# On the local well-posedness of Lovelock and Horndeski theories

## TL;DR

This paper analyzes the hyperbolicity and well-posedness of Lovelock and Horndeski gravity theories, revealing limitations of standard methods and potential issues with their initial value problems in weak-field regimes.

## Contribution

It demonstrates that Lovelock and Horndeski theories generally lack strong hyperbolicity in weak fields, challenging assumptions about their well-posedness.

## Key findings

- Lovelock equations are weakly hyperbolic but not strongly hyperbolic in harmonic gauge.
- Horndeski theories are weakly hyperbolic in any generalized harmonic gauge for weak fields.
- Some Horndeski theories admit gauges with strong hyperbolicity, but not all, raising concerns about their initial value formulations.

## Abstract

We investigate local well-posedness of the initial value problem for Lovelock and Horndeski theories of gravity. A necessary condition for local well-posedness is strong hyperbolicity of the equations of motion. Even weak hyperbolicity can fail for strong fields so we restrict to weak fields. The Einstein equation is known to be strongly hyperbolic in harmonic gauge so we study Lovelock theories in harmonic gauge. We show that the equation of motion is always weakly hyperbolic for weak fields but, in a generic weak-field background, it is not strongly hyperbolic. For Horndeski theories, we prove that, for weak fields, the equation of motion is always weakly hyperbolic in any generalized harmonic gauge. For some Horndeski theories there exists a generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a weak-field background. This includes "k-essence" like theories. However, for more general Horndeski theories, there is no generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a generic weak-field background. Our results show that the standard method used to establish local well-posedness of the Einstein equation does not extend to Lovelock or general Horndeski theories. This raises the possibility that these theories may not admit a well-posed initial value problem even for weak fields.