Dynamical Formation of Apparent Horizons due to Boundary Effect in Vacuum Einstein Gravity

TL;DR

This work proves that an apparent horizon can dynamically form in pure vacuum gravity starting from horizon-free initial data, via a boundary mean-curvature mechanism in a double-null setting. The authors introduce a semi-global evolution in a region $\mathcal{D}_{a,\epsilon}$ with large parameter $a$ and small $\epsilon$, concentrating generalized mean curvature $c=H-|\,\kappa|$ on the evolving boundary to trigger horizon formation. Horizon exclusion on an interior Cauchy segment is achieved through a Corvino–Schoen type gluing to Kerr exterior and a Yau-radius barrier ensuring no MOTS initially, while the characteristic slab imports data that dynamically raise $c$ in the future, yielding a trapped surface as per Yau’s criterion. The analysis blends hyperbolic transport methods in the double-null gauge with scale-critical norms and a novel interior data construction inspired by Yau, avoiding elliptic estimates and providing a new pathway to horizon formation in vacuum GR. The results have implications for the understanding of black-hole creation mechanisms and the interplay between boundary geometry and dynamical horizons in general relativity.

Abstract

We prove that an apparent horizon can form as a result of Einsteinian evolution in pure vacuum spacetime starting from regular initial data free of apparent horizons due to pure boundary effects. We adapt a Cauchy-double-null framework and use the boundary generalized mean curvature condition for the existence of an interior apparent horizon imposed by the author S-T Yau in \cite{yau}. In particular, we prove that the condition of \cite{yau} can be met dynamically starting from a configuration that does not verify the same through a focusing mechanism. This is the first part of a two-part sequence, and in the sequel, we will focus on explicitly constructing the Cauchy data.

Figures (1)

• Figure 1: The schematics of the current framework: concentration of the generalized mean curvature $H-|\kappa|$ while increasing the radius. The initial data is provided on the null hypersurfaces $u=u_{\infty}$ and $\underline{u}=0$, and the interior Cauchy slice $\mathcal{M}_{1}=\widehat{\mathcal{M}}_{int}$. The Cauchy data on the slice $\widehat{\mathcal{M}}_{t=-a}$ is prescribed by gluing data on $\mathcal{M}_{1}$, the induced data on $\mathcal{M}_{2}$ by the Characteristic development on the slab $\mathcal{D}_{a,\epsilon}:=[u_{\infty},-a]\times [0,\epsilon]\times \mathbb{S}^{2}$, and exterior of a Kerr slice data on $\mathcal{M}_{3}$. The data on $\mathcal{M}_{1}$ is set such that it does not contain an apparent horizon. By the solution of the Characteristic semi-global problem in $\mathcal{D}_{a,\epsilon}$, one does not have any apparent horizon in the entire Cauchy slice $\widehat{\mathcal{M}}_{t=-a}$. In particular, $S_{-a,0}=\underline{H}_{0}\cap H_{-a}$ does not verify the condition of yau, while $S_{-a,\epsilon}:=H_{-a}\cap \underline{H}_{\epsilon}$ does. Therefore, $J^{+}(\mathcal{M}_{1})\cap \widehat{M}_{t=-a+\epsilon}$ has an apparent horizon inside.

Theorems & Definitions (51)

• Definition 1
• Definition 2
• Theorem 1.1: Penrose Incompleteness
• Definition 3
• Theorem 1.2
• Remark 1
• Theorem 3.1: Main Theorem
• Lemma 4.1
• proof
• Remark 2