Classification of Oppenheimer-Snyder Collapse: Singular, Bouncing, and Soft-Landing Scenarios

TL;DR

This work extends Oppenheimer-Snyder collapse to general special-static exteriors with two horizons, revealing two physical signatures—a star-surface bounce and an apparent-horizon left vertex—that classify OS outcomes. The RN exterior is shown to generically exhibit both features, yielding a threefold fragmentation of collapse into singular, bouncing, and soft-landing scenarios, with the bounce tied to an inner repulsive core and the left vertex tied to AH dynamics; these results are linked to inner-horizon instability and Penrose’s strong cosmic censorship. In contrast, regular black holes with de Sitter cores exhibit neither bounce nor left vertex, instead undergoing monotone collapse with a soft landing, and a no-go theorem shows Minkowski-core regular BHs must violate the NEC. Altogether, the paper provides a geometry-driven classification framework that connects exterior metric properties, horizon dynamics, and interior fate, with implications for inner-horizon stability and SCCC.

Abstract

We study Oppenheimer-Snyder (OS) gravitational collapse matched to a general static, spherically symmetric exterior spacetime. Unlike the Schwarzschild case, two new features can arise in black holes with two horizons: an apparent-horizon left vertex, a temporary minimum in the apparent-horizon radius during collapse, and a bounce, where the star surface stops collapsing at a nonzero radius and reverses into expansion. We identify the conditions that lead to these two features. For two-horizon exteriors, trapped-region consistency requires that the apparent-horizon turning point occurs no earlier than the surface crossing of the inner horizon. As a concrete example, the OS collapse of the Reissner-Nordström (RN) spacetime shows both effects. In contrast, regular black holes with de Sitter cores show neither: their collapse is smooth and monotonic, and the surface approaches the center only as the proper time goes to infinity. These results naturally classify the OS collapses into three categories: singular, which ends at the center in finite time; bouncing, which reverses at a finite radius; and soft-landing, which reaches the center only asymptotically. We argue that these features are consistent with Penrose's strong cosmic censorship conjecture.

Figures (3)

• Figure 1: Depending on the charges, two possible time orderings between the apparent-horizon turning point and the inner-horizon crossing in RN OS collapse with $m=1$ (left: $q=0.65$; right: $q=0.95$). The black curve is the star-surface trajectory $(R,\,T(R))$, while the red curve is the apparent-horizon trajectory $\bigl(R_{\rm AH}(R),\,T(R)\bigr)$. The condition \ref{['eq:RN_vertex_physcrit']} is a global consistency requirement: it ensures that the relevant inner apparent-horizon evolution connects smoothly to the surface crossing of the inner horizon; otherwise, the turning occurs too early, while the surface is still outside $R_-$.
• Figure 2: RN collapse in the $(R,T)$ plane. The initial radius is $R_0=4.5$, and the inner and outer horizons are set to $(R_{-},R_{+})=(0.5,\,1.5)$. These choices fix the mass and charge as $m=(R_{+}+R_{-})/2=1$ and $q^2=R_{+}R_{-}=0.75$. The collapse features three key radii: the surface bounce radius $R_\ast=0.375$, the apparent-horizon vertex $R_{\rm turn}=0.5=R_{-}$, and the inflection point $R_{\rm infl}=0.75$. The black curve shows the star surface $R(T)$. The blue and red curves denote the apparent horizon and event horizon, respectively. The trapped regions are shaded.
• Figure 3: Collapse diagram in the $(R,T)$ plane for a electrically-charged regular black hole in EMS gravity with $(n,\alpha)=(1,1)$ and $(q,m)=(15,m_{\rm cr}(15))$, starting from $R_0=40$. The vertical lines indicate the horizon radii $R_\pm$ in \ref{['eq:EMS_Rpm_numeric']}. The star surface $R(T)$ (black) exhibits no turning point and thus no bounce. The apparent-horizon curve $R_{\rm AH}(T)$ (pink) is monotone and has no left vertex. The inflection radius $R_{\rm infl}\simeq 5.47$ is defined by $\ddot R=0$: the collapse accelerates in the $R>R_{\rm infl}$ region and decelerates in the $R<R_{\rm infl}$ region, consistent with the soft-landing behavior as $R\to0$.