ebook img

The golden ratio in Schwarzschild-Kottler black holes PDF

0.16 MB·
by  N. Cruz
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview The golden ratio in Schwarzschild-Kottler black holes

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.

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.