Physics Letters B 726 (2013) 533–536

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Physics Letters B

www.elsevier.com/locate/physletb

The gravitational two-body problem in the vicinity of the light ring:Insights from the black-hole–ring toy model

Shahar Hod a,b,∗a The Ruppin Academic Center, Emeq Hefer 40250, Israelb The Hadassah Institute, Jerusalem 91010, Israel

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 July 2013Accepted 26 August 2013Available online 31 August 2013Editor: M. Cvetic

The physical properties of an axisymmetric black-hole–ring system are studied analytically within theframework of general relativity to second order in the dimensionless mass ratio μ ≡ m/M . In particular,we analyze the asymptotic behaviors of the binding energy and the total angular momentum of thetwo-body system in the vicinity of the light ring at R = 3M , where the circular orbit becomes null.We find that both quantities diverge quadratically in μ(1 − 3M/R)−1 at the light ring. The reporteddivergent behavior of the physical quantities stems from the second-order spin–orbit interaction betweenthe black hole and the orbiting object (the dragging of inertial frames by the orbiting ring). It is shownthat this composed black-hole–ring toy model captures some of the essential features of the conservativedynamics of the (astrophysically more interesting) black-hole–particle system. In particular, we show thatboth systems share the same quadratic divergent behavior of the physical quantities near the light ring.

Moreover, we prove that both systems are characterized by the same ratio E(2)(R→3M)

J (2)(R→3M)= 1

3√

3, where E(2)

and J (2) are the divergent second-order (self-interaction) expansion coefficients of the binding energiesand the angular momenta of the systems, respectively.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The gravitational two-body problem has attracted much at-tention over the years from both physicists and mathematicians,see [1–20] and references therein. As is well known, the two-body problem in Newtonian gravity is characterized by a partic-ularly elegant and simple analytic solution [1]. In fact, Newto-nian gravity provides a very accurate mathematical description forthe dynamics of a wide variety of astrophysical binaries in theweak-field limit. However, the dynamics of close binaries com-posed of black holes and superdense neutron stars are charac-terized by strong-gravity effects which cannot be treated properlywithin the limited framework of Newtonian mechanics. Instead,the dynamics of compact binaries in the strong-gravity regimeshould be described by the (mathematically more complicated)equations of the general theory of relativity.

The Einstein equations which describe the dynamics of close bi-naries in the strong-gravity regime are very complex and have nocompact analytic solution. Despite this fact, the two-body problemin general relativity can be tackled using an approximation method

* Correspondence to: The Ruppin Academic Center, Emeq Hefer 40250, IsraelE-mail address: shaharhod@gmail.com.

0370-2693/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physletb.2013.08.070

which is based on a perturbative treatment in powers of the sym-metric mass ratio [20]

μ ≡ Mm

(M + m)2, (1)

where M and m are the masses of the two compact objects.(We use gravitational units in which G = c = 1.)

Two important physical quantities which characterize the cir-cular two-body dynamics are the binding energy and the totalangular momentum of the system. The binding energy Ebinding ofthe two-body system is given by the difference between the totalgravitational energy of the system at infinity and the total gravita-tional energy of the system at a finite separation:

Ebinding(x) = E(x = 0) − E(x). (2)

Here

x ≡ [(M + m)Ω

]2/3(3)

is a convenient invariant (and dimensionless) coordinate con-structed from the characteristic frequency Ω of the circular orbit[20,21]. The binding energy can be expanded in powers of the di-mensionless mass ratio μ:

534 S. Hod / Physics Letters B 726 (2013) 533–536

Ebinding(x)/M =∞∑

k=1

E(k)(x) · μk. (4)

Likewise, the angular momentum J of the two-body system canbe expressed as a power series of the dimensionless mass ratio μ:

J (x)/M2 =∞∑

k=1

J (k)(x) · μk. (5)

In the zeroth-order (μ → 0) approximation the spacetime met-ric is described by the physical parameters of the larger object (thecentral black hole) while the smaller object (the ‘test-particle’) fol-lows a geodesic of the black-hole spacetime. In this test-particlelimit the system is characterized by the well-known relations [2]

E(1)(x) = 1 − 2x√(1 − 3x)

− 1 and J (1)(x) = 1√x(1 − 3x)

. (6)

Realistic astrophysical binaries are often composed of an orbit-ing object whose mass m is smaller but non-negligible as com-pared to the mass M of the central black hole. In these situationsthe zeroth-order (test-particle) approximation (6) is no longer validand one should take into account the gravitational self-force cor-rections to the orbit [7–20]. These corrections (which are second-order in the symmetric mass ratio μ) take into account the finitemass of the orbiting object.

The gravitational self-force has two distinct contributions: (1) Itis responsible for dissipative effects that cause the orbiting parti-cle to emit gravitational waves [6,7]. (2) The self-force due to thefinite mass of the orbiting object is also responsible for conser-vative corrections [of order O (μ2)] to the binding energy and tothe total angular momentum of the two-body system. FollowingRefs. [9–20], in the present study we shall focus on these conser-vative corrections to the orbit.

2. The black-hole–particle system near the light ring

The conservative second-order (self-interaction) contributionsto the binding energy and to the total angular momentum of thetwo-body (black-hole–particle) system, E(2)

particle(x) and J (2)

particle(x),were computed most recently in [20,22]. The mathematical expres-sions of these physical quantities are quite cumbersome [20], buta remarkably simple quadratic divergence of both these quantitieswas observed in the vicinity of the unperturbed light ring [23] atx = 1

3 [20]:

E(2)

particle(z → 0) =(

1

27− 1

12ζ

)· z−2 and

J (2)

particle(z → 0) =(

1

3√

3−

√3

4ζ

)· z−2, (7)

where

z ≡ 1 − 3x. (8)

Here ζ is a “fudge” factor which was introduced in [20]. Using nu-merical computations, the value of this fudge factor was estimatedin [20] to be

ζ ≈ 1. (9)

We would like to point out that the physical quantities E(2)(x)and J (2)(x) satisfy the simple relation [see Eq. (7)]

E(2)

particle(z → 0)

J (2)(z → 0)

= 1

3√

3(10)

particle

in the vicinity of the light ring. It is worth emphasizing that thisrelation holds true regardless of the value of the (numerically com-puted) fudge factor ζ .

It is worth noting that the quadratic divergence of the physi-cal quantities which characterize the dynamics of the black-hole–particle system near the light ring [see Eq. (7)] could only beinferred using numerical computations, see Fig. 2 of [20]. The maingoal of the present Letter is to analyze a simple toy model whichcaptures, at least qualitatively, some important features of the fun-damental two-body problem in general relativity. In particular, wewould like to provide an analytical explanation for the quadratic di-vergence of the self-interaction quantities E(2)(x) and J (2)(x) in thevicinity of the light ring.

3. The black-hole–ring system near the light ring

In a recent paper [24] we proposed to model the conserva-tive behavior of the two-body system using the analytically solv-able model of a stationary axisymmetric ring of particles in orbitaround a central black hole. This composed system was analyzedin detail by Will [19]. We have shown [24] that this toy modelcaptures, at least qualitatively, some important features of the con-servative dynamics of the (astrophysically more relevant) black-hole–particle system [25]. In particular, like the orbiting particle,the rotating ring can drag the generators of the black-hole hori-zon [19].

It is well known [19] that local inertial frames are dragged by anorbiting object. In fact, because of the dragging of inertial framesby the orbiting object, one can have a Schwarzschild-like blackhole with zero angular momentum ( JH = 0) but with a non-zeroangular velocity (ωH �= 0) [see Eq. (12) below]. Specifically, it wasfound in [19] that the angular momentum of a black hole which isperturbed by an orbiting ring is given by [26]

JH = 4M3ωH − 8Mmx3 J (1)(x), (11)

where the leading-order dimensionless angular momentum of thering, J (1)(x), is given by the expression (6). Thus, to first-order inthe symmetric mass ratio μ of the system [see Eq. (1)], a zero an-gular momentum black hole ( JH = 0) in the composed black-hole–ring system is characterized by a non-zero angular velocity ωH:

MωH = 2x3 J (1)(x) · μ. (12)

The relation (12) [and, in particular, the fact that ωH( JH = 0) �= 0]is a direct consequence of the dragging of inertial frames by theorbiting ring [19,27].

The binding energy of the black-hole–ring system, Ebinding(x),can be expanded in the form (4). From the results presentedin [19] for the composed black-hole–ring system one finds aftersome algebra that the leading-order expansion coefficient E(1)

ring(x)[the coefficient of the O (μ) term in the expansion (4)] is givenby the expression (6). In addition, one finds [19] that the O (μ2)

contribution to the energy budget of the black-hole–ring system inthe vicinity of the light ring (z → 0) is dominated by the divergentterm

μE(2)

ring(z → 0) = − 4

27zMωH J (1). (13)

The expression (13) represents a spin–orbit interaction betweenthe spinning black hole (which is characterized by the horizon an-gular velocity ωH) and the rotating ring (which is characterizedby the angular momentum m J (1)). It is worth emphasizing thatωH is linear in the mass m of the orbiting ring [see Eq. (12)].Thus, this spin–orbit interaction term represents a second-order

S. Hod / Physics Letters B 726 (2013) 533–536 535

self-interaction term of order O (μ2). Taking cognizance of Eqs. (6),(12), and (13), we find [28]

E(2)

ring(z → 0) = − 8

243· z−2 (14)

in the vicinity of the light ring.Likewise, the total angular momentum of the black-hole–ring

system, J (x), can be expanded in the form (5). From the resultspresented in [19] for the composed black-hole–ring system onefinds after some algebra that the leading-order expansion coef-ficient J (1)

ring(x) [the coefficient of the O (μ) term in the expan-

sion (5)] is given by (6). In addition, one finds [19] that the O (μ2)

contribution to the angular momentum of the black-hole–ring sys-tem in the vicinity of the light ring (z → 0) is dominated by thedivergent term

μ J (2)

ring(z → 0) = −4

3MωHz−3/2. (15)

Taking cognizance of Eqs. (6), (12), and (15), we find [28]

J (2)

ring(z → 0) = − 8

27√

3· z−2 (16)

in the vicinity of the light ring.It is worth emphasizing that the perturbation expansions (4)

and (5) are valid in the regime E(2)(x) · μ2 E(1)(x) · μ 1[and likewise J (2)(x) · μ2 J (1)(x) · μ 1]. Thus, the divergentbehaviors (14) and (16) are valid in the regime

μ2/3 z 1. (17)

Inspection of Eqs. (14) and (16) reveals that the binding energyand the total angular momentum of the black-hole–ring system di-verge quadratically in μz−1 at the light ring. Remarkably, the phys-ical quantities of the original black-hole–particle system share thissame divergent behavior (that is, a quadratic divergence in μz−1)in the vicinity of the light ring, see Eq. (7).

Moreover, the physical quantities E(2)

ring and J (2)

ring which charac-terize the black-hole–ring system satisfy the simple ratio

E(2)

ring(z → 0)

J (2)

ring(z → 0)= 1

3√

3, (18)

which is identical to the corresponding ratioE(2)

particle(z→0)

J (2)

particle(z→0)= 1

3√

3

[see Eq. (10)] satisfied by the physical quantities of the originalblack-hole–particle system!

4. Summary and discussion

The physical properties of a black-hole–ring system were an-alyzed in the vicinity of the photon orbit at R = 3M , where thecircular orbit of the ring becomes null. We have shown that thissimple toy model may capture some important features of the con-servative dynamics of the (physically more interesting) black-hole–particle system. In particular, this model provides a simple ana-lytic explanation for the quadratic divergence of the self-interactionquantities E(2)(x) and J (2)(x) in the vicinity of the light ring, seeEqs. (14) and (16).

Moreover, we have shown that the black-hole–particle systemand the black-hole–ring system share the same relation

E(2)

particle(R → 3M)

J (2)

particle(R → 3M)= E(2)

ring(R → 3M)

J (2)

ring(R → 3M)= 1

3√

3(19)

between the second-order expansion coefficients.

The present toy model suggests that the second-order spin–orbit interaction between the black hole and the orbiting object[the dragging of inertial frames by the orbiting object, see Eq. (12)]is the main element determining the observed (quadratic) diver-gent behavior of the self-interaction quantities in the vicinity ofthe light ring.

Finally, it is worth pointing out the simple relations [seeEqs. (7), (14), and (16)]

E(2)

particle(R → 3M)

E(2)

ring(R → 3M)= J (2)

particle(R → 3M)

J (2)

ring(R → 3M)= 81

32

(ζ − 4

9

)(20)

between the self-interaction quantities of the black-hole–particlesystem and the corresponding physical quantities of the black-hole–ring system. We note that these ratios would be equal to 1 ifthe fudge factor ζ [see Eq. (9)] equals 68/81.

Acknowledgements

This research is supported by the Carmel Science Foundation.I thank Yael Oren, Arbel M. Ongo and Ayelet B. Lata for helpfuldiscussions.

References

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[10] S. Detweiler, in: L. Blanchet, A. Spallicci, B. Whiting (Eds.), Mass and Motion inGeneral Relativity, Springer, 2011.

[11] L. Barack, Class. Quantum Gravity 26 (2009) 213001.[12] S. Detweiler, Phys. Rev. D 77 (2008) 124026.[13] N. Sago, L. Barack, S. Detweiler, Phys. Rev. D 78 (2008) 124024.[14] T.S. Keidl, A.G. Shah, J.L. Friedman, D. Kim, L.R. Price, Phys. Rev. D 82 (2010)

124012.[15] A. Shah, T. Keidl, J. Friedman, D. Kim, L. Price, Phys. Rev. D 83 (2011) 064018.[16] T. Damour, Phys. Rev. D 81 (2010) 024017.[17] L. Barack, N. Sago, Phys. Rev. Lett. 102 (2009) 191101;

L. Barack, N. Sago, Phys. Rev. D 81 (2010) 084021.[18] M. Favata, Phys. Rev. D 83 (2011) 024027.[19] C.M. Will, Astrophys. J. 191 (1974) 521;

C.M. Will, Astrophys. J. 196 (1975) 41.[20] S. Akcay, L. Barack, T. Damour, N. Sago, Phys. Rev. D 86 (2012) 104041.[21] Note that x reduces to M/R in the test-particle (m → 0) limit, where R is

the Schwarzschild radial coordinate, associated with the central black hole (thelarger object).

[22] It should be emphasized that the authors of Ref. [20] focused on the conserva-tive circular dynamics of the two-body system. It is only in this non-dissipativeregime that the energy and angular momentum of the system are conservedquantities.

[23] The light ring is often referred to as the null circular geodesic of the black-holespacetime.

[24] S. Hod, Phys. Rev. D 87 (2013) 024036.[25] It should be emphasized that this toy model, being axially-symmetric, can-

not describe the most important characteristic of the gravitational two-body problem: the emission of gravitational radiation. Nevertheless, followingRefs. [9–20], in the present study we shall focus on the conservative dynam-ics of the two-body system. That is, following Refs. [9–20] we shall ignore theemission of gravitational waves.

[26] Note that the results presented in [19] are expressed in terms of the irreduciblemass Mir of the black hole. For a ‘bare’ Schwarzschild black hole the irreduciblemass coincides with the mass M of the black hole. For the black-hole–ring sys-tem considered in [19] one finds Mir = M[1 + O (μ2)]. Taking cognizance of

536 S. Hod / Physics Letters B 726 (2013) 533–536

Eq. (6), one finds that the O (μ2) difference between Mir and M does notaffect the leading-order divergent behaviors [see Eqs. (14) and (16) below]of the O (μ2) correction terms. In addition, the results presented in [19] areexpressed in terms of the dimensionless ratio Mir/R , where R is the propercircumferential radius of the ring. The invariant coordinate x defined in (3) re-duces to M/R in the test-particle (m → 0) limit. For finite m values one hasx = M

R [1 + O (μ)]. Taking cognizance of Eq. (6), one finds that the O (μ) differ-ence between x and M/R does not affect the leading-order divergent behaviors[see Eqs. (14) and (16) below] of the O (μ2) correction terms. Finally, we note

that the symmetric mass ratio μ [see Eq. (1)] is closely related to the dimen-sionless mass ratio q ≡ m/M: μ = q[1 + O (q)]. Taking cognizance of Eq. (6),one finds that the leading-order divergent coefficients [see Eqs. (14) and (16)below] of the O (μ2) correction terms would also describe the leading-orderdivergent coefficients of the corresponding O (q2) corrections terms.

[27] In particular, note that a ‘bare’ (unperturbed) Schwarzschild black hole is char-acterized by the relation JH = 4M3ωH, which implies the simple relationωH( JH = 0) = 0.

[28] Here we have used the fact that x → 13 , in the vicinity of the light ring.