# Hamiltonian vs stability in alternative theories of gravity

## TL;DR

This paper clarifies that Hamiltonian boundedness does not necessarily imply stability in alternative gravity theories, emphasizing the coordinate independence of stability criteria and illustrating this with a Horndeski theory example.

## Contribution

It explains why Hamiltonian unboundedness does not imply instability and introduces a correct, coordinate-independent stability criterion for alternative gravity theories.

## Key findings

- Stability is coordinate-independent, unlike Hamiltonian density.
- A Schwarzschild-de Sitter solution in Horndeski theory can be stable within certain parameters.
- Incorrect claims in literature about instability are corrected.

## Abstract

When a Hamiltonian density is bounded by below, we know that the lowest-energy state must be stable. One is often tempted to reverse the theorem and therefore believe that an unbounded Hamiltonian density always implies an instability. The main purpose of this presentation (which summarizes my work with E. Babichev, C. Charmousis and A. Leh\'ebel) is to pedagogically explain why this is erroneous. Stability is indeed a coordinate-independent property, whereas the Hamiltonian density does depend on the choice of coordinates. In alternative theories of gravity, like k-essence or Horndeski theories, the correct stability criterion is a subtler version of the well-known "Weak Energy Condition" of general relativity. As an illustration, this criterion is applied to an exact Schwarzschild-de Sitter solution of a Horndeski theory, which is found to be stable for a given range of its parameters, contrary to a claim in the literature.