Hamiltonian vs stability and application to Horndeski theory

Eugeny Babichev, Christos Charmousis, Gilles Esposito-Farèse, Antoine Lehébel

Published 2018-03-30Version 1

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: this is a coordinate-dependent statement. We discuss this fact in detail, 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.