  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-Nordstr\"om 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.
