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≡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 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 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)=133, 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.