A Family of Horizon-penetrating Coordinate Systems for the Schwarzschild Black Hole Geometry with Cauchy Temporal Functions

TL;DR

This paper introduces a new family of horizon-penetrating coordinate systems for Schwarzschild and Reissner-Nordström black holes, facilitating the study of matter and radiation evolution within these spacetimes.

Contribution

It presents an explicit geometric construction method for horizon-penetrating coordinates with Cauchy temporal functions, applicable to Schwarzschild and Reissner-Nordström black holes.

Findings

Coordinates are well suited for evolving matter and radiation fields.

The construction relies on the Penrose diagram and metric product structure.

Method can be adapted to other spacetimes with similar geometry.

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-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.