The motion of photons around black holes determines the shape of shadow and match the ringdown properties of a perturbed black hole. Observations of shadows and ringdown waveforms will reveal the nature of black holes. In this paper, we study the motion of photons in a general parametrized metric beyond the Kerr hypothesis. We investigated the radius and frequency of the photon circular orbits on the equatorial plane and obtained fitted formula with varied parameters. The Lyapunov exponent which connects to the decay rate of the ringdown amplitude is also calculated. We also analyzed the shape of shadow with full parameters of the generally axisymmetric metric. Our results imply the potential constraint on black hole parameters by combining the Event Horizon Telescope and gravitational wave observations in the future.

光子围绕黑洞的运动决定了阴影的形状，并与受扰的黑洞的衰荡特性相匹配。对阴影和振铃波形的观察将揭示黑洞的本质。在本文中，我们研究了超出克尔假设的一般参数化度量中的光子运动。我们研究了赤道​​平面上光子圆轨道的半径和频率，得到了不同参数的拟合公式。还计算了连接到衰荡幅度衰减率的 Lyapunov 指数。我们还分析了具有一般轴对称度量的完整参数的阴影形状。我们的结果暗示了未来通过结合事件视界望远镜和引力波观测对黑洞参数的潜在约束。