Gravitational Collapse: Generalizing Oppenheimer-Snyder and a Conjecture on Horizon Formation Time

TL;DR

This work extends the Oppenheimer-Snyder model by matching a homogeneous FLRW interior to a broad class of static, spherically symmetric exteriors via coordinate systems tailored to freely falling observers. It develops general surface-evolution equations and analyzes the formation of apparent and event horizons under different interior curvatures ($k=0,1$) and exterior geometries including Schwarzschild, Schwarzschild–AdS/dS, and RN spacetimes, while checking standard energy conditions. A key result is the identification of a minimum initial radius required for black-hole formation and a conjectured universal bound on horizon-formation time: $\ \\Delta T_{ ext{eh}} \le \\frac{19}{6} M$, with the Schwarzschild case saturating the bound; the bound depends on the exterior geometry and the presence of a cosmological constant or charge. The findings illustrate how global spacetime structure, Λ, and electric charge influence collapse dynamics, horizon development, and the viability of OS-type scenarios in GR.

Abstract

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-Lemaitre-Robertson-Walker (FLRW) geometry. Using Painleve-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 ensured a smooth transition between the interior and exterior of the collapsing star. By providing general formulas, we analyzed how apparent and event horizons form during the collapse and checked whether the matter satisfies standard energy conditions. For both k=0 and k=1 cases, we studied explicit examples such as Schwarzschild, Schwarzschild-AdS/dS, and Reissner-Nordstrom (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 Delta T_eh <= 19M/6. The upper bound is saturated by the Schwarzschild black hole.

Figures (5)

• Figure 1: The $k=0$ collapsing of the $m=1$ black holes with a cosmological constant. The black and gray lines show the star's surface radius eq. (\ref{['tsch1']}) for Schwarzschild-dS ($R_+=2.2$) and AdS ($R_+=1.8$), respectively. The dotted line is the usual Schwarzschild black hole with $R_+=2$. The red and blue lines (dS), and pink and purple lines (AdS) show the apparent and event horizons, respectively, obtained by solving eqs. (\ref{['reh1']}), (\ref{['rapsch1']}), and (\ref{['tsch1']}). For the dS case, $R_+ = 2.2$, $R_{0,\text{min}} = 5.54$, and for the AdS case, $R_+ = 1.8$, $R_{0,\text{min}} = 3.56$. Trapped surfaces are marked in yellow and pink for the interior and exterior regions of the star, respectively.
• Figure 2: $k = 0$, $R_0 = 4.5$: The black line shows the star's surface from eq. (\ref{['RNs1']}). The red and blue lines show the inner apparent and event horizons obtained from eqs. (\ref{['RNs1']}), (\ref{['RNrap1']}), and (\ref{['RNreh1']}), respectively. The inner horizon evolution is also given by $T_{\text{ca}}(R)$ from eq. (\ref{['RNreh2']}), and represented by the blue dashed line. All three curves meet at $R_+ = 1.5$ and $R_- = 0.5$. Trapped surfaces are marked in yellow and pink for inside and outside areas of the star, respectively.
• Figure 3: $k = 0$, $R_0 = 4.5$: The black, red, blue, and blue-dashed lines show $T(R)$, $T_{\text{eh}}(R)$, $T_{\text{ap}}(R)$, and $T_{\text{ca}}(R)$ obtained from eqs. (\ref{['RNs1']}), (\ref{['RNrap1']}), (\ref{['RNreh1']}), and (\ref{['RNreh2']}), respectively. All three curves meet at $R_+ = 1.8$, but they do not meet at $R_- = 0.2$. Trapped surfaces are marked in yellow and pink for inside and outside areas of the star, respectively.
• Figure 4: The collapsing of the Schwarzschild of $m=1$. We consider $k = 0$, $R_0 = 5$. The black line represents the star's surface radius (\ref{['tsch']}), while the red and blue lines correspond to the interior apparent and event horizons, respectively, given by eqs. (\ref{['tapsch']}) and (\ref{['tehsch']}). All three curves intersect at $T = (-4 + 5\sqrt{10})/3$ and $R_+ = 2$. Trapped surfaces are marked in yellow and pink for inside and outside areas of the star, respectively.
• Figure 5: The collapsing of the Schwarzschild of $m=1$, with $k = 1$, $R_{\text{max}} = 5$. The black line represents the star's surface radius (\ref{['tschc']}), while the red and blue lines correspond to the apparent and event horizons, respectively, obtained by numerically solving eqs. (\ref{['rapschc']}), (\ref{['rehschc']}), and (\ref{['tschc']}). All three curves intersect at $R_+ = 2$. Trapped surfaces are marked in yellow and pink for inside and outside areas of the star, respectively.