We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also ‘spread’ to future null infinity. However, by imposing certain regularity conditions on the initial data, the lowest-order singularities can be removed. Here we are interested in a generalisation of these results to a rotating black hole background and study the influence of the rotation rate on the properties of the solutions. To this aim, we first construct a conformal compactification of the Kerr solution which yields a suitable representation of the cylinder at spatial infinity. Besides analytical investigations on the cylinder, we numerically solve the wave equation with a fully pseudospectral method, which allows us to obtain highly accurate numerical solutions. This is crucial for a detailed analysis of the regularity of the solutions. In the Schwarzschild case, the numerical problem could effectively be reduced to solving (1 + 1)-dimensional equations. Here we present a code that can perform the full 2 + 1 evolution as required for axisymmetric waves on a Kerr background.

我们通过数值和解析方法研究了克尔背景上共形不变波动方程的轴对称解。我们的主要重点是接近空间无穷大的解的行为，它被适当地表示为一个圆柱体。Schwarzschild 背景下波动方程的早期研究揭示了有关相应解的规律性的重要细节。发现，在圆柱体上，解通常以无限多的阶数产生对数奇点。此外，这些奇点也“传播”到未来的零无穷大。然而，通过对初始数据施加一定的规律性条件，可以去除最低阶的奇点。在这里，我们感兴趣的是将这些结果推广到旋转的黑洞背景，并研究旋转速率对解决方案性质的影响。为此，我们首先构建了 Kerr 解的共形紧缩，它产生了空间无限远圆柱的合适表示。除了对圆柱体的分析研究外，我们还使用全伪谱方法对波动方程进行了数值求解，这使我们能够获得高精度的数值解。这对于详细分析解的规律性至关重要。在 Schwarzschild 案例中，数值问题可以有效地简化为求解 (1 + 1) 维方程。在这里，我们提出了一个代码，可以根据克尔背景上的轴对称波的要求执行完整的 2 + 1 演化。