A Family of Horizon-penetrating Coordinate Systems for the Schwarzschild Black Hole Geometry with Cauchy Temporal Functions
Christian Röken
TL;DR
This work develops a constructive method to build horizon-penetrating coordinate systems for Schwarzschild (and RN up to the Cauchy horizon) that are based on Cauchy temporal functions whose level sets yield smooth, asymptotically flat spacelike Cauchy hypersurfaces spanning exterior and interior regions. The core idea is to reform the Penrose diagram into a diamond and define a family of smooth foliations via functions $V_ u(W)$ subject to precise geometric and causal conditions, guaranteeing that the resulting time coordinate $ u$ is globally temporal and its leaves form a Cauchy foliation. An explicit integrated algebraic sigmoid example provides closed-form expressions for the foliation and the transformed metric, showing regular behavior at the foliation parameter $ u=0$ and ensuring globally hyperbolic evolution across the horizon. The construction is then generalized to Reissner–Nordström up to the Cauchy horizon, with discussion of extensions to Kerr and other spacetimes possessing similar product structures, highlighting both the potential and the limitations of the approach for more complex geometries.
Abstract
We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry featuring time coordinates that are Cauchy temporal functions for which the level sets 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.