K-dynamics: well-posed 1+1 evolutions in K-essence

TL;DR

This paper investigates the well-posedness of the Cauchy problem in K-essence theories, demonstrating conditions under which solutions remain stable and identifying problematic initial data that lead to divergences.

Contribution

It characterizes the class of K-essence theories with well-posed evolutions and analyzes the nature of initial data causing breakdowns, proposing potential solutions.

Findings

Most K-essence theories have well-posed evolutions outside black hole horizons.

Certain initial data near critical collapse cause diverging characteristic speeds.

The study provides criteria to identify and address problematic initial conditions.

Abstract

We study the vacuum Cauchy problem for K-essence, i.e. cosmologically relevant scalar-tensor theories that involve first-order derivative self-interactions, and which pass all existing gravitational wave bounds. We restrict to spherical symmetry and show that there exists a large class of theories for which no breakdown of the Cauchy problem occurs outside apparent black hole horizons, even in the presence of scalar shocks/caustics, except for a small set of initial data sufficiently close to critical black hole collapse. We characterise these problematic initial data, and show that they lead to large or even diverging (coordinate) speeds for the characteristic curves. We discuss the physical relevance of this problem and propose ways to overcome it.