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A Hamiltonian density bounded from below implies that the lowest-energy stateis stable. We point out, contrary to common lore, that an unbounded Hamiltoniandensity does not necessarily imply an instability: Stability is indeed acoordinate-independent property, whereas the Hamiltonian density does depend onthe 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 thecausal cones for all propagating degrees of freedom. We then apply thiscriterion to an exact Schwarzschild-de Sitter solution of a beyond-Horndeskitheory, while taking into account the recent experimental constraint regardingthe speed of gravitational waves. We extract the spin-2 and spin-0 causal conesby analyzing respectively all the odd-parity and the $\ell = 0$ even-paritymodes. Contrary to a claim in the literature, we prove that this solution doesnot exhibit any kinetic instability for a given range of parameters definingthe theory.

Hamiltonian vs stability and application to Horndeski theory