标题： Kerr-MOG黑洞周围的球形光子轨道 显示英文标题

标题： Spherical photon orbits around Kerr-MOG black hole

摘要： 本研究调查了Kerr-MOG黑洞周围的光子轨道。 通过求解哈密顿-雅可比方程，得到了Kerr-MOG黑洞周围光子运动的方程，该方程表示为一个六阶多项式，涉及倾角$v$、旋转参数$u$和表征修改引力的变形参数$\alpha$。 我们发现$\alpha$限制了黑洞的旋转，改变了其引力场，并导致了不同的光子轨道特征。 数值分析表明，极平面（$v=1$）有两个有效轨道：一个在事件视界之外，另一个在事件视界之内；而赤道平面（$v=0$）有四个有效轨道：两个在事件视界之外，两个在事件视界之内。 此外，我们推导出了极平面和赤道平面之间一般光子轨道的精确公式（$0<v<1$）。 在极端情况下，旋转速度显著影响一般光子轨道。 一个缓慢旋转的极端黑洞在事件视界之外有两个一般光子轨道，而一个快速旋转的极端黑洞仅有一个这样的轨道。 在非极端情况下，参数空间$\left(v, u, \alpha \right)$中存在一个临界倾角$v_{cr}$。 低于$v_{cr}$，存在四条一般光子轨道，而高于$v_{cr}$，存在两条轨道。 在临界倾角处，发现三个解：两个位于事件视界之外的光子轨道和一个位于事件视界之内的光子轨道。 此外，结果表明所有轨道都是径向不稳定的。 此外，通过分析光子的碰撞参数，我们认为$\alpha$影响黑洞的观测特性。 显示英文摘要

摘要： This study investigates photon orbits around Kerr-MOG black holes. The equation of photon of motion around the Kerr-MOG black hole is derived by solving the Hamilton-Jacobi equation, expressed as a sixth-order polynomial involving the inclination angle $v$, the rotation parameter $u$, and the deformation parameter $\alpha$ that characterizes modified gravity. We find that $\alpha$ constrains the rotation of the black hole, modifying its gravitational field and leading to distinct photon orbital characteristics. Numerical analysis reveals that the polar plane ($v=1$) has two effective orbits: one outside and one inside the event horizon, while the equatorial plane ($v=0$) has four effective orbits: two outside and two inside the event horizon. Moreover, we derive the exact formula for general photon orbits between the polar and equatorial planes ($0<v<1$). In the extremal case, the rotation speed significantly impacts general photon orbits. A slowly rotating extremal black hole has two general photon orbits outside the event horizon, whereas a rapidly rotating extremal black hole has only one such orbit. In the non-extremal case, a critical inclination angle $v_{cr}$ exists in the parameter space $\left(v, u, \alpha \right)$. Below $v_{cr}$, there are four general photon orbits, while above $v_{cr}$, there are two orbits. At the critical inclination angle, three solutions are found: two photon orbits outside and one inside the event horizon. Additionally, the results indicate that all orbits are radially unstable. Furthermore, by analyzing photon impact parameter, we argue that $\alpha$ influences observational properties of the black hole.