Photon spherical orbits around a Kerr black hole

Abstract Black holes are trapped spacetime regions. Their inexorable gravitational attraction determines that nothing – not even light – may escape from this region. Thus, a black hole cannot be directly observed. It must be studied by the way it influences the motion of matter particles and light rays in its neighborhood. These trajectories are determined (within a certain approximation) by the black hole geometry's geodesics. In particular, null geodesics – those followed by light rays – possess considerable astrophysical interest: they represent how the radiation emitted around the black hole, or incoming from far away sources, is distorted by the black hole. In this work we present the equations of motion of light rays in the Kerr geometry, that describes a rotating black hole, making use of Hamilton-Jacobi theory. Subsequently, we analyze the conditions for the existence of spherical photon orbits around the black hole. These orbits, representing strong light bending, are the key to determine the optical appearance of the black hole. Finally, using computational methods, we exhibit plots illustrating some of these trajectories.

摘要 黑洞是受困的时空区域。其势不可挡的引力作用决定了任何物质——哪怕是光线——都无法从该区域逃逸，因此黑洞无法被直接观测，只能通过其对邻近区域内物质粒子与光线运动的影响开展研究。这些运动轨迹（在特定近似条件下）由黑洞几何的测地线（geodesics）所决定。具体而言，零测地线（null geodesics）——即光线所遵循的轨迹——具有重要的天体物理研究价值：它们刻画了黑洞周围辐射或来自遥远天体的辐射如何被黑洞扭曲。本研究借助哈密顿-雅可比理论（Hamilton-Jacobi theory），推导得出了描述旋转黑洞的克尔几何（Kerr geometry）中光线的运动方程。随后，本研究分析了黑洞周围球形光子轨道的存在条件。这类代表强光偏折的轨道，是确定黑洞光学外观的关键。最后，本研究借助计算方法，展示了部分此类轨迹的绘图结果。