A family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry with Cauchy temporal functions

• Main Author: Röken, C. (Author)
• Format: Article
• Language: English
• Published: Springer 2022
• Subjects: Black hole geometries; Cauchy temporal functions; Global horizon-penetrating coordinate systems; Penrose diagrams
• Online Access: View Fulltext in Publisher

 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. © 2022, The Author(s).