A Maximum Mass-to-Size Ratio in Scalar-Tensor Theo

arXiv:gr-qc/9802049v310Sep1998AMaximumMass-to-SizeRatioinScalar-TensorTheoriesofGravityTooruTSUCHIDA∗,GoKAWAMURA†andKazuyaWATANABE‡DepartmentofPhysics,NiigataUniversity,Niigata950-21,Japan.AbstractWederiveamodifiedBuchdahlinequalityforscalar-tensortheoriesofgravity.Ingen-eralrelativity,Buchdahlhasshownthatthemaximumvalueofthemass-to-sizeratio,2M/R,is8/9forstaticandsphericallysymmetricstarsundersomephysicallyreasonableassumptions.WeformallyapplyBuchdahl’smethodtoscalar-tensortheoriesandobtaintheory-independentinequalities.Afterdiscussingthemassdefinitioninscalar-tensorthe-ories,theseinequalitiesarerelatedtoatheory-dependentmaximummass-to-sizeratio.WeshowthatitsvaluecanexceednotonlyBuchdahl’slimit,8/9,butalsounity,whichwecalltheblackholelimit,incontrasttogeneralrelativity.Next,wenumericallyexaminethevalidityoftheassumptionsmadeinderivingtheinequalitiesandtheapplicabilityofouranalyticresults.Wefindthattheassumptionsaremostlysatisfiedandthatthemass-to-sizeratioexceedsbothBuchdahl’slimitandtheblackholelimit.However,wealsofindthatthisrationeverexceedsBuchdahl’slimitwhenweimposethefurthercondition,ρ−3p≥0,onthedensity,ρ,andpressure,p,ofthematter.1IntroductionEinstein’sgeneralrelativitypostulatesthatgravitationalinteractionsaremediatedbyatensorfield,gμν.Itisalsowell-knownthatelectro-magneticinteractionsaremediatedbyavectorfield,Aμ.Onemaythereforesuspectthatsomeunknowninteractionsmaybemediatedbyscalarfields.Suchtheorieshavebeensuggestedsincebeforetheappearanceofgeneralrelativity.Moreover,ithasbeenrepeatedlypointedoutovertheyearsthatunifiedtheoriesthatcontaingravityaswellasotherinteractionsnaturallygiverisetoscalarfieldscoupledtomatterwithgravitationalstrength.Thismotivationhasledmanytheoreticalphysiciststostudyscalar-tensortheoriesofgravity(scalar-tensortheories)[1],[2],[3],[4].Thescalar-tensortheoriesarenaturalalternativestogeneralrelativity,andgravityismediatednotonlybyatensorfieldbutalsobyascalarfieldinthesetheories.Recently,suchtheorieshavebeenofinterestaseffectivetheoriesofstringtheoryatlowenergyscales[5].Manypredictionsofthescalar-tensortheoriesinstronggravitationalfieldsaresummarizedinRef.[4],[6],[7].Ithasbeenfoundthatawideclassofscalar-tensortheoriescanpassallexperimentaltestsinweakgravitationalfields.However,ithasalsobeenfoundthatscalar-tensortheoriesexhibitdifferentaspectsofgravityinstronggravitationalfieldsincontrastto∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]ythatnonperturbativeeffectsinthescalar-tensortheoriesincreasethemaximummassofanisolatedsystemsuchasaneutronstar[6],[7].Ingeneralrelativity,themass-to-sizeratioofastarhasphysicalsignificance,especiallyforanisolatedsystem.Buchdahlhasobtainedamaximumvalueofthemass-to-sizeratioofastaticandsphericallysymmetricstarunderthefollowingphysicallyreasonableassumptions[8],[9],[10].•Noblackholeexists.•Theconstitutionofthestarisaperfectfluid.•Thedensityatanypointinthestarisapositiveandmonotonouslydecreasingfunctionoftheradius.•Aninteriorsolutionofthestarsmoothlymatchesanexteriorsolution,i.e.,Schwarzschild’ssolution.Buchdahlhasobtainedanupperlimitofthemass-to-sizeratioas2M/R≤8/9.WeshallrefertothisastheBuchdahlinequality.MotivatedbyBuchdahl’stheorem,weshallderiveamodifiedBuchdahlinequalitytoobtainthemaximummass-to-sizeratioinscalar-tensortheories.Wethennumericallyexaminethevalidityoftheassumptionsmadeinderivingtheinequality.Theapplicabilityofouranalyticresultsisalsoexamined.Thispaperisorganizedasfollows.Insection2,wesummarizethebasicequationsinthescalar-tensortheories.Insection3,wederiveamodifiedBuchdahlin-equalityinthescalar-tensortheories,andthenumericalresultsarecomparedwiththeanalyticresultsinsection4.Abriefsummaryisgiveninsection5.2BasicequationsWeshallconsiderthesimplestscalar-tensortheory[1],[4],[11].Inthistheory,gravitationalinteractionsaremediatedbyatensorfield,gμν,andascalarfield,φ.TheactionofthetheoryisS=116πZ√−gφR−ω(φ)φgμνφ,μφ,νd4x+Smatter[Ψm,gμν],(2.1)whereω(φ)isadimensionlessarbitraryfunctionofφ,Ψmrepresentsmatterfields,andSmatteristheactionofthematterfields.Thescalarfield,φ,playstheroleofaneffectivegravitationalconstantasG∼1/φ.Varyingtheactionbythetensorfield,gμν,andthescalarfield,φ,yields,respectively,thefollowingfieldequations:Gμν=8πφTμν+ω(φ)φ2φ,μφ,ν−12gμνgαβφ,αφ,β+1φ(∇μ∇νφ−gμν2φ),(2.2)2φ=13+2ω(φ)8πT−dωdφgαβφ,αφ,β.(2.3)Nowweperformtheconformaltransformation,gμν=A2(ϕ)g∗μν,(2.4)2suchthatG∗A2(ϕ)=1φ,(2.5)whereG∗isabaregravitationalconstant,andwecallA(ϕ)acouplingfunction.Hereafter,thesymbol,∗,denotesquantitiesorderivativesassociatedwithg∗μν.ThentheactioncanberewrittenasS=116πG∗Z√−g∗(R∗−2gμν∗ϕ,μϕ,ν)d4x+Smatter[Ψm,A2(ϕ)g∗μν],(2.6)wherethescalarfield,ϕ,isdefinedbyα2(ϕ)≡dlnA(ϕ)dϕ2=13+2ω(φ).(2.7)Varyingtheactionbyg∗μνandϕyields,respectively,G∗μν=8πG∗T∗μν+2ϕ,μϕ,ν−12g∗μνgαβ∗ϕ,αϕ,β,(2.8)2∗ϕ=−4πG∗α(ϕ)T∗,(2.9)whereTμν∗representstheenergy-momentumtensorwithrespecttog∗μνdefinedbyTμν∗≡2√−g∗δSmatter[Ψm,A2(ϕ)g∗μν]δg∗μν=A6(ϕ)Tμν.(2.10)Theco