Schwarzschild–de Sitter spacetime in regular coordinates with cosmological time

Starting from the Einstein equations in Schwarzschild--de Sitter (SdS) spacetime and imposing Friedmann-Robertson-Walker coordinates at large distances, we find two coordinate systems with time-dependent metrics that are smooth across both the black hole and cosmological horizons. These coordinates require positive cosmological constant for them being regular, and thus they are not de Sitter extensions of the Kruskal--Szekeres or Israel coordinates. One of the coordinate systems was only found in 1999 (Abbassi coordinates), and it has led to conflicting interpretations in the literature; while the other was briefly commented and promptly dismissed as unphysical or incompatible with SdS. We derive that the second solution is equivalent to the first one, and that both are indeed equivalent descriptions of SdS spacetime. We also derive explicit coordinate transformations linking these coordinate systems to the Kottler coordinates and the maximally extended Lake--Israel coordinates. Among other applications, these results, which extend the largely used cosmological and local coordinates, should be useful for further developments on understanding the exact interplay between black holes and the cosmological background, which has been the focus of a number of recent works.