Well-posedness of cubic Horndeski theories

Abstract

We study the local well-posedness of the initial value problem for cubic Horndeski theories. Three different strongly hyperbolic modifications of the ADM formulation of the Einstein equations are extended to cubic Horndeski theories in the weak field regime. In the first one, the equations of motion are rewritten as a coupled elliptic-hyperbolic system of partial differential equations. The second one is based on the BSSN formulation with a generalised Bona-Massó slicing (covering the 1+log slicing) and non-dynamical shift vector. The third one is an extension of the CCZ4 formulation with a generalised Bona-Massó slicing (also covering the 1+log slicing) and a gamma driver shift condition. This final formulation may be particularly suitable for applications in nonlinear numerical simulations.

Challenges to global solutions in Horndeski’s theory

“…Such scenario takes place when l µ ∇ µ r = 0, where l µ is a null vector [37]. In the gauge (29), this is simply α = 0. Consequently, with our current implementation we can explore up to black hole formation.…”

“…The analysis of this condition in specific theories is typically complex, which has hindered drawing straightforward conclusions for Horndeski's theories in the past. Recent works however have begun to explore this issue and, in particular, have uncovered significant restrictions [27][28][29] even locally. We stress the importance of this condition can not be underestimated even in linearized regimes.…”

Gravitational collapse in Einstein dilaton-Gauss–Bonnet gravity

“…This is certainly a sign for concern, however demanding that a theory be well-posed in all possible situations might be unnecessarily restrictive if problems do not occur in scenarios of interest, here in particular for binary BH mergers. Considering the recent results of [17], we also mention there may exist other gauges for which EdGB may have well posed initial value problem for generic small field initial data.…”

Towards numerical relativity in scalar Gauss-Bonnet gravity: 3+1 decomposition beyond the small-coupling limit

“…We remark, however, that the existence of such a term does not rule out the possibility of a well-posed formulation. For instance, in the case of cubic Horndeski gravity a strongly hyperbolic formulation has been found [30] despite the presence of terms quadratic in the second spatial derivatives [see, e.g., Eq. (107) of [30] ].…”

“…[22][23][24] for a review). A proof of well-posedness beyond GR has been recently obtained, but only for the simplest theories (namely, the socalled Bergmann-Wagoner scalar-tensor theories [25,26]), in Einstein-AEther theory [27], and for higher derivative theories such as Horndeski or Lovelock gravity in the weak-field regime [28][29][30]. Clearly, extending such results to a well-motivated, nonperturbative modified theory of gravity would be extremely important.…”