Hamiltonian unboundedness vs stability with an application to Horndeski theory

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 = 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.1 See also [24] for relations between higher dimensional and 4-dimensional scalar-tensor gravity theories. 2 We do not consider here Lorentz-breaking theories [28,29] where equations of motion can be of higher order. Also, in extensions of Horndeski theory [30][31][32][33][34][35][36][37][38][39][40][41][42][43], one may get Euler-Lagrange equations of third order, but by manipulating these equations their order is reduced.

Exact black hole solutions in shift-symmetric quadratic degenerate higher-order scalar-tensor theories

“…These classes also fall into Cases 1-Λ and 2-Λ for the shift-symmetric Horndeski, which are defined by the conditions (67) and (68) after plugging (63). Note that in the model (88) c t = c. The stability of hairy black holes in the shift-symmetric Horndeski and GLPV theories with the scalar field profile φ(t, r) = qt + ψ(r) has been discussed in the literature [24,[43][44][45][46][47]. The odd-parity perturbations about the Schwarzschild[-(anti-)de Sitter] solutions in the Horndeski theories were analyzed in Refs.…”

Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations

“…As pointed out in Refs. [29,34], an unbounded Hamiltonian in a specific coordinate system does not necessarily mean instability of the system. This is because a Hamiltonian is not a scalar quantity, and thus there may exist a coordinate system where the Hamiltonian is bounded below.…”

Rotating black holes in higher order gravity

“…Since these solutions acquire a metric similar to that of GR while having arXiv:1903.05519v1 [hep-th] 13 Mar 2019 a non trivial scalar field, they have been widely called stealth solutions 1 . They can be mapped via disformal transformations to stealth solutions of DHOST theories with unitary speed of gravitational waves [16], and are free of ghost and gradient instabilities [17]. For spherically symmetric stealth solutions in DHOST theories see [18][19][20].…”

“…2 Note, for illustration we take timelike geodesics, should a spacelike congruence be required, substitute m 2 → −m 2 in the derivation. This has now reduced the parameter space to an overall scaling, m, and a "relative energy" η, constrained to lie in η ∈ [η c , 1]; the upper limit coming from Θ ≥ 0, and the lower limit from R ≥ 0 in (17). At first sight, it appears we have four distinct solutions coming from the choice of signs in (14), however, an interesting restriction occurs when η = 1 or η c .…”