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The Heun functions and their applications in astrophysics. DenitsaStaicovaandPlamenFiziev 6 1 0 2 y a M 7 1 AbstractTheHeunfunctionsareoftencalledthehypergemeotrysuccessorsofthe 21stcentury,becauseof the wide numberof their applications.In this proceeding ] we discuss their application to the problem of perturbations of rotating and non- h p rotatingblackholesandhighlightsomerecentresultsontheirlate-timering-down - obtainedusingthosefunctions. h t a m 1 The Heun Functions [ 2 v The Heun functions are gaining popularity due to the vast number of their appli- 1 cations. The Heun project, a site dedicated to gathering scientists working in this 2 area, has already accumulated more than 500 articles on the theory and the ap- 0 plicationsofthose functions.Amongthe topicsare theSchro¨dingerequationwith 4 0 anharmoicpotential,theTeukolskylinearperturbationtheoryfortheSchwarzschild . andKerrmetrics,transversablewormholes,quantumRabimodels,confinementof 1 grapheneelectronsindifferentpotentials,quantumcriticalsystems,crystallinema- 0 6 terials,three-dimensionalatmosphericandoceanwaves,singlepolymerdynamics, 1 economics,geneticse.t.c(seethebibliographysectionin[10]). : ThegeneralHeunfunctionisdefinedasthelocalsolutionofthefollowingsecond v i orderFuchsianordinarydifferentialequation(ODE)[5,6]: X r d2 g d e dH(z) ab z q a H(z)+ + + + − H(z)=0 (1) dz2 z z 1 z a dz z(z 1)(z a) (cid:20) − − (cid:21) − − InstituteforNuclearResearchandNuclearEnergy, Bulgarian Academy of Sciences, bul. “Tsarigrasko shose” 72, Sofia 1784, Bulgaria, e-mail: [email protected] SofiaUniversityFoundationforTheoreticalandComputationalPhysicsandAstrophysics, 5JamesBourchierBlvd.,1164Sofia,BulgariaandJINR,Dubna,141980MoscowRegion,Russia e-mail:[email protected] 1 2 DenitsaStaicovaandPlamenFiziev normalizedto1 atz=0Here e =a +b g d +1.Thisequationposses4reg- ularsingularities:z=0,1,a,¥ andit gene−rali−zesthe hypergeometricfunction,the Lame´ function,the Mathieufunction,the spheroidalwave functionsetc. Itsgroup ofsymmetriesisoforder192. Forcomparison,thehypregeometricdifferentialequationhas3regularsingular- itiesz(z 1)d2w(z)+[c (a+b+1)z]dw(z) abw(z)=0withgroupofsymmetries − dz2 − dz − oforder24. Recallingthedefinitionofirregularsingularity: Definition1. For an ODE of the form: P(x)y (x)+Q(x)y(x)+R(x)y(x)=0, the ′′ ′ point x is singular if Q(x)/P(x) or R(x)/P(x) diverge at x = x . If the limits 0 0 lim Q(x)(x x ) and lim R(x)(x x )2 exist and are finite then the point x→x0 P(x) − 0 x→x0 P(x) − 0 x isregularsingularity,otherwise,itisirregularoressentialsingularity.Thepoint 0 x =¥ istreatedthesamewayunderthechangex=1/z. 0 ThegeneralHeunfunctionhas4regularsingularities,fromwhichunderthepro- cess called confluence of singularities, one obtains 4 different types of confluent Heunfunctionswithfewersingularitiesbutofhighers-rank(SeeFig.1forillustra- tion). FortheconfluentHeunfunctionwhichwewillusebelow,thisprocessmeansthe redefinitionofb =b a,e =e a,q=qaandtakingthelimita ¥ .Thisgivesusthe → followingODE: d2 d g d ab qz q H(z) e H(z) − + H(z)=0 (2) dz2 − −z 1− z dz − z 1 z (cid:18) − (cid:19) (cid:18) − (cid:19) InMaplenotations,thedefaultformofthesolutionofthistypeofODEisdenoted asHeunC(a ,b ,g ,d ,h ,z)whichweadopt.ToobtainfromMaple’sdefaultformEq. 1, one needs to set a = (e 2 4q )1/2,b =g 1,g = 1+d ,d = a b + (1/2)d e +(1/2)e g ,h =− 0(1−/2)d0g (1/2)e0−g +q +−1/2an0dvisev−ers0a(0the 0 0 0 0 0 0 0 0 0 − − “ ”subscriptdenotestheparametersinEq.2. 0 2 Applications oftheHeun functions inastrophysics 2.1 TeukolskyAngular equationand TeukolskyRadialequation In the frame of the Teukolsky linear perturbations theory, the late-time ringing of a black hole due to a perturbation of different spin is described by one Mas- terequation.UnderthesubstitutionY (t,r,q ,f )=ei(w t+mf )S(q )R(r)(wherewhere m=0, 1, 2)thisequationsplitsintwosecondorderODEsoftheconfluentHeun type–T±he±TeukolskyAngularEquation(TAE): d d (m+su)2 du 1−u2 duSlm(u) + (aw u)2+2aw su+E−s2− 1 u2 Slm(u)=0, (3) (cid:16)(cid:0) (cid:1) (cid:17) (cid:18) − (cid:19) andtheTeukolskyRadialEquation(TRE): TheHeunfunctionsandtheirapplicationsinastrophysics. 3 GeneralHeunfunction(GHE) Singularities: regular={0,1,a,∞} d2 γ δ ǫ d αβz−q dz2H(z)+"z+z−1+z−a#dzH(z)+z(z−1)(z−a)H(z)=0 ConfluentHeunfunction(CHE) Singularities: regular={0,1},irregular={∞1} ddz22H(z)+"γz+z−δ1+ǫ#ddzH(z)+ qz−−α1β−zq!H(z)=0 BiconfluentHeunfunction(BHE) DoubleconfluentHeunfunction(DHE) Singularities: regular={0},irregular={∞2} Singularities: regular={},irregular={−11,11} ddz22H(z)+"−2z−β+1+zα#ddzH(z)+ γ−α−2−(1+α2)zβ+δ!H(z)=0 ddz22H(z)−α(+z2+z1+)2α(zz2−−1)22z3ddzH(z)+δ+(z(−2α1+)3γ(z)z++1)β3z2H(z)=0 TriconfluentHeunfunction(THE) Singularities: regular={},irregular={∞3} ddz22H(z)−(γ+3z2)ddzH(z)+(α+βz−3z)H(z)=0 Fig.1 AschemeofthedifferentconfluentODEsobtainablefromtheODEofthegeneralHeun function(inMaple’snotations).Thesubscriptnexttotheirregularsingularitiesistheirrank d2Rl,m(r)+(1+s) 1 + 1 dRl,m(r)++ K2 dr2 (cid:18)r−r+ r−r−(cid:19) dr (r−r+)(r−r−)− is 1 + 1 K l 4isw r Rl,m(r) =0 (4) (cid:18)r−r+ r−r−(cid:19) − − !(r−r+)(r−r−) where D =r2 2Mr+a2 =(r r )(r r ), K = w (r2+a2) ma, l =E + s(s+1)+a2w −2+2amw and u=−co−s(q )−. Here r =−M √M2 −a2 are the inn−er ± ± − and outer horizon of the rotating black hole. Being interested in electromagnetic perturbationswefixthespintos= 1. Inthissystem,theunknownquan−titiesarethecomplexfrequenciesw giving l,m,n usthespectrumandtheconstantofseparationE whichfora=0isE=l(l+1) l,m,n (for s= 1). The only physical parameters of the system, in agreement with the − No-HairTheorem,aretherotationalparameteraandthemassoftheblackholeM, whichweherefixtoM=1/2. Thesingularitiesofthetwoequationsareasfollows:fortheTREr=r –regular and r=¥ – irregular.For the TAE, the regularsingularities are: q = ±p and the irregularisagainq =¥ . ± 2.2 Boundary conditions In order to find the spectrum, we need to solve the central two-point connection problem,imposingappropriateboundaryconditionsontwo ofthesingularpoints. 4 DenitsaStaicovaandPlamenFiziev Detailsontheboundaryconditions,aswellasonthewholeapproachandtheexplicit valuesoftheparameters,canbefoundin[1,2,3,7,4,8,9].Inbrief,werequire: 1. OntheTAE: a. Quasi-normal modes (QNMs): we require angular regularity. This translate intothefollowingdeterminant. W[S1,S2]= HHeeuunnCC′((aa11,,bb11,,gg11,,dd11,,hh11,,((ccooss((pp//66))))22))+ HeunC′(a 2,b 2,g2,d2,h 2,(sin(p /6))2)+p=0 (5) HeunC(a 2,b 2,g2,d2,h 2,(sin(p /6))2) wheredetailsontheparameterscanbeseenin[1,7,4,8] b. Jet modes: A qualitatively new boundary condition has been used in [8] to obtain the so-called primary jet modes. The condition was that of angular singularitywhichtranslatesintopolynomialconditionforthesolutionsofthe TAE,i.e.: d b +g + +N+1=0 a 2 D N+1(m )=0 whereD (m )istridiagonaldeterminant[3]. N+1 2. OntheTRE: a. Black hole boundaryconditions:For any m, the solution R is valid for fre- 2 quenciesforwhich´ (w ) ( ma ,0)andalsothat:sin(arg(w )+arg(r))<0. 6∈ −2Mr+ b. Quasi-boundboundaryconditions:Foranym,thesolutionR isvalidforfre- 1 quenciesforwhich´ (w ) ( ma ,0)andalsothat:sin(arg(w )+arg(r))>0. 6∈ −2Mr+ 2.3 Numericalresults Thesodescribedboundaryconditionsleadtoatwo-dimensionalspectralsystemon theunknownsw andE.BecauseofthecomplexityoftheconfluentHeunfunctions, we use an algorithm developed by the team to find the roots of the system. The numerical results give different spectra of discrete complex frequencies some of whichcanbeseenonFig.2.Aspartofourstudy,weexaminedhowthosespectra changewithintroductionofrotation(a=0),uptothelimita M,andwetestedthe 6 → numericalstabilityoftheso-obtainedfrequencies,inordertoensuretheyrepresent physicalquantitiesandnota numericalartifact(an examplecanbe seenon Fig.2 b).) The physically interesting result are the qualitatively different spectra (Fig. 2 a) ), dependingon the boundaryconditionsimposed on the system, which can be used as an independenttool to discover the nature of the physicalobject emitting electromagneticorgravitationalwaves. TheHeunfunctionsandtheirapplicationsinastrophysics. 5 Fig.2 Examplesofthedifferentspectraobtainedfromthespectralsystem.a)Complexplotofthe first7modesintheQNMs(crosses)andprimaryjetmodes(diamonds)b)QNMs(point-line)and thenon-physicalspuriousmodes(solidlines)fora=[0,M] 3 Conclusion InthisproceedingwediscussedtheapplicationoftheHeunfunctionstotheproblem ofquasinormalmodesofrotatingandnon-rotatingblackholes.Wepresentedsome ofourlatestnumericalresults,keytowhichisthedevelopmentofthetheoryofthe Heunfunctionsandtheirnumericalimplementation. Acknowledgements This article was supported by the Foundation ”Theoretical and Computa- tionalPhysicsandAstrophysics”,bytheBulgarianNationalScientificFundundercontractsDO-1- 872,DO-1-895,DO-02-136,andSofiaUniversityScientificFund,contract185/26.04.2010,Grants oftheBulgarianNuclearRegulatoryAgencyfor2013,2014and2015. References 1. Fiziev P.P.,Class.QuantumGrav.232447-2468(2006) 2. Fiziev P.P.,Class.QuantumGrav.27135001(2010) 3. Fiziev P.P.,J.Phys.A.:MathTheor.43035203(2010) 4. FizievP.,StaicovaD.,Phys.Rev.D84,127502(2011) 5. Heun K.,Math.Ann.33161,(1889) 6. Ronveaux,A.,ed.Heun’sDifferentialEquations.Oxford,England:OxfordUniversityPress, 1995. 7. StaicovaD.R.,FizievP.P.AstrophysSpaceSci332:385-401(2011) 8. StaicovaD.,FizievP.BulgarianAstronomicalJournal23,83,(2015) 9. StaicovaD.,FizievP.AstrophysicsandSpaceScience,358:10(2015) 10. TheHeunProjecthttp://theheunproject.org/bibliography.html