类空无穷远圆柱附近共形不变波动方程的全伪谱解。III：零无穷远处的非球面史瓦西波和奇点
Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: Nonspherical Schwarzschild waves and singularities at null infinity
摘要：
我们从球对称解的情况下，在Schwarzschild背景上的共形不变的波动方程的数值和分析考虑，在类中讨论。量子重力34，045005（2017），一般的情况下，非对称的解决方案。我们的方法的一个关键要素是现代标准表示类空无限作为一个圆柱体。通过球谐分解，我们将四维波动方程化为一族二维方程。这些方程可以用来研究在圆柱体的行为，在那里的解决方案变成有无穷多个订单的对数奇点。我们推导出的正则性条件，可以施加在初始数据，以避免第一奇异项。然后，我们证明了完全伪谱时间演化方案可以应用于该问题，从而对非对称解进行高精度的数值重建。我们特别感兴趣的行为的解决方案在未来的零无穷大，我们数值计算表明，奇点传播从圆柱到零无穷大。所观察到的数值行为是一致的，类似的对数奇异性分析发现的气缸。最后，我们证明，即使在低阶奇异的解决方案，可以获得高精度的坐标变换，转换成光滑的解决方案与对数奇点的功能。
英文摘要：
We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Class. Quantum Grav. 34, 045005 (2017), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the four-dimensional wave equation to a family of two-dimensional equations. These equations can be used to study the behaviour at the cylinder, where the solutions turn out to have logarithmic singularities at infinitely many orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that the fully pseudospectral time evolution scheme can be applied to this problem leading to a highly accurate numerical reconstruction of the nonsymmetric solutions. We are particularly interested in the behaviour of the solutions at future null infinity, and we numerically show that the singularities spread from the cylinder to null infinity. The observed numerical behaviour is consistent with similar logarithmic singularities found analytically on the cylinder. Finally, we demonstrate that even solutions with singularities at low orders can be obtained with high accuracy by virtue of a coordinate transformation that converts functions with logarithmic singularities into smooth solutions.
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