Marginally bound circular orbits in the composed black-hole-ring system
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Regular Article - Theoretical Physics
Marginally bound circular orbits in the composed black-hole-ring system Shahar Hod1,2,a 1 2
The Ruppin Academic Center, 40250 Emeq Hefer, Israel The Hadassah Academic College, 91010 Jerusalem, Israel
Received: 8 September 2020 / Accepted: 6 October 2020 © The Author(s) 2020
Abstract The physical and mathematical properties of the non-linearly coupled black-hole-orbiting-ring system are studied analytically to second order in the dimensionless angular velocity Mir ωH of the black-hole horizon (here Mir 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 mass ratio m/Mir , the shift mb /mb in the orbital frequency of the marginally bound circular geodesic that characterizes the composed curved spacetime. Interestingly, our analytical results for the frequency shift mb in the composed black-hole-orbiting-ring toy model agree qualitatively with the recently published numerical results for the corresponding frequency shift in the physically related (and mathematically much more complex) black-hole-orbiting-particle system. In particular, the present analysis provides evidence that, at order O(m/Mir ), the recently observed positive shift in the angular frequency of the marginally bound circular orbit is directly related to the physically intriguing phenomenon of dragging of inertial frames by orbiting masses in general relativity.
1 Introduction Geodesic orbits of test particles in curved spacetimes are of central importance in black-hole physics [1–14]. 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 blackhole spacetime is of special importance in astrophysics and general relativity [1–14]. This physically interesting geodesic represents the innermost circular orbit of a massive particle which is energetically bound to the central black hole.
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For a test particle of proper mass m, the marginally bound circular geodesic is characterized by the simple energy relation [1–4] E(rmb ) = m,
(1)
where E is the energy of the particle as measured by asymptotic observers. Interestingly, the marginally bound circular geodesic (1) marks the boundary between bound orbits, which are characterized 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, as nicely demonstrated in [5,6], the critical (marginally bound) circular geodesic (1) plays a key 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 innermost bound spherical orbit (IBSO) [6,7]]. An important gauge-invariant physical quantity that characterizes the motion of particles along
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