For a vacuum initial data set of the Einstein field equations it is possible to carry out a conformal rescaling or conformal compactification of the data giving rise to an initial data set for the Friedrich vacuum conformal equations. When will the data development with respect to the conformal equations of this set be a conformal extension of a type D solution? In this work we provide a set of necessary and sufficient conditions on a set of initial data for the conformal equations that guarantees that the data development of the conformal equations has a subset that is conformal to a vacuum type D solution of the Einstein’s equations. In particular we find the conditions under which this vacuum solution corresponds to the Kerr solution. Using our results we are able to show that there are no obstructions to extend the Petrov type of the physical spacetime to the unphysical spacetime if the conformal data are hyperboloidal.

中文翻译：

---

D型保形初始数据

对于爱因斯坦场方程的真空初始数据集，可以对数据进行共形重标度或共形紧缩，从而产生弗里德里希真空共形方程的初始数据集。关于该集合的共形方程的数据开发何时会成为 D 类解的共形扩展？在这项工作中，我们为共形方程的一组初始数据提供了一组必要和充分条件，以保证共形方程的数据开发具有与爱因斯坦方程的真空 D 型解共形的子集。特别地，我们找到了该真空解决方案对应于克尔解决方案的条件。