Hamiltonian vs stability and application to Horndeski theory

TL;DR

This paper clarifies the relationship between Hamiltonian boundedness and stability in field theories, introduces a coordinate-independent stability criterion, and applies it to a specific Horndeski theory solution, confirming its stability under certain conditions.

Contribution

It corrects misconceptions about Hamiltonian boundedness and stability, and develops a new coordinate-independent stability criterion applied to Horndeski theories.

Findings

Hamiltonian density's unboundedness does not imply instability.

The stability criterion depends on causal cone orientations.

The Schwarzschild-de Sitter solution in beyond-Horndeski theory is stable within certain parameters.

Abstract

A Hamiltonian density bounded from below implies that the lowest-energy state is stable. We point out, contrary to common lore, that an unbounded Hamiltonian density does not necessarily imply an instability: Stability is indeed a coordinate-independent property, whereas the Hamiltonian density does depend on the choice of coordinates. We discuss in detail the relation between the two, starting from k-essence and extending our discussion to general field theories. We give the correct stability criterion, using the relative orientation of the causal cones for all propagating degrees of freedom. We then apply this criterion to an exact Schwarzschild-de Sitter solution of a beyond-Horndeski theory, while taking into account the recent experimental constraint regarding the speed of gravitational waves. We extract the spin-2 and spin-0 causal cones by analyzing respectively all the odd-parity and the $\ell = 0$ even-parity modes. Contrary to a claim in the literature, we prove that this solution does not exhibit any kinetic instability for a given range of parameters defining the theory.