We describe the first axisymmetric numerical code based on the generalized harmonic formulation of the Einstein equations which is regular at the axis. We test the code by investigating gravitational collapse of distributions of complex scalar field in a Kaluza-Klein spacetime. One of the key issues of the harmonic formulation is the choice of the gauge source functions, and we conclude that a damped wave gauge is remarkably robust in this case. Our preliminary study indicates that evolution of regular initial data leads to formation both of black holes with spherical and cylindrical horizon topologies. Intriguingly, we find evidence that near threshold for black hole formation the number of outcomes proliferates. Specifically, the collapsing matter splits into individual pulses, two of which travel in the opposite directions along the compact dimension and one which is ejected radially from the axis. Depending on the initial conditions, a curvature singularity develops inside the pulses.
Weyl证明了四维真空爱因斯坦方程的一般静态轴对称解是由三维平坦空间中拉普拉斯方程的单个轴对称解给出的。Weyl的构造在这里推广到任意维数的$D\ge 4$。本文给出了含有D-2正交交换非零Killing矢量场的D维真空Einstein方程的通解，它可以用三维平坦空间中Laplace方程的D-3独立轴对称解表示，也可以用二维平坦空间中Laplace方程的D-4独立解表示.给出了新的解决方案的明确的例子。其中包括一个五维的渐近平坦的“黑环”，它的事件视界的拓扑结构为S^1 x S^2，由一个圆盘形式的圆锥奇点保持平衡。
General-relativistic perturbation equations for the dynamics of elastic deformable astronomical bodies expanded in terms of generalized spherical harmonics
