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Eur.Phys.J.CmanuscriptNo. (willbeinsertedbytheeditor) The golden ratio in Schwarzschild–Kottler black holes NormanCruza,1,MarcoOlivaresb,2,J.R.Villanuevac,3 1DepartamentodeFísica,FacultaddeCiencia,UniversidaddeSantiagodeChile,AvenidaEcuador3493,EstaciónCentral,Casilla307,Santiago 2,Chile. 2FacultaddeIngeniería,UniversidadDiegoPortales,AvenidaEjércitoLibertador441,Casilla298–V,Santiago,Chile. 3InstitutodeFísicayAstronomía,UniversidaddeValparaíso,AvenidaGranBretaña1111,Casilla5030,Valparaíso,Chile 7 1 0 thedateofreceiptandacceptanceshouldbeinsertedlater 2 n a Abstract Inthispaperweshowthatthegoldenratioispresent the parameter characterizing such a conical defect is less J intheSchwarzschild–Kottlermetric.Fornullgeodesicswith than 10−9. The study of geodesics is also comprehensive. 1 maximalradialacceleration,theturningpointsoftheorbits SomepropertiesofthemotionoftestparticlesonSchwarzschild– 1 are in the golden ratio F =(√5 1)/2. This is a general Kottlerspacetimescanbefoundin[11].Timelikegeodesics − ] resultwhichisindependentofthevalueandsignofthecos- forpositivecosmologicalconstantwereinvestigatedin[12], c mologicalconstantL . using only the method of an effective potential in order to q - foundtheconditionsfortheexistenceofboundorbits.Anal- r PACS 02.30.Gp,04.20.-q,04.20.Fy,04.20.Gz,04.20.Jb, g ysis of the effective potential for radial null geodesics in 04.70.Bw [ Reissner–NordströmdeSitterandKerrdeSitterspacetimes wasperformedin[13],whereassomepropertiesoftheReissner– 1 v Nordströmblackholeandnakedsingularityspacetimeswith 6 1 INTRODUCTION anon–zerocosmologicalconstantcanbefoundin[14].Null 6 geodesicsinachargedanti–deSitterspacetimewasstudied 1 Thepresenceofanon–zerovacuumenergy(thecosmolog- byVillanuevaetal.[15].Podolsky[16]investigatedallpos- 3 ical constantL ) in the main models of theoretical physics 0 sible geodesic motions for extreme Schwarzschild de Sit- suchasthesuperstringandthestandardEinsteincosmolog- . terspacetimes.ThemotionofmassiveparticlesintheKerr 1 icalmodelshavemotivatedconsiderationofsphericalsym- 0 andKerranti–deSittergravitationalfieldswasinvestigated metricspacetimeswithnon–zerovacuumenergyinorderto 7 in [17], where the geodesic equations are derived by solv- 1 studythewell-knowneffectspredictedbyGeneralRelativ- ingtheHamilton-Jacobipartialdifferentialequation.Equa- : ityforplanetaryorbitsandmasslessparticlesinthecontext v torialcircularorbitsintheKerrdeSitterspacetimeswasper- i oftheSchwarzschildspacetime[1],whichcanbefound,for X formedbyStuchlíkandSlaný[18].Astudywhichincluded example,in[2–6],amongothers.Thisstudyinvolvesdeter- nullgeodesicsandtimelikegeodesicsinSchwarzschildanti– r miningthegeodesicstructureofKottlerspacetimes[7]and a deSitterspacetimeswasconductedin[19]. then using a classical test to proof the influence of L . In thissense, the literaturedealingwith the applicationofthe Themainpurposeofthisarticleistoshowageneralbe- classical test of general relativity is extensive. To mention haviorofnon–radialnullgeodesics,commontoSchwarzschild, afew,thebendingoflightwasexaminedbyLake[8],who SchwarzschilddeSitter,andSchwarzschilanti–deSitterspace- found that the cosmological constant produces no change times. This general property does not depend on the value in this effect; Kraniotis and Whitehouse [9] obtained the of the cosmological constant and appears in the ratio be- compactcalculationoftheperihelionprecessionofMercury tween the apastron and periastron of two non–radial pho- bymeansofgenus–2Siegelschemodularforms.Bothtests tons,whichpossessthesameconstantofmotionE buttheir were applied by Freire et. al. [10] in the Schwarzschild– movementsareallowedinregionsseparatedbytheeffective Kottlerspacetimeplusaconicaldefect,sotheyobtainedthat potentialbarrieroftheequivalentonedimensionalproblem ae-mail:[email protected] forthe radialcoordinater. We have foundthat thisratio is be-mail:[email protected] the golden ratio F =(√5 1)/2. We have solved explic- − ce-mail:[email protected] itly,intermsoftheJacobiellipticfunctions,non–radialnull 2 geodesicsinSchwarzschild–antideSitterandSchwarzschild– 2 NULLGEODESICS deSitterspacetimes. Asastartingpoint,wewillconsiderthemostgeneralmetric It is well known that F appears quite frequently in bi- forastatic,sphericallysymmetricspacetimewithacosmo- ology, where many growth patterns exhibit the Fibonacci logicalconstantL ,whichreads numbers in which the next number is the sum of the pre- dr2 ds2= f(r)dt2+ +r2(dq 2+sin2q df 2), (1) vious2numbers(1,1,2,3,5,8,13,21,etc.).TheFibonacci − f(r) sequenceisconnectedwiththegoldenratio.Whatisofin- where f(r)isthelapsefunctiongivenby terest in biology is the existence of systems that can grow f(r)=1 2M L r2. (2) and evolve. Nevertheless, in non–equilibriumphase transi- − r − 3 Fromthislapsefunctionanddependingonthevalueof tions,whichappearsforexampleincondensedmatter,itis thecosmologicalconstant,wecanstudythelocationofthe possible to find this number. S. Dammer et al. [20] inves- horizonsbyanalyzingseparatelythethreedifferentconfig- tigated the properties of a direct bond percolation process urationsseparately: foracomplexpercolationparameter p.Theyfoundthatfor p= F , 1+F ,and2,thesurvivalprobabilityofacluster 1. Schwarzschildcase (L =0):Asthecosmologicalcon- − canbecomputedexactly. stant vanishes, the spacetime allows a unique horizon (theeventhorizon),whichislocatedat IthasbeenpointedoutbyM.Livio[21]thatthegolden r =2M. (3) + ratio appears in the physics of black holes. A well-known 2. Anti-deSittercase(L = 3 <0):whenthecosmolog- result [22] is the infinity discontinuity of the specific heat −ℓ2 icalconstantisnegative,thespacetimeallowsa unique atsomevaluesoftheangularmomentumandthechargeof horizon(theeventhorizon),whichmustbetherealpos- Kerr–Newmanblackholes.Thespecificheatchangesfrom itivesolutiontothecubicequation negativetopositiveforKerrblackholeswhentheratioa= J/M satisfies a 0.68M. This last value is very close to r3+ℓ2r 2Mℓ2=0, (4) thevalueoftheg≃oldenratioF =0.618033...,butisnotex- − anditsresultis[19] actlythesame.Inthestudyofphotongeodesicsingravita- tionalfieldsdescribedbygeneralrelativity,thegoldenratio 4ℓ2 1 3√3M r = sinh arcsinh . (5) + has been reported by Coelho et al. [23]. In that work, the r 3 "3 ℓ !# circular photon orbits in the Weyl solution describing two 3. de Sitter case (L >0): When a positive cosmological Schwarzschild black holes were considered. It was found constantsatisfiesL <1/9M2,thespacetimeallowstwo that as the distance betweenthe two blackholes increases, horizons(theeventhorizonr andthecosmologicalhori- + photonorbitsapproachoneanotherandmergewhenM = K zon r ), which are obtained from the cubic equation F L, where M is the Komar mass of each black hole. In ++ K [12] thecontextofsupersymmetry,Hubschetal.[24]foundthat 3 6M thegoldenratiocontrolledchaosinthedynamicsassociated r3 r+ =0. (6) with some supersymmetricLagrangians.Also, F has been −L L reportedinhigherdimensionalblackholes[25,26] Therefore, by defining Q = arccos( 3M√L )/3, their − expressionsaregivenby In thispaperwe reporthowthe goldenratio appearsin 1 r = √3sinQ cosQ , (7) the rather simple field of Schwarzschild black holes with + √L − a cosmological constant. Their appearance in the geodesic (cid:16) (cid:17) and structureofblackholesandtheirassociationwithageneral 2 behaviorofnullparticleswasquitesurprisingforus. r++= √L cosQ . (8) The geodesic motion of photons in a spacetime described Ourpaperisorganizedasfollows:InSection2,wede- by (1)–(2) can be obtained by solving the Euler-Lagrange rive the geodesic equations of motion for non–radial pho- equationsassociatedwiththismetric(see[2,3,19],forin- tons using the variationalproblemassociated with the cor- stance): responding spacetime metric. Using the effective potential ¶ L relatedtotheequivalentone–dimensionalproblemforther P˙ =0, (9) q− ¶ q coordinate,wefoundaNewtontypelawofforce,evaluating where P =¶ L/¶ q˙ is the generalizedconjugate momen- q the points where the maximum acceleration r¨ occurs. Ex- tumofthecoordinateq.Recallingthatformasslessparticles plicitsolutionsarefoundforthiscaseintermsofJacobiinte- (ds)2=2L =0,theLagrangianisgivenby grals.Inthesesolutionsthegoldenratioisexplicitlyshown. dt 1 1 1 1 Finally,inSection3wediscussourresults. L = f(r)t˙2+ f−1(r)r˙2+ r2q˙2+ r2sin2q f˙2,(10) −2 2 2 2 3 whereadotrepresentsthederivativewithrespecttoanaffine V (r) parameter, t , along the geodesic. Clearly (t,f ) are cyclic eff coordinates,sotheircorrespondingconjugatemomentaare conservedgivingaplacethefollowingexpressions P = f(r)t˙= √E, (11) t − − and P f =r2sin2q f˙ =L, (12) whereE andLareconstantsofmotion.Sincethemetric(1) Λ < 0 isasymptoticallyflatonlywhenL =0,theconstantofmo- tionEcanbeassociatedwiththeenergyfortheSchwarzschild Λ = 0 case.On theotherhand,since themotionisconfinedtoan invariant plane, without loss of generality we can choose Λ > 0 q =p /2so q˙ =0. Therefore,using(11) and(12) into Eq. r r r r (10), we obtain the equation of motion for the unidimen- + m ++ sionalequivalentproblem V (r) r˙2=E V (r). (13) eff eff − whereV definesaneffectivepotentialgivenby eff L2 f(r) V = . (14) eff r2 InFig.1,weplottheeffectivepotentialasafunctionofthe radialcoordinatefortheSchwarzschildcaseL =0,theSchwarzschild anti–de Sitter case L <0, and the Schwarzschild de Sitter r r a p caseL >0. Differentiation of the equation (13) with respect to the affine parameter t allows us to find a Newton type law of effectiveforcefortheradialcoordinategivenby r d(Veff) 2L2(r 3M) r+ rm r¨= = − . (15) − dr r4 Fig.1 Thefigureshowsthetypicaleffective potentialfornon-radial This radial acceleration is an indication of the variation of photonsinthethreecases:L =0(Schwarzschildcase),L <0(SAdS theradialcoordinateduetothecurvatureofthephotontra- case)andL >0(SdScase).Themaximumsarecoincidentatrm=3M, jectory. For radial photons with L=0 this acceleration is regardlessofthevalueofthecosmologicalconstantL . zero,aswecanseefromtheaboveequation.Noticethatthe .. above expression is independent of the cosmological con- r(r) stant L , which implies that the location of the maximum ofthe effectivepotentialr =3M is commonforthe three .. m r spacetimes (see Fig.1). In other words, the zero effective c forceonthephotonsisindependentofL . Also,noticethat theradialaccelerationhasamaximumatr =4M,equalto c 0 L2 r¨ = . (16) c 128M3 In Fig.2 we show the radial acceleration r¨ as a function of theradialcoordinater. Whenthephotonspossessthemaximumradialeffective acceleration,theirimpactparameterb=L/√E becomes 1 L −21 bF = , (17) r 32M2 − 3 (cid:18) (cid:19) 3M 4M whereaswhenthephotonspossesszeroradialacceleration, theirenergiesread Fig. 2 The plot shows the radial acceleration r¨ as a function of the 1 L −12 coordinater.Itpossessesamaximumatr=4Mandgoestozeroatr= b = . (18) 3M.Soforr<3M,thephotonspossessanegativeradialacceleration, 0 27M2 − 3 whichcorrespondstothefactthatthephotonsarefallingintotheevent (cid:18) (cid:19) Fromthetwolastequations,itisnothardtoprovethatb0< horizon bF . . Ourgoalisperformadescriptionoftheorbitsofthefirst andsecondkind,whichrepresenttheorbitsforphotonswith 4 b <b<¥ ,sotheeffectivepotentialimposestheexistence 2.1 Thegoldenmotion 0 of a turningpoint,r for orbitsof the first kind, andr for a p orbitsof the second kind (see rightpanelof Fig.1). There- As previously mentioned, when the motion of photons is fore,westartconsideringthezerosofEquation(13),which characterizedbyanimpactparameterequaltobF ,Eqs.(22) obligesustosolvethecubicequation (23) and (24) imply that rp = 4M, ra = 4MF and rn = P (r) r3 B2r+2MB2=0, (19) 4M/F .Therefore,fororbitsofthefirstkindr>4M,and 3 − ≡ − theequationofmotion(28)becomes whereBistheanomalousimpactparameterdefinedbythe du 2 1 F 1+F relation[19] =2M u u+ u . (29) 1 1 L df 4M− 4M 4M − (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) = + . (20) Performingthechangeofvariablesuggestedin[2,19], B2 b2 3 F NoticethatintheSchwarzschildcasetheanomalousimpact u=u 1 (1+cosc ) (u=u when c =p ),(30) p p − 2 parametercoincideswiththeusualimpactparameter.Also, (cid:20) (cid:21) weobtainthefollowingquadrature fromEqs.(17),(18)and(20)itisnothardtoseethatB = 0 dc 2 2F +1 c F +1 √27M=√3rm andBF =√32M=√2rc,so,bydefining df = 2 1−ksin2 2 ,with k=2F +1.(31) ¡ = 2√3B, X = 1arccos B0 , (21) T(cid:18)here(cid:19)fore,thesoluti(cid:16)onfortheang(cid:17)ularcoordinatef isgiven 3 3 − B by (cid:18) (cid:19) theturningpointsaregivenby 1 c f = K(k) F ,k , (32) r =¡ cosX , (22) a − 2 p whereFh(y ,k)ist(cid:16)heinc(cid:17)oimpleteellipticintegralofthefirst ¡ r = √3sinX cosX , (23) kind, K(k) F(p /2,k) is the complete elliptic integral of a 2 − the first kin≡d, and a =(5/64)1/4. Therefore,invertingthis whereas(cid:16)theotherrootofth(cid:17)ecubicpolynomial(withoutphys- lastequation,andreturningtotheoriginalvariable,weob- icalmeaning)isgivenby ¡ taintheequationoftheorbitofthefirstkind rn=−2 √3cosX +sinX . (24) r(f )= 1 F cn24(KM(k) af ), (33) An im(cid:16)portant and novel(cid:17)result is found when we con- − − wherecn(u) cn(u,k)istheJacobiellipticcosinefunction. sidertheratiobetweentheturningpoints(22) and(23)de- ≡ Additionally,fororbitsofthesecondkindwehavethat finedby r 4MF ,andtheequationofmotion(28)isgivenby z (b,M)= ra = √3tanX −1. (25) ≤du 2 1 F 1 rp 2 df =2M u−4M u+4M u−4MF . (34) Therefore, when massless particles are close to having a I(cid:18)nthis(cid:19)case, iti(cid:18)spossible(cid:19)to(cid:18)obtaina(cid:19)n(cid:18)easy quadra(cid:19)tureper- maximum radial acceleration, their impact parameter b formingthefollowingchangeofvariable: bF ,andthenweobtaintheidentity → u= 1 1+F secc , (35) F = lim z (b,M)= √3tan 1arccos 27 1,(26) such4tMhat(cid:16)u=uawhe2n(cid:17)c =0,andu ¥ whenc p .This b→bF 2 "3 −r32!#−2 substitution reduces Eq. (29) to the→same form a→s Eq. (31) whereF =0.618034...=1/(1+F )isthegoldenratio.An withthesamevalueofk,butnowitmustbewrittenas 1 c important corollary of the previous statement is obtained f = F ,k , (36) iwnhtehneLSc=ha3rzBschi2l,dadnedSthitetererfcoarese,.itFirsomnotEhqa.r(d17to),sbeFe →from¥ wherae th(cid:16)e 2zero(cid:17)of f is now at the apoastron ra = 4MF . F− Therefore, the trajectory can be obtained by inverting this Eqs. (7)–(8) that r =4M and r =4MF , i.e., the hori- ++ + lastequation,resultingin zonsareinthegoldenratio. 4M Also,wedefinethex ratioas r(f )= 1+F nc(af ), (37) r √3tanX +1 where nc(y )=1/cn(y ), and cn(y ) cn(y ,k) is the Ja- x (b,M)= n = , (27) ≡ cobi elliptic cosine function. In Fig. 3 we have plotted the −rp 2 and thus x =1+F =1/F when b bF . Notice that the orbitsof the first and second kind for photonswith impact twolastdefinitionsmakeitpossibleto→writethepolynomial parameterb=bF . (19)as P (r)= r r r z r (r+x r ), so, usingEqs. 3 p p p | − || − | (12)–(13), and then introducing the new variable u=1/r, 3 FINALREMARKS theequationofmotionreads du 2 up up In this paper we have studied the motion of massless par- =2M u u u +u , (28) −df p− z − x ticles in a backgrounddescribedby Schwarzschild–Kottler (cid:18) (cid:19) (cid:12) (cid:12)(cid:18) (cid:19) whereu =1/r .(cid:12) (cid:12)(cid:12) (cid:12) metric, whose general form is given by Eqs. (1)–(2). It is p p(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) 5 futurepossibilityofstudyinggravitationwithfractalgeom- etry,thegeometryofnature. Acknowledgements N.C.andM.O.acknowledgesthehospitalityof InstitutodeFísicayAstronomíaofUniversidaddeValparaíso,where partofthisworkwasdone.ThisresearchwassupportedbyCONICYT through Grant FONDECYT No. 1140238 (NC) and No. 11130695 (JRV) r + References ra rp 1. K. Schwarzschild, Über das Gravitations–feld eines MassenpunktesnachderEinsteinschenTheorie,Sitzungs- ber.Preuss.Akad.Wiss.3189-196(1916). 2. S. Chandrasekhar The mathematical theory of black holes,Oxforduniversitypress1993. 3. R.Adler,M.BazinandM.SchifferIntroductiontogen- eralrelativity,McGraw-Hill1975. 4. B. Shutz A first course in general relativity, Cambridge universitypress1990. 5. G.W.GibbonsandM.Vyska,TheApplicationofWeier- strassellipticfunctionstoSchwarzschildNullGeodesics, Fig.3 Thepolarplotshowsthenullgeodesicofthefirstandsecond kind.Thesetrajectoriescorrespondtothemotionofphoton,whichpos- Class.Quant.Grav.29065016(2012). sessanimpactparameter b=bF ,suchthatrp=4M andra=F rp, 6. S.Cornbleet,Elementaryderivationoftheadvanceofthe whereF =0,618034...isthegoldenratio. perihelionofaplanetaryorbit,Am.J.Phys.61,7(1993). 7. F.Kottler,ÜberdiephysikalischenGrundlagenderEin- steinschen RelativitÃd’tstheorie, Annalen der Physik 56 given as a solution to the Einstein equations, and is com- 410(1918). pletelydeterminedbyitsmassMandthecosmologicalcon- 8. K.Lake,Bendingoflightandthecosmologicalconstant stantL . Herewe havepresenteda reviewofthe spacetime Phys.Rev.D65087301(2002). andthecorrespondingequationsoftheangularmotion,with- 9. KraniotisG V andS. B. Whitehouse,Compactcalcula- outanyrestrictiononthevalueofL . tionoftheperihelionprecessionofMercuryingeneralrel- An importantfeaturefor this class of spacetime occurs ativity, the cosmological constant and Jacobi’s inversion whentheaccelerationoftheradialcoordinateisconsidered. problem,Class.QuantumGrav.204817-4835(2003). In such a situation, photonswith maximum radial acceler- 10. W. H. C. Freire, V. B. Bezerra and J. A. S. ation have an impact parameter bF , and then their return Lima,Cosmologicalconstant,conicaldefectandclassical pointsareinthegoldenratio.Thisresultprovestobeinde- testsofgeneralrelativity,Gen.Rel.Grav.331407(2001). pendent of the value of the cosmological constant, and al- 11. Z. Stuchlík and S. Hledík, Some properties of the lowsustoexpressthegoldenratioF asalimitofthefunc- Schwarzschildde Sitter and Schwarzschildanti–deSitter tion (26), i.e., F =limb bF z (b,M), where z is the ratio spacetimes,Phys.Rev.D60044006(1999). → between the apoastron and periastron distances. Thus, the 12. M. J. Jaklitsch , C. Hellaby and D. R. Matravers, Par- goldenratio,whichcharacterizesthefractalstructureofna- ticlemotioninthesphericallysymmetricvacuumsolution ture,alsoappearsinthegeodesicstructureofblackholes,in with positive cosmological constant, Gen. Rel. Grav. 21 particular in the movements of null particles and indepen- 941(1989). dently of the value and sign of the cosmological constant 13. Z.StuchlíkandM.Calvani,Nullgeodesicsinblackhole L . metrics with non–zero cosmological constant, Gen. Rel. Theunderstandingofgravitationalfieldsisstronglylinked Grav.23507(1991). to geometry:Newton’stheoryis developedon a three- di- 14. Z. Stuchlík and S. Hledík, Properties of the Reissner– mensionalplane space in Euclidean geometry.The change Nordström spacetimes with a nonzero cosmologicalcon- thatEinsteinmadewasenormousininterpretingspacetime stant,ActaPhys.Slov.52,363(2002). asa curvedmanifold,i.e.,adescriptionofgravitythrough 15. J.R.Villanueva,J.Saavedra,M.OlivaresandN.Cruz, Riemann’sgeometry.Inthisway,whenwefindthe golden Photonsmotioninchargedanti–deSitterblackholes,As- ratiointhegeodesicstructureofblackholes,itgivesusthe trophys.SpaceSci.344437(2013). 6 16. J.Podolsky,ThestructureoftheextremeSchwarzschild 22. P. C. Davis, The thermodynamic theory of black holes deSitterSpacetimeGen.Rel.Grav.311703(1999). Proc.R.Soc.Lond.A.353,499-521(1977). 17. G. V. Kraniotis, Precise relativistic orbits in Kerr 23. F.S.CoelhoandC.A.R.Herdeiro,RelativisticEuler’s space–timewithacosmologicalconstant,Class.Quantum three–bodyproblem,opticalgeometry,andthegoldenra- Grav.214743–69(2004). tio,Phys.Rev.D80104036(2009). 18. Z. Stuchlík and P. Slaný, Equatorialcircular orbits in 24. T. Hubsch and G. A. Katona, Golden ratio controlled the Kerr–de Sitter spacetimes, Phys. Rev. D 69 064001 chaos in supersymmetric dynamics, Int. J. Mod. Phys. A (2004). 281350156(2013). 19. N.Cruz,M.OlivaresandJ.R.Villanueva,Thegeodesic 25. J. A.Nieto,Alink betweenblackholesandthe golden structure of the Schwarzschild anti-de Sitter black hole, ratio(2011)[arXiv:1106.1600] Class.QuantumGrav.221167-1190(2005). 26. J. A. Nieto, E. A. León, V. M. Villanueva, Higher– 20. S. Dammer, S. R. Dahmen and H. Hinrichsen, Di- dimensionalchargedblack holesas constrainedsystems, rected Percolation and the Golden Ratio (2001) [arXiv: Int.J.Mod.Phys.D221350047(2013). 0106396]. 21. M.Livio,TheStoryofPHI,theWorld’sMostAstonish- ingNumber,BroadwayBooks2002.