Matching McVittie spacetimes

Gravitational collapse and bubble evolution in the asymptotical Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe is an intriguing and intricate problem. We systematically analyze dynamics for contact \sch-FLRW (McVittie) spacetimes, focusing on their general junction conditions and introducing a novel function to simplify the extrinsic curvature and surface stress-energy tensor. We explore both static and dynamic scenarios, including special cases such as Schwarzschild, FLRW, and Einstein-Straus configurations by using our general framework. Numerical calculations further reveal the evolution of the concentric McVittie spacetimes with various initial conditions, offering deep insights into the interplay between the McVittie mass parameter and initial peculiar velocity. These results provide a unified perspective for understanding gravitational collapse and bubble evolution in cosmology and astrophysics.