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

The physical properties of an axisymmetric black-hole-ring system are studied analytically within the framework of general relativity to second order in the dimensionless mass ratio $μ\equiv m/M$. In particular, we analyze the asymptotic behaviors of the binding-energy and the total angular-momentum of the two-body system in the vicinity of the light ring at $R=3M$, where the circular orbit becomes null. We find that both quantities diverge {\it quadratically} in $μ(1-3M/R)^{-1}$ at the light ring. The reported divergent behavior of the physical quantities stems from the second order spin-orbit interaction between the black hole and the orbiting object (the dragging of inertial frames by the orbiting ring). It is shown that this composed black-hole-ring toy model captures some of the essential features of the conservative dynamics of the (astrophysically more interesting) black-hole-particle system. In particular, we show that both systems share the {\it same} quadratic divergent behavior of the physical quantities near the light ring. Moreover, we prove that both systems are characterized by the {\it same} ratio ${{E^{(2)}(R\to 3M)}\over{J^{(2)}(R\to 3M)}}={{1}\over{3\sqrt{3}}}$, where $E^{(2)}$ and $J^{(2)}$ are the divergent second order (self-interaction) expansion coefficients of the binding-energies and the angular-momenta of the systems, respectively.