Saddle-point dynamics of a Yang-Mills field on the exterior Schwarzschild spacetime

TL;DR

This paper analyzes the evolution of a spherically symmetric SU(2) Yang-Mills field outside a Schwarzschild black hole, revealing how solutions approach unstable static states and describing the universal phases of their dynamics.

Contribution

It demonstrates the saddle-point dynamics near an unstable static solution, identifying universal phases and the role of this solution as an intermediate attractor.

Findings

Unstable static solutions act as intermediate attractors.

Universal phases include ringdown, exponential departure, and decay.

Solutions remain smooth and do not all scatter to infinity.

Abstract

We consider the Cauchy problem for a spherically symmetric SU(2) Yang-Mills field propagating outside the Schwarzschild black hole. Although solutions starting from smooth finite energy initial data remain smooth for all times, not all of them scatter since there are non-generic solutions which asymptotically tend to unstable static solutions. We show that a static solution with one unstable mode appears as an intermediate attractor in the evolution of initial data near a border between basins of attraction of two different vacuum states. We study the saddle-point dynamics near this attractor, in particular we identify the universal phases of evolution: the ringdown approach, the exponential departure, and the eventual decay to one of the vacuum states.