Visualizing spacetimes via embedding diagrams

0.62 MB·English

by  Stanislav Hledik

Visualizing spacetimes via embedding diagrams
Stanislav Hledík, Zdeneˇk Stuchlík and Alois Cipko
InstituteofPhysics,FacultyofPhilosophy&Science,SilesianUniversityinOpava,
CZ-74601Opava,CzechRepublic
Abstract. It is hard to imagine curved spacetimes of General Relativity. A simple but powerful way how to achieve this
is visualizing them via embedding diagrams of both ordinary geometry and optical reference geometry. They facilitate
to gain an intuitive insight into the gravitational field rendered into a curved spacetime, and to assess the influence of
7
parameterslikeelectricchargeandspinofablackhole,magneticfieldorcosmologicalconstant.Opticalreferencegeometry
0
and related inertial forces and their relationship to embedding diagrams are particularly useful for investigation of test
0
particlesmotion. Embedding diagramsof staticand sphericallysymmetric, orstationary andaxiallysymmetricblack-hole
2
andnaked-singularity spacetimesthuspresentausefulconcept forintuitiveunderstanding ofthesespacetimes’nature. We

n concentrateongeneralwayofembeddinginto3-dimensionalEuclideanspace,andgiveasetofillustrativeexamples.
a
Keywords: Blackholes,nakedsingularities,ordinarygeometry,opticalreferencegeometry,embeddingdiagram
J
PACS: 04.20.-q,97.60.Lf,04.20.Dw,95.30.Sf

1
1
INTRODUCTION

2
v
Theanalysisofembeddingdiagrams[1,2,3,4,5,6,7,8]rankamongthemostfundamentaltechniquesthatenable
7
understandingphenomenapresentin extremelystronggravitationalfieldsof blackholesandothercompactobjects.
3
2 Theinfluenceofcharge,spin,magneticfieldorcosmologicalconstantonthestructureofspacetimescansuitablybe
1 demonstratedbyembeddingdiagramsof2-dimensionalsectionsoftheordinarygeometry(t consthypersurfaces)
0 into3-dimensionalEuclideangeometryandtheirrelationtothewellknown‘Schwarzschildth=roat’[9].
7
Propertiesofthemotionofbothmassiveandmasslesstestparticlescanbeproperlyunderstoodintheframeworkof
0
opticalreferencegeometryallowingintroductionoftheconceptofgravitationalandinertialforcesintheframeworkof
/
h generalrelativityinanaturalway[10,11,12,13,14,15,16,17,18],providingadescriptionofrelativisticdynamics
p inaccordwithNewtonianintuition.
-
o Theopticalgeometryresultsfromanappropriateconformal(3 1)splitting,reflectingcertainhiddenpropertiesof
+
r the spacetimesunderconsiderationthroughtheir geodesicstructure. Letus recall the basic propertiesof the optical
t
s geometry.Thegeodesicsofthe opticalgeometryrelatedtostatic spacetimescoincidewith trajectoriesoflight,thus
a being‘opticallystraight’[17, 19].Moreover,thegeodesicsare‘dynamicallystraight,’becausetestparticlesmoving
:
v alongthemareheldbyavelocity-independentforce[12];theyarealso‘inertiallystraight,’becausegyroscopescarried
i alongthemdonotprecessalongthedirectionofmotion[13].
X
Some fundamental properties of the optical geometry can be appropriately demonstrated namely by embedding
r
a diagrams of its representative sections [10, 3, 5]. Because we are familiar with the Euclidean space, 2-dimensional
sectionsoftheopticalspaceareusuallyembeddedintothe3-dimensionalEuclideanspace.Insphericallysymmetric
static spacetimes, the central planes are the most convenient for embedding – with no loss of generality one can
consider the equatorial plane, choosing the coordinate system such that θ π/2. In Kerr–Newman backgrounds,
=
the most representative section is the equatorial plane, which is the symmetry plane. This plane is also of great
astrophysicalimportance,particularlyinconnectionwiththetheoryofaccretiondisks[6].
In spherically symmetric spacetimes (Schwarzschild [17], Schwarzschild–deSitter [1], Reissner–Nordström [3],
and Reissner–Nordström–deSitter [4]), an interesting coincidence appears: the turning points of the central-plane
embedding diagrams of the optical space and the photon circular orbits are located exactly at the radii where the
centrifugal force, related to the optical space, vanishes and reverses sign. The same conclusion holds for the Ernst
spacetime[5].
However,intherotatingblack-holebackgrounds,thecentrifugalforcedoesnotvanishattheradiiofphotoncircular
orbits[20,21].Ofcourse,thesamestatementistrueiftheserotatingbackgroundscarryanon-zeroelectriccharge.
Throughoutthepaper,geometricalunits(c G 1)areused.
= =
OPTICALGEOMETRYAND INERTIAL FORCES
The notions of the optical reference geometry and the related inertial forces are convenient for spacetimes with
symmetries, particularly for stationary (static) and axisymmetric (spherically symmetric) ones. However, they can
beintroducedforageneralspacetimelackinganysymmetry[18].
Assuming a hypersurfaceglobally orthogonalto a timelike unit vector field n and a scalar field 8 satisfying the
conditionsn n 0, nκn 1, n nκ n 8, the 4-velocityu ofa test particleof restmass m can be
κ λ µ κ λ κ λ λ
∇ = =− ˙ ≡ ∇ =∇
uniquelydecomposedas
uκ γ(nκ vτκ). (1)
= +
Here,τ isaunitvectororthogonalton,v isthespeed,andγ (1 v2) 1/2.
−
= −
Introducing,basedontheapproachofAbramowiczetal.[18],aprojected3-spaceorthogonaltonwiththepositive
definite metric h giving the so-called ordinary projected geometry, and the optical geometry h by conformal
κλ ˜κλ
rescaling
h e 28h , h g n n , (2)
˜κλ= − κλ κλ= κλ+ κ λ
theprojectionofthe4-accelerationa hλuµ u canbeuniquelydecomposedintotermsproportionaltothezeroth,
first, andsecondpowersofv, respecκ⊥ti=velyκ,an∇dµtheλvelocitychangev (e8γv) uµ. Thus,we arriveata covariant
,µ
˙ =
definitionofinertialforcesanalogoustotheNewtonianphysics[18,22]
ma G (v0) C (v1) Z (v2) E (v), (3)
κ⊥= κ + κ + κ + κ ˙
wheretheterms
G m 8 m8 , C mγ2vnλ( τ τ ), Z m(γv)2τλ τ , E mvτ , (4)
κ =− ∇κ =− ,κ κ =− ∇λ κ−∇κ λ κ =− ˜ ∇˜λ˜κ κ =− ˙˜κ
correspondtothegravitational,Coriolis–Lense–Thirring,centrifugalandEulerforce,respectively.Here,τ istheunit
˜
vectoralongτ intheopticalgeometryand isthecovariantderivativewithrespecttotheopticalgeometry.
∇˜
InthesimplecaseofstaticspacetimeswithafieldoftimelikeKillingvectorsn ξ ∂/∂t,wearedealingwith
(t)
≡ =
spacecomponentsthatwewilldenotebyLatinindicesinthefollowing.Themetriccoefficientsoftheopticalreference
geometryaregivenbytheformula
h e 28h , e28 g ξµξν g . (5)
˜ik = − ik =− µν (t) (t)=− tt
Intheopticalgeometry,wecandefinea3-momentumofatestparticleanda3-forceactingontheparticle[18,23]
pi e28pi, f e28f , (6)
˜ = ˜i = i
where pi, f arespacecomponentsofthe4-momentum p andthe4-force f.Insteadofthefullspacetimeform,the
i
equationofmotiontakesthefollowingformintheopticalgeometry,
mf pk p 1m2 8, (7)
˜i = ˜ ∇˜k ˜i+2 ∇˜i
where represents the covariant derivative components with respect to the optical geometry. We can see directly
∇˜k
that photon trajectories (m 0) are geodesics of the optical geometry. The first term on the right-hand side of (7)
=
correspondstothecentrifugalforce,thesecondonecorrespondstothegravitationalforce.
InthecaseofKerr–Newmanspacetimes,detailedanalysis(see,e.g.,[21])leadstodecompositionintogravitational,
Coriolis–Lense–Thirringandcentrifugalforces(4)
G m8 m1∂ ln gt2φ−gttgφφ , (8)
κ ,κ κ
= − =− 2 " gφφ !#
Cκ = mA2((cid:127)−(cid:127)LNRF)√gφφ ∂κ gtφgφ−φ1/2 +(cid:127)LNRF∂κ√gφφ , (9)
1 h (cid:16) (cid:17)g2 i
Z mA2((cid:127) (cid:127) )2g ∂ ln φφ , (10)
κ = −2 − LNRF φφ κ" gt2φ−gttgφφ!#
respectively; we also denote ι e 8n. The Euler force will appear for (cid:127) const only, being determined by (cid:127)
uλ (cid:127) (see [11]). Here we sh=all−concentrate on the inertial forces acting6=on the motion in the equatorial pl˙an=e
λ
∇
(θ π/2)andtheirrelationtotheembeddingdiagramoftheequatorialplaneoftheopticalgeometry.
=
Clearly,bydefinition,thegravitationalforceisindependentoftheorbitingparticlevelocity.Ontheotherhand,both
the Coriolis–Lense–Thirringandthe centrifugalforcevanish(atanyr) if(cid:127) (cid:127) , i.e.,if theorbitingparticleis
LNRF
=
stationaryattheLNRFlocatedattheradiusofthecircularorbit;thisfactclearlyillustratesthattheLNRFareproperly
chosen for the definition of the optical geometry and inertial forces in accord with Newtonian intuition. Moreover,
bothforcesvanishatsomeradiiindependentlyof(cid:127).Inthecaseofthecentrifugalforce,namelythispropertywillbe
imprintedintothestructureoftheembeddingdiagrams.
EMBEDDING DIAGRAMS
The propertiesof the (optical reference) geometry can convenientlybe represented by embeddingof the equatorial
(symmetry)planeintothe3-dimensionalEuclideanspacewith lineelementexpressedin thecylindricalcoordinates
(ρ,z,α)inthestandardform.Theembeddingdiagramischaracterizedbytheembeddingformulaz z(ρ)determin-
=
ingasurfaceintheEuclideanspacewiththelineelement
dz 2
dℓ2 1 dρ2 ρ2dα2 (11)
(E)=" +(cid:18)dρ(cid:19) # +
isometrictothe2-dimensionalequatorialplaneoftheordinaryortheopticalspacelineelement[24]
dℓ2 h dr2 h dφ2. (12)
rr φφ
= +
Theazimuthalcoordinatescanbeidentified(α φ)whichimmediatelyleadstoρ2 h ,andtheembeddingformula
φφ
≡ =
isgovernedbytherelation
dz 2 dr 2
h 1. (13)
rr
dρ = dρ −
(cid:18) (cid:19) (cid:18) (cid:19)
Itisconvenienttotransfertheembeddingformulaintoaparametricformz(ρ) z(r(ρ))withr beingtheparameter.
=
Then
dz dρ 2
h . (14)
rr
dr =±s − dr
(cid:18) (cid:19)
The sign in this formula is irrelevant, leading to isometric surfaces. Because dz/dρ (dz/dr)(dr/dρ), the turning
=
points of the embedding diagram, giving its throats and bellies, are determined by the condition dρ/dr 0. The
embeddingdiagramcanbeconstructediftherealityconditionh (dρ/dr)2 0holds. =
rr
− ≥
EXAMPLES OFEMBEDDING DIAGRAMS
In this section we present graphic representations of embedding diagrams for various spacetimes with their brief
description.Forfulltreatment,seethereferenceprovidedatthebeginningofeachsubsection.
Schwarzschild spacetime withnonzero cosmologicalconstant
The influence of cosmological constant on the structure of Schwarzschild–deSitter spacetime is treated in full
technicaldetailsbyStuchlíkandHledík[1].Puttingy 3M2/3,t t/M,r r/M,whereMisthemassparameter
≡ → →
and3denotesthecosmologicalconstant,thelineelementinstandardSchwarzschildcoordinatesreads
ds2 1 2 yr2 dt2 1 2 yr2 −1dr2 r2(dθ2 sin2θdφ2). (15)
=− −r − + −r − + +
(cid:18) (cid:19) (cid:18) (cid:19)
Embedding diagrams of the ordinary geometry are given by the formula z z(r), which can be obtained by
=
integratingtherelation
dz 2 yr3 1/2
+ . (16)
dr = r 2 yr3
(cid:18) − − (cid:19)
In the case of Schwarzschild–deSitter spacetimes, the embedding can be constructed for complete static regions
betweentheblack-hole(r )andcosmological(r )horizons.Recallthatthestaticregionexistsfor y 1/27only.In
h c
≤
thecaseofSchwarzschild–anti-deSitterspacetimes,thestaticregionextendsfromtheblack-holehorizontoinfinity.
However,wecanseedirectlyfromEq.(16)thattheembeddingdiagramsoftheordinaryspacecanbeconstructedin
alimitedpartofthestaticregion,locatedbetweentheblack-holehorizonr andr ( 2/y)1/3.
h e(ord)
= −
Embeddingdiagramsofopticalreferencegeometryaregivenbyparametricformulasz z(r),ρ ρ(r):
= =
dz r 4r−9−yr4 1/2, ρ r 1 2 yr2 −1/2. (17)
dr = r 2 yr3 r(r 2 yr3) = −r −
− − (cid:20) − − (cid:21) (cid:18) (cid:19)
Theembeddingformulaz z(ρ)canthenbeconstructedbyanumericalprocedure.Further,itcanbeshown[1]that
=
‘turningradii’oftheembeddingdiagramsaregivenbytheconditiondρ/dr 0.Since
=
dρ 1 3 1 2 yr2 −3/2, (18)
dr = −r −r −
(cid:18) (cid:19)(cid:18) (cid:19)
we cansee thattheturningradiusdeterminingathroatoftheembeddingdiagramoftheopticalgeometryislocated
justatr 3,correspondingtotheradiusofthephotoncircularorbit;itisexactlythesameresultasthatobtainedin
=
the pureSchwarzschildcase. The radiusof the photoncircularorbitis importantfromthe dynamicalpointof view,
becausethecentrifugalforcerelatedtotheopticalgeometryreversesitssignthere[17,23].Abovethephotoncircular
orbit, the dynamicsis qualitativelyNewtonianwith the centrifugalforce directedtowardsincreasingr. However,at
r 3,thecentrifugalforcevanish,andatr <3M itisdirectedtowardsdecreasingr. Thephotoncircularorbit,the
=
throatofthe embeddingdiagramoftheopticalgeometry(dρ/dr 0),andvanishingcentrifugalforce,allappearat
=
theradiusr 3.
=
In the case of Schwarzschild–deSitter spacetimes containing the static region, the embeddability of the optical
geometryis restrictedbothfrombelow,andfromabove.Usinga numericalprocedure,theembeddingdiagramsare
constructedforthesamevaluesof y asinthecaseoftheordinarygeometry.
InthecaseofSchwarzschild–anti-deSitterspacetimes,theembeddabilityoftheopticalgeometryisrestrictedfrom
belowagain;with y thelimitshiftstor 0,alongwiththeradiusoftheblack-holehorizon.Theembedding
→−∞ →
3 3
y=-0.01 y=-10-4
2.5 y=10-6 2.5 y=-0.03 y=-0.003 y=0
y=10-4
2 2
y=0.002 y=0 y=0.002
logz1.51 y=0.03 y=0.02 y=-10-6 logz1.51 (a) (b)
y=-0.002 y=0.02
0.5 yy==--00..0032 0.5 y=0.03
0 0
0 0.5 1 l1o.g5r 2 2.5 3 0.6 0.8 1 1l.o2gρ1.4 1.6 1.8 (c) (d)
FIGURE1. Leftpanel:QualitativefeaturesoftheembeddingdiagramsoftheordinarygeometryoftheSchwarzschild–deSitter
andSchwarzschild–anti-deSitterspacetimesinalog-logdiagram.Onecanimmediatelyseehowthediagramswithy 0‘peeloff’
6=
thepureSchwarzschilddiagram(y 0,boldcurve).Allsectionswithy 0arecomplete(i.e.,themaximumthatcanbeembedded
= 6=
intoEuclideanspaceisshown),exceptuninterestinglowerpartsofthethroats.Thediagramsclearlyindicatemodificationsofthe
spacetimestructurecausedbythepresenceofacosmologicalconstant.Middlepanel:Sameasleftpanelbutforopticalgeometry.
Rightpanel: Exampleofembedding diagramsoftheopticalreferencegeometryoftheSchwarzschild–deSitterspacetimes.The
pureSchwarzschildcaseistakenforcomparisonin(a).Thediagramsaregivenfor(b) y 10 6,(c)y 0.002,and(d)y 0.03.
−
= = =
TheyaresimilartothepureSchwarzschildcase,becausetheregionnearthecosmologicalhorizonbeingofhighlydifferentcharacter
is‘cutoff’bythelimitofembeddability.Takenfrom[1].
diagramsareconstructedbythenumericalprocedureforthesamevaluesofyasfortheordinaryspace.Thesediagrams
have a special property, not present for the embedding diagrams in the other cases. Namely, they cover whole the
asymptoticpartoftheSchwarzschild–anti-deSitterspacetime,butinarestrictedpartoftheEuclideanspace.Thisis
clearfromtheasymptoticbehaviorofρ(r).Forr ,thereisρ ( y) 1/2.Clearly,withdecreasingattractive
−
→+∞ ∼ −
cosmological constant the embedding diagram is deformed with increasing intensity. The circles of r const are
concentratedwithanincreasingdensityaroundρ ( y) 1/2asr . =
−
= − →∞
BasicfeaturesoftheembeddingdiagramsoftheSchwarzschild–deSitterspacetimesareillustratedinFigure1.
Interior uniform-density Schwarzschild spacetimewithnonzero cosmologicalconstant
Theinfluenceofthecosmologicalconstantonthestructureofinterioruniform-densitySchwarzschildspacetimeis
treatedinfulltechnicaldetailsbyStuchlíketal.[2].
ThelineelementinstandardSchwarzschildcoordinatesreads
ds2 e28(r)dt2 e29(r)dr2 r2(dθ2 sin2θdϕ2), (19)
=− + + +
where
r2 −1/2 1 2M 3 9M 3M 3R3
e9(r)= 1−a2 , e8(r)= Ae−9(R)−Be−9(r), a2 = R3 + 3 , A= 6M 3R3, B= 6M−3R3. (20)
(cid:18) (cid:19) + +
HereM denotestotalmass,ρ(uniform)massdensityandRexternalradiusoftheconfiguration;3isthecosmological
constant.Examplesofembeddingdiagrams(forrepulsivecosmologicalconstant)areshowninFigure2.
y=0.008
y=0.03
y=0
y=0.015 y=0.005
y=0
x=2.45
x=2.45
Ordinarygeometry Opticalgeometry
FIGURE 2. Axial sections of the Schwarzschild–de Sitter (3>0) case. The dashed curve represents pure Schwarzschild
spacetime, the dots mark the position of the edge of the internal spacetime (i.e., r R xM). The embedding diagrams of
= =
theoptical referencegeometryshow theexistenceofinternalcircularnullgeodesics whichplayakeyroleforthepossibilityof
neutrinotrapping.
Reissner–Nordström spacetimewithnonzero cosmologicalconstant
Thissectionisbasedon[25],wherefull-detailtreatmentcanbefound.Putting y 13M2,e Q/M, t t/M,
≡ 3 ≡ →
r r/M,whereMisthemassparameter,Qistheelectricchargeand3isthecosmologicalconstant,thelineelement
→
instandardSchwarzschildcoordinatesreads
2 e2 2 e2 −1
ds2 1 yr2 dt2 1 yr2 dr2 r2 dθ2 sin2θdφ2 . (21)
=− −r +r2 − + −r +r2 − + +
(cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
AnexampleofclassificationbasedonpropertiesofembeddingdiagramsofopticalreferencegeometryisinFigure3.
H
HH (cid:0)
HH (cid:0)
00..22 HH (cid:0)
00 ddSS--NBSH--OO11 (cid:8)yh (maH(cid:8)x)Hj(cid:8)*6?d@I(cid:0)(cid:9)S@-NSd-SO-2NS-O3 AKAA
--00..22 (cid:8)(cid:8) @ A
yy A(cid:8)dS-BH-O1
--00..44(cid:8) AdS-N@S-O3 A
(cid:8)(cid:8) yh (min) yextr (opt) @ A
--00..66 AdS-NS-O1 @ notembeddable
@
00..2255 00..55 00..7755 11 11..2255 11..55 11..7755
ee22
FIGURE 3. An example classification of Reissner–Nordström spacetimes according to properties of embedding diagrams of
opticalreferencegeometry.Takenfrom[25].
Ernst spacetime
ThestaticErnstspacetime[26,5]istheonlyexactsolutionofEinstein’sequationsknowntorepresentthespacetime
ofasphericallysymmetricmassivebodyorblackholeofmass M immersedinanotherwisehomogeneousmagnetic
field.Ifthemagneticfielddisappears,thegeometrysimplifiestotheSchwarzschildgeometry.Therefore,sometimes
theErnstspacetimeiscalledmagnetizedSchwarzschildspacetime.Usuallyitisbelievedthatforextendedstructures
like galaxies both the effects of general relativity and the role of a magnetic field can be ignored. However, in the
caseofactivegalacticnucleiwithahugecentralblackholeandanimportantmagneticfield,theErnstspacetimecan
representsomerelevantpropertiesofthegalacticstructures.Therefore,itcouldevenbeastrophysicallyimportantto
discussandillustratebasicpropertiesoftheErnstspacetime.
The Ernst spacetime has the important property that it is not asymptotically flat. Far from the black hole, the
spacetimeiscloselyrelatedtoMelvin’smagneticuniverse,representingacylindricallysymmetricspacetimefilledwith
anuniformmagneticfieldonly.TheErnstspacetimeisaxiallysymmetric,anditsstructurecorrespondstothestructure
oftheSchwarzschildspacetimeonlyalongitsaxisofsymmetry.Offtheaxis,thedifferencesareveryspectacular,and
weshalldemonstratethemfortheequatorialplane,whichisthesymmetryplaneofthespacetime.
ThelineelementoftheErnstspacetimereads
ds2 32 1 2M dt2 1 2M −1dr2 r2dθ2 r2sin2θ dφ2, (22)
= − − r + − r + + 32
" (cid:18) (cid:19) (cid:18) (cid:19) #
where M M G/c2isthemass, B B G1/2/c2isthestrengthofthemagneticfield,3 1 B2r2sin2θ.
cgs cgs
≡ ≡ ≡ +
The dimensionless product BM 1 in astrophysically realistic situations. Really, BM 2 10 53B M .
− cgs cgs
≪ ∼ ×
For example, in case of a black hole with M 109M (typical for AGN), the value BM 1 corresponds to
B 1011Gauss,whichisunrealistic.Someillu=strativee⊙mbeddingdiagramsarecollectedinFi∼gure4.
cgs
∼
B 0.2>B
c
=
B 0.08<B
c
=
FIGURE4. Leftcolumn:ordinarygeometry,rightcolumn:opticalgeometryofErnstspacetime.Itcanbeproved[5]thatacritical
magneticfieldBc 0.0947exists.ForB>Bc,neitherthroatsnorbelliesandnocircularphotonorbitsexist.ForB<Bc,thethroat
≈
andthebellydevelop,correspondingtotheinnerunstableandouterstablephotoncircularorbit.
Kerr–Newmanspacetimes
Wefocusontheresultsconcerningtheopticalreferencegeometry[21].Itcanbeshown[21]thattheturningpoints
oftheembeddingdiagramsarereallylocatedattheradiiwherethecentrifugalforcevanishesandchangessign.Thus,
wecanconcludethatitisexactlythispropertyofopticalgeometryembeddingsforthevacuumsphericallysymmetric
spacetimes (see [10, 3, 1]) survivesin the Kerr–Newmanspacetimes. However,photoncircular orbitsare displaced
fromtheradiicorrespondingtotheturningpointsoftheembeddingdiagrams.
Afterastraightforward,butrathertedious,discussion,thefullclassificationcanbemadeaccordingtotheproperties
oftheembeddingdiagrams(see[21]).
Examples of typical black-hole embedding diagrams of classes labeled in Stuchlík et al. [21] as BH –BH are
1 4
presentedinFigure5.
12
12 12 12
10
0.12
8
z 6
6 6 6 0.1
0.04 0.06
4 0.05
0.02
0
2 7.15 7.2 7.25 7.3
0
7 7.03 7.06 0
05 6 7 8ρ 9 10 11 05 7 9 11 05 7 7.119 7.1114 05 7 9 11
(a) (b) (c) (d) (e)
FIGURE5. (a),(b):EmbeddingdiagramoftheKerr–NewmanblackholesofthetypeBH1,constructedfora2 0.16,e2 0.16.
= =
Theringsinthe3Ddiagramrepresentphotoncircularorbits.Bothcorotating(grayspotin2Ddiagram)andcounterrotating(black
spot in 2D diagram) orbits are displaced from the throat of the diagram where the centrifugal force vanishes. This isa general
propertyoftherotatingbackgrounds.(c)–(e):EmbeddingdiagramsoftheblackholesclassesBH –BH .(c)ClassBH (a2 0.7,
2 4 2
e2 0.16);(d)classBH3(a2 0.75,e2 0.16);(e)classBH4(a2 0.81,e2 0.16).Takenfrom[21]. =
= = = = =
CONCLUDINGREMARKS
Embeddingdiagramsoftheopticalgeometrygiveanimportanttoolofvisualizationandclarificationofthedynamical
behavior of test particles moving along equatorial circular orbits: we imagine that the motion is constrained to the
surface z(ρ) (see [3]). The shape of the surface z(ρ) is directly related to the centrifugal acceleration. Within the
upwardslopingareasoftheembeddingdiagram,thecentrifugalaccelerationpointstowardsincreasingvaluesofr and
the dynamicsof test particles has an essentially Newtoniancharacter. However,within the downward sloping areas
oftheembeddingdiagrams,thecentrifugalaccelerationhasaradicallynon-Newtoniancharacterasitpointstowards
decreasingvaluesofr. Such a kindof behaviorappearswhere the diagramshavea throator a belly.Atthe turning
pointsofthediagram,thecentrifugalaccelerationvanishesandchangesitssign.
ACKNOWLEDGMENTS
ThepresentworkwassupportedbythegrantMSM4781305903.
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