The late-time behaviour of the solutions of the Fackerell–Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell–Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price’s law relevant to the Fackerell–Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell–Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.

对 Fackerell-Ipser 方程（这是电磁场强度张量的自旋零分量的波动方程）的解对亚极端 Kerr 时空外部通信域闭合的后期行为进行了数值研究。在克尔家族中，也考虑了史瓦西背景的情况。使用穿透水平的压缩双曲面坐标，这允许在事件视界和未来零无穷大处观察解的行为。对于初始数据，采用具有紧凑支撑且稳态或非稳态的纯多极构型。研究发现，利用这些初始数据，Fackerell-Ipser 方程的解在后期收敛到已知的静态解（最多为常数因子）或收敛为零。当接近极限时，解表现出准正规振铃，最后出现幂律衰减。从初始数据的各种参数值的数值数据中提取表征场变量球谐分量幂律衰减的指数，并根据结果提出与 Fackerell-Ipser 方程相关的 Price 定律。某些守恒能量和角动量电流用于验证基础数学模型的数值实现。在构建这些电流时，还考虑了 Fackerell-Ipser 方程的离散对称性，该方程是赤道反射和复共轭的乘积。