Marginally bound circular orbits in the composed black-hole-ring system

TL;DR

This paper analytically investigates the properties of a black-hole-ring system, focusing on the shift in marginally bound orbit frequency due to the ring's influence, and finds qualitative agreement with numerical results for similar systems.

Contribution

It provides the first analytical calculation of the frequency shift of marginally bound orbits in a black-hole-ring system, linking it to frame dragging effects.

Findings

Analytical frequency shift matches numerical results qualitatively.

Positive frequency shift is related to frame dragging.

Results are valid to first order in ring-to-black-hole mass ratio.

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 $M_{\text{ir}}\omega_{\text{H}}$ of the black-hole horizon (here $M_{\text{ir}}$ 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 mass ratio $m/M_{\text{ir}}$, the shift $\Delta\Omega_{\text{mb}}/\Omega_{\text{mb}}$ in the orbital frequency of the {\it marginally bound} circular geodesic that characterizes the composed curved spacetime. Interestingly, our analytical results for the frequency shift $\Delta\Omega_{\text{mb}}$ 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 $O(m/M_{\text{ir}})$, 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.