Gravitational instability of a kink

We study the equilibria of a self-gravitating scalar field in the region outside a reflecting barrier. By introducing a potential difference between the barrier and infinity, we create a kink which cannot decay to a zero-energy state. In the realm of a small amplitude, the kink decays to a known static solution of the Einstein-Klein-Gordon equation. However, for larger kinks the static equilibria are degenerate, forming a system with two energy levels. The upper level is unstable and, under small perturbations, decays to the lower energy stable equilibrium. Under large perturbations, the unstable upper level undergoes collapse to a black hole. The equilibrium of the system provides a remarkably simple and beautiful illustration of a turning point instability.