Formation of trapped surfaces for the spherically symmetric Einstein-Yang-Mills system with non-trivial incoming data

Technical Summary: Formation of Trapped Surfaces for the Spherically Symmetric Einstein–Yang–Mills System

Problem Statement
This paper addresses the dynamical question of gravitational collapse within the spherically symmetric Einstein–Yang–Mills (EYM) system with a purely magnetic $SU(2)$ gauge group. While the static theory of this system is well-understood—featuring globally regular particle-like solutions (Bartnik-McKinnon) and unstable black holes with non-trivial interior fields—the dynamical evolution from regular characteristic data remains complex. Specifically, the authors investigate the quantitative conditions under which regular initial data on intersecting null hypersurfaces evolve to form a trapped surface.

A primary challenge in this system, distinguishing it from the Einstein–Maxwell or Einstein–scalar field models, is the presence of non-Abelian gauge fields. These fields introduce a competition between focusing mechanisms (kinetic energy) and defocusing mechanisms (magnetic potential). Furthermore, unlike the electric charge in the Einstein–Maxwell system, the magnetic "charge" aspect $Q$ in the EYM system is dynamical and coupled to the radiative component of the field, preventing the treatment of charge as a fixed parameter.

Methodology and Framework
The analysis is conducted in a double-null gauge on a spherically symmetric spacetime $(M, g)$. The metric is expressed as $g = -4\Omega^2 du dv + r^2 d\sigma_{S^2}$, and the Yang-Mills connection is reduced via a purely magnetic ansatz, leaving a single scalar degree of freedom $w(u,v)$. The system reduces to a coupled set of wave and transport equations for the area radius $r$, the lapse function $\Omega$, and the scalar field $w$.

The authors employ a singular characteristic method, extending the techniques developed by An and Lim for the Einstein–Maxwell–scalar field system to the non-Abelian setting. The proof strategy relies on a bootstrap argument to establish a trapped surface formation criterion. The core of the argument involves analyzing the evolution of the normalized mass concentration parameter:
$$\eta(u) = \frac{m(u, v_2) - m(u, v_1)}{r(u, v_2)}$$
where $m$ is the Hawking mass. The goal is to show that under specific initial conditions, $\eta(u)$ grows sufficiently to violate the condition for the absence of trapped surfaces (specifically, showing that $2m/r > 1$ at a specific point).

Key technical steps include:

1. Monotonicity and Bounds: Establishing a monotonicity formula for the radial width and deriving a priori bounds on the magnetic charge aspect $Q$. Unlike the static case, $Q$ must be controlled dynamically throughout the spacetime region.
2. Handling Competing Terms: The proof carefully separates the focusing contribution of the kinetic energy $(\partial w)^2$ from the defocusing contribution of the magnetic potential $Q^2/r^2$ in the Raychaudhuri and cross-focusing equations.
3. Non-Trivial Incoming Data: A significant methodological departure from previous works (such as those by An and Lim) is that the incoming initial hypersurface is not assumed to be Minkowskian. The data on the incoming null cone $C$ are allowed to be geometrically and gauge-theoretically non-trivial. The proof must propagate the subextremality condition and control accumulated errors without relying on an exact background solution.

Key Results
The paper establishes a rigorous trapped-surface formation theorem (Theorem 1.2) for the spherically symmetric, purely magnetic $SU(2)$ Einstein–Yang–Mills system.

• Initial Data Conditions: The theorem requires smooth characteristic data on two intersecting null hypersurfaces ($C$ and $C$) satisfying:
  • Regularity and no-initial-trapping conditions.
  • Magnetic subextremality on the incoming cone: $m(u, v_1) \ge |Q|(u, v_1)$.
  • A bound on the initial magnetic aspect: $\varepsilon = \sup_{C \cup C} Q^2/r^2 < 1$.
  • A sufficiently small initial relative radial width $\delta_0$.
• The Criterion: A trapped surface is guaranteed to form if the initial mass concentration $\eta_0$ exceeds a specific lower bound determined by the magnetic subextremality parameter $\varepsilon$, the radial width $\delta_0$, and a parameter $\omega \in (0, 2/3)$. The bound is given by:
  $$\eta_0 > \max \left\{ \frac{13\varepsilon}{\omega} + g_\omega(\delta_0), \quad \frac{9}{2^{1+\omega/2}(1+\delta_0)^2}\delta_0^{1-\omega/2} + g_\omega(\delta_0) \right\}$$
  where $g_\omega$ is an explicitly computable function derived from the differential inequalities governing the system.
• Outcome: If these conditions are met, there exists a point $(u_{tr}, v_{tr})$ in the development such that $\partial_u r < 0$ and $\partial_v r < 0$, indicating the formation of a trapped surface.

Significance and Claims
The authors position this work as the first part of a program toward proving weak cosmic censorship for the spherically symmetric Einstein–Yang–Mills system.

• Novelty in Data: The result is significant because it allows for non-trivial incoming data. Previous trapped-surface formation results for non-Abelian systems often required trivial data on the incoming cone. This paper demonstrates that the collapse mechanism is robust even when the incoming region carries non-trivial geometry and gauge fields.
• Structural Difficulties: The paper highlights that the non-Abelian nature of the Yang–Mills field introduces new structural difficulties, specifically the dynamical coupling of the "charge" $Q$ and the competition between focusing and defocusing terms. The proof successfully navigates these by separating the mass increment from the magnetic potential effects.
• Connection to Static Theory: The theorem connects the classical existence theory of static "colored" black holes (which are unstable) with a quantitative dynamical mechanism showing how trapped regions can arise from regular characteristic data.
• Scope: The authors explicitly state that this paper constitutes the "collapse component" of the broader cosmic censorship program. It relies on local existence and continuation theories established in a companion work [19] (specifically the $H^1$ extension principle) to ensure the solution exists up to the formation of the trapped surface.

The paper does not claim to resolve the full weak cosmic censorship conjecture (which would require analyzing the global structure of the spacetime beyond the trapped surface), but rather provides the necessary quantitative criterion for the formation of the trapped region itself in the presence of non-trivial non-Abelian fields.