Marginally bound circular orbits in the composed black-hole-ring system
aa r X i v : . [ g r- q c ] J a n Marginally bound circular orbits in the composed black-hole-ringsystem
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: January 22, 2021)
Abstract
The physical and mathematical properties of the non-linearly coupled black-hole-orbiting-ringsystem are studied analytically to second order in the dimensionless angular velocity M ir ω H ofthe black-hole horizon (here M ir is the irreducible mass of the slowly rotating central black hole).In particular, we determine analytically, to first order in the dimensionless ring-to-black-hole massratio m/M ir , the shift ∆Ω mb / Ω mb in the orbital frequency of the marginally bound circular geodesicthat characterizes the composed curved spacetime. Interestingly, our analytical results for the fre-quency shift ∆Ω mb in the composed black-hole-orbiting-ring toy model agree qualitatively with therecently published numerical results for the corresponding frequency shift in the physically related(and mathematically much more complex) black-hole-orbiting-particle system. In particular, thepresent analysis provides evidence that, at order O ( m/M ir ), the recently observed positive shift inthe angular frequency of the marginally bound circular orbit is directly related to the physicallyintriguing phenomenon of dragging of inertial frames by orbiting masses in general relativity. . INTRODUCTION Geodesic orbits of test particles in curved spacetimes are of central importance in black-hole physics [1–8]. They provide valuable information on the physical parameters (mass,charge, angular momentum) of the central black hole. In particular, the marginally bound circular orbit of a curved black-hole spacetime is of special importance in astrophysics andgeneral relativity [1–8]. This physically interesting geodesic represents the innermost circularorbit of a massive particle which is energetically bound to the central black hole.For a test particle of proper mass m , the marginally bound circular geodesic is charac-terized by the simple energy relation [1–4] E ( r mb ) = m , (1)where E is the energy of the particle as measured by asymptotic observers. Interestingly, themarginally bound circular geodesic (1) marks the boundary between bound orbits, which arecharacterized by the sub-critical energy relation E < m , and unbound circular orbits with
E > m which, given a small outward perturbation, can escape to infinity. In particular, asnicely demonstrated in [5, 6], the critical (marginally bound) circular geodesic (1) plays akey role in the dynamics of star clusters around super-massive black holes in galactic nuclei.[The critical orbit (1) is sometimes referred to in the physics literature as the innermostbound spherical orbit (IBSO) [6, 7]].An important gauge-invariant physical quantity that characterizes the motion of particlesalong the marginally bound circular geodesic is the orbital angular frequency Ω mb of theparticles as measured by asymptotic observers. For a test-particle moving in the spinlessspherically symmetric Schwarzschild black-hole spacetime, this physically important orbitalfrequency is given by the simple dimensionless relation [2–4] M ir Ω mb = 18 . (2)Here M ir is the irreducible mass [9] of the central black hole.In recent years, there is a growing physical interest in calculating the O ( m/M ir ) cor-rections to the orbital frequency (1) of the marginally bound circular orbit in non-linearlycoupled black-hole-particle systems (see the physically interesting work [10] and referencestherein). To this end, one should take into account the gravitational self-force correctionsto the geodesic orbits of the particles [11–23].2he gravitational self-force has two distinct physical contributions to the dynamics of acomposed black-hole-particle system:(1) It is responsible for non-conservative physical effects, like the emission of energy andangular momentum in the form of gravitational waves [11].(2) The composed black-hole-particle system is also characterized by conservative gravita-tional self-force effects that preserve the total energy and angular momentum of the systembut shift the orbital frequency of the marginally bound orbit.Computing the gravitational self-force (GSF) correction ∆Ω mb to the zeroth-order fre-quency (1) of the marginally bound circular orbit is a highly non-trivial task. Intriguingly,Barack at. al. [10] have recently used sophisticated numerical techniques in the composedSchwarzschild-black-hole-orbiting-particle system in order to compute the characteristic shift∆Ω mb in the orbital frequency of the marginally bound orbit which is caused by the conser-vative part of the GSF.In particular, Barack at. al. [10] have found the ( positive ) dimensionless value∆Ω mb Ω mb = c · η + O ( η ) with c ≃ . η ≡ mM ir (4)is the dimensionless ratio between the mass of the orbiting particle and the mass of thecentral Schwarzschild black hole. The physical importance of the result (3) of [10] stemsfrom the fact that it provides gauge-invariant information about the strong-gravity effectsin the highly curved region ( r ≃ M ir ) of the black-hole spacetime.The main goal of the present compact paper is to use analytical techniques in orderto gain some physical insights on the intriguing O ( m/M ir ) increase [see Eq. (3)] in theorbital frequency of the marginally bound circular orbit as recently observed numericallyin the physically important work [10]. In particular, we shall analyze a simple black-hole-orbiting-ring toy model which, as we shall explicitly show below, captures some of theessential physical features of the (astrophysically more interesting and mathematically muchmore complex) black-hole-orbiting-particle system in general relativity. As nicely proved byWill [24], the composed black-hole-orbiting-ring toy model is amenable to a perturbativeanalytical treatment to second order in the dimensionless angular velocity M ir ω H of thecentral slowly rotating black hole. 3 I. THE ORBITAL FREQUENCY OF THE MARGINALLY BOUND CIRCULARORBIT IN THE COMPOSED BLACK-HOLE-ORBITING-RING SPACETIME
In the present paper we would like to gain some analytical insights into the conservativepart of the O ( m/M )-shift in the orbital frequency Ω mb of the marginally bound orbit that hasrecently been computed numerically in the highly interesting work [10]. To this end, we shalluse the analytically solvable model of an axisymmetric ring of matter which orbits a centralslowly spinning black hole [24]. In particular, we shall use this simplified axisymmetrictoy model (which, due to its symmetry, has no dissipative effects) in order to model theconservative part of the dynamics of the mathematically more complex black-hole-orbiting-particle system [25].We expect the composed axisymmetric black-hole-orbiting-ring system to capture, at leastqualitatively, the essential physical features that characterize the conservative dynamics ofthe composed black-hole-orbiting-particle system. In particular, both the orbiting particlein the black-hole-particle system and the orbiting ring in the black-hole-ring system dragthe generators of the central black-hole horizon [24].The physically intriguing general relativistic effect of dragging of inertial frames by anorbiting object is reflected, both in the black-hole-particle system and in the black-hole-ringsystem, by a non-linear spin-orbit interaction term of order ω H · j in the total gravitationalenergy of the composed systems (here ω H is the angular velocity of the black-hole horizonand j is the angular momentum per unit mass of the orbiting ring).Interestingly, and most importantly for our analysis, the main mathematical advantage ofthe black-hole-orbiting-ring system over the physically more interesting (but mathematicallymore complex) black-hole-orbiting-particle system stems from the fact that the spin-orbitinteraction term in the black-hole-ring system is known in a closed analytical form to secondorder in the dimensionless angular velocity M ir ω H of the central black hole [24] [see Eq. (10)below].In a very interesting work, Will [24] has analyzed the total gravitational energy and thetotal angular momentum of a stationary physical system which is composed of an axisym-metric ring of particles of proper mass m which orbits a central slowly rotating black holeof an irreducible mass M ir . In particular, it has been proved in [24] that the composed ax-4symmetric black-hole-orbiting-ring system is characterized by the total angular momentum J total ( x ) = mj + 4 M ω H − mjx , (5)where x ≡ M ir R (6)is the dimensionless ratio between the irreducible mass of the black hole and the propercircumferential radius of the ring, j ( x ) = M ir [ x (1 − x )] / · [1 + O ( M ir ω H )] (7)is the angular momentum per unit mass of the orbiting ring, and ω H is the angular velocityof the black-hole horizon.Since the first term on the r.h.s of (5) represents the angular momentum J ring of theorbiting ring of mass m , one concludes [24] that the last two terms in (5) represent theangular momentum J H = 4 M ω H − mjx (8)which is contributed by the slowly spinning central black hole as measured by asymptoticobservers. In particular, it is interesting to point out that, while the first term in (8)represents the usual relation between the angular momentum and the angular velocity of aslowly rotating Kerr black hole, the second term on the r.h.s of (8) is a direct consequenceof the dragging of inertial frames caused by the orbiting ring [24].A simple inspection of the compact expression (8) reveals the physically important factthat, unlike vacuum Schwarzschild black holes, a zero angular momentum ( J H = 0) blackhole in the non-linearly coupled black-hole-orbiting-ring system is characterized by the non -zero horizon angular velocity ω H ( J H = 0) = 2 x M · mj . (9)In addition, it has been explicitly proved in [24] that, to second order in the angular ve-locity of the black-hole horizon, the composed axisymmetric black-hole-orbiting-ring systemis characterized by the total gravitational energy E total ( x ) = m − m Φ( x ) + M ir + 2 M ω − ω H mj Ψ( x ) − m x πM ir ln (cid:16) M ir xr (cid:17) (10)5s measured by asymptotic observers. Here we have used the dimensionless radially depen-dent functions Φ( x ) ≡ − − x (1 − x ) / ; Ψ( x ) ≡ x − x − x . (11)The various terms in the energy expression (10), which characterizes the composed black-hole-orbiting-ring system, have the following physical interpretations [24]: • The first term in the energy expression (10) represents the proper mass of the ring. • In order to understand the physical meaning of the second term in the energy ex-pression (10), it is worth pointing out that, in the small- x regime (large ring radius, R ≫ M ir ), this term can be approximated by the compact expression [see Eqs. (6),(10), and (11)] − M ir m/ R · [1 + O ( M ir /R )], which is simply the sum of the potentialand rotational Newtonian energies of the ring in the background of the central com-pact object. Thus, this term represents the leading order (linear in the mass m of thering) interaction between the central black hole and the orbiting ring. • In order to understand the physical meaning of the third and fourth terms in theenergy expression (10), it is worth pointing out that a slowly spinning bare (isolated)Kerr black hole is characterized by the simple mass-angular-velocity relation M Kerr = M ir + 2 M ω + O ( M ω ). Thus, the third and fourth terms in (10) can be identifiedas the contribution of the slowly spinning central black hole to the total energy of thesystem. Interestingly, taking cognizance of Eq. (9) one learns that due to the generalrelativistic frame dragging effect, which is caused by the orbital motion of the ring,the fourth term in (10) contains a self-interaction contribution [of order O ( m /M ir )]to the total energy of the composed black-hole-orbiting-ring system. • The fifth term in the energy expression (10) is a non-linear spin-orbit interactionbetween the slowly spinning central black hole and the orbiting ring. This energy termplays a key role in our composed black-hole-orbiting-ring toy model system since it isexpected to mimic, at least qualitatively, the physically analogous non-linear spin-orbitinteraction in the original black-hole-orbiting-particle system. Taking cognizance ofEq. (9) one learns that, due to the intriguing general relativistic phenomenon of framedragging, the spin-orbit interaction term in (10) contains a non-linear contribution6o the total energy of the composed black-hole-orbiting-ring system which is of order O ( m /M ir ). • The sixth term in the energy expression (10) is the gravitational self-energy of thering [26] (not discussed in [24]), where r ≪ R is the half-thickness of the ring. Thisenergy contribution represents the inner interactions between the many particles thatcompose the axisymmetric ring. Since our main goal in the present paper is to presenta simple analytical toy-model for the physically more interesting (and mathematicallymore complex) two-body system in general relativity, which is characterized by a single orbiting particle, we shall not consider here this many-particle energy contribution.This approximation allows one to focus the physical attention on the general relativistic frame-dragging effect which characterizes both the black-hole-orbiting-particle systemand the black-hole-orbiting-ring system.Taking cognizance of Eqs. (7), (9), (10), and (11), one finds the compact functionalexpression E total ( x ) = M ir + m · h − x (1 − x ) / + 8 x ( − x )(1 − x ) · η + O ( η ) i (12)for the total gravitational energy of the non-linearly coupled black-hole-orbiting-ring system.In the decoupling R/M ir → ∞ limit, in which the ring is located at spatial infinity, thesystem is characterized by the presence of two non-interacting physical objects: (1) a bare(unperturbed) Schwarzschild black hole of mass M = M ir [27], and (2) a ring of proper mass m . Thus, the total energy of the black-hole-ring system in the R/M ir → ∞ limit is given bythe simple expression [see Eq. (12) with x → E total ( R/M ir → ∞ ) = M + m with M = M ir . (13)Energy conservation implies that the marginally bound orbit of the composed black-hole-orbiting-ring system is characterized by the same total gravitational energy [28] E total ( x = x mb ) = M ir + m (14)as measured by asymptotic observers. Substituting the relation (14) into Eq. (12), one findsthe simple expression x mb = 14 · h · η + O ( η ) i (15)7or the O ( m/M ir )-corrected location of the marginally bound circular orbit in the composedblack-hole-orbiting-ring system.Substituting the dimensionless radial coordinate (15) of the marginally bound orbit intothe functional expression [24] M ir Ω = x / · h − x / · M ir ω H + O [( M ir ω H ) ] i (16)for the dimensionless orbital frequency of the axisymmetric orbiting ring and using Eqs. (7)and (9) [29], one obtains the O ( m/M ir )-corrected expression M ir Ω mb = 18 · h · η + O ( η ) i (17)for the characteristic orbital frequency of the marginally bound circular geodesic in thecomposed black-hole-orbiting-ring system. III. SUMMARY
We have analyzed the physical and mathematical properties of a composed black-hole-orbiting-ring system. In particular, we have proposed to use this analytically solvable con-servative [25] system as a simple toy model for the conserved dynamics of the astrophysicallymore interesting (and mathematically more complex) black-hole-orbiting-particle system ingeneral relativity.Our main goal was to provide a simple qualitative analytical explanation for the increase in the orbital frequency of the marginally bound circular geodesic that has recently beenobserved numerically in the physically important work [10]. To this end, we have used thenon-trivial spin-orbit interaction between the central black hole and the orbiting ring, whichis known in a closed analytical form to second order in the dimensionless angular velocity M ir ω H of the black-hole horizon, in order to capture the essential physical features of asimilar non-linear spin-orbit interaction which is expected to characterize the conservativedynamics of the black-hole-orbiting-particle system.Interestingly, the analytically derived expression [see Eqs. (2) and (17)]∆Ω mb Ω mb = 1332 · η + O ( η ) (18)for the dimensionless O ( m/M ir )-shift in the orbital frequency of the marginally bound cir-cular geodesic in the composed black-hole-orbiting-ring system provides the correct order of8agnitude (with the correct sign) for the corresponding shift in the orbital frequency of themarginally bound circular geodesic of the physically more interesting black-hole-orbiting-particle system.This qualitative agreement suggests that the observed shift (3) in the characteristic orbitalfrequency of the marginally bound circular geodesic is mainly determined by the generalrelativistic effect of dragging of inertial frames by orbiting objects (the non-linear spin-orbitinteraction between the orbiting object and the generators of the central black-hole horizon). ACKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I thank Yael Oren, ArbelM. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.9
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