−2 = 0 is exactly Eq. (35), it can be checked that the solutions to s are either s = 1or s = 2. Theassociatedrecurrencerelationobtained − 1+ 9+4A+ (m ) generallyis s=1 −s=1 m+ = q . (44) (m+s)(m+s+3) A s=−2 2 am+2 = (m+s+2)(m+s+3−) 2am, (38) Summarizing: ifEq.(43)issatisfiedforanynatural p 2, − therealwaysexistmodes,characterizedbyEqs.(42)and(≥44), withm =0,2,4...,anda1 =a3 =a5 =...= 0. Letusanalyze whichguaranteethefinitenessofξ+asalinearcombinationof firsttheinnerregion.Itisclearinthiscasethattheassociated solutionsfors=1ands= 2,associatedwithagiveneigen- ab0oufnodrasry=co−n2dimtiounsts.bFernoumll,Eaqs.(a38c)o,nosneequcelenacrelyofseoensethoaftothuer frequencyωm−s=1 thatjusttak−esintoaccountaspectsofthein- nerphaseofthesystem. Inthiscase,oneisabletocomeout powerseriesgivenbyEq.(37)doesnotconverge. Therefore, with two arbitrary constants of integration, that would then in order to satisfy the finiteness anywhere of ξ, we have to guarantee that the additional boundary conditions raised by impose that the series be truncated somewhere, rendering it the stratification, Eqs. (28) and (32), be satisfied. It is im- actuallyapolynomial.Hence mediatetoseethatasimilarreasoningasaboveensuesifone A =(m+1)(m+4). (39) choosesnowωm+ ascomingfromaspectsoftheouterregion, −m,s=1 givennowbyEqs=s1.(36) and(41). For thiscase, we will find bFyromEqE.q(.39(3)]6)a,roeneposseseisbltehattojuthstisdirsecgrieoten.freFqruoemnciEeqss[.gi(v3e6n) nEoqws. a(4m2)−s=a1ndan(d44a),mw+s=it−h2tahsesococinadteitdiownigthivωenm+sb=1y,Easq.g(i4v3e)n, brey- and (39), forhaving the frequencyof the fundamentalmode placingA+ (m )byA (m+ ). SinceΓ andρ aregiven s=1 −s=1 −s=1 s=1 ± ± (m = 0) positive, one should have Γ 4/3. Summing up, quantities,oneseesthattheonlypossibleeigenfrequenciesto − thephysicallyrelevantsolutiontothisc≥asejustleavesoutan systemshouldsatisfy9+4A = (2p+1)2. Thisconstraint ±s=1 arbitraryconstantofintegration,asrequiredduetothescaling is uniquelyimposeddue to the extra boundaryconditionsto lawpresenttoξfromEq.(31). the problem and is very restrictive. We have just shown a Letusnowanalyzetheouterregion.Thisisthemostphysi- simpleexamplewheresomeoftheaspectsimprintedbystrat- callyinterestingregionsincetheproblemsatthestar’scenter ificationraise. Whenevertherearetwoarbitrarysolutionsto areabsentandthereforeinprincipleonecouldevenhavetwo ξ in a given phase, it will be always possible to satisfy the linearly independentsolutions to ξ. Due to the finiteness of constraints(28)and(32). 6 PereiraandRueda 4. SYSTEMSWITHANELECTROMAGNETICSTRUCTURE concerning the stability of the stratified charged case are to- tallyanalogoustotheneutralone,obtainedbysimplymaking Nowweattempttogiveafurtherstepinourclassicalgen- thereplacement . eralization, by endowing the phases (as well as the surface Q P→P of discontinuity) with an electromagnetic structure. Just for clarity,letusworkwithasystemthatexhibitsjustanelectric 5. STRATIFIEDSYSTEMSINGENERALRELATIVITY field. Thefirstpointtobetakenintoaccountistheadditional Nowwegeneralizetheanalysisofstratifiedsystemstogen- electricforcepresentinthesystem.Thiswouldhavethesame eral relativity. From the classical analysis, we have learned structureasthegravitationalforceandthereforeitsgeneraliza- that surface quantities must also be inserted into the gener- tionisstraightforward.Nowoneshoulddefinealsoadistribu- alized equation of hydrostatic equilibrium. Therefore, in a tionalsolutionto the chargedensity. Thesurfaceforceasso- certain sense, we must find the proper generalization of the ciatedwiththesurfacetensionshouldhavethesameformas surface forcesin generalrelativity. This will notbe difficult previously,butnowshouldalsotakeintoaccountthepresent bearing in mind the thin shell formalism, as we shall see in electricaspects. Thepressureinthiscasewouldalsochange thesequel. Suchaformalismstatesthatinordertosearchfor due to the presence of the electric field and its jump over a distributional solutions to general relativity, one has to con- surfaceofdiscontinuitycouldstillbekeptfree. sideranenergy-momentumtensoratasurfaceofdiscontinu- From the (distributional)Maxwell equationsin the spheri- ity, thatwe shallname Σ. Itis preciselythis surfacecontent callysymmetriccaseonehasthat thatleadstothejumpofquantitiesthatarerelatedtophysical observables,suchastheextrinsiccurvature. We nowoutline Q(r) r E(r)= , Q(r)(cid:17)4π ρ (r¯)r¯2dr¯, (45) theformalismsuccinctly. Letusworkjustin thespherically r2 Z0 c symmetriccase,whereΣisdefinedasΦ=r R(τ)=0,with − τthethepropertimeofanobserverontheaforesaidhypersur- whereρ (r) isthe chargedensityatr. Theassociated “force c face. Assumethatthemetricsintheregionsaboveandbelow density” is dF~ /dv = ρ E(r)rˆ. Therefore, the equation of el c Σ (w.r.t. to the normalvectorto it), described by the coordi- hydrostaticequilibriumnowreads natesystemsxµ (cid:17)(t ,r ,θ ,ϕ ),respectively,aregivenby ± ± ± ± ± dP 2 +ρ(r)g(r) ρ (r)E(r) PQδ(r R)=0. (46) ds2 =e2α (r )dt2 e2β (r )dr2 r2dΩ2, (50) dr − c − R − ± ± ± ±− ± ± ±− ± ± where Therefore,likethegravitationalfield,theelectricfieldalso presents a jump at any surface of discontinuity (at r = R) dΩ2 =dθ2 +sin2θ dϕ2. (51) endowed with surface charges. We write the charge-density ± ± ± ± Assume that the (three dimensional) hypersurface Σ be de- as scribedbythe(intrinsic)coordinatesya (cid:17)(τ,θ,ϕ)suchthatat ρ (r)=ρ (r)θ(R r)+ρ+(r)θ(r R)+σ δ(r R), (47) thehypersurfacet = t (τ),θ = θandϕ = ϕ,besidesobvi- c −c − c − c − ouslyr = R(τ). I±nord±ertor±enderthep±rocedureconsistent, whilethedistributionalelectricfieldis one has±to impose primarily that the intrinsic metric to Σ is E(r)= E (r)θ(R r)+E+(r)θ(r R), [E(R)]+ =4πσ . (48) unique.Thisfixesthecoordinatetransformationsxµ = xµ(ya). − − − − c Thisisthegeneralizationofthecontinuityofthegr±avitat±ional By substituting now Eqs. (10), (45), (47) and (48) into potential across a surface harboring surface degrees of free- Eq. (46), we have that the surface tension at equilibrium dom.Now,ifthejumpoftheextrinsiccurvatureisnon-null,it shouldread isautomaticallyguaranteedtheexistenceofasurfaceenergy- momentumtensor(Poisson2004)thatinthesphericallysym- R G 1 = [P(R)]++ [M2(R)]+ [Q2(R)]+. (49) metriccasecanalwaysbecastasSa =diag(σ, , ),with PQ 2 − 16πR3 −− 16πR3 − (see,e.g.,Lobo&Crawford2005) b −P −P Notice that the existence of a surface mass would lead to 1 + [M2(R)]+ > 0,while[Q2(R)]+ couldinprinciplebeany. The σ= e 2β+R˙2 , (52) appearan−ceofthelasttermin−Eq.(49)isconsistentwith the −4πR(cid:20)p − (cid:21) − expectedandlongagoknowncontributionofelectricdouble- layers on the surface tension and surface energy of metals, = σ + 1 Rα′(e−2β+R˙2)+R¨R+β′RR˙2 +, (53) asrecalledbyFrenkel(1917)in hisseminalwork. Theexis- P −2 8πR" √e 2β+R˙2 # tenceofsuchsurfaceelectricfieldsiswell-knowninmaterial − − sciencesandithasbeendeterminedexperimentallyfromthe where generically A (cid:17) ∂A/∂r and A˙ (cid:17) dA/dτ. Finally, the ′ photoelectricphenomenonbymeasuringtheamountofwork discontinuity of the extrinsic curvature is the generalization donebyelectronstoescapefromthemetal’ssurface.Thereis ofthediscontinuityofthegravitationalfieldacrossasurface avastliteratureontheroleofelectricdouble-layersonsurface with nontrivial degrees of freedom. The case of interest to phenomenain metals and contact surfaces, and we refer the beanalyzedhereisthestaticandstable(uponradialdisplace- reader for instance to Huang&Wyllie (1949); Israelachvili mentsofΣ)oneR˙ =R¨ =0,i.e. anequilibriumpoint. (2011),andreferencestherein,forfurtherdetailsonthissub- Letusseenowhowthegeneralizationofthesurfaceforces ject. appearinthisformalism.Firstofall,weknowthat Since in the presence of an electric field the hydrostatic equilibriumequationand the surface tension changes, it can Tµν =Tµ+νθ(r−R)+Tµ−νθ(R−r)+eaµebνSabδ(r−R), (54) becheckedthatEq.(30) keepsthesame functionalform. In drawing this conclusion, it was also assumed that the total where eaµ (cid:17) ∂ya/∂xµ and Sab = hacSbc, hab (cid:17) eµaeνbgµν. No- charge of the system is a constant. This also means that ticethateaµ isitself definedasa distribution. Forhab, itdoes ∆Q = 0. One also sees immediately that the main results notmatterthesideofΣonetakestoevaluateit,sinceitmust Radialstabilityinstratifiedstars 7 beunique. Letusconstrainourselvesfirsttothe caseofper- potential.Besides,itcanbeshownthatinsuchacase fectfluids(locallyneutral)oneachsideofΣ. Oneseesfrom 1 Eq.(54)andthecoordinatetransformationsatΣthat σ= [M(R)]+, (63) 4πR2 − T0 (cid:17)ρ=ρ+θ(r R)+ρ θ(R r)+σδ(r R), (55) 0 − − − − where the above quantities are in cgs units. In Eq. (63) one T1 (cid:17) P= P+θ(r R) P θ(R r) (56) recognizes the jump of the gravitational field g(r) = φ′ = 1 − − − − − − GM(r)/r2atr=R,asexactlygivenbyEq.(8). Therefore, and FromEq.(56T),22w=enTo33ti(cid:17)ce−tPhatt=th−ePre−arPeδn(ort−asRso).ciatedsur(f5ac7e) σ2{α′+(R)+α′−(R)}≃ G[M8π2R(R4)]+− (64) stresses. This is exactly what we advanced in the classical . (65) casewithheuristicargumentsandobtainedhereasageneral SubstitutingEq.(65)intoEq.(62),weseethattheterminside consequence of distributional solutions to general relativity. thecurlybracketsofthelatterequationisnull[seeEqs.(53) Let us search formally for solutions to Einstein’s equations and (11)]. Hence, the remaining term in front of the delta withtheenergy-momentumgivenbyEqs.(55)–(57)withthe functionisexactly2 /Raswealreadyadvancedandexpected ansatz P [seeEq.(5)]. ds2 =e2αdt2 e2βdr2 r2(dθ2+sin2θdϕ2). (58) Nowweareinapositiontotalkaboutperturbationsinthe − − general relativistic scenario. When they take place, metric The distributional nature of Eq. (58) will be evidenced by and fluid quantities change at a given spacetime point from Eq.(54). Asthesolution,wehave theirstatic counterparts. Itis customarytoassume thatsuch 2m(r) r departures are small, what allows us to work perturbatively. e−2β =1 , m(r)(cid:17)4π ρ(r¯)r¯2dr¯. (59) The primarytask is to find suchchangesfromthe system of − r Z 0 equations coming from relativistic hydrodynamics and gen- Notice from the above equation that ρ is given by Eq. (55) eralrelativity. Nevertheless, these solutionsare alreadyvery andthereforee 2β isadistributiongenerallydiscontinuousat well-known(Misneretal.1973). Ourultimatetaskissimply − R. Forαwehave,though togeneralizethemtothedistributionalcase. Theequationgoverningtheevolutionofthefluiddisplace- e2β mentsoneachsideofΣisthegeneralrelativisticEulerequa- α = [4πPr3+m(r)], (60) ′ r2 tion,relatedtotheorthogonalprojectionofTµν;ν (perfectflu- ids)ontouµ,i.e. wherePisgivenbyEq.(56). FromEqs.(59)and(60),wesee that α is a distribution with no Dirac delta functions terms. (ρ+P)uν uµ (cid:17)(ρ+P)aν =(gµν uµuν)P , (66) ′ ;µ ,µ Therefore, it implies that [α]+ = 0. In other words, the − function α is generally continu−ous though not differentiable wherethelabels“ ′′foreachtermintheaboveequationwere ± omittedjustnottooverloadthenotation. atR. Nevertheless, fromthe conservationlaw of the energy- momentumtensorgivenbyEq.(54),wealsohave Inthehydrostaticcasewehaveonlythatut = e−α±0. When perturbationsarepresent(Misneretal.1973)± α (ρ +P )= P . (61) ′ ′ WenoticethatTµν;ν =±0,±taking±into−ac±countEq.(54),would ut± =e−α± =e−α±0(1−δα±), (67) give us in principle terms dependent upon Heaviside func- where δα is the change of the static solution α0 in the pres- tions, Dirac delta functionsand their derivatives. The terms ence of perturbationsat a givenspacetime point. For the ur associated with the Heaviside functions are null due to the component,usingthenormalizationconditionuµu±µ = 1,on±e validity of Einstein’s equations on each side of Σ. The nul- showsthat(Misneretal.1973) ± lity of the remaining terms is associated with identities that thesurfaceenergy-momentumtensorhastosatisfy(see,e.g., ur =e−α±0ξ˙˜ , (68) Mansouri&Khorrami 1996). It is not difficult to show that ± ± withξ˙˜ (cid:17) ∂ξ˜ /∂t . Justforcompleteness,uθ = uϕ = 0. For suchidentitiesareautomaticallysatisfiedwhenonetakesinto ± ± ± the components of uµ given by Eqs. (67) an±d (68±), the left- accountEqs.(52)and(53)(see,e.g.,Lobo&Crawford2005). handsideofEq.(66)±givesastheonlynontrivialcomponent Thisshowsthatthethin-shellformalismisconsistentandthe surIfnacoerdteerrmtosmpuutstEinqd.e(e6d1)beintatkheenfaosrmthethaafotrwesoauidldeaqluloatwionuss. ar± =−e−2β±a±r,with to consider Eqs. (55) and (56), we mandatorily should add −a±r =α′0±+δα′±+e2(β±0−α±0)ξ¨˜±. (69) surfaceterms. Thecorrectwayofdoingitis andtheassociatedequationofmotion ddPr =−(ρ+P)α′+ 2RPδ(r−R)+(cid:26)[P(R)]+−+ (ρ±+P±)(−a±r)=−∂∂Pr±. (70) σ σ [αe β]+ ± 2[α′+(R)+α′−(R)]+ R − 4′π−R −(cid:27)δ(r−R). (62) FromEq.(69),Eq.(70)canbecastas actNlyowtowEeq.s(h5o)wanthdatthiunsthiteiscliatssspicroalpleirmgiet,nEerqa.l(i6za2t)iorend.uFcierssteoxf- (ρ±0 +P±0)e2(β±0−α±0)ξ¨˜± =−∂∂Pr±± −(ρ±+P±)α′±. (71) all,noticethatinsuchalimit,α=φ(r),φ(r)thegravitational Therefore,intermsofdistributionsEq.(71)reads 8 PereiraandRueda (ρ0+P0)e2(β0−α0)ξ¨˜ =−∂∂Pr −(ρ+p)α′+ 2RPδ(r−R)+(cid:20)σR − [α4′eπ−Rβ]+− +[P(R)]+− σ + 2 ne2(β+0−α+0)ξ¨˜++e2(β−0−α−0)ξ¨˜−+α′++α′−o(cid:21)δ(r−R), (72) where ρ and P are given by Eqs. (55) and (56), respectively, wherewehaveusedEq.(24). Whenthelasttermontheleft- andnow handside ofaboveequationis expanded,by usingEqs. (60) and(74),Eq.(80)canbefurthersimplifiedto ξ˜(r,t)=ξ˜+(r+,t+)θ(r+ R)+ξ˜ (r ,t )θ(R r ). (73) − − − − − − 2η2∆σ [coshβ]+ 2 Notice that in Eq. (72) we are considering jumps and sym- ∆ − = P [Peβα′]+ ξ˜+[∆Peβ0]+. metrizations of quantities defined in the presence of pertur- R − " 4πR2 # "R2 − −# − bations. As we stated previously, such perturbationschange (81) slightly the value of the physical quantities with respect to OneseesfromEq.(81)thatEq.(32)isrecoveredintheclas- theirhydrostaticvalues. ThesquarebracketstermofEq.(72) sical limit by recalling that β = M(r)/r, which implies that is the proper generalization of the curly brackets term in the last term on the left-hand side of the above equation is Eq. (20). Naturally the reasoning for the eigenfrequencies 4σ(g++g−)/(2R). Besides, in thislimitwetake P 0and of ξ˜ in the general relativistic is the same as in the classical eβ 1fortheremainingterms. → → case. The same can be said aboutthe continuity,thoughnot The case where electromagnetic interactions are also differentiability,ofξãtanysurfaceofdiscontinuity. present is also of interest since its associated energy- Now,inordertohavethepropergeneralizationofEq.(30), momentumtensorisanisotropic.Thisnaturallyinfluencesthe weshouldevaluateEq.(72)atr+ξãndthensubtractitfromits equationofhydrostaticequilibrium,sinceitnowbecomes evaluationat r concerningthe static solution. In order to do 2 it properly, one should take into account the general results α(P+ρ)= P (P P), (82) ′ ′ t − − r − for the case of Lagrangian displacements coming from the standardprocedure(see,e.g.,Misneretal.1973),butnowin where P, Pt and ρ are the resultant radial pressure, tangen- thesenseofdistributions,byrecallingthat∆A(cid:17) A(r+ξ˜,t) tialpressureandenergydensityofthefluid,respectively. For A0(r,t),whereA0concernsthequantityAatequilibrium. − the electromagnetic fields, clearly (Pt − P) is solely related It is not difficult to see that we have the following re- tothem. Duetotheaforementionedaspects,thelattershould sults in the general relativistic distributional case (see, e.g., alsoinfluencethedynamicsoftheradialperturbations,aswe Misneretal.1973,forthetreatmentofacontinuoussystem): shallshowinthenextsection. ∆β= α ξ˜, (74) 6. ELECTROMAGNETICINTERACTIONSINSTRATIFIEDSYSTEMS − ′0 WITHINGR e2β0 ∆α′=4πre2β0 δP+2δβP0 + r δβ+α′0ξ˜, (75) We considernow the inclusion of electromagneticinterac- tionswithinthescopeofstratifiedsystemsingeneralrelativ- (cid:2) (cid:3) e−β0 r2eβ0ξ˜ ′ ity. Animportantcommentatthislevelisinorder. Sincewe ∆P=−γP0 (cid:16)r2 (cid:17) +δβ, (76) are dealing with electromagnetic fields in stars, it would be  1 ∂  σ ∂ξ˜+ ∂ξ˜ mmoedreiar.eaNsoenvaebrtlheetloesass,susimnceethteheMkanxowwelleldegqeuaotfiotnhseinstmruactteurrieasl ∆ρ= (ρ +P ) (r2ξ˜) + 0 + − constituting the stars is not yet precise, it is difficult to as- − 0 0 "r2∂r # (cid:18) 2 (cid:26) ∂r ∂r (cid:27) sess their realistic dielectric properties. Since working with 2σ ξ˜ dP Maxwell equations in the absence of material media gives +∆σ+ 0 +[P0]+ξ˜ δ(r R) 0ξ˜, (77) usupperlimitstothefieldsundernormalcircumstances,this R − (cid:19) − − dr seemstobeagoodfirsttooltoevaluatetherelevanceandef- ρ +P ∂P γ(cid:17) 0 0 , (78) fectsofelectromagnetisminstars. Weshalladoptforthetime P0 ∂ρ!s=const beingtofollowthisapproach. Theenergy-momentumtensor 2σ ξ˜ [coshβ ]+ ofeachlayerofthesystemwearenowinterestedshouldalso ∆σ=− R0 −ξ˜ [eβ0P0]+−+ 4πR20 −!. (79) havetheelectromagneticone6,i.e. FαβF WestressthatEq.(76)assumesadiabaticprocesses,inwhich 4πT(em) = F F gαβ+g αβ, (83) oneconsidersP= P(ρ)andwehavemadeuseof[ξ˜]+ =0for µν − µα νβ µν 4 theaboveequations. − where we defined F (cid:17) ∂ A ∂ A . Solving Einstein- We shall seek for solutions to the perturbations as ξ˜ = Maxwellequationsoµnνeachlµayeνr−ofaν sµtratified systemleads eiωtξ(r) with ω the same for all the phases the system may ustothefollowingequilibriumcondition(Bekenstein1971) have. We just need to worry about the Dirac delta function term, since it gives us the desired boundary condition valid ∂P = Q(r)Q′(r) α (ρ+P), (84) for the separationof each two phases. The termsin frontof ∂r 4πr4 − ′Q theHeavisidefunctionsbydefaultshallbetheonesfoundin where continuousmedia. Itisnothardtoseethatthesurfaceterms r Q(r) attheendshouldsatisfythecondition Q(r)(cid:17)Z 4πr2ρceβQdr, E(r)=eαQ+βQ r2 , (85) ∆σ ∆[αeβ]+ 0 R (2η2+1)− 4π′R − =0, (80) 6WerestrictouranalysestotheMaxwellLagrangian,−FµνFµν/4(cid:17)−F/4. Radialstabilityinstratifiedstars 9 e−βQ =1− 2mrQ(r) + Qr22(r) (86) coNinoctiidceewthitahtEthqe.c(4la6s)s,ibcaylrleicmailtlintogEthqa.t(E8c4l)asc(ra)n=beQschlaos(wr)n/rto2 and, from Eq. (85), Q (r) = 4πr2ρ . Besides, we recall e2βQ Q2(r) ′clas c α = 4πr3P+m (r) (87) that when brought to cgs units, the term Q2/r (here in geo- ′Q r2 " Q − r # metric units) becomes Q2/(c2r), which is null in the classic and non-relativisticlimit,aswellasanypressuretermontheright- r Q2 1 r Q2 handsideoftheaforementionedequation. mQ(r)(cid:17)Z 4πr2ρdr+ 2r + 2Z r2 dr. (88) Now, consider the analysis of a charged system consti- 0 0 tutedoftwoparts,connectedbyasurfaceofdiscontinuity(at Eqs. (85), (86), (87) and (88) are the charge, the radial and r = R) which hostssurface degreesof freedom,such as an ± timecomponentsofthemetricandtheenergyofthesystemup energydensity,chargedensityandasurfacetension. Itsgen- toaradialcoordinater,respectively.Besides,inEq.(85),we eralizationtoanarbitrarynumberoflayersisimmediatesince alsoexhibitedtheelectricfield E(r)inthecontextofgeneral eachsurfaceofdiscontinuityisonlysplitbytwophases. The relativity, obtained by means of the definition F (cid:17) E(r) = properdescriptionofthechargedensityinthiscasewouldbe tr ∂ A . Westressthatρ isthephysicalchargedensityofthe givenbythegeneralizationofEq.(47). Therefore,onewould r 0 c s−ystem, defined in terms of the four-currentby jµ (cid:17) e−αQuµ, haveatequilibriumthat where uµ is the four-velocityof the fluid with respect to the coordinatesystem(t,r,θ,ϕ)(seeLandau&Lifshitz1975,for Q(r)= Q (r )θ(R r )+Q+(r+)θ(r+ R). (89) − − − − − furtherdetails). FromEq.(84)oneseesthatinthescopeofgeneralrelativ- andforQ(r),aDiracdeltashallrise,duetoρ . ′ c itytheeffectofthechargeisnotmerelytocounterbalancethe Letusdefinethedistribution gravitationalpull. Forcertaincases, itcouldevencontribute toit. Thereasonforthatisduetothecontributionoftheelec- ρ¯c (cid:17)ρceβQ (cid:17)ρ¯+cθ(r−R)+ρ¯−cθ(R−r)+σ¯cδ(r−R), (90) tromagneticenergytothefinalmassofthesystem,asclearly givenbyEq.(88).NoticefromEq.(88)thatwehaveassumed where ρ¯±c= ρ¯±c(r±). From the above definition, we have that thatthe mass at the originis null, in order to avoid singular- Q = 4πr2ρ¯ . It implies that the total charge is the same as ′ c ities there. More generically,onecouldassume pointor sur- theoneassociatedwithρ¯ inanEuclideanspace. Therefore, c face mass contributions in Eq. (88) by conveniently adding all classical results aproposof the charge densities and total Dirac delta functions in ρ. Finally, we stress that Eq. (85) chargesthat we deducedin the previoussections ensue here canindeedbeseenasthegeneralizationofthechargeingen- forρ¯ . c eral relativity, since it takes into account the nontrivial con- We seek now for the distributional generalization of tribution coming from the spacetime warp due to its energy- Eq.(84). Thiscanbeeasilydonebyfollowingthesamerea- momentumcontent. soningfromtheprevioussection,whichfinallyleadsusto ddPr = Q(rr2)ρ¯c−(ρ+P)α′Q+2PRQδ(r−R)+[P(R)]+−+ σ2Q[α′Q+(R)+α′Q−(R)]+ σRQ − [α′Q4eπ−RβQ]+− − 2σ¯Rc2 [Q+(R)+Q−(R)]δ(r−R),   (91) where we are assuming surface quantities with the subindex place in our charged system. This case is more involved “Q” arerelatedto thechargedversionsof Eqs.(52)and(53) than the neutral case since the charged particles also feel [see also Eqs. (86) and (87)]. It is easy to show that in the an electric force. The equation describing the evolution of classicallimitEq.(49)naturallyraises,implyingthatinsuch the displacements can be shown to be generalized to (see alimitthecurlybracketsinEq.(91)isnull. Anninos&Rothman2002,forthedynamicsoftheradialper- We consider now the case where radial perturbationstake turbationsinagivenphase) (ρ0+P0)e2(βQ0−αQ0)ξ¨˜ =−∂∂Pr −(ρ+P)α′Q+ Q(rr2)ρ¯c + 2PRQδ(r−R)+(cid:20)σRQ − [α′Q4eπ−RβQ]+− +σ2Q ne2(β+Q0−α+Q0)ξ¨˜++e2(β−Q0−α−Q0)ξ¨˜−+α′Q++α′Q−o− 2σ¯Rc2 [Q+(R)+Q−(R)]+[P(R)]+−(cid:21)δ(r−R). (92) ForthechangeinQ(r),itcanbeshown(seeBekenstein1971) r+ξãndsubtractitfromEq.(91).Thisisduetothedefinition thatinthecomovingframetherearenocurrents. Thismeans of the Lagrangiandisplacementof a givenphysicalquantity, that the Lagrangian displacements of Q are null, ∆Q = 0. intrinsicallyrelatedtothenotionofcomovingobserverswith Eq. (92) takes into account the values of the physical quan- thefluid,whonaturallycoulddescribeitsthermodynamics. titiesinthepresenceofperturbationsatr. Inordertoobtain In order to simplify Eq. (92), we have that in the gener- thegeneralizationofEq.(30),weshouldevaluateEq.(92)at alized charged case (see Anninos&Rothman 2002, for the treatmentinaphaseofachargedsystem) 10 PereiraandRueda ∆β = α ξ˜, (93) Q − ′Q0 ∆α′Q =4πre2βQ0δP+2δβQP0− 8Qπr204+ e2rβQ0δβQ+α′Q0ξ˜+ 4πe2βQr0ρ¯cQ0ξ˜− 2πRσ¯cξ˜ Q+0e2β+0 +Q−0e2β−0 δ(r−R), (94) e−βQ0 r2eβQ0ξ˜′  (cid:16) (cid:17) ∆P= γP +δβ , (95) − 0 (cid:16)r2 (cid:17) Q  1 ∂ dP Qρ¯ σ ∂ξ˜+ ∂ξ˜ 2σ ξ˜ σ¯ ξ˜ ∆ρ= (ρ +P ) (r2ξ˜) 0 c ξ˜+ Q0 + − +∆σ + Q0 +[P ]+ξ˜ c [Q +Q ] δ(r R), − 0 0 "r2∂r #−" dr − r2 # (cid:18) 2 (cid:26) ∂r ∂r (cid:27) Q R 0 − − 2R2 + − (cid:19) − (96) ρ +P ∂P γ(cid:17) 0 0 , (97) P ∂ρ! 0 s=const 2σ [coshβ ]+ ∆σQ =−" RQ0 +[eβQ0P0]+−+ 4πRQ20 −#ξ˜. (98) Theadditional“0”subindexinaphysicalquantitymeansthat ourobjectivetosystematicallyapplyourformalismhere,but itsvalueatequilibriumwastaken.WejuststressthatEq.(97) simplytoderiveandexpoundit. Itisclearthatforpreciseand isthegeneralrelativisticdefinitionoftheadiabaticindexand realisticnumericalstabilitycalculations,itwouldbeidealto assumestheexistenceofanequationofstatelinkingthepres- havethemicrophysicalknowledgeofthepropertiesofthein- sureandthedensityofthesystem,P = P(ρ). Inthissense,it terfacialsurfaces.However,thisadifficultproblemwhichhas generalizesΓasdefinedbyEq.(3). beenelusive even in the mostadvancedfield of materialsci- By seeking for solutions ξ˜ = eiωtξ(r), one can see that ence,wherelaboratorydataofthematerialsurfaceproperties Eq. (92) just gives meaningful boundary conditions when a are accessible, but still with the lack of a complete physical frequencyωisthesameforallthephasespresentinthesys- theory for their explanation (Israelachvili 2011). Thus, the tem. We emphasize this is an universal property of the ap- measurementofthestar’seigenmodesbecomesofmajorrel- proachdevelopedhere,duetothesurfacedegreesoffreedom evance,sinceitcouldgiveinformationnotonlyonthestar’s andthewell-posednessoftheproblemofradialperturbations bulk structure but also on the possible existence of interior in stratified systems. The associated boundaryconditionris- interfaces and their associate microphysicaland electromag- ing from this analysis leads us to the conclusion that gener- neticphenomena. ically ξ(r) is not differentiable at a surface of discontinuity, Theradialinstabilitiesshowninouranalysesshouldbein- thoughcontinuous.Itcanbeshownthattheassociatedbound- terpretedanalogouslyasforcontinuousstarssince,eveninthe aryconditiontobetakenintoaccounthereisfunctionallythe stratified case, a globalset of eigenfrequenciesraises. Thus, sameasEq.(80)[orEq.(81)],wherenowthemetricandsur- stratifiedstarswouldeitherimplodeorexplodewhentheyare facequantitiesshouldberelatedtothechargedcase. radiallyunstable.Thoughwewereconcernedonlywithradial perturbations,itisofinteresttoinvestigatetheadditionalos- 7. CONCLUSIONS cillation modes owing to nonradial perturbations. The only In this article we have developed a formalism for assess- point to be added, with respect to continuous stars, is the ing the stability of an stratified star against radial perturba- properredefinitionofthesurfacesofdiscontinuitywhensuch tions. We have derived the relevant equations defining this perturbationstakeplace. Thisisclearlyaricherscenariothat boundary-value problem for both neutral and charged stars insertadditionaldegreesoffreedomintothesystem,leading andinNewtonianandEinstein’sgravity. Itmakesuseofthe to the appearance of additional modes such as the gravita- generalizedtheoryofdistributions,sinceweassumedthatthe tional ‘g-modes’ (see, e.g., Reisenegger&Goldreich 1992). surfaces of discontinuity are thin, that they host surface de- Such an analysis, however, is as a second step that goes be- grees of freedom, and that the phases separated by them do yondthegoalofthepresentwork,andthatweareplanningto notmix.Weshowedthatthoughthephasesmaybeverydiffer- investigateelsewhere. entamongstthemselves,whenperturbationstake place,they leadtothenotionofasetofeigenfrequenciesdescribingthe whole system, instead of an independent set for each phase. We are grateful to Professor Thibault Damour for the dis- Besides, ourformalismgaveusas a consequencethe proper cussions in the various occasions of the International Rel- additionalboundaryconditionstobetakenintoaccountwhen ativistic Astrophysics (IRAP) PhD-Erasmus Mundus Joint working with stratified systems. Such boundary conditions Doctorate Schools held in Nice. We are likewise grate- encompasssurfacedegreesoffreedominthesurfacesofdis- ful to Professor Luis Herrera. J.P.P. acknowledges the sup- continuityandgenericallymodifythesetofeigenfrequencies portgivenby the ErasmusMundusJointDoctorateProgram with respect to their continuouscounterpart. This should be within the IRAP PhD, under the Grant Number 2011–1640 a genericfingerprintofstratifiedsystemswith nontrivialsur- fromEACEAoftheEuropeanCommission. J.A.R.acknowl- face degrees of freedom. Our analyses are relevant for the edges the supportby the InternationalCooperation Program assessmentofthestabilityofrealisticstarmodels,sincethey CAPES-ICRANet financed by CAPES – Brazilian Federal ensue the precise notion of boundaryconditions. It was not Agency for Support and Evaluation of Graduate Education