Dynamics of Apparent Horizon in Collapse

• The apparent horizon is defined as the locus where the expansion of outgoing null geodesics vanishes, marking the boundary between trapped and untrapped regions.
• Its evolution, governed by local mass functions and spacetime foliation, is critical in predicting whether gravitational collapse results in a black hole or a locally naked singularity.
• A positive slope in the local expansion series (χ(0) > 0) signals the emergence of outgoing null geodesics, potentially leading to observable naked singularities.

The dynamics of the apparent horizon describe how the outer boundary of the trapped region—formed by marginally trapped surfaces (where the expansion of outgoing null geodesics vanishes)—evolves in time as massive matter clouds undergo gravitational collapse. In general relativity, the apparent horizon (AH), unlike the event horizon, is a quasi-local construct sensitive to the spacetime foliation and responds dynamically to the matter flux and geometry inside a collapsing cloud. The timing and evolution of the AH in relation to the formation of the spacetime singularity are central to determining whether gravitational collapse produces a black hole or a naked singularity. Recent rigorous analysis has established precise criteria for the causal structure of the resulting singularity in spherical symmetry, elucidating the fundamental role of AH dynamics in deciding the end state of collapse (Koushiki et al., 20 Aug 2025).

1. Apparent Horizon and Trapped Surfaces in Spherical Collapse

In spherically symmetric spacetimes with the metric:

$$ds^2 = -e^{2\nu} dt^2 + e^{2\psi} dr^2 + R^2 d\Omega^2,$$

(where $R = r\, a(r,t)$ is the physical area radius), the formation of a trapped surface is associated with the condition that both null expansions are negative—signaling that all outgoing and ingoing null geodesics are converging. The apparent horizon is defined as the locus where the expansion of the outgoing null congruence $\Theta_l$ vanishes:

$\Theta_l = g^{ij} \partial_i R \partial_j R = 0.$

This condition is equivalent to:

$$G - H = 1 - \frac{F(r, t)}{R} = 0 \implies F/R = 1,$$

where $F(r,t)$ is the Misner–Sharp mass function (the generalized mass encapsulated within the comoving radial coordinate $r$ at time $t$). The AH thus demarcates the boundary inside which all outgoing light rays are trapped ($F/R > 1$), and outside which at least some can escape ($F/R < 1$).

2. Local Temporal Structure: Singularity and AH Curves

Dynamical gravitational collapse is characterized by tracking two fundamental curves:

• The singularity curve $t_s(r)$, defined by $a(r, t_s(r)) = 0$ (i.e., $R = 0$), denoting when each shell reaches the central singularity.
• The apparent horizon curve $t_{AH}(r)$, determined by the AH condition $F/R = 1$. Both are expanded near the center as:

$$t_s(r) = t_{s0} + r\,\chi(0) + O(r^2), \qquad t_{AH}(r) = t_{AH,0} + r\,\Psi(0) + O(r^2).$$

The critical result is that, at the center ($r=0$), $t_{s0} = t_{AH,0}$: the formation of the central singularity and the AH coincide in time. However, the slopes of these curves—the coefficients $\chi(0)$ and $\Psi(0)$—determine the subtle local structure of their evolution and, ultimately, the visibility of the singularity.

3. Outgoing Null Geodesics and the Visibility Criterion

The presence of outgoing future-directed null geodesics emanating from the central singularity is the defining characteristic of a locally naked singularity. Geodesics are governed by:

$\frac{dt}{dr} = e^{\psi - \nu},$

which can be recast near the singularity in terms of a suitable parameter $u = r^\alpha$ ($\alpha > 1$), capturing the local behavior:

$x_0 = \lim_{t\to t_s,\, r\to 0} \frac{R}{u},$

where $x_0 > 0$ signals that null geodesics can escape from the singularity. The explicit calculation yields:

$$x_0 = \left( \frac{3}{2} \sqrt{\mathcal{M}_0}\,\chi(0) \right)^{2/3},$$

where $\mathcal{M}_0$ is the central value of the Misner–Sharp mass. Therefore, a necessary and sufficient condition for outgoing null geodesics to exist is:

$\chi(0) > 0.$

Since $\Psi(0) = \chi(0)$ at $r=0$, this is equivalently the condition that the AH curve is increasing in $r$ at the center.

4. Dynamical Interplay Between AH Formation and Singularity

In "standard" black hole formation—such as in the Schwarzschild or Oppenheimer–Snyder scenarios—trapped surfaces and thus the AH form strictly before the appearance of the singularity, ensuring that outgoing geodesics are always contained within the trapped region and cannot escape. In the generic situation analyzed, however, the formation of the central singularity and central AH is simultaneous, and only the behavior of higher-order terms ($\chi(0)$ and beyond) dictates whether light can escape. If the first nontrivial term ($\chi(0)$) is positive, a family of null geodesics emerges from the singularity, producing a locally naked singularity. If $\chi(0)\leq 0$, the trapped region covers the singularity from the outset.

If all lower-order terms vanish, higher-order terms in the Taylor expansion of $t_s(r)$ and $t_{AH}(r)$ must be investigated. The sign of the first non-vanishing term remains the decisive factor for the visibility of the singularity.

5. Implications for Cosmic Censorship and Gravitational Collapse Outcomes

The causal structure determined by AH dynamics has direct implications for the cosmic censorship hypothesis. The condition:

$\chi(0) > 0$

is both necessary and sufficient for the formation of a locally naked singularity (i.e., observable at least from the arbitrarily close vicinity of the central singularity), whereas $\chi(0) \leq 0$ ensures censorship by keeping the entire singularity within a trapped region—effectively producing a black hole. Thus, AH dynamics, governed by the interplay between mass function evolution, initial matter profiles, and local geometry, provides a framework for predicting the outcome of generic gravitational collapse under the weak energy condition and $\mathcal{C}^2$ regularity.

Condition	Outcome	Key Parameter
$\chi(0) > 0$	Locally naked singularity	Slope of $t_s(r)$ at $r=0$
$\chi(0) \leq 0$	Black hole (censored)	Slope of $t_s(r)$ at $r=0$

A plausible implication is that the nature of the end state for a generic type-I matter field collapse can, in principle, be tuned by engineering initial density and pressure profiles that control $\chi(0)$ and the timing of trapped surface formation.

6. Summary

The dynamics of the apparent horizon in gravitational collapse directly govern the causal structure of the final singularity by determining when and where a trapped region forms relative to the singularity. The simultaneous formation of the AH and singularity at the center necessitates analysis of their local expansions: a positive slope ($\chi(0) > 0$) yields families of escaping null geodesics and hence a naked singularity, while non-positive slope guarantees the singularity remains hidden behind the trapped region. These results, for spherically symmetric collapse with general type-I matter and generic regular initial data, supply precise necessary and sufficient conditions on AH dynamics for the (non-)visibility of spacetime singularities (Koushiki et al., 20 Aug 2025). This analysis refines our understanding of the end states of collapse beyond the traditional global or event horizon picture and connects quasi-local geometric dynamics with fundamental questions in classical gravity.