We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry that feature time coordinates, which are specific Cauchy temporal functions, i.e., the level sets of these time coordinates are smooth, asymptotically flat, spacelike Cauchy hypersurfaces. Coordinate systems of this kind are well suited for the study of the temporal evolution of matter and radiation fields in the joined exterior and interior regions of the Schwarzschild black hole geometry, whereas the associated foliations can be employed as initial data sets for the globally hyperbolic development under the Einstein flow. For their construction, we formulate an explicit method that utilizes the geometry of—and structures inherent in—the Penrose diagram of the Schwarzschild black hole geometry, thus relying on the corresponding metrical product structure. As an example, we consider an integrated algebraic sigmoid function as the basis for the determination of such a coordinate system. Finally, we generalize our results to the Reissner–Nordström black hole geometry up to the Cauchy horizon. The geometric construction procedure presented here can be adapted to yield similar coordinate systems for various other spacetimes with the same metrical product structure.

我们为史瓦西黑洞几何引入了一个新的视界穿透坐标系，它具有时间坐标，它们是特定的柯西时间函数，即这些时间坐标的水平集是光滑的、渐近平坦的、类空间的柯西超曲面。这种坐标系非常适合研究史瓦西黑洞几何形状的外部和内部连接区域中物质和辐射场的时间演化，而相关的叶状结构可以用作全球双曲线发展的初始数据集在爱因斯坦流下。对于它们的构造，我们制定了一种明确的方法，该方法利用了史瓦西黑洞几何的彭罗斯图的几何和固有的结构，从而依靠相应的度量产品结构。作为一个例子，我们考虑一个集成的代数 sigmoid 函数作为确定这种坐标系的基础。最后，我们将我们的结果推广到 Reissner-Nordström 黑洞几何直到柯西视界。此处介绍的几何构造过程可适用于为具有相同度量乘积结构的各种其他时空产生类似的坐标系。