Resonant Dynamics and the Instability of the Box Minkowski Model

TL;DR

This paper demonstrates that the box Minkowski model with Dirichlet boundary conditions is generically unstable to black hole formation from small perturbations, using resonant system analysis and weakly nonlinear perturbation theory.

Contribution

It introduces a detailed resonant system for the model, revealing additional resonant terms and conserved quantities, and analyzes their impact on instability and singularity formation.

Findings

Generic solutions become singular in finite time.

Additional resonant interactions do not prevent singularity formation.

Interaction coefficients are simple, making it a useful toy model.

Abstract

We revisit the box Minkowski model [Phys. Rev. Lett. 109, 221101 (2012)] and provide a strong argument that, subject to the Dirichlet boundary condition, it is unstable toward black hole formation for arbitrarily small generic perturbations. Using weakly nonlinear perturbation theory, we derive the resonant system, which compared to systems with the anti-de Sitter asymptotics, has extra resonant terms, and study its properties, including conserved quantities. We find that the generic solution of the resonant system becomes singular in finite time. Surprisingly, the additional resonant interactions do not significantly affect the singular evolution. Furthermore, we find that the interaction coefficients take a relatively simple form, making this a particularly attractive toy model of turbulent gravitational instability.