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新的f(R黑洞
吕文东;奚萍;胡倩
【摘要】f(R引力是一个直接拓展广义相对论的修正引力理论,它的拉格朗日量是一个仅含曲率标量R的任意函数f(R.在F(r=1+αr的条件下(F(r≡(df(R(r/(dr和αr是一个对广义相对论小的修正量,导出了度规f(R引力理论中场方程的精确球对称真空解.此外,考虑了这个黑洞背景时空中的标量场扰动.用六阶WKB(Wentzel-Kramers-Brillouin方法,讨论了拟正则模和这个黑洞的参数之间的关系,得出这个黑洞是稳定的结论.%f(Rgravityisastraightextensionofgeneralrelativity,inwhichtheLagrangianisanarbitraryfunctionofthecurvaturescalarR.Wederiveanexactsphericallysymmetricvacuumsolutiontothefieldequationsofthemetricf(RgravitywiththeconditionF(r=1+αr(F(r≡(df(R(r/(drandthetermαrindicatesaminorcorrection(togeneralrelativity.Moreover,weconsiderthescalarperturbationsonthisblackholebackgroundspacetimes.Usingthesix-orderWKB(Wentzel-Kramers-Brillouinmethod,wediscusstherelationsbetweenquasinormalmodesofthisblackholeanditsparameterswhichshowthattheblackholeisstable.
【期刊名称】《上海师范大学学报（自然科学版）》【年(卷,期】2018(047004【总页数】6页(P406-411
【关键词】f(R引力;黑洞解;六阶WKB方法

【作者】吕文东;奚萍;胡倩
【作者单位】上海师范大学数理学院,上海200234;上海师范大学数理学院,上海200234;上海师范大学数理学院,上海200234【正文语种】中文【中图分类】P142

1IntroductionIn1998,theobservationsforSupernovaetypeIa(SNelashowthatouruniverseisinthestageofacceleratedexpansion.Thisobservableconsequenceisconfirmedbythesubsequentobservations,suchascosmicmicrowavebackground(CMB,largescalestructure(LSSandbaryonacousticoscillations(BAO.Incontextofgeneralrelativity(GR,thisphenomenonisexplainedbyaspatiallyhomogeneousandgravitationallyrepulsiveenergycomponent,dubbeddarkenergy.However,itiswellknownthattherearesomepuzzlesinthesedarkenergymodelssuchasthefine-tuneproblem,thecoincidenceproblem,andthenatureofdarkenergyproblem.Thus,thecosmicaccelerationmayoriginatefromsomemodificationofgravitytoGRsuchasLovelocktheories[1],stringtheory[2]orf(Rtheories[3].Inf(Rtheories,thecurvaturescalarRoftheLagrangianintheEinsteinHilbertactionisreplacedbyanarbitraryfunctionofR,f(R,whichisoneofthesimplestmodificationstoGR.TheLagrangianforGRisaspecialcaseof
f(Rgravitytheories,f(R=R-2Λ(Λisthecosmologicalconstant..DifferentfromGR,f(Rtheoriescanreproducethetwoacceleratedexpansionsofcosmologicalhistory:inflationandlate-timecosmicacceleration.Nevertheless,subjectedtolocaltestsofgravityandcosmologicalconstraints,theviablef(RtheoriesmustclosetoGR(f(R=R2Λ,inwhichf(Rmustsatisfytheconditionf(R=T+δ(R,andδ(Rmustbesmallinrecentera[4].Accordingtothisrequirement,Cramesetal.[5]proposedthatfoundaf(Rglobalmonopolesolutionunderthiscondition.Thenthissolutionisextensivelyinvestigated.Thestronglensingeffectforthef(Rglobalmonopolewasdiscussed[6].Arotatingf(Rglobalmonopolesolution[7]wasconsideredbythecoordinatetransformation.Moreover,thestabilityofthef(RblackholewithasoliddeficitanglewasstudiedbyWentzel-Kramers-Brillouin(WKBmethod[8].However,theinvestigationofthesphericallysymmetricvacuumsolutioninthemetricf(Rgravityanditsdynamicalpropertyislacking.Inthispaper,weconsiderthesphericallysymmetricvacuumsolutiontothefieldequationsofthemetricf(RgravitywithF(r=1+αranditsdynamicalproperty.Firstly,wederiveanewblackholesolutiontothefieldequationofthef(Rgravityinthemetricformalismanalytically.Secondly,weuseoneofsemi-analyticmethodsthesix-orderWKBmethod[9]tocalculatequasinormalmodes(QNMsofthisblackholeforscalarperturbations.TherelationsbetweenQNMsofthef(Rblackholeanditsparametersarediscussed.Basedontheseinformation,weconcludethatthenewf(Rblackholeisstable.
2Anewblackholesolutioninthemetricf(Rgravityandscalarperturbations2.1Anewf(RblackholesolutionInthiswork,wediscussthesphericallysymmetricstaticemptyspacesolutioninthemetricf(RgravityundertheconditionF(r=1+αrcorrespondingLagrangedensityisL=f(R,(1wheref(RisanarbitraryfunctionofR.Varyingequation(1withrespecttothemetrictensorgμv,weobtainthefieldequation

μvF(R+gμv□F(R=0,
(2withistheRiccitensorand□=μμ(isthecovariantderivative.Takingthetraceofthisfieldequation(2,f(Rcanbeexpressedas□F(R].
(3Thus,thefieldequation(2canberewrittenas
□F(R-μvF(R=0,
(4Itisobviousthatthefieldequation(4inthemetricf(RgravitytheoriesconsistsofF(Randitsderivatives.So,equation(4maybesolved,ifwehave
aknownF(R.Inthestaticsphericallysymmetricspacetimes,theRicciscalarR=R(r,ThusF(RcanbewrittenasF(R(r=F(r.Obviously,asF(r=1,thef(RgravityreproducesGR.F(rcanbeexpressedasF(r=1+Γ(r,
(5whereΓ(rmustbesmallinrecenteraandisanarbitraryfunctionthatcontainstheinformationofthemodificationofthegravitytoGR.Here,weadoptΓ(r=αrwhereαisaconstantasshownin[9].
Thegeneralstaticmetricwithsphericallysymmetrycanbewrittenasds2=B(rdr2-A(rdr2-r2(dθ2+sin2θdφ2.
(6Thedynamicalequations(4inthestaticsphericallysymmetricspacetimescanbeexpressedas(7
(8Here,weadopttheformF(rasequation(5.ForF(r=1+αr,thesystemofthefieldequations(7-(8canbereducedasU(r=U0,(9
(10whereU(r=A(rB(randU0istheintegralconstant.Theexactsolutionto
theequation(10isasfollows:
(11whereC1andC2aretheintegralconstants.Thus,A(r=U0B-1(r.Itisworthtonotethatthetermsincludingthefactorof|αr|inB(rareminorcorrections.Onecanscalethevariablesas
(12Thesphericallysymmetriclineelement(6canberewrittenas(13where(14
(152.2ScalarperturbationsNow,weconsiderconcretelythebehaviorsofscalarperturbationsinablackholeinf(Rtheoriesofgravityabove.ThepropagationofamasslessscalarfieldisdescribedbytheKlein-GordonequationμμΦ=0(μ=0,1,2,3.
(16Thenweseparatevariablesbysetting

(17whereYlm(θ,φaretheusualsphericalharmonics.SubmittingEq.(17toEq.(16,weobtain
(18whereωdenotesquasinormalfrequenciesandr*isthetortoisecoordinate,
(19Thus,whenrtendstotheeventhorizonofthef(Rblackhole,r*tendsto-∞.Asrapproaches+∞,r*approaches+∞.TheeffectivepotentialVinEq.(18is
(20wherelistheangularharmonicindex.Forperturbationswithl≥0,wecanshowexplicitlythattheeffectivepotentialisdefinitelypositive.Itvanishesattheeventhorizonandatr→∞,whichcorrespondstor*→-∞andr*→∞.So,thewavefunctionsassolutionstoEq.(18canbeplanewaveasfollows
(213WKBmethodandnumericalresultTheWKBapproximativemethodwasfirstlyappliedbySchutzandWill[10]totheproblemofscatteringaroundblackholes.Thismethodisbasedon
matchingoftheasymptoticWKBsolutionsatspatialinfinityandtheeventhorizonwiththeTaylorexpansionnearthetopoftheeffectivepotentialbarrierthroughthetwoturningpoints.Inanotherword,WKBmethodcandeterminethecomplexfrequenciesofablackholeifitsperturbationequationcanbedescribedbyaSchrödinger-likeequation.TherearethreeWKBcomputationalschemes:thelowestapproximation[10],thethird-orderimprovements[11]andthesixth-ordercorrections[9].Theaccuracyofthesixth-ordermethodisbetterthanthoseoftheformertwoforasmallermultipleindexl.Here,weadoptthesixth-orderWKBmethodtonumericallycalculatethesolutionstothebasicequationforresonantperturbationsofanewf(Rblackhole.TheQNMsisasfollows[9]
(22OurnumericalresultsofQNMsforscalarperturbationsarelistedinTable1-3.Asareminder,theoscillatingquasi-periodandthedampingtimescaleareshowninthesetables.InTable1,thecomplexfrequenciesvarywiththeparameterrs.Therealpartsofthequasinormalfrequenciesdecreasewithrsincreasing.Buttheimaginarypartsincreasewithrs.Itmeansthatthelargerrsis,morequicklythef(Rblackholeoscillatesandtheoscillationofthisblackholedecays.InTable2,therelationbetweenthefrequenciesofQNMsandtheparameterαislisted.Wefindthattheperiodoftheoscillationsincreaseswithαincreasing.Whilethedampingrategraduallydecreasesasαincreases.Itcorrespondstothecasethatwithanincreaseofα,theoscillationandthedecayoftheperturbationbothdecrease.InTable
3,itisshownthatthequasinormalfrequenciesvarywithdifferentl.Therealpartsandtheimaginarypartsofthequasinormalfrequenciesbothincreaseaslincreases.Itistosaythatthelargerlis,themoreslowlytheperturbationoscillatesandtheoscillationdecays.ItisworthtonotethequasinomalfrequenciesdonotvarywiththeparameterU0orβ.Itistosaythat,fordifferentU0orβ,therealpartandtheimaginarypartofthequasinormalfrequenciesareconstant.Fromtheseresults,weconcludethatthenewf(Rblackholeisstableunderthescalarperturbations.Table1QNMsofaf(RblackholewithU0=1,α=0.000001,β=0.0000012andl=2forscalarperturbationswithdifferentrsrsn=0n=1n=2n=310.967281-0.193532i0.927691-0.591252i0.860769-1.017397i0.786411-1.479766i20.483639-0.096766i0.463844-0.295625i0.430383-0.508697i0.393204-0.739881i30.322425-0.064510i0.309228-0.197083i0.286921-0.339130i0.262135-0.493252i40.241818-0.048382i0.231921-0.147812i0.215190-0.254347i0.196601-0.369938iTable2QNMsofaf(Rblackholewithrs=1,U0=1,β=0.0000012andl=2forscalarperturbationswithdifferentααn=0n=1n=2n=31×10-60.967281-0.193532i0.927691-0.591252i0.860769-1.017397i0.786411-1.479766i2×10-60.967278-0.191531i0.927688-0.591250i0.860766-1.017394i0.786409-1.479762i3×10-60.967275-0.191530i0.927685-0.591249i0.860764-1.017391i0.786406-1.479757i4×10-60.967272-0.191530i0.927682-0.591247i0.860761-1.017388i0.786404-1.479753iTable3QNMsofaf(Rblackholewithrs=1,U0=1,α=0.000001and
β=0.0000012forscalarperturbationswithdifferentlln=0n=1n=2n=310.220933-0.201632i0.178058-0.689056i0.383646-0.952811i2.084840-0.739348i20.585817-0.195523i0.528940-0.613035i0.462027-1.084330i0.444187-1.590330i30.967281-0.193532i0.927691-0.591252i0.860769-1.017397i0.786411-1.479766i41.350728-0.193001i1.321338-0.584574i1.267178-0.992018i1.196855-1.422766i4ConclusionInthiswork,weobtainasphericallysymmetricsolutioninthecontextofthemetricf(RgravitywithF(r=1+αr.Usingthesix-orderWKBmethod,wecalculatequasinormalfrequenciesofthescalarperturbationsinthisnewf(Rblackholespacetime.Therelationsbetweenthequasinormalfrequenciesandtheparametersofthisspacetimesareobtained.Itisshownthattherealpartsofthequasinormalfrequenciesdecreasewhiletheimaginarypartsincreasewithrsincreasing;withanincreaseofα,theoscillationandthedecayoftheperturbationbothdecrease;thelargerlis,themoreslowlytheperturbationoscillatesandtheoscillationdecays.ThemostinterestingresultisthattherealpartandtheimaginarypartofthequasinormalfrequencieskeepconstantfordifferentU0orβ.Fromtheseresults,weconcludethatthenewf(Rblackholeisstable.References:

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