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 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.

对非线性耦合黑洞轨道环系统的物理和数学性质进行了无量纲角速度\（M _ {\ text {ir}} \ omega _ {\ text {H}}的二阶分析研究。黑洞视界的\）（此处\（M _ {\ text {ir}} \）是缓慢旋转的中心黑洞的不可约质量）。特别地，我们以无量纲的环对黑洞质量比\（m / M _ {\ text {ir}} \）进行解析确定，从而得出位移\（\ Delta \ Omega _ {\ text { mb}} / \ Omega _ {\ text {mb}} \）的边界有限的圆形测地线的轨道频率，它表征了合成的弯曲时空。有趣的是，我们对频移的分析结果合成的黑洞环玩具模型中的\（\ Delta \ Omega _ {\ text {mb}} \）与最近发布的与物理相关的频率偏移的数学结果定性一致（并且在数学上更多）黑洞轨道粒子系统。特别地，本分析提供了证据，证明在阶次\（O（m / M _ {\ text {ir}}）\）下，最近观察到的边际约束圆轨道角频率的正向位移与正弦波直接相关。在广义相对论中绕轨道绕惯性运动拖动惯性系的物理现象。