Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes

0.17 MB·

by  Ronaldo S. S. Vieira

Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric
spacetimes
Ronaldo S. S. Vieira,1,2,3,∗ Wl odek Klu´zniak,2,3,† and Marek Abramowicz2,3,4,‡
1Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas,
Universidade de S˜ao Paulo, 05508-090, S˜ao Paulo, SP, Brazil
2Copernicus Astronomical Center, ul. Bartycka 18, PL-00-716, Warszawa, Poland
3Institute of Physics, Faculty of Philosophy and Science,
Silesian University in Opava, Bezruˇcovo n´am. 13, CZ-74601 Opava, Czech Republic
4Physics Department, Gothenburg University, SE-412-96 G¨oteborg, Sweden
(Dated: February 1, 2017)
Thesumofsquaredepicyclicfrequenciesofnearlycircularmotion(ωr2+ωθ2)inaxiallysymmetric
configurations of Newtonian gravity is known to depend both on the matter density and on the
7
angular velocity profile of circular orbits. It was recently found that this sum goes to zero at
1
the photon orbits of Schwarzschild and Kerr spacetimes. However, these are the only relativistic
0
configurationsforwhichsuchresultexistsintheliterature. Here,weextendtheaboveformalism in
2
order to describe the analogous relation for geodesic motion in arbitrary static, axially symmetric,
n asymptotically flatsolutionsofgeneralrelativity. Thesumofsquaredepicyclicfrequenciesisfound
Ja tovanishatphotonradiiofvacuumsolutions. Inthepresenceofmatter,weobtainthatωr2+ωθ2>0
forperturbedtimelikecirculargeodesicsontheequatorialplaneifthestrongenergyconditionholds
0 forthematter-energyfluidofspacetime;invacuum,theallowedregionfortimelikecirculargeodesic
3
motion is characterized by the inequality above. The results presented here may be of use to shed

light on general issues concerning the stability of circular orbits once they approach photon radii,
]
c mainly the ones corresponding to stable photon motion.
q
- PACSnumbers: 04.20.-q,04.20.Cv,95.30.Sf
r
g
[
I. INTRODUCTION Eq. (1), valid for Newtonian gravity, to equatorial cir-

1 cular geodesics in static, axially symmetric spacetimes.
v In Section II we summarize the formalism to obtain the
General relativity (GR) introduces many additional
3 epicyclic frequencies for nearly circular motion in static,
featureswhicharenotpresentinNewtoniangravity. For
7
axiallysymmetricspacetimes,followingthederivationby
8 instance, the radialand verticalepicyclic frequencies are
[5]. Section III contains the results of our work, namely
8 notequalinsphericallysymmetricspacetimes,afactfirst
0 pointedoutby[1]. Thedifferencebetweenthesefrequen- the relationbetween the sum ωr2+ωθ2 andthe Ricci ten-
. cies in Schwarzschild spacetime generates an innermost sor, as well as the particular case of vacuum GR space-
1
times and the relation between the allowed regions for
0 stable circular orbit (at which the radial epicyclic fre-
circular geodesics (in terms of the sign of ω2+ω2) and
7 quencyvanishes),whichdeterminestheinnerrimofthin r θ
1 accretion discs around black holes. Recently, Amster- thestrongenergyconditionforthespacetimeenergycon-
v: damski et al. [2] found outside Newtonian Maclaurin tent. We present our conclusions in Section IV.
i spheroids a minimum radius inside which there are no
X
circular orbits, a phenomenon which was believed to ex-
r ist only in GR. Related to that, Klu´zniak and Rosin´ska II. STATIC, AXIALLY SYMMETRIC
a SPACETIMES
[3] presented a relation between the sum of the squared
epicyclic frequencies ω (radial) and ω (vertical), the
r θ
density ρ of the background matter and the orbital fre- Let η =∂/∂t and ξ =∂/∂ϕ be the timelike and az-
quency Ω of the original circular orbit (see also [4]), imuthalKillingvectorfieldsofastatic,axiallysymmetric
spacetime. The line element can be written, in a coordi-
ω2+ω2 =2Ω2+4πGρ. (1) nate system adapted to these Killing vector fields, as
r θ
Theyalsofoundthatthe(formal)expressionforthesum ds2 =gttdt2+gϕϕdϕ2+grrdr2+gθθdθ2. (2)
ofthesquaredepicyclicfrequenciesgoestozeroatphoton
The metric coefficients depend only on R and θ. The
orbits of Kerr spacetime [3].
metric signature is takenas (+ – – –) and the spacetime
The purpose of the present paper is to extend
isassumedasymptoticallyflat,sog →η (Minkowski
µν µν
metric) at spatial infinity. We also impose equatorial-
plane symmetry. In particular, g =0 on the equa-
µν,θ
torial plane (defined by θ =π/2). We must stress that
∗[email protected]
†[email protected] the formalismpresentedinthis paperis validforsmooth
‡[email protected] metrics; the vertical stability criteria for circular orbits
2
in razor-thin disks were analyzed in [6, 7]. some invariant function of R , which must reduce to
µν
We adopt the conventions of [5]. In particular, the 4πGρintheNewtonianlimit. ThisscalarfunctionofR
µν
specific angular momentum ℓ=−u /u of timelike cir- would give the dependence of ω2+ω2 on the matter–
ϕ t r θ
cular geodesics on the equatorial plane is given by energy content of spacetime. Also, we know from the
ℓ2(r)=−gtt /gϕϕ, where uµ is the 4-velocity of the Schwarzschild case [3] that the squared sum of epicyclic
,r ,r
geodesic. The corresponding angular velocity Ω is given frequenciesformallyvanishesattheradiusofthecircular
by Ω2(r)=−g /g and these quantities are related photonorbit,hereaftercalledphoton radius r . Wethus
tt,r ϕϕ,r ph
by expect that the sum ω2+ω2 would involve two terms:
r θ
the first term being related to the local matter-energy
ℓ2(r)=r˜4Ω2(r), (3) content of the spacetime (i.e. the Ricci tensor) and the
second term vanishing at the photon radius. This sec-
where ond term must reduce to 2Ω2 in the Newtonian limit,
r˜2 =−gϕϕ (4) according to Eq. (1).
g Equatorial-planesymmetry implies thatthe Ricciten-
tt
sor R is diagonal at θ = π/2. Since the epicyclic fre-
µν
istheradialcoordinateinopticalgeometry[8,9]. Thera- quencies do not depend on partial derivatives of g and
rr
dial(ωr) and vertical(ωθ) epicyclic frequencies ofnearly gθθ, these terms should not appear in the corresponding
circular orbits, measured at infinity, are given by [5] combination of Ricci tensor coefficients. On the other
hand, R =4πGρ in the Newtonian limit, which im-
(g )2 tt
ω2 =− tt gtt +ℓ2(r)gϕϕ , (5) plies that this term should appear in the expression for
r 2|grr|h ,rr ,rri ω2+ω2. If we look for linear combinations of the coef-
r θ
ficients R , the curvature-like contribution must be of
µν
the form R +Ω2(r)R in order to cancel g and g
(g )2 tt ϕϕ rr θθ
ω2 =− tt gtt +ℓ2(r)gϕϕ . (6) derivatives. We see next that this is indeed the correct
θ 2|g | ,θθ ,θθ
θθ h i expression; in fact we have on the equatorial plane, by
direct computation,
In terms of dℓ2/dr, the frequency ω can be written as
r
R +Ω2(r)R = ∂ Γr +Ω2∂ Γr +
1 dℓ2(r) tt ϕϕ r tt r ϕϕ
ω2 = g . (7) (cid:16) (cid:17)
r 2r˜4grr ϕϕ,r dr + ∂θΓθtt+Ω2∂θΓθϕϕ (13)
(cid:16) (cid:17)
We may write dℓ2/dr in terms of dr˜2/dr and +2Γr Γϕ −Γt .
dΩ2(r)/dr: tt(cid:16) rϕ rt(cid:17)
Equation (6) for ω2 allows us to write
dℓ2(r) dr˜2 dΩ2(r) θ
dr =2r˜2Ω2(r) dr +r˜4 dr . (8) ωθ2 =∂θΓθtt+Ω2∂θΓθϕϕ. (14)
The other two terms of (13) can be written as
Combining Eqs. (7) and (8), we obtain
g dΩ2(r)
ωr2 = ggϕϕ,rΩ2(r)r˜12 ddr˜r2 + g2ϕgϕ,rdΩd2r(r). (9) ∂rΓrtt+Ω2∂rΓrϕϕ = 2ϕgϕr,rr dr (15)
rr rr
and
1g 1 dr˜2
2Γr Γϕ −Γt = ϕϕ,rΩ2(r) . (16)
III. RELATION BETWEEN EPICYCLIC tt rϕ rt 2 g r˜2 dr
(cid:16) (cid:17) rr
FREQUENCIES AND CURVATURE
Wecanthenseparatethe contributionofdr˜2/dr appear-
ing in (9) with the help of Eqs. (15) and (16) in two
We adopt the following conventions for the Riemann
(R ν) and Ricci (R ) tensors: terms,
αβµ µν
1g 1 dr˜2
Rαβµν =∂αΓνβµ−∂βΓναµ+ΓσβµΓνασ−ΓσαµΓνβσ (10) ωr2 = 2 gϕϕ,rΩ2(r)r˜2 dr +
rr
g dΩ2(r)
Rµν =Rαµνα, (11) +(cid:20)2Γrtt(cid:16)Γϕrϕ−Γtrt(cid:17)+ 2ϕgϕr,rr dr (cid:21). (17)
where the Christoffel symbols Γµνα are given by The curvaturetermRtt+Ω2(r)Rϕϕ is theneasilyrec-
ognized from Eq. (13) in the combination ω2+ω2, ac-
r θ
Γµ = 1gµσ g +g −g . (12) cording to Eqs. (14), (15), (16), and (17). The final ex-
να 2 σα,ν σν,α να,σ pression is written in terms of R , Ω(r) and r˜as
(cid:0) (cid:1) µν
According to the Newtonian relation(1), we expect that g 1 dr˜2
ω2+ω2 = R +Ω2(r)R + ϕϕ,rΩ2(r) . (18)
the relativistic expression for ωr2+ωθ2 will depend on r θ h tt ϕϕi 2grr r˜2 dr
3
The first term on the right-hand side is zero when 0.010
the spacetime is Ricci flat, while the second term van-
0.008
ishes at photon radii (dr˜2/dr =0, see [10]) and when
Ω2(r)=0 (the same radius at which ℓ2 =0 and the ra- 0.006
dial acceleration of a static observer is null, because
g =0 at this radius). The existence of such radius Ω2LΘ 0.004
tt,r +
whereΩ2(r)=0isacharacteristicoffewspacetimes,for Ω2r 0.002
instance the Reissner–Nordstro¨m metric in the naked- 2HM
0.000
singularity regime [11] and its generalizations [12]. This
behavior is also present in the Kehagias-Sfetsos naked -0.002
singularityspacetime[9,10]andintheno-horizonparam-
-0.004
eter region of what would be otherwise a regular black
hole spacetime [13, 14]. 2 4 6 8 10
r(cid:144)M
Equation(18)reducestotheNewtonianresult[Eq.(4)
of [3]; see also Eq. (1)] in the appropriate limit: FIG.1: Thequantityωr2+ωθ2asafunctionoftheradialcoor-
Rtt ≈4πGρ, Rϕϕ ≈0, r˜≈r, grr ≈−1, gϕϕ,r ≈−2r. dinate r for the Schwarzschild metric (solid black line). The
parameter M is the Schwarzschild mass. As we know from
[3],thisquantitygoestozeroatthephotonradiusrph =3M
(markedbyaverticalgray line intheFigure). Ourapproach
A. Vacuum (Ricci-flat) spacetimes
extends this result: the Schwarzschild metric is a particular
case of a vacuum spacetime in GR. We haveωr2+ωθ2 >0 for
In Ricci-flat spacetimes (vacuum spacetimes for gen- r>rph and ωr2+ωθ2<0 for r<rph [see Eq. (19)].
eral relativity), Eq. (18) reduces to
g 1 dr˜2
ωr2+ωθ2 = 2ϕgϕ,rΩ2(r)r˜2 dr ; (19) h=−uϕ and conservedenergy E =ut of timelike circu-
rr
largeodesics[Eq.(11)of[15]]instatic,axiallysymmetric
this expression vanishes at photon radii. This simple spacetimes with equatorial-plane symmetry allows us to
statement has tremendous consequences. First of all, we show that dℓ2/dr and dh2/dr have the same sign in al-
muststressthatinthiscaseω2+ω2 =0preciselyatpho- lowed regions for this kind of motion, since [10]
r θ
ton radii, being positive in allowed regions for circular dℓ2 1 dh2
orbits (dr˜/dr >0 and Ω2(r)>0) and – formally – neg- = 1−v2 , (20)
ative in forbiden regions (dr˜/dr <0 or Ω2(r)<0). In dr E2(cid:2) (cid:3) dr
particular,there are no timelike circular geodesicsin the wherevisthespeedoftheparticleasmeasuredbyalocal
regionwhereω2+ω2 ≤0. Thisistheproperextensionof static observer. We can also show from these arguments
r θ
the Newtonian result [3] with ρ=0, Eq. (1), to geodesic that [10, 15]
motion in static, Ricci-flat spacetimes (in particular to
dr˜2 −1
test-particle motion in vacuum GR). We exemplify the h2(r)=r˜4g . (21)
tt,r
behavior of ω2+ω2 in vacuum spacetimes by plotting it (cid:16) dr (cid:17)
r θ
as a function of radius for the Schwarzschild metric in Therefore, since h2 →∞ when r approaches the photon
Fig. 1. radius, ℓ2 grows as r →r . Moreover,circular geodesic
ph
Apart from the degenerate case in which the expres- motionisforbiddenifdr˜2/dr <0(theregioninrbetween
sions for both ω2 and ω2 are zero at the photon radius, astable andanunstable photonorbit),since inthis case
r θ
the generalsituationisthatoneofthesquared“frequen- h2 <0.
cies”is positiveandthe other isnegative. By continuity, However, the case of radially stable photon orbits is
there will be a finite region around the photon radius in subtler and more interesting (in this case timelike cir-
which one of the squared frequencies is negative. There- cular motion is allowed only for r < r ). Although in
ph
fore,inRicci-flatspacetimestherewillalwaysbearegion this situation ℓ2(r) grows with r until reaching from the
of unstable timelike circular geodesics between the pho- left the photon radius, correspondingto ω2 >0, the fact
r
tonradiusandtheradiusoftheclosestmarginallystable that the expression for ω2+ω2 vanishes at photon or-
r θ
orbit; the region of stability will never reach the photon bits means that the value of ω2 must become negative
θ
radius. as the circular orbit approaches the photon orbit. The
This phenomenon also follows from the behavior of orbits become vertically unstable as they approach this
ℓ2(r) in the case of unstable photon orbits (in this case photonradius (althoughthey areradiallystable because
timelike circular motion is allowed only for r > r ), of Rayleigh’s criterion [15]). This result means that off-
ph
sinceℓ2(r)growsasr approachesthephotonradiusfrom equatorial motion has a fundamental importance to the
the right, which corresponds to radial instability of the stability analysis of circular equatorial geodesic motion
corresponding circular orbits [5]. Let us remark that invacuumspacetimes,sheddinglightontherecentques-
the relation between the conserved angular momentum tions raised in the literature concerning the behavior of
4
circular geodesics in vacuum multipole solutions of Ein- 0.015
stein’s equations [16]. In particular, for the mentioned
multipole solutions, the radial stability analysis is not
sufficient to classify circular geodesic motion in regions 0.010
nearthecentralcompactobjectwhenradiallystablepho-
ton orbits are present. Therefore, even if circular orbits Ω2LΘ
+
areradiallystablein this region,they willeventuallybe- Ω2r 0.005
come vertically unstable as r grows in the direction of 2HM
the photon orbit.
0.000
The Ricci-flat condition is in fact too stringent. It is
notnecessarythatthespacetimeisgloballyRicciflat;all
theaboveargumentsarealsovalidlocallyundertheonly
-0.005
requirement that R =0 in a radial interval containing 2 4 6 8 10
µν r(cid:144)M
the photon radius.
FIG.2: Thequantityωr2+ωθ2asafunctionoftheradialcoor-
dinater fortheReissner-Nordstr¨ommetric(solid blackline).
B. Background matter and energy conditions Weadoptedq=Q/M =0.8. Theverticalgraylinerepresents
the photon radius rph =2.485M. We see that ωr2+ωθ2 >0
If R +Ω2(r)R 6=0 at the photon radius, there is foreverytimelikecirculargeodesic,aspredictedfromEq.(18)
tt ϕϕ
notaclearrelationanymorebetweenthe signofω2+ω2 since the Reissner-Nordstr¨om metric satisfies the strong en-
r θ ergycondition,Eq.(22). Thereisalsoaregioninsidethepho-
and the allowed regions for timelike equatorial circu-
ton radius (where circular motion is not allowed) for which
lar geodesics. In an arbitrary static, axially symmetric
theformal expression ωr2+ωθ2 is positive.
spacetime, there is no “first-principles” argument which
prohibits the existence of (unstable) circular timelike
geodesics with ω2+ω2 <0.
r θ
We might conclude that this is a purely general rela- background matter guarantees that ω2 +ω2 > 0 in the
r θ
tivisticeffect,sinceitdoesnotappearinNewtoniangrav- allowed region for timelike circular motion. As a con-
ity[3]. Thisapparentdifferencecomesfromthefactthat sequence the strong energy condition is, in our context,
thedensityofmatterisalwayspositive,ρ≥0. Anappro- the appropriate relativistic extension to the positivity of
priategeneralizationofthisrequirementmustbeimposed massinNewtoniangravity. Thebehaviorofω2+ω2 asa
r θ
in our case, in order to maintain the well-behaved prop- functionoftheradialcoordinateisillustratedinFig.2for
erties of the energy-momentum tensor. The established the case of the Reissner-Nordstro¨m metric with charge-
conditions for regular behaviorof the matter and energy to-mass parameter q =0.8.
content of spacetime are the energy conditions [17]. We Moreover,ifR +Ω2(r)R >0atthephotonradius,
tt ϕϕ
find here that the strong energy condition [17] for the we do not have anymore the characterization of the al-
background matter, lowed regions for circular motion in terms of ω2+ω2,
r θ
presentedinthe formerSectionforRicci-flatspacetimes.
RµνXµXν ≥0 for all timelike vectors Xµ, (22) Infact,inthiscaseω2+ω2 >0atthephotonradius,and
r θ
thereforeitispossibletohaveaninnerregionofstability
implies that all circular timelike geodesics have
which reaches a radially stable photon orbit (this situ-
ω2+ω2 >0 for their perturbed motion. In fact, writing
r θ ation indeed happens for many naked singularity space-
the tangent unit vector to the aforementioned geodesics
times [10–12] and spacetimes without horizons [13, 14]).
as
On the other hand, although ω2+ω2 > 0 in the region
r θ
uα =A ηα+Ω(r)ξα (23) outside unstable photon orbits, the corresponding circu-
largeodesicsareradiallyunstableaccordingtoRayleigh’s
(cid:2) (cid:3)
with A=A(r)>0, we have the equality criterion. Therefore, Eq. (18) does not tell us about the
stability of circular orbits without additional assump-
1
R +Ω2(r)R = R uµuν. (24) tions about the spacetime structure.
tt ϕϕ A2 µν Precise conditions for the existence of an instability
The“circularvectorfield”uα istimelikeinregionswhere region between the photon radius and the radius of the
marginally stable orbit can be obtained when spherical
r˜ increases with r (dr˜/dr > 0), spacelike where r˜ de-
symmetry is assumed. In this case ω2 =Ω2, which is
creases with r (dr˜/dr < 0) and null at photon orbits. θ
We have then the following picture: if R uµuν ≥0 for positiveinallowedregionsforcirculartimelikegeodesics.
µν We can therefore write Eq. (18) as an equation for ω2:
all timelike circular vector fields of the form (23), then r
therearenotimelikecirculargeodesicswhoseexpressions g 1 dr˜2
forthe “epicyclicfrequencies”satisfyωr2+ωθ2 ≤0. Since ωr2 = Rtt+Ω2(r) Rϕϕ−1 + 2ϕgϕ,rΩ2(r)r˜2 dr . (25)
the strong energy condition (22) implies the above re- h (cid:0) (cid:1)i rr
sult, we obtain that the strong energy condition for the The condition to guarantee that the stability region
5
reaches the photon orbit is therefore IfR 6=0,itisalsopossibletofindarelationbetween
µν
thesignofω2+ω2 andtheallowedregionsforequatorial
R +Ω2(r) R −1)>0 (26) r θ
tt ϕϕ circular orbits, if the strong energy condition is satisfied
(cid:0)
at the photon radius r , since from Eq. (25) it forthespacetimematter-energycontent. Namely,inthis
ph
is equivalent to have ω2 >0 at the photon ra- case ω2+ω2 >0 in allowed regions for timelike circular
r r θ
dius. If R +Ω2(r) R −1)<0 at r , then there geodesics. Moreover,the formalismpresentedhere is the
tt ϕϕ ph
will be an instabilit(cid:0)y region between rph and the relativisticgeneralizationoftheanalogousresultinNew-
next marginally stable circular geodesic, whereas if tonian gravity [3, 4] for static, axially symmetric config-
R +Ω2(r) R −1)=0 at r the criterion is incon- urations. The dependence of ω2+ω2 on the Ricci ten-
tt ϕϕ ph r θ
clusive. The(cid:0)aboveinequalitycanalsobewritteninterms sorshowsthepropergeneralizationofthematterdensity
of spacetime invariants as termintheNewtonianequationforourcase[seeEqs.(1)
and (18)], the strong energy condition being the correct
Rµν −ξµξν uµuν >0, (27) relativistic generalization of the positivity of mass (as it
h i
wasrecentlyfoundin[7]forrelativisticrazor-thindisks).
evaluated at the photon radius.
Thus, as in many other qualitative results for timelike
According to Eq. (25) and Rayleigh’s criterion, we ex-
geodesic motionin Lorentzianmanifolds [17], the results
pect R +Ω2(r) R −1)≤0 atradiicorrespondingto
tt ϕϕ obtainedherehaveadeeprelationwithenergyconditions
radially unstable(cid:0)photon orbits in spherically symmetric
for the backgroundspacetime.
spacetimes. Bythesameargument,theradialregionjust
Modified theories of gravity do not have, necessarily,
insideastablephotonorbitcorrespondstostabletimelike
circulargeodesics,andthusR +Ω2(r) R −1)≥0at (22) as an energy condition; the properties of geodesic
tt ϕϕ
motionmaynotbedirectlyconnectedtothepropertiesof
radii corresponding to radially stable ph(cid:0)oton orbits.
matterinthese theories. However,eveninthese theories
the above arguments are valid, since they only depend
IV. CONCLUSIONS on the properties of the metric and of the Ricci tensor,
which are geometrically well defined quantities.
Circular orbits have been studied since the beginnings A well known result relates the radial stability of the
of the astrophysical applications of Newtonian gravity photonorbitandthe propertiesofnearbytimelike circu-
and general relativity. Nevertheless, it was only re- lar geodesics. Let rph be the radius of the photon orbit,
cently that their qualitative properties in general rela- and assume this orbit is not marginally stable. From
tivity received proper attention. We presented in this Rayleigh’sstability criterionforcirculargeodesics[5,15]
paperclosed-formexpressionsforthe sumofthe squared we have the following correspondence between the prop-
epicyclicfrequenciesofperturbedtimelikeequatorialcir- erties of photon orbits and of nearby timelike circular
cular geodesics in static, axially symmetric, asymptot- geodesics: If the photon orbit is radially unstable, time-
ically flat spacetimes; these expressions are written in likecirculargeodesicsareallowedonlyforr >rphandare
termsoftheRiccitensorandofaquantitywhichvanishes radially unstable in a neighbourhood of rph. If the pho-
at photon orbits. For Ricci-flat spacetimes, the present tonorbitisradiallystable,timelikecirculargeodesicsare
framework establishes the existence of an instability re- allowed only for r <rph and are radially stable locally.
gion around each photon radius. Although this result This behavior is seen in many spacetimes with photon
is a consequence of Rayleigh’s criterion near the radius orbits and no horizon[10–14]. Our findings demonstrate
of a radially unstable photon orbit, the same criterion that the full linear stability analysis near stable photon
implies that circular geodesics are radially stable in the orbits relies heavily on the properties of off-equatorial
inner region near the radius of a radially stable photon motion and, by Eqs. (18) and (24), on the local proper-
orbit. The mentioned regionof instability means, in this tiesoftheRiccitensor. Ingeneralrelativity,thepresence
case, that the formula for ω2 is negative in this inner of a region of stability up to the stable photon orbit is
θ
region, in the vicinity of the photon radius. Therefore, therefore strongly dependent on the local properties of
althoughω2 is positive in this region,motion is unstable matter.
r
under off-equatorialperturbations if we getclose enough It was shown in [3] that the sum ω2+ω2 vanishes at
r θ
to the radius of the stable photon orbit. This simple the photon orbits of Kerr spacetime. The question of
statement has a profound impact on the analysis of cir- whether the formalism presented here extends to rotat-
culargeodesicsinquasi-spherical,multipolarvacuumso- ing,stationaryspacetimesremainsanopenproblem;our
lutions of Einstein’s equations [16]: radial perturbations results are a starting point to tackle this more general
are not sufficient to analyze circular motion near an ex- case. Based on the results for Kerr spacetime [3], it is
isting inner photon orbit. If off-equatorial perturbations reasonable to suppose that the quantity ω2 +ω2 might
r θ
are also considered, these geodesics will eventually be- vanish at photon radii of stationary vacuum spacetimes
come unstable before reaching the inner photon radius (for both prograde and retrograde orbits), a hypothesis
(andthereforeahypotheticthinaccretiondiscinthisin- that deserves a thorough investigation. We also conjec-
nerregionwillhavea‘gap’betweenitsmarginallystable turethatthestrongenergyconditionshouldbesufficient
circular orbit and the inner photon radius). toensurethepositivityofω2+ω2 forallcirculartimelike
r θ
6
geodesics in stationary, axially symmetric spacetimes. No.2013/10/M/ST9/00729. M.A. wassupported by the
PolishNCN GrantNo.2015/19/B/ST9/01099. R.S.S.V.
thanks the financial support from “S˜ao Paulo Research
Acknowledgments Foundation” (FAPESP), Grants No. 2010/00487-9,
No. 2013/01001-0, and No. 2015/10577-9, and the
W.K. acknowledges support from the Pol- hospitality of Nicolaus Copernicus Astronomical Center,
ish NCN Grants No. 2013/08/A/ST9/00795 and where most of this researchwas developed.
[1] M. F. Shirokov, General Relativity http://adsabs.harvard.edu/abs/2015GReGr..47..132G.
and Gravitation 4, 131 (1973), URL [10] R. S. S. Vieira, J. Schee, W. Klu´zniak,
http://adsabs.harvard.edu/abs/1973GReGr...4..131S. Z. Stuchl´ık, and M. Abramowicz, Phys.
[2] P. Amsterdamski, T. Bulik, D. Gondek- Rev. D 90, 024035 (2014), 1311.5820, URL
Rosin´ska, and W. Klu´zniak, Astron. Astro- http://adsabs.harvard.edu/abs/2014PhRvD..90b4035V.
phys. 381, L21 (2002), astro-ph/0012547, URL [11] D. Pugliese, H. Quevedo, and R. Ruffini, Phys.
http://adsabs.harvard.edu/abs/2002A%26A...381L..21A. Rev. D 83, 024021 (2011), 1012.5411, URL
[3] W. Klu´zniak and D. Rosin´ska, Mon. Not. http://adsabs.harvard.edu/abs/2011PhRvD..83b4021P.
R. Astron. Soc. 434, 2825 (2013), URL [12] Z. Stuchl´ık and S. Hled´ık, Acta Physica
http://adsabs.harvard.edu/abs/2013MNRAS.434.2825K. Slovaca 52, 363 (2002), 0803.2685, URL
[4] J. Binney and S. Tremaine, Galactic Dynamics (Prince- http://adsabs.harvard.edu/abs/2008arXiv0803.2685S.
ton Univ.Press, Princeton, NJ, 2008), 2nd ed. [13] A. Garc´ıa, E. Hackmann, J. Kunz, C. L¨ammerzahl,
[5] M. A. Abramowicz and W. Klu´zniak, Astrophys. and A. Mac´ıas, Journal of Mathematical
Space Sci. 300, 127 (2005), astro-ph/0411709, URL Physics 56, 032501 (2015), 1306.2549, URL
http://adsabs.harvard.edu/abs/2005Ap%26SS.300..127A. http://adsabs.harvard.edu/abs/2015JMP....56c2501G.
[6] R. S. S. Vieira and J. Ramos-Caro, Ce- [14] K. Boshkayev, E. Gasper´ın, A. C. Guti´errez-
lestial Mechanics and Dynamical Astron- Pin˜eres, H. Quevedo, and S. Toktarbay, Phys.
omy 126, 483 (2016), 1606.06349, URL Rev. D 93, 024024 (2016), 1509.03827, URL
http://dx.doi.org/10.1007/s10569-016-9705-0. http://adsabs.harvard.edu/abs/2016PhRvD..93b4024B.
[7] R. S. S. Vieira, J. Ramos-Caro, and A. Saa, [15] P. S. Letelier, Phys. Rev. D 68,
Phys. Rev. D 94, 104016 (2016), URL 104002 (2003), gr-qc/0309033, URL
http://link.aps.org/doi/10.1103/PhysRevD.94.104016. http://adsabs.harvard.edu/abs/2003PhRvD..68j4002L.
[8] M. A. Abramowicz, Mon. Not. R. As- [16] J. L. Hernandez-Pastora, L. Herrera, and J. Os-
tron. Soc. 245, 733 (1990), URL pino, Phys. Rev. D 88, 064041 (2013), 1309.2455, URL
http://adsabs.harvard.edu/abs/1990MNRAS.245..733A. http://adsabs.harvard.edu/abs/2013PhRvD..88f4041H.
[9] K.Goluchova´,K.Kulczycki,R.S.S.Vieira,Z.Stuchl´ık, [17] R. Wald, General Relativity (The University of Chicago
W. Klu´zniak, and M. Abramowicz, General Relativ- Press, Chicago, IL, 1984).
ity and Gravitation 47, 132 (2015), 1511.01345, URL