Regular frames for spherically symmetric black holes revisited

TL;DR

This paper unifies different regular coordinate frames for spherically symmetric black holes, showing that Kruskal-Szekeres and Lemaître frames originate from the same root and providing a comprehensive scheme for horizon-regular coordinates.

Contribution

It demonstrates the common origin of Kruskal-Szekeres and Lemaître frames and develops a general scheme for regular coordinate frames across black hole horizons.

Findings

Unified the Kruskal-Szekeres and Lemaître frames under a common framework.

Linked transformation parameters to the energy of observers.

Derived a smooth limit for the homogeneous metric under the horizon.

Abstract

We consider a space-time of a spherically symmetric black hole with one simple horizon. As a standard coordinate frame fails in its vicinity, this requires continuation across the horizon and constructing frames which are regular there. Up to now, several standard frames of such a kind are known. It was shown in literature before, how some of them can be united in one picture as different limits of a general scheme. However, some types of frames (the Kruskal-Szekeres and Lema\^{\i}tre ones) and transformations to them from the original one remained completely disjoint. We show that the Kruskal-Szekeres and Lema\^{\i}tre frames stem from the same root. Overall, our approach in some sense completes the procedure and gives the most general scheme. We relate the parameter of transformation $e_{0}$ to the specific energy of fiducial observers and show that in the limit $% e_{0}\rightarrow 0$ a homogeneous metric under the horizon can be obtained by a smooth limiting transition.