Gravitational collapse: Generalizing Oppenheimer-Snyder and a conjecture on horizon formation time

We generalize the Oppenheimer-Snyder model of gravitational collapse by considering a broader class of static, spherically symmetric exterior spacetimes, with an interior geometry described by a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) geometry. Using Painlev\'e-Gullstrand (PG) coordinates for the spatially flat interior geometry ($k=0$) and a Novikov-like coordinate system for the spatially closed geometry ($k=1$), we ensure a smooth transition between the interior and exterior of the collapsing star. By providing general formulas, we analyze how apparent and event horizons form during the collapse and examine whether the matter satisfies standard energy conditions. For both $k=0$ and $k=1$ cases, we study explicit examples such as Schwarzschild, Schwarzschild--AdS/dS, and Reissner-Nordstr\"om (RN) black holes, taking into account the effects of the cosmological constant and electric charge. These factors significantly influence the collapse process and can impose constraints on the physical parameters. Our analysis leads to two important results: first, to form a black hole, there is a minimum or critical initial radius for the star to begin collapsing. Second, we propose a conjecture of an inequality regarding the event horizon formation time, starting from the critical radius, namely $\mathrm{\ensuremath{\Delta}}{T}_{\mathrm{eh}}\ensuremath{\le}19M/6$. The upper bound is saturated by the Schwarzschild black hole.