In this paper, the neutral particle trajectory around a rotating black hole in the four-dimensional Kerr space–time is studied. The equation of motion is derived by using the Kerr metric within the slowly rotating and weak-field limit. The stability conditions of the orbital motion of the neutral particle are deduced. For small amplitude of motion, the approximate solutions of the resulting differential equations are obtained by the perturbation method of Lindstedt and Poincare. These are then expressed as power series for some small perturbation parameters. The angular frequencies are also found in the corresponding order of the perturbation. By using angular frequencies, an expression of the perihelion advance is derived. To find the orbital equation similar to Newtonian orbit, the coordinate system is transformed into isotropic coordinates. The Newtonian orbital equation is derived and hence the eccentricity of the orbit is obtained. The resulting expression of circular motion shows the relationship between energy and angular momentum of a spinning particle in the Kerr field.

中文翻译：

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克尔场中的中性粒子轨迹

本文研究了四维克尔时空中围绕旋转黑洞的中性粒子轨迹。运动方程是通过在缓慢旋转和弱场限制内使用克尔度量推导出来的。推导出中性粒子轨道运动的稳定条件。对于小幅度的运动，所得微分方程的近似解是通过 Lindstedt 和 Poincare 的微扰方法获得的。然后将它们表示为一些小扰动参数的幂级数。角频率也按照相应的扰动顺序找到。通过使用角频率，推导出近日点提前的表达式。为了找到类似于牛顿轨道的轨道方程，将坐标系转换为各向同性坐标。推导出牛顿轨道方程，从而获得轨道的偏心率。由此产生的圆周运动表达式显示了克尔场中旋转粒子的能量和角动量之间的关系。