Overspinning a Kerr black hole: the effect of self-force

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by  Marta Colleoni

Overspinning a Kerr black hole: the effect of self-force
Marta Colleoni and Leor Barack
Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK
(Dated: January 30, 2015)
We study the scenario in which a massive particle is thrown into a rapidly rotating Kerr black
holeinanattempttospinitupbeyonditsextremallimit,challengingweakcosmiccensorship. We
work in black-hole perturbation theory, and focus on non-spinning, uncharged particles sent in on
equatorialorbits. Wefirstidentifythecompleteparameter-spaceregioninwhichoverspinningoccurs
when back-reaction effects from the particle’s self-gravity are ignored. We find, in particular, that
overspinningcanbeachievedonlywithparticlessentinfrominfinity. Gravitationalself-forceeffects
maypreventoverspinningbyradiatingawayasufficientamountoftheparticle’sangularmomentum
(“dissipativeeffect”),and/orbyincreasingtheeffectivecentrifugalrepulsion,sothatparticleswith
suitable parameters never get captured (“conservative effect”). We analyze the full effect of the
self-force, thereby completing previous studies by Jacobson and Sotiriou (who neglected the self-
5 force) and by Barausse, Cardoso and Khanna (who considered the dissipative effect on a subset of
1 orbits). Our main result is an inequality, involving certain self-force quantities, which describes a
0 necessary and sufficient condition for the overspinning scenario to be overruled. This “censorship”
2 conditionisformulatedonacertainone-parameterfamilyofgeodesicsinanextremalKerrgeometry.
We find that the censorship condition is insensitive to the dissipative effect (within the first-order
n
self-force approximation used here), except for a subset of perfectly fine-tuned orbits, for which
a
J a separate censorship condition is derived. We do not obtain here the self-force input needed to
evaluate either of our two conditions, but discuss the prospects for producing the necessary data
9
using state-of-the-art numerical codes.
2

]
c I. INTRODUCTION true also for a spinning test particle dropped from rest
q at infinity along the symmetry axis of an extremal Kerr
-
r black hole, with its spin aligned along the axis. In this
g Thecosmiccensorshipconjecture[1]hasovertheyears case,itistherepulsionforcefromspin-spincouplingthat
[ becomeacornerstoneofclassicalgeneralrelativity. Inits
prevents suitable particles from ever entering the black
weakversionitstates,inessence,thatcurvaturesingular-
1 hole.
ities arising in solutions to the Einstein’s field equations
v However, later work has demonstrated that over-
must be cloaked behind event horizons, so that they are
0 extremality is achievable when the initial black hole is
3 prevented from being in causal contact with distant ob-
taken to be nearly extremal—if back-reaction effects on
3 servers. Despite being strongly motivated on physical
theparticle’strajectoryareignored. Thiswasfirstshown
7 grounds, the conjecture’s precise extent of validity re-
by Hubeny [5] for a nearly extremal Reissner-Nordstr¨om
0 mains unclear. A notable counterexample involves finely
. black hole, and more recently by Jacobson and Sotiriou
1 tuned initial conditions [2]. The formulation of the con-
[6]foranearlyextremalKerrblackhole(“overcharging”
0 jecture may be refined to exclude such examples [3].
and “overspinning” scenarios, respectively). The nearly
5
1 In a 1974 paper [4] Wald proposed a simple but pow- extremalKerr-Newmancasewassubsequentlystudiedin
: erful framework for testing weak cosmic censorship, us- Ref. [7]. In all cases, all orbits identified as capable of
v ing the gedanken experiment of a particle thrown into a driving the black hole beyond the extremal limit lie very
i
X Kerr–Newman black hole. If parameters can be chosen close, in the relevant parameter space, to the separatrix
r such that the post-capture mass Mf, charge Qf and spin between orbits that are captured by the black hole and
a Jf satisfy Mf2 < (Jf/Mf)2 +Q2f, then a naked singular- ones that are scattered off it. In Hubeny’s analysis of
ity would presumably form, in direct violation of weak a radially falling electric charge, electrostatic repulsion
censorship. Whether the equations of classical general only marginally fails to prevent the particle from falling
relativitypermitsuchaprocesshassincebeensubjectof intothehole: Theparticle’sradialvelocityuponcrossing
much investigation. It is usually assumed that the par- (whatwouldhavebeen)theeventhorizonisproportional
ticle’s energy and electric charge are much smaller than to the ratio η˜(cid:28)1 between the particle’s energy and the
those of the black hole, which then places the problem black hole’s mass. The amount of post-capture excess
within the realm of black-hole perturbation theory. charge, Q −M , isfoundtobequadraticin η˜. Similarly,
f f
In [4] Wald showed that the over-extremality sce- in Ref. [6]’s analysis of equatorial-plane captures, over-
nario is ruled out when the configuration is that of a spinningparticlesclearthepeakoftheeffectivepotential
pointlike test particle captured by an extremal Kerr– barrier with radial velocities ∝ η˜, and the post-capture
Newman black hole. Electrostatic and centrifugal repul- excess spin, J −M2, is quadratic in η˜.
f f
sion, he showed, would prevent a particle carrying suf- This suggests strongly that back-reaction effects can-
ficient charge and/or angular momentum from entering not be ignored and may well change the outcome of the
the black hole. The same conclusion was shown to hold gedanken experiment. Heuristically, effects of the (elec-
2
tromagnetic and/or gravitational) self-force enter the showed that dissipation averts the overspinning for some
analysis in two ways. First, the dissipative piece of the but not all of Jacobson–Sotiriou’s orbits. For sufficiently
self-force continually removes some of the particle’s en- smallη˜,thedissipativeeffectisalwaysnegligibleandcan-
ergy and angular momentum, sending them to infinity not prevent overspinning. This result highlights the im-
and down the event horizon in gravitational waves. In portanceofaccountingforthefulleffectofGSF.Toreach
the Kerr case, dissipative effects may accumulate as the a definitive conclusion necessitates an actual calculation
particle“hovers”abovethepeakoftheeffectivepotential of the full local GSF acting on the captured particles.
onanearlycircularorbit. Second,theconservativepiece Inthepastfewyears,rigorousmethodsforGSFcalcu-
of the self-force might supply just the right amount of lationsinKerrspacetimehaveadvancedenoughtoallow
additionalrepulsiveforcetopreventwould-beovercharg- a more systematic and complete treatment of the over-
ing/overspinning particles from ever entering the black spinning problem. The program initiated with this pa-
hole. For particles sent in from infinity in the Kerr case, per revisits the problem from this new vantage point. It
thissecondeffectmaybeformulatedintermsofashiftin seeks to obtain a more conclusive answer to the ques-
the critical impact parameter for capture: If the gravita- tion of whether it is indeed the self-force that provides
tional self-force (GSF) shifts the critical impact parame- the mechanism by which black holes protect themselves
ter inward by a sufficient amount (for a given energy-at- from being overspun.
infinity), then would-be overspinning particles may end Our current paper lays the necessary groundwork.
up being scattered away rather than captured. Concentrating on equatorial orbits, we first identify the
There have been several recent attempts to quan- complete“window”intheparameterspaceinwhichover-
tify the effect of back-reaction in the problem. Focus- spinningoccursiftheGSFisignored. Wethenformulate
ing on the Reissner-Nordstr¨om case, Isoyama, Sago and a condition for this window to be eliminated by the ef-
Tanaka [8] argued that the full effect can be properly fect of the full GSF. The condition takes the form of
taken into account by considering the quasi-equilibrium an inequality that is required to hold for each member
configuration of a charged particle placed precisely on of a certain 1-parameter family of geodesics, and it in-
the capture–scatter separatrix. An exact solution is volves the GSF calculated along such orbits. Here we
knownforthisconfiguration—thestaticdoubleReissner- do not obtain the necessary GSF data, but we discuss
Nordstr¨om spacetime—and the authors calculated that methods for computing it (numerically) using existing
its total energy is always greater than its total charge. codes. With collaborators we have began work to obtain
They have also established that radiative losses during the GSF data, and we intend to present the results in a
thefinalplungearenegligible,henceconcludingthat(un- follow-up paper.
der the assumption that the true capture system does The rest of this introduction summarizes our analysis
indeed go through a quasi-equilibrium state) the final (alsoinrelationtopreviouswork)anddescribesitsmain
configuration cannot be a naked singularity. results.
Inalaterwork,Zimmerman,VegaandPoisson[9]took
up the challenge of directly calculating the charged par-
ticle’s trajectory including the full effect of the electro- A. This work: overview and results
magnetic self-force. Analyzing numerically a large sam-
ple of orbits within the domain identified by Hubeny, Jacobson and Sotiriou [6] assumed that overspinning
the authors found no example of successful overcharg- occurs if two conditions are met: (i) the geodesic trajec-
ing: All particles with a combination of charge and en- tory of the test particle is timelike at the horizon, and
ergy suitable for overcharging the black hole were found (ii) J > M2. The first condition is very lax. It allows
f f
to be repelled before reaching the horizon. This anal- for low-energy orbits that are deeply bound to the black
ysis, however, neglected the potentially important ef- hole and confined to the immediate neighborhood of the
fect of back-reaction from the gravitational perturba- horizon. The physics of such orbits becomes very sub-
tion sourced by the particle’s electromagnetic energy- tle, especially when self-gravity and finite-size effects are
momentum. A complete analysis would require calcu- included [consider that deeply-bound orbits below the
lation of the corresponding GSF, but techniques for cal- innermost stable circular orbit (ISCO) plunge into the
culating self-forces in the coupled problem are only now black hole within an amount of proper time that shrinks
starting to be developed [10, 11]. to zero in the extremal limit [14]], so one would prefer-
In that respect, the Kerr setup provides a cleaner en- ably avoid such orbits as candidates (see, however, Hod
vironment, in which the perturbative problem is purely [15] for a heuristic treatment). Jacobson and Sotiriou
gravitational (at the obvious cost of abandoning spheri- acknowledge this issue, and to address it they supple-
cal symmetry). Barausse, Cardoso and Khanna [12, 13] menttheiranalysiswithtwospecificnumericalexamples
studied the dissipative GSF effect in the Kerr over- of overspinning orbits that are sent in from afar. They
spinningproblem,focussingonultra-relativisticparticles stop short of determining the full range of overspinning
on equatorial orbits. Using analytic arguments backed orbits when deeply bound orbits are disallowed.
by a numerical calculation of the energy and angular- Ourcaptureconditionwillbemorestringent,andmore
momentum carried away in gravitational waves, they inthespiritofRef.[9]: Wewillsendourparticleinfrom
3
“sufficiently far” (this condition will be made precise in the extremal Kerr limit. Overspinning can be ruled out
thenextsection), anddeemit“captured”ifithasnoin- if and only if the condition is met for each member of
nerradialturningpointoutsidetheblackhole. Thus,for this one-parameter family. The condition for fine-tuned
alegitimatecapturewedemandthattheparticle“clears” orbitsrequires, inadditiontotheconservativeGSF,also
the peak of the effective potential on its inward journey. knowledge of the fluxes of energy and angular momen-
In Sec. II we revisit the overspinning problem in the tum radiated to infinity by particles on unstable circular
geodesicapproximation. Weidentifythepreciseregionin orbits, in the extremal Kerr limit. Overspinning can be
theparameterspaceofequatorialorbitsaroundanearly- ruledoutifandonlyiftheconditionismetforanyvalues
extremalKerrblackhole(excludingdeeplyboundorbits) of the initial and final energies.
where overspinning occurs, and give analytic expressions In Sec. V we propose an alternative form of the over-
for the boundaries of that region. The overspinning win- spinningconditions,basedontheframeworkofthe“first
dow is illustrated in Figs. 2 and 3. Perhaps unexpect- law of binary black-hole mechanics”, as recently applied
edly, we find that only particles sent in from infinity are to orbits in Kerr [16]. The alternative form, given (for
capable of overspinning the black hole. This fact has the generic case) in Eq. (104), involves only perturba-
somehowgoneunnoticedinpreviouswork,tothebestof tive quantities calculated along (unstable) circular or-
our knowledge. We find that for any given value of the bits, which should be more easily computable with ex-
particle’s energy at infinity, there exists an open range isting GSF codes. This simplified formulation relies on
oforbitalangularmomentaandparticle’srestmassesfor explicit expressions, given in [16], for the ADM-like en-
whichoverspinningoccurs. Thatonlyorbitscomingfrom ergy and angular momentum of circular-orbit configu-
infinity are potential overspinniners is somewhat fortu- rations in Kerr, including leading-order self-interaction
itous,becauseforsuchorbitsitisstraightforwardtoiden- terms. Theunderlyingtheoreticalframeworkisyettobe
tify the system’s total [Arnowitt–Deser–Misner (ADM)] firmly established and tested, however, so we regard the
energy and angular momentum even when GSF effects simplifiedcondition(104)assomewhatlessrigorousthan
are included. our direct condition (69). Our stance is that it would be
desirableto evaluate both formsof thecondition, forthe
WethenturntoanalyzetheGSFeffect. InSec.IIIwe
sake of establishing confidence in the result.
first review essential results from GSF theory, and then
In Sec. VI we discuss the numerical input required for
discussthedeterminationofthe“criticalorbit”thatsep-
evaluatingouroverspinningconditions,andtheprospects
arates (in the relevant parameter space) between plung-
for obtaining it through adaptation of existing codes.
ing and scattered orbits. We do this in two steps. First,
Evaluation of the direct conditions (69) and (81) in-
we ignore the dissipative piece of the GSF, and calculate
volves GSF calculations along unbound orbits on a
the correction due to the conservative GSF to the criti-
nearly-extremal Kerr background, which has not been
cal value of the angular momentum for a fixed value of
attempted so far. However, we think the basic computa-
the energy-at-infinity. Then we restore dissipation and
tionalinfrastructureforsuchcalculationsiswellinplace.
consider its consequences. Under the full GSF, all criti-
cal orbits merge into a “global attractor” that takes the Section VII summarizes our results and speculates on
system adiabatically along a sequence of quasi-circular whatanumericalevaluationofourcensorshipconditions
unstable orbits ending at the ISCO, before plunging into might yield. The Appendices contain some of the details
the black hole. By fine-tuning the initial value of the of calculations done in Secs. III and IV: a derivation of
angular momentum (for a given initial energy), an orbit the ADM energy and angular momentum for the system
can be made to evolve arbitrary far along the global at- under consideration; a calculation of the GSF-induced
tractor. Wemakeaformaldistinctionbetween“generic” shift in the critical value of the angular momentum; and
and “fine-tuned” captured orbits, based on how the dif- an evaluation of radiative loses during the final plunge
ference between the initial and final values of the par- into the black hole.
ticle’s specific energy scales with the particle’s mass µ Throughout this paper we set G = c = 1 and use the
(in a procedure whereby µ is taken to zero while holding metric signature (−,+,+,+).
fixedtheinitialspecificenergyandangularmomentum).
Genericorbitsareonesforwhichthatdifferencevanishes
for µ → 0 (this includes, for example, all of the orbits II. OVERSPINNING ORBITS IN THE
considered in Refs. [12, 13]); fine-tuned orbits are ones GEODESIC APPROXIMATION
for which the difference does not vanish even for µ→0.
With this preparatory work in place, we move on, in Our initial configuration features a Kerr black hole of
Sec.IV,toobtaintheoverspinningconditionasmodified mass M and angular momentum J = aM < M2. A
by the full GSF. The end result are two inequalities, one pointlike test particle of rest mass µ (cid:28) M is sent in on
for generic orbits [Eq. (69)] and another for fine-tuned a geodesic of the background Kerr geometry. As in [6],
ones [Eq. (81)], which describe conditions for overspin- we restrict attention to prograde orbits in the equatorial
ningtobeavertedundertheeffectofthefullGSF.Inthe plane, so that the orbital angular momentum is aligned
genericcase,theconditioninvolvesonlytheconservative with the spin of the black hole. (Intuitively, this con-
piece of the GSF, evaluated along critical geodesics in figuration seems most favourable for a successful over-
4
spinning.) We denote the particle’s specific energy and Substituting from Eq. (4) and solving for E and L in
angular momentum by E and L, respectively; these are terms of the circular-orbit radius, r = R, gives E =
constants of the geodesic motion. For the geodesic ap- E (R) and L=L (R), with
c c
proximation to make sense, we must assume µE (cid:28) M
and µL (cid:28) J. Then, clearly, overspinning could only be 1−2R˜−1+a˜R˜−3/2
E (R)= , (6)
possible,inprinciple,iftheblackholeisnearlyextremal. c (cid:112)
1−3R˜−1+2a˜R˜−3/2
We write
R˜1/2(1−2a˜R˜−3/2+a˜2R˜−2)
L˜ (R)= . (7)
a/M =1−(cid:15)2, (1) c (cid:112)1−3R˜−1+2a˜R˜−3/2
where (cid:15)(cid:28)1.1 Here an overtilde denotes a-dimensionalization using M,
Belowwestudytheoverspinningscenariointheabove i.e.,R˜ :=R/M,a˜:=a/M andL˜ :=L/M;weshalladopt
setup,butwebeginwithasurveyofsomeessentialprop- this notation throughout the rest of the paper. Timelike
ertiesoftimelikeequatorialgeodesicsoftheKerrmetric. circular orbits exist only for R > R (a), the radius of
ph
a photon’s unstable circular orbit (“light ring”). R (a)
ph
is the (unique) root of 1−3R˜−1+2a˜R˜−3/2 greater than
A. Relevant results for Kerr geodesics theeventhorizon’sradius, R˜ (a)=1+(1−a˜2)1/2. The
eh
angularvelocityΩ:=uφ/utofanycirculargeodesicorbit
Let uα denote the particle’s four-velocity. In Boyer- reads
Lindquist coordinates {t,r,θ,φ} we have uθ ≡0, and
Ω˜(R)=(a˜+R˜3/2)−1. (8)
u˙ =0, u˙ =0, (2)
t φ
The number of stationary points of V and their loca-
+
where an overdot denotes differentiation with respect to tion depend on L. There are none outside the black hole
propertime. Thetwoequalitiesexpresstheconservation whenLisbelowacertaincriticalvalueLisco(a),andthere
of energy E = −ξ(αt)uα = −ut and angular momentum aretwoforL>Lisco(a): amaximumrepresentinganun-
L = ξα u = u , where ξα := ∂α and ξα := ∂α are stable circular orbit, and, further out, a minimum rep-
(φ) α φ (t) t (φ) φ resenting a stable one. The critical value L (a) marks
Killing vectors associated with the time-translation and isco
theinnermoststablecircularorbit(ISCO).Itisgivenby
rotational symmetries of the Kerr background. The pair
L = L (R ), where the ISCO radius R is found
{E,L} parametrizes the family of equatorial geodesics isco c isco isco
by solving Eqs. (5) simultaneously with ∂2V (L,r) = 0.
(up to initial conditions). r +
TheISCOmayalsobesaidtorepresenttheouterbound-
The normalization u uα = −1 now gives the radial
α ary of the region of unstable circular orbits.
equation of motion, which we write in the form
The radii of unstable circular geodesic orbits span the
r˙2 =B(r)[E−V−(L,r)][E−V+(L,r)]. (3) intervalRph(a)<R<Risco(a). This1-parameterfamily
oforbitswillfeaturedominantlyinouranalysis, because
HereristheBoyer-Lindquistradiusoftheorbit,B(r):= it defines the capture–scatter threshold where much of
1+a2(r+2M)/r3, and (for MaL(cid:54)=0) the relevant physics occurs. Members of the family may
be parametrized by either E or L, both being mono-
(cid:32) (cid:114) (cid:33) tonically decreasing functions of R between R (where
2MaL Br3[L2(r−2M)+r∆] ph
V±(L,r):= Br3 1± 1+ 4M2a2L2 , E,L → ∞) and Risco for any a˜ < 1. This monotonicity
can be readily established from Eqs. (6) and (7). Hence,
(4) the radius R itself is also a valid parameter.
with ∆ := r2−2Mr+a2. For prograde orbits, the po-
To each unstable circular orbit there correspond non-
tential V is manifestly negative definite, so the factor
− circular homoclinic-like geodesic orbits [17] that join the
B(r)(E −V ) in Eq. (3) is manifestly positive definite.
− circular orbit asymptotically in either their infinite past
Thus, V plays the role of an effective potential for the
+ or their infinite future, or both. Nearly-homoclinic or-
radial motion, which is allowed for E ≥ V (L,r), with
+ bits exhibit a “zoom-whirl” behavior [18]: an episode
an equality identifying radial turning points.
of prolonged rotation (“whirl”) about the location of
Stationary points of V+(L,r) outside the black hole, the associated unstable circular orbit. We will see that
when they exist, correspond to circular orbits. These all orbits relevant to the overspinning problem fall in
satisfy the simultaneous conditions that category. Based on the correspondence with homo-
clinicorbits,unstablecircularorbitsmaybedividedinto
E =V , ∂ V =0 (circular orbits). (5)
+ r + “bound”(E <1)and“unbound”(E ≥1). Theradiusof
the innermost bound circular orbit (IBCO) is obtained
by solving E (R)=1, giving R˜ =[1+(1−a˜)1/2]2.
c ibco
Figure 1 illustrates the range of stable and unstable
1 Note,toavoidconfusion,that[6]hasinsteada/M =1−2(cid:15)2. circular orbits, and the location of the various special
5
unstablecircularorbitsenclosedbetweenthem. Alsope-
culiar is the fact that the ratio of coordinate differences
(R˜ −R˜ )/(R˜ −R˜ ) diverges as (cid:15) → 0. A closer
isco eh ibco eh
look reveals [19] that the light ring, IBCO and ISCO re-
main separated from the horizon, and from each other,
when examined on a Boyer-Lindquist t=const slice: On
that slice, the proper radial distance between the light-
ring and the horizon is finite, and so is the distance be-
tween any fixed-E unstable circular orbit and the light
ring. The proper radial distance between the ISCO
and any fixed-E unstable circular orbit diverges on the
t=const slice; the geometry of the t=const hypersurface
appears to “stretch” infinitely around the ISCO location
[19]. The situation, however, is rather different when ex-
amined on a horizon-crossing time slice. As emphasized
recently by Jacobson [14], on any such slice, the light
ring, IBCO and ISCO all actually coincide with horizon
generators. From that perspective, they—and all unsta-
ble orbits in between them—are “at the same place” in
FIG. 1. Timelike circular equatorial geodesics around a
nearly extremal Kerr black hole, shown here for a = 0.99M. the extremal limit.
The plot shows specific angular momentum versus Boyer- These subtleties will not affect our analysis directly: (cid:15)
Lindquistradius. Orbitswithr>R (magenta)arestable, will be kept small but nonzero, and the strict ordering
isco
whilethesewithr<R (blue)areunstable. Alsoindicated (9) will therefore apply on any time slice. However, we
isco
are the innermost bound circular orbit (IBCO, E = 1) and musttakenoteofthedegeneracyofRasaparameterfor
the photon orbit (“light ring”, E,L → ∞). In the extremal unstable circular orbits when (cid:15) → 0. The energy E, on
limit,a→M,theradiiR ,R andR allcoincidewith
isco ibco ph the other hand, remains a good parameter even in this
the horizon radius Reh. limit, spanning the entire range ∞ > E > √1 . We will
3
thus generally adopt E for labelling unstable circular or-
bits. Given E, the angular momentum L (R(E)), which
c
orbits mentioned, in a particular example (a˜ = 0.99). we henceforth write as L (E), is obtained by substitut-
c
We note the ordering ingEqs.(1)and(14)inEq.(7)andthenexpandingin(cid:15).
The result is
R <R <R <R , (9)
eh ph ibco isco
L˜ (E)=2E+(6E2−2)1/2(cid:15)+O((cid:15)2). (16)
c
which applies for any a˜<1.
Let us now specialize to a near-extremal Kerr back- We note that to determine the O((cid:15)) term here required
ground with spin as in Eq. (1). One finds the explicit values of both ρ and ρ of Eq. (15).
1 2
√
R˜ =1+ 2(cid:15)+O((cid:15)2), (10)
eh
R˜ =1+(cid:112)8/3(cid:15)+O((cid:15)2), (11) B. Exclusion of deeply bound orbits
ph
R˜ =(1+(cid:15))2 (exact), (12)
ibco Heuristically,ifweassumeourpointparticlerepresents
R˜ =1+(2(cid:15))2/3+O((cid:15)4/3). (13) acompactobject—say,aSchwarzschildblackhole—then
isco
its effective proper “diameter” is ∼ µ. Below it will be-
The function E (R) in Eq. (6) can be inverted perturba- come clear that a successful overspinning requires µ∼(cid:15),
c
tively in (cid:15) to obtain the radius of an arbitrary unstable and so relevant objects have proper diameters ∼(cid:15). Now
circular orbit in terms of its energy E. We find consider placing such an object in a deeply bound or-
bit with an outer turning point at r < R [and with
R˜ =1+(cid:15)ρ (E)+(cid:15)2ρ (E)+O((cid:15)3), (14) isco
1 2 L > L (E)]. Such an object (it can be checked) will
c
plunge through the horizon within a proper time of O((cid:15))
where the first two coefficients, needed below, read
(at most), comparable to its own “light-crossing time”.
√
2 2E 2(2E4−E2+1) Itisnotclearwhethertheobjectcanbemadetoinitially
ρ = √ , ρ = . (15)
1 3E2−1 2 (3E2−1)2 “fit” in its entirety outside the hole. At the very least, it
is not clear if the simple model of a point particle and a
Equation (12) is the special case of (14) with E = 1, stationary horizon provides a faithful description of the
giving ρ =2 and ρ =1. physics in this case.
1 2
Itfollows that, in theextremallimit (cid:15)→0, theBoyer- To avoid such subtleties, we wish to exclude deeply
Lindquist radii of the light-ring and the ISCO both co- bound orbits from our analysis. We achieve this by re-
incide with the horizon radius, and so do the radii of all quiringthat,iftheorbitpossessesanouterradialturning
6
point at some r =r , then wefindthat(20)isalwaysviolatedfor(cid:15)< 4 .] Thisrules
out 27
out the ISCO itself, and it immediately rules out also all
rout >Risco((cid:15)). (17) orbitswith{E >Eisco,L=Lisco},forwhichW >Wisco.
Orbitswith{E <E ,L }canpotentiallysatisfyEq.
isco isco
It can be checked that, under this condition, the proper-
(20), but they are always deeply bound in the sense of
time interval along any timelike equatorial geodesic con-
failing to satisfy Eq. (17): For any E < E , the orbit
isco
necting r = r to r = R is finite (nonzero) even in
out eh hasanouterradialturningpointataradiusr <R .
out isco
the limit (cid:15) → 0 (taken with fixed E,L). The condition
TheupshotisthatorbitswithL=L arealloutside
isco
(17) demands that eligible particles must clear the peak
OS. For orbits with L < L we would need to require
isco
of the effective potential (when such a peak exists) as
E < E in order for W to be sufficiently negative.
isco
they plunge into the black hole.
But, once again, such orbits are excluded on account of
their being deeply bound. We conclude that orbits with
L≤L are all outside OS.
isco
C. Overspinning domain
Let us focus therefore on orbits with L > L . For
isco
such an orbit to be in OS, we require that (given E,η,(cid:15))
Given the restriction (17), a necessary and sufficient L is bounded from above by L (E) and simultaneously
c
condition for a falling particle of specific energy E to be from below via Eq. (20):
captured by the black hole is
(cid:15)2+2ηE+η2E2 <ηL˜ <ηL˜ (E;(cid:15)). (22)
c
L<L (E). (18)
c
We have made here the (cid:15) dependence of L explicit, for
c
A captured particle would overspin the black hole if and clarity. The span of the permissible range is η∆ :=
L
only if −(cid:15)2−η[2E−L˜ (E;(cid:15))]−η2E2, or, using Eq. (16),
c
(cid:112)
(M +µE)2 <aM +µL. (19) η∆ =−(cid:15)2+η(cid:15) 6E2−2−η2E2, (23)
L
Using a˜ = 1−(cid:15)2 and introducing the small mass ratio wherewehaveomittedtermsofO(η(cid:15)2). OSispopulated
η :=µ/M, this condition becomes if and only if we can find E,η,(cid:15) for which ∆L >0.
A few conclusions can be drawn immediately. First,
(cid:15)2+ηW +η2E2 <0, (20) consideringη∆ inEq.(23)asaquadraticfunctionofη,
L
we find it has a maximum value
where we have introduced2
(cid:15)2(E2−1)
maxη∆ = . (24)
W :=2E−L˜. (21) η L 2E2
ThisispositiveonlyforE >1. Therefore,allorbitswith
Note that Eq. (20) sets a lower bound on L (for given
E ≤1 fall outside OS. Bound orbits cannot overspin.
E,η,(cid:15)), while Eq. (18) sets an upper bound. Also note
Second, for any E > 1, we can obtain ∆ > 0 by
that Eq. (20) implies the necessary condition W <0 for L
choosing the mass ratio η from within the interval
overspinning to occur.
Our goal now is to identify the complete domain in
(cid:15)η (E)<η <(cid:15)η (E), (25)
− +
the space of {η,E,L} for which the conditions (18) and
(20)aresimultaneouslysatisfied,assuming(cid:15)(cid:28)1andthe where
condition(17). Foreasyreference,letuscallthisdomain
1 (cid:104)(cid:112) (cid:112) (cid:105)
“OS”, for “overspinning”. η = √ 3E2−1± E2−1 . (26)
±
2E2
WefirstshowthatorbitswithL≤L allfalloutside
isco
OS. To this end, consider first the ISCO itself, where Inotherwords,overspinningcanbeachievedforanyE >
W =2Eisco−L˜isco =:Wisco. UsingEqs.(6),(7)a√nd(13) 1, as long as η satisfies (25). Sincethecondition∆L >0
weobtainW =−cˆ(cid:15)4/3+O((cid:15)2),wherecˆ=21/3 3>0. is both necessary and sufficient, the converse also holds:
isco
Thus, W is negative as required, but it can be easily All orbits in OS satisfy E >1 with Eq. (25).
isco
checked that (20) is always violated for sufficiently small Third, from Eq. (25) it follows that η must be chosen
(cid:15): Replacing W → −cˆ(cid:15)4/3 in Eq. (20) and considering to be of O((cid:15)) (assuming E (cid:28) 1/(cid:15)). One can check that
the left-hand side as a quadratic function of η, we find η has a maximal value of
+
this function is positive definite for any (cid:15) < (2E /cˆ)3.
isco (cid:112)
[SinceEisco isboundedfrombelowbyEisco((cid:15)=0)= √1 , maxη+ = 3/2, (27)
3 E
√
(cid:112)
obtainedforE =2/ 3. Therefore,therangeη ≥ 3/2(cid:15)
liesoutsideOS.Thebandwidthofadmissiblemassratios,
2 Heuristically, W/2 may be interpreted as the specific energy in for given E and (cid:15), is
a co-rotating frame with Ω˜ = 1/2, i.e., the common angular
(cid:112)
velocityofallunstablecirculargeodesicsintheextremallimit. ∆ :=(cid:15)η −(cid:15)η =(cid:15) 2(E2−1)/E2, (28)
η + −
7
W
W_
W
W
W
FIG. 2. Domain of mass ratios η for which overspinning is FIG. 3. The overspinning window, shown in the plane of
possible in the geodesic approximation. η is shown divided E,W (whereW =2E−L/M)forseveralvaluesofη/(cid:15). Note
by the near-extremality parameter (cid:15)=(1−a/M)1/2, and E W isshowndividedby(cid:15). TheboundariesW aregiveninEq.
±
is the particle’s specific energy. The boundaries η (E) are (30). ThelowerboundaryW (E)(whichdoesnotdependon
± −
given in Eq. (26). Overspinning is not possible for E < 1 or η)arisesfromtherequirementthattheparticleiscapturedby
η > (cid:112)3/2(cid:15). However, for any value E > 1 there is a range the black hole. The upper boundary W+(E,η/(cid:15)) comes from
ofη forwhichtheblackholemaybeoverspun. Thishappens therequirementthatthefinalobjectisanover-extremalblack
iftheparticle’sangularmomentumischosenfromwithinthe hole. Overspinning is possible with any E >1, provided η is
range indicated in Eq. (29). chosen from within the range shown in Eq. (25).
√ III. SELF-FORCE PRELIMINARIES
which is maximal for E = 2. Figure 2 depicts the
permissible range of η/(cid:15) as a function of E.
Because the width of the overspinning window is of
Fourth, from Eqs. (24) and (28) we learn that an E =
O(η), self-gravity effects may potentially close this win-
const(> 1) slice of OS has maximal dimensions ∆ ∼
L dow,andtheymustthereforebeincludedintheanalysis.
(cid:15)2/η ∼ (cid:15) and ∆ ∼ (cid:15). OS is thus a narrow “tube” in
η Specifically,theGSFmodifiesthecapturecondition(18)
the {E,L,η} parameter space, of a cross section ∼ (cid:15)×
by changing the functional relation L (E) at O(η). It
(cid:15), whose boundary is tangent to the surface of unstable c
alsomodifiestheoverspinningcondition(19)bydissipat-
circular orbits, L=L (E).
c ing away some of the system’s initial energy and angular
Tosummarize,wehavefoundthatOSisanarrowtube- momentum. In this section we introduce relevant results
like region of the {E,L,η} space, described by E > 1, from the theory of self-forced motion. In Sec. IV we will
Lc(E;(cid:15))−∆L(E,η;(cid:15))<L<Lc(E;(cid:15)) and (cid:15)η−(E)<η < then use these results to derive conditions for capture
(cid:15)η+(E),where∆Landη±aregiveninEqs.(23)and(26), and overspinning under the full GSF effect.
respectively. A neater description of the OS window is
obtained in terms of the quantity W defined in Eq. (21):
Rearranging Eq. (22) and using (16), we find A. Equation of motion with self-force
(cid:15)W (E)<W <(cid:15)W (E,η/(cid:15)), (29) There now exists a rigorous formulation of the equa-
− +
tionsofmotionforcompactobjectsincurvedspacetime,
valid through first post-geodesic order in perturbation
where
theory—see [20–22] and references therein, and [23, 24]
(cid:18) (cid:19) for recent reviews. The formulation applies in situations
(cid:112) (cid:15) η
W =− 6E2−2, W =− + E2 . (30) where all lengthscales associated with the compact ob-
− + η (cid:15)
ject are much smaller than the typical curvature radius
ofthebackgroundgeometry. Themotionofthecompact
ThisdomainisillustratedinFigure3forasampleofη/(cid:15) object is then determined via a systematic procedure of
values. To overspin a black hole of given M and (cid:15) (cid:28) 1, matched asymptotic expansions, and interpreted as an
one should pick an E greater than 1, choose any η from acceleratedmotioninthebackgroundspacetime,subject
within the interval (25), and then choose W (hence L) toaneffectiveGSF(∝η2). Oneoftheresultsisthatthe
from within the interval (29). internal structure of the object does not affect the self-
8
acceleration at O(η) (except, if the particle is spinning, From Eq. (34) we have
through the familiar Mathisson–Papapetrou spin term).
The GSF formalism should be applicable in our setup, Eˆ(τ)=E +∆E(τ), Lˆ(τ)=L +∆L(τ), (37)
∞ ∞
since we work under the assumption µE (cid:28) M. The
introduction of the small background-related parameter where
(cid:15) should not pose a problem, because the background’s (cid:90) τ (cid:90) τ
curvature radius remains much larger than µE even in µ∆E(τ)=− F dτ, µ∆L(τ)= F dτ. (38)
t φ
the limit (cid:15)→0, and even as the particle approaches the −∞ −∞
horizon. We will indeed proceed under the assumption
In principle, the coupled set (35) with (37) determines
thatthestandardfirst-orderGSFformalismisapplicable
theself-acceleratedorbit,giventheinitialvaluesE ,L
∞ ∞
anywhere along the particle’s trajectory until it crosses
and a method for calculating the GSF along the orbit.
the horizon.
The equation of motion, including the leading-order
GSF, may be written in the form
B. Dissipative and Conservative pieces of the
self-force
µuˆβ∇ uˆα =Fα. (31)
β
Hereuˆα istheparticle’sfour-velocity,tangenttothe(ac- The quantities ∆E(τ) and ∆L(τ) encapsulate both
celerated)trajectoryinthebackgroundspacetime(Kerr, conservative and dissipative effects of the GSF. This ter-
in our case) and normalized using minology refers to a splitting of the GSF in the form
gαβuˆαuˆβ =−1, (32) Fα =Fcαons+Fdαiss, (39)
where g is the background (Kerr) metric. The covari-
αβ where the first and second terms are the self-forces ex-
ant derivative in (31) is taken with respect to g , and
αβ erted, respectively, by the “time-symmetric” and “time-
Fα is the first-order GSF, proportional to µ2. The GSF
antisymmetric”piecesofthe(regularized)metricpertur-
is normal to the four-velocity, g uˆαFβ =0, so that the
αβ bation(cf.[25]foramoreprecisedefinition). Forgeodesic
restmassµremainsconstant. MethodstocomputeFαin motion in the equatorial plane of a Kerr black hole, Fα
Kerr spacetime are reviewed in [25]. In should be noted
canbethoughtofasafunctionofonly r andr˙ alongthe
thatFα itselfisagauge-dependentnotion: Afull,gauge-
orbit. The particular time symmetry of such geodesics
invariantinformationaboutthemotioniscontainedonly
then implies [25]
in the combination of the GSF and the metric perturba-
tion with which it is associated [26]. Fα (r,r˙)= 1(cid:2)Fα(r,r˙)+s Fα(r,−r˙)(cid:3), (40)
Now consider a particle sent in along the equator of cons 2 (α)
the Kerr black hole, i.e. with θ =π/2 and uˆθ =0 at the Fα (r,r˙)= 1(cid:2)Fα(r,r˙)−s Fα(r,−r˙)(cid:3) (41)
initial moment, where hereafter τ is proper time along diss 2 (α)
theself-acceleratedorbiting . Inanyreasonablegauge,
αβ (no summation over α), where s = −1 = s and
the component Fθ would vanish from symmetry and the (t) (φ)
s =+1. Thisgivesasimpleprescriptionforconstruct-
motion will remain equatorial. Let us then define (r)
ing Fα and Fα along geodesics, given the full GSF.
cons diss
Eˆ :=−uˆ , Lˆ :=uˆ , (33) For circular orbits we have Fα(r,r˙) = Fα(r,−r˙),
t φ
meaning Ft,Fφ are purely dissipative while Fr is purely
inanalogywithE andLofthegeodesiccase. Here,how- conservative. In general, however, each component has
ever, Eˆ and Lˆ are not constants of the motion. Rather, both dissipative and conservative pieces. Of particular
Eq. (31) tells us they evolve (slowly) according to interest to us will be nearly-circular orbits with |r˙|(cid:28)1.
Along such orbits we may write, to leading order in |r˙|,
dEˆ dLˆ
µ =−F , µ =F , (34)
dτ t dτ φ Fα (cid:39)r˙Fα(r), Fα (cid:39)Fα(r) (42)
cons 1 diss 0
where Fα = gαβFβ. With these definitions, Eq. (32) for α=t,φ, and
produces the radial equation of motion
Fr (cid:39)Fr(r), Fr (cid:39)r˙Fr(r), (43)
r˙2 =B(r)(cid:16)Eˆ−V (Lˆ,r)(cid:17)(cid:16)Eˆ−V (Lˆ,r)(cid:17), (35) cons 0 diss 1
− +
where Fα and Fα are some functions of r only.
0 1
whose form is identical to that of Eq. (3)—except that Equations (40)–(43) are applicable, at leading order
here Eˆ and Lˆ are slow functions of τ along the orbit. in η, even for an orbit that is slowly evolving under the
The results of the previous section lead us to focus GSF effect. In that case the GSF depends also on the
attentiononparticlessentinfrominfinity,i.e.,oneswith instantaneous self-acceleration, but that dependence ap-
r(τ →−∞)→∞. For such particles, we define pears only at subleading order in η. At leading order,
Eqs. (40)–(43) maintain their form at each point along
E :=Eˆ(τ →−∞), L :=Lˆ(τ →−∞). (36) the orbit.
∞ ∞
9
The GSF integrals ∆E and ∆L can be related, in cer- D. Critical orbits
tain situations, to asymptotic fluxes of energy and angu-
lar momentum in gravitational waves. This was estab- In the geodesic case we have introduced the function
lishedrigorouslyinRef.[27]foratrajectorystartingand L (E), which we now interpret as the critical value of
c
ending at infinity.3 A similar balance relation has been the angular momentum for a given energy: Geodesic or-
argued to hold also for adiabatic inspiral orbits around bits with L > L (E) scatter back to infinity, while ones
c
a black hole, subject to a suitable averaging over many with L < L (E) fall into the black hole. This type of
c
orbital periods [28, 29]. In both scenarios, the contribu- criticalbehaviorcarriesovertotheGSFcase,thoughra-
tion from Fcαons to the integrals ∆E and ∆L (taken from diation losses then introduce a subtlety, since orbits that
τ =−∞toτ =+∞)vanishesatleadingorder,byvirtue are initially scattered may fall into the black hole at a
oftheorbitalsymmetryexpressedinEq.(40). Thisguar- subsequent approach. However, we may still speak of a
anteesthattheradiatedfluxesbalancetheworkdoneby critical threshold for an immediate capture, which sepa-
the dissipative piece of the self-force alone, as expected. rates (in the space of initial conditions) between orbits
that scatter at first approach and orbits that do not. A
detailed analysis of this critical behavior was given in
C. ADM energy and angular momentum Ref. [30] for orbits in Schwarzschild spacetime (working
in the first-order GSF approximation, as here), and in
the following discussion we assume the same qualitative
Ouranalysisinthenextsectionwillrequireknowledge
behavior applies in the Kerr case too.
ofthetotal,conservedADMenergyandangularmomen-
In particular, we assume there exists a critical value
tumcontentsofthespacetimeintheabovesetup. Specif-
L = L (E ) that separates between the two pos-
ically, we will need expressions for E and L in ∞ ∞,c ∞
ADM ADM sible outcomes. The initial conditions {E ,L (E )}
terms of E and L (and the two masses, M and µ), ∞ ∞,c ∞
∞ ∞ thus define a one-parameter family of “critical orbits”.
correct through O(µ2). A subtlety is that ADM quan-
Let us denote by Eˆ (τ;E ) and Lˆ (τ;E ) the func-
tities are most conveniently evaluated in a “center-of c ∞ c ∞
tions Eˆ(τ) and Lˆ(τ) corresponding to a critical orbit
mass” system (and, at the required order, would include
a contribution from the black hole’s “recoil” motion), with a given E∞ [so that Eˆc(τ → −∞;E∞) = E∞ and
whereas E and L are components of the particle’s Lˆ (τ → −∞;E ) = L (E )]. Unlike in the geodesic
∞ ∞ c ∞ ∞,c ∞
four-velocity, defined in a coordinate system centered case where critical geodesics of different E are disjoint,
around the black-hole. in the GSF case all critical orbits join a global attrac-
Inoursetup,theADMquantitiesaremosteasilyeval- tor, which is the perfectly fine-tuned orbit that evolves
uated on a hypersurface of constant t (cid:28) −M, where radiatively along the sequence of unstable circular or-
the binary separation is r (cid:29) M. In the limit t → −∞ bits starting at the light ring and ending at the ISCO,
(r → ∞), the gravitational interaction energy vanishes where it plunges into the black hole. Figure 1 in Ref.
and does not contribute to E . Working at that [30] illustrates the evolution of the critical orbit along
ADM
limit, we assume that, for the purpose of calculating the attractor, and see also Fig. 4 below.
ADM quantities, the black hole–particle system may be Let us define the “GSF correction”
replaced with that of two relativistic pointlike particles
δL (τ;E ):=Lˆ (τ;E )−L (E ), (46)
in flat spacetime. E is then simply the sum of the c ∞ c ∞ c ∞
ADM
two relativistic energies in the center-of-mass frame, and
and then
L issimilarlythesumoftwoangularmomenta(with
ADM
respect to the center of mass), plus the spin of the black δL (E ):=δL (τ →−∞;E ). (47)
∞ ∞ c ∞
hole.
δL is the GSF-induced shift in the critical value of L
AppendixAgivesthedetailsofthiscalculation,which ∞ ∞
at a fixed E . It may also be interpreted in terms of a
is straightforward. The result is ∞
GSF correction to the critical impact parameter. We as-
(cid:20) (cid:21) sumethatthedifferenceδL (τ;E )remainssmall[O(η)]
1 c ∞
EADM =M 1+ηE∞− 2η2(E∞2 −1) +o(η2), (44) duringtheapproach,whichshouldbethecaseinanyrea-
sonablegauge. However,clearly,thatdifferenceceasesto
remain small as the critical orbit joins the global attrac-
torandevolvesalongit; thenthemeaningofδL (τ;E )
L =M(cid:0)a+ηL −η2L E (cid:1)+o(η2). (45) c ∞
ADM ∞ ∞ ∞ as a small GSF correction is lost.
Forour analysisof overspinning orbits in the next sec-
tion, we will require an explicit expression for δL (E )
∞ ∞
intermsofGSFquantities. Itisinstructivetoderivethis
relationfirstwiththedissipativepieceoftheGSFturned
3 The configuration considered in Ref. [27] had no black hole in
it, but the authors argue convincingly that a similar conclusion off, i.e. replacing the full GSF with its conservative piece
would hold also in the black hole case, if fluxes down the event (in which case the global attractor disappears, and crit-
horizonwereaccountedforinthebalanceequation. ical orbits of different E remain disjoint). Let us call
∞
10
the resulting quantity δLcons(E ). As a second step we with a Lyapunov exponent λ = M−1(3E2 − 1)1/2 at
∞ ∞ ∞
will restore dissipation and consider its effect. leading order in (cid:15) (and ignoring the small effect of the
GSF). The choice τ = −λ−1logη, for example, gives
c
δr(τ ) ∼ η, and the quasi-circular piece of ∆Wcons does
c
1. Conservative GSF effect not contribute to δLcons at leading order in η.
∞
Our discussion assumes that ∆E(τ ) and ∆L(τ )
c c
With dissipation turned off, the critical orbit becomes [hencealso∆W(τc)]areO(η)quantities,i.e.thattheac-
exactly stationary at τ → ∞, where it joins an un- cumulatedGSF-inducedpositionalshiftintheorbitdur-
stable (nongeodesic) circular orbit of radius Rˆ(E ) = ing the approach is a small, O(η) quantity. This should
∞
R(E )+δR. Here R(E ) is the geodesic relation given bethecaseinanyreasonablegauge. Underthisassump-
∞ ∞
in Eq. (14), and δR is a conservative GSF correction. tion, we note that the value of the integral ∆W remains
To obtain δLcons we first substitute Eˆ and Lˆ from Eq. unchanged,atleadingorderinη,ifinEq.(50)wereplace
∞
the integration along the actual, GSF-perturbed orbit,
(37)intotheradialequationofmotion(35),replacingL
∞
withL (E )+δLcons(E ),whereL (E )isthegeodesic with an integration along the critical geodesic of energy
c ∞ ∞ ∞ c ∞
E . This can be exploited to simplify actual calcula-
relationgiveninEq.(16). Wethendemanddr/dτ =0as ∞
well as d2r/dτ2 = 0 at r = Rˆ through O(η). At leading tions: To compute δLc∞ons at leading order in η requires
onlyanevaluationoftheGSFalongafixedgeodesic,and
order in (cid:15) this yields two algebraic equations for the two
there is no need to consider the back-reaction from the
O(η) unknowns δLcons and δR, given E and the GSF.
∞ ∞ GSF on the orbital trajectory.
The solution is
δLcons(E )=2M∆Econs(∞)−∆Lcons(∞), (48)
∞ ∞ 2. Full GSF effect
and δR(E ) = O((cid:15))O(η). Here ∆Econs and ∆Lcons
∞
are the same as ∆E and ∆L of Eq. (38), but with Now restore dissipation. The fine-tuned critical orbit
F →Fcons, andwiththeGSFintegralsevaluatedalong no longer settles into a strictly stationary motion, but
α α
thecriticalorbitwithenergy-at-infinityE . Theprecise rather it continues to evolve radiatively, in an adiabatic
∞
dependenceofδRontheGSFwillnotbeneeded,butwe fashion, through a sequence of unstable circular orbits
note that the O((cid:15)η) GSF correction to the radius of the of decreasing energies (hence increasing radii). With a
critical circular orbit is reassuringly small compared to perfect fine-tuning, the orbit can reach the ISCO be-
the O((cid:15)) radial distance to the light ring. fore plunging into the black hole—a scenario illustrated
To simplify the appearance of subsequent equations, in Fig. 4. A relation between the degree of fine-tuning
let us from now on use units in which M = 1. This, in and the amount of energy loss was derived in Ref. [30]
particular, makes our “tilde” notation redundant (with (for the Schwarzschild case): Rewriting their Eq. (124)
L˜ = L, etc.) and µ becomes interchangeable with η. in terms of angular momentum, we have the scaling
Recalling our W notation from Eq. (21), we rewrite Eq. δL/L∞,c ∼ exp[(Ef −Ei)/η], where δL (not to be con-
(48) as fusedwiththeGSFshiftδL∞)isanysmallperturbation
inthevalueoftheinitialangularmomentumoffthecrit-
δLcons(E )=∆Wcons(∞), (49) ical value L , and E −E is the resulting change in
∞ ∞ ∞,c f i
specific energy as the orbit progresses along the global
where ∆Wcons represents the conservative piece of attractor. To achieve an O(1) change in the specific
energy requires an “exponentially delicate” fine-tuning,
∆W(τ):=2∆E(τ)−∆L(τ)
δL/L ∼exp(−1/η).
(cid:90) τ ∞,c
= −η−1 (2F +F )dτ. (50) For our analysis we do not require knowledge of the
t φ perfectlyfine-tunedangularmomentumatthatlevel. We
−∞
need L through O(η) only. Fine-tuning at O(η) [cor-
∞,c
The quantity ∆Wcons(∞) is the limit τ → ∞ of
responding to δL = o(η)] guarantees only E − E =
f i
∆Wcons(τ). Doesthislimitactuallyexist? Theansweris
O(ηlnη). Therefore,forthepurposeofdeterminingL
∞,c
positive,sincebothFcons andFcons vanishexponentially
t φ through O(η), it is sufficient to restrict attention to the
fast in τ as the orbit approaches the limiting circular or-
early part of the critical orbit, where ∆E and ∆L (spe-
bit at τ →∞.
cific values) are still O(ηlnη) at most, and have not yet
To make this last statement more precise, let us split
accumulated O(1) changes. This observation assists us
the τ integral into an “approach” piece, (cid:82)τc , and a
−∞ in Appendix B, where we derive an expression for the
(cid:82)∞
“quasi-circular”piece, ,withτ chosensothatδr(τ ), leading-order full-GSF correction δL (E ).
τc c c ∞ ∞
where δr(τ) := r(τ)−Rˆ, is already very small. For a Our main result in Appendix B is that
smallδr wehavetheformFcons (cid:39)r˙F (r)[Eq.(42)]and
t 1t δL (E )=∆W(τ )+O((cid:15))O(ηlnη), (51)
similarly for Fcons. Thus (cid:82)∞Fconsdτ (cid:39) −F (Rˆ)δr(τ ), ∞ ∞ c
φ τc t 1t c
and similarly for the φ component. A local analysis of inanalogywiththe“nodissipation”case,Eq.(49). Here,
Eq. (35) near the limiting circular orbit gives δr ∼e−λτ, ∆W(τ ) is the full-GSF integral shown in Eq. (50), eval-
c