On the hyperbolicity of the most general Horndeski theory

TL;DR

This paper investigates the hyperbolic nature of equations in the most general Horndeski gravity theory, revealing that only a restricted subset maintains strong hyperbolicity, which is crucial for well-posedness of the equations.

Contribution

It proves that only a specific class of Horndeski theories, namely k-essence coupled to Einstein gravity, are strongly hyperbolic in weak field regimes, while more general theories are not.

Findings

Einstein-dilaton-Gauss-Bonnet gravity is not strongly hyperbolic.

Only k-essence coupled to Einstein gravity is strongly hyperbolic.

Adding more general Horndeski terms results in weak hyperbolicity.

Abstract

In this paper we study the hyperbolicity of the equations of motion for the most general Horndeski theory of gravity in a generic "weak field" background. We first show that a special case of this theory, namely Einstein-dilaton-Gauss-Bonnet gravity, fails to be strongly hyperbolic in any generalised harmonic gauge. We then complete the proof that the most general Horndeski theory which, for weak fields, is strongly hyperbolic in a generalised harmonic gauge is simply a "k-essence" theory coupled to Einstein gravity and that adding any more general Horndeski term will result in a weakly, but not strongly, hyperbolic theory.