The golden ratio in Schwarzschild-Kottler black holes

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by  N. Cruz

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
+
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