# Hyperbolicity in Spherical Gravitational Collapse in a Horndeski Theory

## TL;DR

This study numerically investigates spherical gravitational collapse in a modified gravity theory, revealing that the equations can change from hyperbolic to elliptic, affecting the well-posedness of the evolution problem.

## Contribution

It demonstrates the potential for mixed-type equations in EdGB gravity, highlighting the need for new approaches to ensure well-posedness in spherical collapse scenarios.

## Key findings

- Equations change from hyperbolic to elliptic in certain regions.
- No singularities or discontinuities at the elliptic boundary.
- Implication for well-posed formulations of EdGB gravity.

## Abstract

We numerically study spherical gravitational collapse in shift symmetric Einstein dilaton Gauss Bonnet (EdGB) gravity. We find evidence that there are open sets of initial data for which the character of the system of equations changes from hyperbolic to elliptic type in a compact region of the spacetime. In these cases evolution of the system, treated as a hyperbolic initial boundary value problem, leads to the equations of motion becoming ill-posed when the elliptic region forms. No singularities or discontinuities are encountered on the corresponding effective "Cauchy horizon". Therefore it is conceivable that a well-posed formulation of EdGB gravity (at least within spherical symmetry) may be possible if the equations are appropriately treated as mixed-type.