Explicit construction of Penrose diagrams for black hole to white hole transition with spacelike thin shells

Abstract

Abstract In this article, we explicitly construct the coordinates associated with the Penrose diagram in spacetimes connected via a spacelike thin shell in the following two examples: the generalized black-to-white hole bounce with mass difference and the Schwarzschild-to-de Sitter transition. We point out the issue of the first junction condition in the Penrose diagram constructed by cutting and pasting analytically known metrics with spherical symmetry by a static spacelike thin shell. With the goal of a global conformal coordinate chart associated with the corresponding Penrose diagram without discontinuity at the thin shell, we give a procedure consisting of three conformal transformations that serve different purposes. The first two of them are used to generate a continuous coordinate patch covering the entire thin shell, and therefore, the Penrose diagram can be constructed properly by patches with overlapping. The third transformation removes any coordinate singularity reintroduced by the first two transformations at the event horizons.